Research ArticleInfluence of Quasiperiodic Gravitational Modulation onConvective Instability of Liquid-Liquid Polymerization Front
Saadia Assiyad1 Karam Allali1 and Mohamed Belhaq2
1Department of Mathematics University of Hassan II-Casablanca FST PO Box 146 Mohammedia Morocco2Department of Mechanics University of Hassan II-Casablanca Casablanca Morocco
Correspondence should be addressed to Karam Allali allalihotmailcom
Received 11 August 2015 Accepted 7 October 2015
Academic Editor Fazal M Mahomed
Copyright copy 2015 Saadia Assiyad et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The influence of quasiperiodic gravitational modulation on convective instability of polymerization front with liquid monomerand liquid polymer is studied The model includes the heat equation the concentration equation and the Navier-Stokes equationsunder the Boussinesq approximationThe linear stability analysis of the problem is carried out and the interface problem is derivedUsing numerical simulations the convective instability threshold is determined and the boundary of the convective instability isobtained for different amplitudes and frequencies ratio
1 Introduction
Frontal polymerization is the process of polymer productionin propagating reaction fronts [1ndash4] In the absence of vibra-tion the influence of convective instability on polymerizationfront when the monomer is liquid and the polymer is solidwas studied in [5] while the case of liquid polymer wasconsidered in [6] The influence of periodic gravitationalmodulation on the convective instability in the case of liquid-solid polymerization front was studied [7] and it was shownthat the propagation of polymerization reaction front isstrongly affected by the amplitude and the frequency of vibra-tions In particular the polymerization front can be stableor unstable depending on the values of vibration parametersThe influence of periodic vibrations on convective instabilityof reaction front was also studied in the case of liquids [8]and it was concluded that for small vibration amplitudesthe reaction front remains stable and it loses its stability forsufficiently large amplitude of vibrations
Recent works were devoted to the influence of a qua-siperiodic (QP) gravitational modulation on the convectiveinstability of reaction front For instance the influenceof the QP gravitational modulation on reaction front wasexamined in the case of porousmedia described by the Darcyequation [9] On the other hand the case of liquid-solid
polymerization front was considered in [10] using theNavier-Stokes equations instead of the Darcy equation and it wasrevealed that both the amplitudes and the frequencies ratioinfluence the stability domain of the polymerization frontSpecifically for appropriate values of vibration amplitudesand increasing values of the frequencies ratio a stabilizingeffect is observed The effect of the wave number on thereaction front was also examined showing that increasing thewave number widens the stability domain
The present work studies the effect of QP gravitationalmodulation on the convective instability of the polymeriza-tion front but in the case of liquid-liquid frontal polymer-ization This case is different from the previous one [10] inthe sense that in [10] the equation of motion is consideredonly after the reaction zone since the polymer is in the solidphase In the present work instead the equation of motion isconsidered before and after the reaction zone because boththe monomer and the polymer are liquids
The next section presents the frontal polymerizationmodel Section 3 develops the perturbation analysis whilethe interface problem is examined in Section 4 The linearstability analysis is discussed in Section 5 and numericalinvestigations are carried out in Section 6 The last sectionconcludes the work
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 484562 11 pageshttpdxdoiorg1011552015484562
2 Mathematical Problems in Engineering
2 Frontal Polymerization Model
We consider a polymerization process with a liquid reactantand a liquid product by assuming that the reaction front prop-agates in the direction opposite to the direction of gravity Inthis case the model of the frontal polymerization is given bythe system of equations
120597119879
120597119905+ (V sdot nabla) 119879 = 120581Δ119879 + 119902119882
120597120572
120597119905+ (V sdot nabla) 120572 = 119882
120597V120597119905
+ (V sdot nabla) V = minus1
120588nabla119901 + ]ΔV
+ 119892 (1 + 119887 (119905)) 120573 (119879 minus 1198790) 120574
div (V) = 0
(1)
with the following boundary conditions
119911 997888rarr +infin 119879 = 119879119894 120572 = 0 V = 0 (2)
119911 997888rarr minusinfin 119879 = 119879119887 120572 = 1 V = 0 (3)
The gradient divergence and Laplace operators aredefined by
nablaV = (120597V120597119909
120597V120597119910120597V120597119911)
div k = 120597k1
120597119909+120597k2
120597119910+120597k3
120597119911
ΔV =1205972V1205972119909
+1205972V1205972119910
+1205972V1205972119911
(4)
where the variables (119909 119910 and 119911) are the spatial coordinatessuch that minusinfin lt 119909 119910 119911 lt +infin 119879 is the temperature 120572 is theconcentration of the reaction product V is the velocity 119901 isthe pressure 120581 is the coefficient of thermal diffusivity 119902 is theadiabatic temperature heat release 120588 is an average value of thedensity ] is the coefficient of kinematic viscosity 120574 is the unitvector in the upward direction 120573 is the coefficient of thermalexpansion 119892 is the gravitational acceleration and 119887(119905) is theQP acceleration acting on the fluid which is given by 119887(119905) =1205821sin(1205901119905) + 120582
2sin(1205902119905) such that 120582
1 1205822are the amplitudes
and 1205901 1205902are the incommensurate frequencies of the QP
gravitational modulation The quantity 1198790is a mean value of
temperature 119879119894is the initial temperature and 119879
119887= 119879119894+ 119902
is the temperature of the reacted mixture We consider one-step reaction of zero order where the reaction rate is definedas follows
119882 = 119896 (119879) 120601 (120572) 120601 (120572) =
1 if 120572 lt 1
0 if 120572 = 1
(5)
The temperature dependence of the reaction rate is givenby the Arrhenius law [11]
119896 (119879) = 1198960exp(minus 119864
1198770119879) (6)
where 1198960is the preexponential factor 119864 is the activation
energy and 1198770is the universal gas constant We assume
that the two liquids are incompressible and the diffusivitycoefficient is very small comparing to the thermal diffusivitycoefficient such that the diffusivity will be neglected in theconcentration equation
To obtain the dimensionless model we introduce thedimensionless spatial variables as
1199091=1199091198881
120581
1199101=1199101198881
120581
1199111=1199111198881
120581
1199051=1199051198882
1
120581
1199011=
119901
11988821120588
1198881=
119888
radic2
V1=
V1198881
120579 =119879 minus 119879119887
119902
1198882=2119896012058111987701198792
119887
119902119864exp(minus 119864
1198770119879119887
)
(7)
where 119888 defines the stationary reaction front velocity andcan be calculated asymptotically for large Zeldovich number[12] For simplicity we keep the same notation for the othervariables and pressure System (1) with the two boundaryconditions (2)-(3) can be written in the form
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 + 119885 exp( 120579
119885minus1 + 120575120579)120601 (120572) (8)
120597120572
120597119905+ (V sdot nabla) 120572 = 119885 exp( 120579
119885minus1 + 120575120579)120601 (120572) (9)
120597V120597119905
+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119875119877 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (120579 + 120579
0) 120574
(10)
div (V) = 0 (11)
with the following conditions at infinity
119911 997888rarr +infin 120579 = minus1 120572 = 0 V = 0
119911 997888rarr minusinfin 120579 = 0 120572 = 1 V = 0
(12)
and 119875 = ]120581 is the Prandtl number 119877 = 1198921205731199021205812(]1198883) is the
Rayleigh number119885 = 11990211986411987701198792
119887is the Zeldovich number and
Mathematical Problems in Engineering 3
120575 and 1205790are given respectively by 120575 = 119877
0119879119887119864 and 120579
0= (119879119887minus
1198790)119902 and 120583 = 2120581120590119888
2Next we perform the linear stability analysis to tackle the
interface problem
3 Approximation of Infinitely NarrowReaction Zone
To study the interface problem analytically we use a singularperturbation analysis where the reaction zone is infinitelynarrow and the reaction term is neglected outside the zone[13] In this way the problem can be reduced to an interfaceproblem
To perform a formal asymptotic analysis 120598 = 119885minus1 is
considered as a small parameter The new independent var-iable is given by 119911
1= 119911 minus 120577(119909 119910 119905) where 120577(119909 119910 119905) defines the
location of the reaction zone Introducing new functions 1205791
1205721 V1 and 119901
1as
120579 (119909 119910 119911 119905) = 1205791(119909 119910 119911
1 119905)
120572 (119909 119910 119911 119905) = 1205721(119909 119910 119911
1 119905)
V (119909 119910 119911 119905) = V1(119909 119910 119911
1 119905)
119901 (119909 119910 119911 119905) = 1199011(119909 119910 119911
1 119905)
(13)
The system of (8)ndash(11) can be written in the form (index 1 forthe new function is omitted)120597120579
120597119905minus120597120579
1205971199111
120597120577
120597119905+ (V sdot nabla) 120579
= Δ120579 + 119885 exp( 120579
119885minus1 + 120575120579)120601 (120572)
120597120572
120597119905minus120597120572
1205971199111
120597120577
120597119905+ (V sdot nabla) 120572 = 119885 exp( 120579
119885minus1 + 120575120579)120601 (120572)
120597V120597119905
minus120597V1205971199111
120597120577
120597119905+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (120579 + 120579
0) 120574
120597V119909
120597119909minus120597V119909
1205971199111
120597120577
120597119909+120597V119910
120597119910minus120597V119910
1205971199111
120597120577
120597119910+120597V119911
1205971199111
= 0
(14)
where Δ nabla and 119876 are given by
Δ =1205972
1205971199092+
1205972
1205971199102+
1205972
12059711991121
minus 21205972
1205971199091205971199111
120597120577
120597119909minus 2
1205972
1205971199101205971199111
120597120577
120597119910
+1205972
12059711991121
((120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus120597
1205971199111
(1205972120577
1205971199092+1205972120577
1205971199102)
nabla = (120597
120597119909minus
120597
1205971199111
120597120577
120597119909120597
120597119910minus
120597
1205971199111
120597120577
120597119910120597
1205971199111
)
119876 = 119875119877
(15)
To approximate the jump conditions and resolve theinterface problem we use the matched asymptotic expansionby seeking the outer solution of problem (14) in the form
120579 = 1205790+ 1205981205791+ sdot sdot sdot
120572 = 1205720+ 1205981205721+ sdot sdot sdot
V = V0+ 120598V1+ sdot sdot sdot
119901 = 1199010+ 1205981199011+ sdot sdot sdot
(16)
Introducing the stretched coordinate 120578 = 1199111120598minus1 where 120598 =
119885minus1 the inner solution can be approximated in the following
form
120579 = 1205981205791+ sdot sdot sdot
120572 = 0+ 1205981+ sdot sdot sdot
V = V0+ 120598V1+ sdot sdot sdot
119901 = 1199010+ 1205981199011+ sdot sdot sdot
120577 = 1205770+ 1205981205771+ sdot sdot sdot
(17)
Substituting the inner and outer solutions in (14) leads to
Order 120598minus2
119875(1 + (1205971205770
120597119909)
2
+ (1205971205770
120597119910)
2
)1205972V0
1205971205782= 0 (18)
Order 120598minus1
(1 + (1205971205770
120597119909)
2
+ (1205971205770
120597119910)
2
)12059721205791
1205971205782
+ exp( 1205791
1 + 1205751205791
)120601 (0) = 0
(19)
minus1205970
120597120578
1205971205770
120597119905minus1205970
120597120578(V0119909
1205971205770
120597119909+ V0119910
1205971205770
120597119910minus V0119911)
= exp( 1205791
1 + 1205751205791
)120601 (0)
(20)
minus120597V0119909
120597120578
1205971205770
120597119905minus V0119909
120597V0
120597120578
1205971205770
120597119909minus V0119910
120597V0
120597120578
1205971205770
120597119910+ V0119911
120597V0
120597120578
= 1199050
1205971199010
120597120578+ 119875(119860
1205972V1
1205971205782+ 1198650
120597V0
120597120578)
(21)
minus120597V0119909
120597120578
1205971205770
120597119909minus120597V0119910
120597120578
1205971205770
120597119910+120597V0119911
120597120578= 0 (22)
4 Mathematical Problems in Engineering
Order 1205980
120597V0
120597119905minus120597V1
120597120578
1205971205770
120597119905minus120597V0
120597120578
1205971205771
120597119905+ V0119909(120597V0
120597119909minus120597V1
120597120578
1205971205770
120597119909
minus120597V0
120597120578
1205971205771
120597119909) + V1119909
120597V0
120597120578
1205971205770
120597119909+ V1119910
120597V0
120597120578
1205971205770
120597119910
+ V0119910(120597V0
120597119910minus120597V1
120597120578
1205971205770
120597119910minus120597V0
120597120578
1205971205771
120597119910) + V0119911
120597V1
120597120578+ V1119911
sdot120597V0
120597120578= minusnabla01199010+ 1199051
1205971199010
120597120578+ 1199050
1205971199011
120597120578+ 119875(119860
1205972V2
1205971205782
+ 1199053
1205972V1
1205971205782+ 1198650
120597V1
120597120578+ 1199054
1205972V0
1205971205782+ 1198651
120597V0
120597120578+ Δ1V0)
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120574120579
0
(23)
120597V0119909
120597119909minus120597V1119909
120597120578
1205971205770
120597119909minus120597V0119909
120597120578
1205971205771
120597119909+120597V0119910
120597119910minus120597V1119910
120597120578
1205971205770
120597119910
minus120597V0119910
120597120578
1205971205771
120597119910+120597V1119911
120597120578= 0
(24)
Order 1205981
120597V1
120597119905minus (
120597V2
120597120578
1205971205770
120597119905+120597V1
120597120578
1205971205771
120597119905+120597V0
120597120578
1205971205772
120597119905) + V0119909(120597V1
120597119909
minus120597V1
120597120578
1205971205771
120597119909minus120597V0
120597120578
1205971205772
120597119909minus120597V2
120597120578
1205971205770
120597119909) + V1119909(120597V0
120597119909
minus120597V0
120597120578
1205971205771
120597119909minus120597V1
120597120578
1205971205770
120597119909) minus V2119909
120597V0
120597120578
1205971205770
120597119909+ V0119910(120597V1
120597119910
minus120597V1
120597120578
1205971205771
120597119910minus120597V0
120597120578
1205971205772
120597119910minus120597V2
120597120578
1205971205770
120597119910) + V1119910(120597V0
120597119910
minus120597V0
120597120578
1205971205771
120597119910minus120597V1
120597120578
1205971205770
120597119910) minus V2119910
120597V0
120597120578
1205971205770
120597119910+ V0119911
120597V2
120597120578
+ V1119911
120597V1
120597120578+ V2119911
120597V0
120597120578= 1199050
1205971199012
120597120578+ 1199051
1205971199011
120597120578minus nabla01199011+ 1199052
sdot1205971199010
120597120578+ 119875(119860
1205972V3
1205971205782+ 1199053
1205972V2
1205971205782+ 1198650
120597V2
120597120578+ 1198651
120597V1
120597120578
+ Δ1V1+ 1198652
120597V0
120597120578) + 119876 (1 + 120582
1sin (120590
1119905)
+ 1205822sin (120590
2119905)) 1205790120574
(25)
120597V1119909
120597119909minus120597V2119909
120597120578
1205971205770
120597119909minus120597V1119909
120597120578
1205971205771
120597119909minus120597V0119909
120597120578
1205971205772
120597119909+120597V1119910
120597119910
minus120597V2119910
120597120578
1205971205770
120597119910minus120597V1119910
120597120578
1205971205771
120597119910minus120597V0119910
120597120578
1205971205772
120597119910+120597V2119911
120597120578= 0
(26)
Order 1205982
120597V2119909
120597119909minus120597V3119909
120597120578
1205971205770
120597119909minus120597V2119909
120597120578
1205971205771
120597119909minus120597V1119909
120597120578
1205971205772
120597119909minus120597V0119909
120597120578
1205971205773
120597119909
+120597V2119910
120597119910minus120597V3119910
120597120578
1205971205770
120597119910minus120597V2119910
120597120578
1205971205771
120597119910minus120597V1119910
120597120578
1205971205772
120597119910
minus120597V0119910
120597120578
1205971205773
120597119910+120597V3119911
120597120578= 0
(27)
where
Δ1=
1205972
1205971199092+
1205972
1205971199102
nabla0= (
120597
120597119909120597
120597119910 0)
119865119894= minus2(
120597120577119894
120597119909
120597
120597119909+120597120577119894
120597119910
120597
120597119910) minus (
1205972120577119894
1205971199092+1205972120577119894
1205971199102) 119868
119894 = 0 1 2
1199050= (
1205971205770
1205971199091205971205770
120597119910 minus1)
119905119894= (
120597120577119894
120597119909120597120577119894
120597119910 minus 1) 119894 = 1 2
1199053= 2(
1205971205771
120597119909
1205971205770
120597119909+1205971205770
120597119910
1205971205771
120597119910)
1199054= (
1205971205771
120597119909)
2
+ (1205971205771
120597119910)
2
+1205971205770
120597119909
1205971205772
120597119909+1205971205770
120597119910
1205971205772
120597119910
119860 = (1 + (1205971205770
120597119909)
2
+ (1205971205770
120597119910)
2
)
(28)
and 119868 is the identity operator The matching conditions as120578 rarr plusmninfin are given by
V0sim V0
10038161003816100381610038161199111=plusmn0
(29)
V1sim (
120597V0
1205971199111
100381610038161003816100381610038161003816100381610038161199111=plusmn0
)120578 + V1
10038161003816100381610038161199111=plusmn0
(30)
V2sim1
2(1205972V0
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0
)1205782+ (
120597V1
1205971199111
100381610038161003816100381610038161003816100381610038161199111=plusmn0
)120578
+ V2
10038161003816100381610038161199111=plusmn0
(31)
V3sim1
6(1205973V0
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0
)1205783+1
2(1205972V1
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0
)1205782
+ (120597V2
1205971199111
100381610038161003816100381610038161003816100381610038161199111=plusmn0
)120578 + V3
10038161003816100381610038161199111=plusmn0
(32)
Mathematical Problems in Engineering 5
As 120578 rarr +infin
1205791asymp 1205791
10038161003816100381610038161199111=+0
+ (1205971205790
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
)120578 0997888rarr 0 (33)
As 120578 rarr minusinfin
1205791sim 1205791
10038161003816100381610038161199111=minus0
0997888rarr 1 (34)
From (18) we obtain
1205972V0
1205971205782= 0 (35)
Consequently we can conclude that V0(120578) is a linear
function of 120578 and identically constant since the velocity isbounded Equation (29) becomes
V0
10038161003816100381610038161199111=+0
= V0
10038161003816100381610038161199111=minus0
(36)
120597V0
120597120578= 0 (37)
From (36)-(37) we deduce that the first term in theexpression of the velocity V
0is continuous at the front By
substituting (37) into (21) we find
1198751198601205972V1
1205971205782+ 1199050
1205971199010
120597120578= 0 (38)
120597V0119909
120597119909minus120597V1119909
120597120578
1205971205770
120597119909+120597V0119910
120597119910minus120597V1119910
120597120578
1205971205770
120597119910+120597V1119911
120597120578= 0 (39)
Differentiating (39) with respect to 120578 one obtains
1205972V1119909
1205971205782
1205971205770
120597119909+1205972V1119910
1205971205782
1205971205770
120597119910minus1205972V1119911
1205971205782= 0 (40)
As a result (38) is a vectorial equation with three com-ponents We multiply the first component by 120597120577
0120597119909 the
second by 1205971205770120597119910 and the third by minus1 and adding we have
119875119860(1205972V1119909
1205971205782
1205971205770
120597119909+1205972V1119910
1205971205782
1205971205770
120597119910minus1205972V1119911
1205971205782) + 119860
1205971199010
120597120578= 0 (41)
Considering the two equations (38) and (40) we obtain
1205971199010
120597120578= 0 (42)
1205972V1
1205971205782= 0 (43)
From the boundedness of the velocity and (30) the con-tinuity of the first order term of the outer expansion and ofthe first derivative of the zero-order term we can write
120597V0
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
=120597V0
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
V1
10038161003816100381610038161199111=+0
= V1
10038161003816100381610038161199111=minus0
(44)
Differentiating (23) once and (26) twice with respect to 120578and using the three equations (37) (42) and (43) we get
1199050
12059721199011
1205971205782+ 119875119860
1205973V2
1205971205783= 0 (45)
1205973V2119909
1205971205783
1205971205770
120597119909+1205973V2119910
1205971205783
1205971205770
120597119910minus1205973V2119911
1205971205783= 0 (46)
As above we multiply the three components of (45)respectively by 120597120577
0120597119909 120597120577
0120597119910 and minus1 and adding we get
119875119860(1205973V2119909
1205971205783
1205971205770
120597119909+1205973V2119910
1205971205783
1205971205770
120597119910minus1205973V2119911
1205971205783) + 119860
12059721199011
1205971205782= 0 (47)
From the previous equation and using (45)-(46) we have
12059721199011
1205971205782= 0
1205973V2
1205971205783= 0
(48)
Knowing that the velocity is bounded and considering(31) we obtain the jump conditions in the following forms
1205972V0
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=+0
=1205972V0
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
120597V1
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
=120597V1
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
V2
10038161003816100381610038161199111=+0
= V2
10038161003816100381610038161199111=minus0
(49)
Differentiating (25) twice and (27) three times withrespect to 120578 and taking into account (37) (42) (48) we obtain
1199050
12059731199012
1205971205783+ 119875119860
1205974V3
1205971205784
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120574
12059721205791
1205971205782= 0
1205974V3119909
1205971205784
1205971205770
120597119909+1205974V3119910
1205971205784
1205971205770
120597119910minus1205974V3119911
1205971205784= 0
(50)
119875119860(1205974V3119909
1205971205784
1205971205770
120597119909+1205974V3119910
1205971205784
1205971205770
120597119910minus1205974V3119911
1205971205784) + 119860
12059731199012
1205971205783
minus 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905))
12059721205791
1205971205782= 0
(51)
From the last equation and using (50) we can write
11986012059731199012
1205971205783minus 119876 (1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905))
12059721205791
1205971205782
= 0
(52)
1205740
12059721205791
1205971205782=1205974V3
1205971205784 (53)
6 Mathematical Problems in Engineering
where 1205740is defined by
1205740= (minus
1205971205770
120597119909
119877
1198602 minus1205971205770
120597119910
119877
1198602119877
1198602minus119877
119860)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905))
(54)
Now integrating (53) with respect to 120578 using the system(32)ndash(34) lead to
1205973V0
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
minus1205973V0
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=+0
= minus1205740
1205971205790
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
(55)
As a first conclusion the velocity jump conditions acrossthe reaction front are given by (36) (44) (49) and (55)
From (20) (22) (37) we conclude that 0is a monotonic
function satisfying 0 lt 0lt 1 Therefore the reaction is of
zero order and 120601(0) equiv 1 Multiplying (19) by 120597120579
1120597120578 and
integrating the result yield
(1205971205791
120597120578)
210038161003816100381610038161003816100381610038161003816100381610038161003816+infin
minus (1205971205791
120597120578)
210038161003816100381610038161003816100381610038161003816100381610038161003816minusinfin
= 2119860minus1int
1205791|1199111=minus0
minusinfin
exp( 120591
1 + 120591120575) 119889120591
(56)
Subtracting (19) from (20) and integrating the result oneobtains
1205971205791
120597120578
100381610038161003816100381610038161003816100381610038161003816+infin
minus1205971205791
120597120578
100381610038161003816100381610038161003816100381610038161003816minusinfin
= minus119860minus1(1205971205770
120597119905+ 119904) (57)
where 119904 is given by
119904 = V0119909
1205971205770
120597119909+ V0119910
1205971205770
120597119910minus V0119911 (58)
As a second conclusion the temperature jump conditionsacross the reaction front are given by the two last equations(56)-(57) By using the matching conditions above andtruncating the expansion as
120579 asymp 1205790
1205791
10038161003816100381610038161199111=minus0
asymp 119885120579|1199111=+0
120577 asymp 1205770
V asymp V0
(59)
the jump conditions can be written as
(120597120579
1205971199111
)
2100381610038161003816100381610038161003816100381610038161003816+0
minus (120597120579
1205971199111
)
2100381610038161003816100381610038161003816100381610038161003816minus0
= 2119885(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot int
1205791|1199111=minus0
minusinfin
exp ( 120591
119885minus1 + 120591120575) 119889120591
120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
minus120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
= minus(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot (120597120577
120597119905+ (V119909
120597120577
120597119909+ V119910
120597120577
120597119910minus V119911)
100381610038161003816100381610038161003816100381610038161199111=+0
)
V119911
10038161003816100381610038161199111=+0
= V119911
10038161003816100381610038161199111=minus0
120597V119911
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
=120597V119911
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
1205972V119911
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=+0
=120597V119911
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
1205973V119911
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
minus1205973V119911
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=+0
= minus119877(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot ((1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
minus 1)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905))
120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
(60)
4 The Interface Problem and Perturbation
To study the propagation of polymerization front with aliquid reactant and liquid product the equation of motionhas been considered before and after the reaction zone Thischanges the jump conditions and influences the stabilityconditions of the frontal polymerization process Notice thatin the case of liquid-solid polymerization front the equationof motion is considered after the reaction zone The originalsystem (2) (3) (8) (9) leads to the following interfaceproblem
In the liquid monomer 119911 gt 120577
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 (61)
120572 = 0 (62)
Mathematical Problems in Engineering 7
120597V120597119905
+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579
0) 120574
(63)
div (V) = 0 (64)
In the liquid polymer 119911 lt 120577
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 (65)
120572 = 1 (66)
120597V120597119905
+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579
0) 120574
(67)
div (V) = 0 (68)
At the interface 119911 = 120577
120579|120577minus0 = 120579|120577+0
120597120579
120597119911
10038161003816100381610038161003816100381610038161003816120577=minus0minus120597120579
120597119911
10038161003816100381610038161003816100381610038161003816120577=+0= (1 + (
120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot (120597120577
120597119905+ (V119909
120597120577
120597119909+ V119910
120597120577
120597119910minus V119911)
10038161003816100381610038161003816100381610038161003816120577
)
(120597120579
120597119911)
2100381610038161003816100381610038161003816100381610038161003816120577minus0
minus (120597120579
120597119911)
21003816100381610038161003816100381610038161003816100381610038161205771+0
= minus2119885(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot int
120579|120577
minusinfin
exp ( 120591
119885minus1 + 120575120591) 119889120591
V119911
1003816100381610038161003816120577=minus0 = V119911
1003816100381610038161003816120577=+0
120597V119911
120597119911
10038161003816100381610038161003816100381610038161003816120577=minus0=120597V119911
120597119911
10038161003816100381610038161003816100381610038161003816120577=+0
1205972V119911
1205971199112
100381610038161003816100381610038161003816100381610038161003816120577=minus0
=1205972V119911
1205971199112
100381610038161003816100381610038161003816100381610038161003816120577=+0
1205973V119911
1205971199113
100381610038161003816100381610038161003816100381610038161003816120577=minus0
minus1205973V119911
1205971199113
100381610038161003816100381610038161003816100381610038161003816120577=+0
= minus119877(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot ((1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
minus 1)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (
120597120579
120597119911)
10038161003816100381610038161003816100381610038161003816120577=+0
(69)
with the conditions at infinity
119911 = minusinfin 120579 = 0 V = 0
119911 = +infin 120579 = minus1 V = 0
(70)
5 Stability Analysis
The interface problem analyzed in Section 4 has a travellingwave solution in the following form
(120579 (119909 119910 119911 119905) 120572 (119909 119910 119911 119905) V)
= (120579119904(119911 minus 119906119905) 120572
119904(119911 minus 119906119905) 0)
(120579119904 (119911 minus 119906119905) 120572119904 (119911 minus 119906119905))
=
(0 1) 1199112lt 0
(exp (minus1199061199112) minus 1 0) 119911
2gt 0
(71)
where 1199112= 119911minus119906119905 and 119906 is the speed of the stationary reaction
frontThis solution is a basic and stationary solution of inter-face problem (61)ndash(70) written in the moving coordinatesThen (61) (63) (65) and (67) can be replaced by
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 + 119906
120597120579
1205971199112
120597V120597119905
+ (Vnabla) V
= minusnabla119901 + 119875ΔV + 119906120597120579
1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (120579 + 120579
0) 120574
(72)
The other equations remain unchanged To study thestability of reaction front we seek the solution of the problemas follows
120579 = 120579119904+ 120579
119901 = 119901119904+ 119901
V = V119904+ V
(73)
where 120579 119901 and V are respectively small perturbation oftemperature pressure and velocity
8 Mathematical Problems in Engineering
Subtituting (73) into (64) (72) one obtains the first-orderterms as follows
In the liquid monomer 1199112gt 120585
120597120579
120597119905= Δ120579 + 119906
120597120579
1205971199112
minus V1199111205791015840
119904 (74)
120597V120597119905
= minusnabla119901 + 119875ΔV + 119906120597120579
1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579120574
(75)
div (V) = 0 (76)
In the liquid polymer 1199112lt 120585
120597120579
120597119905= Δ120579 + 119906
120597120579
1205971199112
(77)
120597V120597119905
= minusnabla119901 + 119875ΔV + 119906120597V1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579120574
(78)
div (V) = 0 (79)
where 120585 is defined by 120585 = 120577 minus 119906119905We note
(120579 V119911) =
(1205791 V1199111) for 119911
2lt 120585
(1205792 V1199112) for 119911
2lt 120585
(80)
and consider the perturbation as follows
120579119894= 120579119894(1199112 119905) exp (119895 (119896
1119909 + 1198962119910))
V119911119894= V119911119894(1199112 119905) exp (119895 (119896
1119909 + 1198962119910))
120585 = 1205981(119905) exp (119895 (119896
1119909 + 1198962119910))
(81)
where 119896119894 119894 = 1 2 and 120598
1denote respectively the wave
numbers (in 119909 and 119910 directions) and the amplitude of theperturbation and 119895
2= minus1 Then jump conditions (69) are
linearized by taking into account the fact that
120579|120585=plusmn0 = 120579119904(plusmn0) + 120585120579
1015840
119904(plusmn0) + 120579 (plusmn0)
120597120579
1205971199112
10038161003816100381610038161003816100381610038161003816120585=plusmn0
= 1205791015840
119904(plusmn0) + 120585120579
10158401015840
119904(plusmn0) +
120597120579
1205971199112
100381610038161003816100381610038161003816100381610038161003816120585=plusmn0
(82)
Then the higher-order terms are given as
1205792
100381610038161003816100381610038161199112=0minus 1205791
100381610038161003816100381610038161199112=0= 119906120585 (83)
1205971205792
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
minus1205971205791
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
= minus1199062120585 minus
120597120585
120597119905+ V119911
10038161003816100381610038161199112=0 (84)
1199062120585 +
1205971205792
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
= minus119885
1199061205791
100381610038161003816100381610038161199112=0 (85)
V2119911
10038161003816100381610038161199112=0= V1119911
10038161003816100381610038161199112=0 (86)
120597V1199112
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
=120597V1119911
1205971199112
100381610038161003816100381610038161003816100381610038161199112=0
(87)
1205972V2119911
12059711991122
1003816100381610038161003816100381610038161003816100381610038161199112=0
=1205972V1119911
12059711991122
1003816100381610038161003816100381610038161003816100381610038161199112=0
(88)
1205973V2119911
12059711991132
1003816100381610038161003816100381610038161003816100381610038161199112=0
=1205973V1119911
12059711991132
1003816100381610038161003816100381610038161003816100381610038161199112=0
(89)
Substituting (81) into (83)ndash(89) leads to
1205792(0 119905) minus 120579
1(0 119905) = 119906120598
1(119905)
1205791015840
2(0 119905) minus 120579
1015840
1(0 119905) = minus120598
1 (119905) 1199062minus 1205981015840
1(119905) + V
1 (0 119905)
1205981 (119905) 119906
2+ 1205791015840
2(0 119905) = minus
119885
1199061205791 (0 119905)
V(119894)2(0 119905) = V(119894)
1(0 119905) 119894 = 0 1 2 3
(90)
where
V(119894) =120597119894V120597V119894
1205791015840=120597120579119894
120597119911
1205981015840
1(119905) =
1198891205981(119905)
119889119905
(91)
After simplification (75) and (78) become
120597
120597119905ΔV119911minus 119906
120597
1205971199112
ΔV119911= 119875ΔΔV
119911+ 119876(1 + 120582
1sin (120590
1119905)
+ 1205822sin (120590
2119905) (
1205972
1205971199092+
1205972
1205971199102))120579
(92)
Substituting (71) (87) (88) into (74) (77) (92) andintroducing the variable 119908 = V10158401015840 minus 1198962V where 119896 = radic1198962
1+ 11989622
one obtains two systems of equations as follows
1199112lt 120585
120597119908
120597119905minus 1199061199081015840minus 119875 (119908
10158401015840minus 1198962119908)
= minus1198761198962(1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579
119908 = V10158401015840 minus 1198962V
120597120579
120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 0
(93)
Mathematical Problems in Engineering 9
0 5 10 15 20 25 300
50
100
150
200
250
300
350
Rc
1205901
1205902 = radic212059011205902 = radic71205901
1205902 = 0
1205902 = radic371205901
Figure 1 Critical Rayleigh number as a function of 1205901for different
frequencies ratio and for 1205902= 0 120582
1= 1205822= 5 119875 = 05 119885 = 8 and
119896 = 07
1199112gt 120585
120597119908
120597119905minus 1199061199081015840minus 119875 (119908
10158401015840minus 1198962119908)
= minus1198761198962(1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579
119908 = V10158401015840 minus 1198962V
120597120579
120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911
2) V
(94)
with the following boundary conditions
1205792(0 119905) minus 120579
1(0 119905) = 119906120598
1(119905) (95)
1205791015840
2(0 119905) minus 120579
1015840
1(0 119905) = minus119906
21205761(119905) minus 120576
1015840
1(119905) + V
1(0 119905) (96)
1205791015840
2(0 119905) +
119885
1199061205791 (0 119905) = minus119906
21205761 (119905) (97)
V(119894)1(0 119905) = V(119894)
2(0 119905) 119894 = 0 1 2 3 (98)
Combining the three boundary conditions (95)ndash(97)yields
1205791015840
1(0 119905) +
119885
1199061205791(0 119905) =
1
119906(1205792(0 119905) minus 120579
1(0 119905))
1015840
119905
minus V1(0 119905)
(99)
6 Numerical Results
The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix
Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved
Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590
1for different frequencies ratio and for
0 25 50 75 100 125 1500
5
10
15
20
25
Rc
1205822
1205821 = 20
1205821 = 50
1205821 = 80
1205821 = 150
Figure 2 Critical Rayleigh number as a function of 1205822for different
values of 1205821 1205902= radic2120590
1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07
0 25 50 75 100 125 150 175 2000
50
100
150
200
250
Rc
1205822
1205902 = radic371205901
1205902 = radic1712059011205902 = radic71205901
1205902 = 1radic712059011205902 = 1radic171205901
1205902 = 1radic371205901
Figure 3 Critical Rayleigh number as a function of 1205822for different
frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590
1= 5
1205902= 0 In the absence of gravitational modulation (120590
1= 1205902=
0) we find the same result (119877119888asymp 27) as in [8 14] It can be
observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the
frequency 1205901 the frequencies ratio 120590
21205901has no effect on
the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590
2= radic37120590
1) comparing
to the periodic modulation case (1205902= 0)
The critical Rayleigh number as a function of the ampli-tude 120582
2is shown in Figure 2 for different values of 120582
1 We can
conclude that for a fixed value of 1205822and increased values of
1205821 the polymerization front loses its stability monotonically
revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation
Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582
2for different frequencies ratio
It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582
2 For
large values of 1205822(1205822ge 100) the effect of the frequencies
ratio on the convective instability boundary is insignificant
10 Mathematical Problems in Engineering
0 25 50 75 100 125 150
k = 15
k = 2
k = 1
0
50
100
150
200
250
300
350
Rc
1205822
Figure 4 Critical Rayleigh number as a function of 1205822for different
values of number wave 1205902= radic7120590
1 1205901= 10 119875 = 05 119885 = 8 and
1205821= 5
0 25 50 75 100 125 1500
255075
100125150175200
Rc
1205822
1205902 = radic21205901
1205902 = radic71205901
Figure 5 Critical Rayleigh number as a function of 1205822for two
different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and
1205901= 10
Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7
the influence of the wave number on the critical Rayleighnumber versus 120582
2 The plots indicate that for large values of
the wave number the reaction front gains stabilityThe influence of 120582
2on the stability of the reaction front is
given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect
7 Conclusion
In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590
1the
critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies
ratio and for a given amplitude 1205822an increase of the ampli-
tude1205821destabilizes the reaction front Also the reaction front
becomes more stable by increasing the wave number
Appendix
The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911
119894= 119894ℎ where ℎ is the
space step and 119894 is a positive integerThe discretization in timeis given by 119905
119894= 119894120591 where 120591 is sufficiently small time step The
variables of discretization are 119908119895119894= 119908(119911
119894 119905119895) V119895119894= V(119911
119894 119905119895)
and 120579119895119894= 120579(119911119894 119905119895) One obtains the following
In the liquid monomer 119911 lt 120577
(minus1
ℎ2) 120579119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ) 120579119895+1
119894
+ (minus1
ℎ2minus119906
ℎ) 120579119895+1
119894+1=1
120591120579119895
119894
(minus119875
ℎ2)119908119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119908119895+1
119894+1=1
120591119908119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 120579119895
119894
(1
ℎ2) V119895+1119894minus1
+ (minus2
ℎ2minus 1198962) V119895+1119894
+ (1
ℎ2) V119895+1119894+1
= 119908119895
119894
(A1)
In the liquid polymer 119911 gt 120577
(minus1
ℎ2)119879119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ)119879119895+1
119894
+ (minus1
ℎ2minus119906
ℎ)119879119895+1
119894+1=1
120591119879119895
119894+ 119906 exp (minus119906119911
119894) 119881119895
119894
(minus119875
ℎ2)119882119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119882119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
119894+1=1
120591119882119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 119879119895
119894
(1
ℎ2)119881119895+1
119894minus1+ (minus
2
ℎ2minus 1198962)119881119895+1
119894+ (
1
ℎ2)119881119895+1
119894+1= 119882119895
119894
(A2)
From the jump conditions we have also the followingsystems
(minus1
ℎ2) 120579119895+1
1
+ (1
120591+
2
ℎ2+ 1198962+119906
ℎ+ (
1198881
1198882
)(1
ℎ2+119906
ℎ)) 120579119895+1
0
Mathematical Problems in Engineering 11
= (minus1
120591) 120579119895
0minus (
1
1199061205911198882
)(1
ℎ2+119906
ℎ) (119879119895
0minus 120579119895
0)
minus (V0
1198882
)(1
ℎ2+119906
ℎ)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
1
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198882
1198881
)1198790minus
1
1199061205911198881
(119879119895
0minus 120579119895
0) minus
V0
1198881
)
(A3)
(1
120591+
2
ℎ2+ 1198962+119906
ℎ+
1198882
ℎ21198881
)119879119895+1
0+ (
minus1
ℎ2minus119906
ℎ)119879119895+1
1
= (minus1
120591)119879119895
0minus 119906 exp (minus119906ℎ) minus ( 1
119906120591ℎ21198881
) (119879119895
0minus 120579119895
0)
minusV0
ℎ21198881
(A4)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
0
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198881
1198882
)1205790minus
1
1199061205911198882
)(119879119895
0minus 120579119895
0minusV0
1198882
)
(A5)
with
1198881= (
1
ℎminus119885
119906minus
1
119906120591)
1198882= (minus
1
ℎ+
1
119906120591)
(A6)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974
[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011
[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014
[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994
[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997
[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996
[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001
[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010
[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012
[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014
[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969
[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961
[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938
[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
2 Frontal Polymerization Model
We consider a polymerization process with a liquid reactantand a liquid product by assuming that the reaction front prop-agates in the direction opposite to the direction of gravity Inthis case the model of the frontal polymerization is given bythe system of equations
120597119879
120597119905+ (V sdot nabla) 119879 = 120581Δ119879 + 119902119882
120597120572
120597119905+ (V sdot nabla) 120572 = 119882
120597V120597119905
+ (V sdot nabla) V = minus1
120588nabla119901 + ]ΔV
+ 119892 (1 + 119887 (119905)) 120573 (119879 minus 1198790) 120574
div (V) = 0
(1)
with the following boundary conditions
119911 997888rarr +infin 119879 = 119879119894 120572 = 0 V = 0 (2)
119911 997888rarr minusinfin 119879 = 119879119887 120572 = 1 V = 0 (3)
The gradient divergence and Laplace operators aredefined by
nablaV = (120597V120597119909
120597V120597119910120597V120597119911)
div k = 120597k1
120597119909+120597k2
120597119910+120597k3
120597119911
ΔV =1205972V1205972119909
+1205972V1205972119910
+1205972V1205972119911
(4)
where the variables (119909 119910 and 119911) are the spatial coordinatessuch that minusinfin lt 119909 119910 119911 lt +infin 119879 is the temperature 120572 is theconcentration of the reaction product V is the velocity 119901 isthe pressure 120581 is the coefficient of thermal diffusivity 119902 is theadiabatic temperature heat release 120588 is an average value of thedensity ] is the coefficient of kinematic viscosity 120574 is the unitvector in the upward direction 120573 is the coefficient of thermalexpansion 119892 is the gravitational acceleration and 119887(119905) is theQP acceleration acting on the fluid which is given by 119887(119905) =1205821sin(1205901119905) + 120582
2sin(1205902119905) such that 120582
1 1205822are the amplitudes
and 1205901 1205902are the incommensurate frequencies of the QP
gravitational modulation The quantity 1198790is a mean value of
temperature 119879119894is the initial temperature and 119879
119887= 119879119894+ 119902
is the temperature of the reacted mixture We consider one-step reaction of zero order where the reaction rate is definedas follows
119882 = 119896 (119879) 120601 (120572) 120601 (120572) =
1 if 120572 lt 1
0 if 120572 = 1
(5)
The temperature dependence of the reaction rate is givenby the Arrhenius law [11]
119896 (119879) = 1198960exp(minus 119864
1198770119879) (6)
where 1198960is the preexponential factor 119864 is the activation
energy and 1198770is the universal gas constant We assume
that the two liquids are incompressible and the diffusivitycoefficient is very small comparing to the thermal diffusivitycoefficient such that the diffusivity will be neglected in theconcentration equation
To obtain the dimensionless model we introduce thedimensionless spatial variables as
1199091=1199091198881
120581
1199101=1199101198881
120581
1199111=1199111198881
120581
1199051=1199051198882
1
120581
1199011=
119901
11988821120588
1198881=
119888
radic2
V1=
V1198881
120579 =119879 minus 119879119887
119902
1198882=2119896012058111987701198792
119887
119902119864exp(minus 119864
1198770119879119887
)
(7)
where 119888 defines the stationary reaction front velocity andcan be calculated asymptotically for large Zeldovich number[12] For simplicity we keep the same notation for the othervariables and pressure System (1) with the two boundaryconditions (2)-(3) can be written in the form
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 + 119885 exp( 120579
119885minus1 + 120575120579)120601 (120572) (8)
120597120572
120597119905+ (V sdot nabla) 120572 = 119885 exp( 120579
119885minus1 + 120575120579)120601 (120572) (9)
120597V120597119905
+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119875119877 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (120579 + 120579
0) 120574
(10)
div (V) = 0 (11)
with the following conditions at infinity
119911 997888rarr +infin 120579 = minus1 120572 = 0 V = 0
119911 997888rarr minusinfin 120579 = 0 120572 = 1 V = 0
(12)
and 119875 = ]120581 is the Prandtl number 119877 = 1198921205731199021205812(]1198883) is the
Rayleigh number119885 = 11990211986411987701198792
119887is the Zeldovich number and
Mathematical Problems in Engineering 3
120575 and 1205790are given respectively by 120575 = 119877
0119879119887119864 and 120579
0= (119879119887minus
1198790)119902 and 120583 = 2120581120590119888
2Next we perform the linear stability analysis to tackle the
interface problem
3 Approximation of Infinitely NarrowReaction Zone
To study the interface problem analytically we use a singularperturbation analysis where the reaction zone is infinitelynarrow and the reaction term is neglected outside the zone[13] In this way the problem can be reduced to an interfaceproblem
To perform a formal asymptotic analysis 120598 = 119885minus1 is
considered as a small parameter The new independent var-iable is given by 119911
1= 119911 minus 120577(119909 119910 119905) where 120577(119909 119910 119905) defines the
location of the reaction zone Introducing new functions 1205791
1205721 V1 and 119901
1as
120579 (119909 119910 119911 119905) = 1205791(119909 119910 119911
1 119905)
120572 (119909 119910 119911 119905) = 1205721(119909 119910 119911
1 119905)
V (119909 119910 119911 119905) = V1(119909 119910 119911
1 119905)
119901 (119909 119910 119911 119905) = 1199011(119909 119910 119911
1 119905)
(13)
The system of (8)ndash(11) can be written in the form (index 1 forthe new function is omitted)120597120579
120597119905minus120597120579
1205971199111
120597120577
120597119905+ (V sdot nabla) 120579
= Δ120579 + 119885 exp( 120579
119885minus1 + 120575120579)120601 (120572)
120597120572
120597119905minus120597120572
1205971199111
120597120577
120597119905+ (V sdot nabla) 120572 = 119885 exp( 120579
119885minus1 + 120575120579)120601 (120572)
120597V120597119905
minus120597V1205971199111
120597120577
120597119905+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (120579 + 120579
0) 120574
120597V119909
120597119909minus120597V119909
1205971199111
120597120577
120597119909+120597V119910
120597119910minus120597V119910
1205971199111
120597120577
120597119910+120597V119911
1205971199111
= 0
(14)
where Δ nabla and 119876 are given by
Δ =1205972
1205971199092+
1205972
1205971199102+
1205972
12059711991121
minus 21205972
1205971199091205971199111
120597120577
120597119909minus 2
1205972
1205971199101205971199111
120597120577
120597119910
+1205972
12059711991121
((120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus120597
1205971199111
(1205972120577
1205971199092+1205972120577
1205971199102)
nabla = (120597
120597119909minus
120597
1205971199111
120597120577
120597119909120597
120597119910minus
120597
1205971199111
120597120577
120597119910120597
1205971199111
)
119876 = 119875119877
(15)
To approximate the jump conditions and resolve theinterface problem we use the matched asymptotic expansionby seeking the outer solution of problem (14) in the form
120579 = 1205790+ 1205981205791+ sdot sdot sdot
120572 = 1205720+ 1205981205721+ sdot sdot sdot
V = V0+ 120598V1+ sdot sdot sdot
119901 = 1199010+ 1205981199011+ sdot sdot sdot
(16)
Introducing the stretched coordinate 120578 = 1199111120598minus1 where 120598 =
119885minus1 the inner solution can be approximated in the following
form
120579 = 1205981205791+ sdot sdot sdot
120572 = 0+ 1205981+ sdot sdot sdot
V = V0+ 120598V1+ sdot sdot sdot
119901 = 1199010+ 1205981199011+ sdot sdot sdot
120577 = 1205770+ 1205981205771+ sdot sdot sdot
(17)
Substituting the inner and outer solutions in (14) leads to
Order 120598minus2
119875(1 + (1205971205770
120597119909)
2
+ (1205971205770
120597119910)
2
)1205972V0
1205971205782= 0 (18)
Order 120598minus1
(1 + (1205971205770
120597119909)
2
+ (1205971205770
120597119910)
2
)12059721205791
1205971205782
+ exp( 1205791
1 + 1205751205791
)120601 (0) = 0
(19)
minus1205970
120597120578
1205971205770
120597119905minus1205970
120597120578(V0119909
1205971205770
120597119909+ V0119910
1205971205770
120597119910minus V0119911)
= exp( 1205791
1 + 1205751205791
)120601 (0)
(20)
minus120597V0119909
120597120578
1205971205770
120597119905minus V0119909
120597V0
120597120578
1205971205770
120597119909minus V0119910
120597V0
120597120578
1205971205770
120597119910+ V0119911
120597V0
120597120578
= 1199050
1205971199010
120597120578+ 119875(119860
1205972V1
1205971205782+ 1198650
120597V0
120597120578)
(21)
minus120597V0119909
120597120578
1205971205770
120597119909minus120597V0119910
120597120578
1205971205770
120597119910+120597V0119911
120597120578= 0 (22)
4 Mathematical Problems in Engineering
Order 1205980
120597V0
120597119905minus120597V1
120597120578
1205971205770
120597119905minus120597V0
120597120578
1205971205771
120597119905+ V0119909(120597V0
120597119909minus120597V1
120597120578
1205971205770
120597119909
minus120597V0
120597120578
1205971205771
120597119909) + V1119909
120597V0
120597120578
1205971205770
120597119909+ V1119910
120597V0
120597120578
1205971205770
120597119910
+ V0119910(120597V0
120597119910minus120597V1
120597120578
1205971205770
120597119910minus120597V0
120597120578
1205971205771
120597119910) + V0119911
120597V1
120597120578+ V1119911
sdot120597V0
120597120578= minusnabla01199010+ 1199051
1205971199010
120597120578+ 1199050
1205971199011
120597120578+ 119875(119860
1205972V2
1205971205782
+ 1199053
1205972V1
1205971205782+ 1198650
120597V1
120597120578+ 1199054
1205972V0
1205971205782+ 1198651
120597V0
120597120578+ Δ1V0)
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120574120579
0
(23)
120597V0119909
120597119909minus120597V1119909
120597120578
1205971205770
120597119909minus120597V0119909
120597120578
1205971205771
120597119909+120597V0119910
120597119910minus120597V1119910
120597120578
1205971205770
120597119910
minus120597V0119910
120597120578
1205971205771
120597119910+120597V1119911
120597120578= 0
(24)
Order 1205981
120597V1
120597119905minus (
120597V2
120597120578
1205971205770
120597119905+120597V1
120597120578
1205971205771
120597119905+120597V0
120597120578
1205971205772
120597119905) + V0119909(120597V1
120597119909
minus120597V1
120597120578
1205971205771
120597119909minus120597V0
120597120578
1205971205772
120597119909minus120597V2
120597120578
1205971205770
120597119909) + V1119909(120597V0
120597119909
minus120597V0
120597120578
1205971205771
120597119909minus120597V1
120597120578
1205971205770
120597119909) minus V2119909
120597V0
120597120578
1205971205770
120597119909+ V0119910(120597V1
120597119910
minus120597V1
120597120578
1205971205771
120597119910minus120597V0
120597120578
1205971205772
120597119910minus120597V2
120597120578
1205971205770
120597119910) + V1119910(120597V0
120597119910
minus120597V0
120597120578
1205971205771
120597119910minus120597V1
120597120578
1205971205770
120597119910) minus V2119910
120597V0
120597120578
1205971205770
120597119910+ V0119911
120597V2
120597120578
+ V1119911
120597V1
120597120578+ V2119911
120597V0
120597120578= 1199050
1205971199012
120597120578+ 1199051
1205971199011
120597120578minus nabla01199011+ 1199052
sdot1205971199010
120597120578+ 119875(119860
1205972V3
1205971205782+ 1199053
1205972V2
1205971205782+ 1198650
120597V2
120597120578+ 1198651
120597V1
120597120578
+ Δ1V1+ 1198652
120597V0
120597120578) + 119876 (1 + 120582
1sin (120590
1119905)
+ 1205822sin (120590
2119905)) 1205790120574
(25)
120597V1119909
120597119909minus120597V2119909
120597120578
1205971205770
120597119909minus120597V1119909
120597120578
1205971205771
120597119909minus120597V0119909
120597120578
1205971205772
120597119909+120597V1119910
120597119910
minus120597V2119910
120597120578
1205971205770
120597119910minus120597V1119910
120597120578
1205971205771
120597119910minus120597V0119910
120597120578
1205971205772
120597119910+120597V2119911
120597120578= 0
(26)
Order 1205982
120597V2119909
120597119909minus120597V3119909
120597120578
1205971205770
120597119909minus120597V2119909
120597120578
1205971205771
120597119909minus120597V1119909
120597120578
1205971205772
120597119909minus120597V0119909
120597120578
1205971205773
120597119909
+120597V2119910
120597119910minus120597V3119910
120597120578
1205971205770
120597119910minus120597V2119910
120597120578
1205971205771
120597119910minus120597V1119910
120597120578
1205971205772
120597119910
minus120597V0119910
120597120578
1205971205773
120597119910+120597V3119911
120597120578= 0
(27)
where
Δ1=
1205972
1205971199092+
1205972
1205971199102
nabla0= (
120597
120597119909120597
120597119910 0)
119865119894= minus2(
120597120577119894
120597119909
120597
120597119909+120597120577119894
120597119910
120597
120597119910) minus (
1205972120577119894
1205971199092+1205972120577119894
1205971199102) 119868
119894 = 0 1 2
1199050= (
1205971205770
1205971199091205971205770
120597119910 minus1)
119905119894= (
120597120577119894
120597119909120597120577119894
120597119910 minus 1) 119894 = 1 2
1199053= 2(
1205971205771
120597119909
1205971205770
120597119909+1205971205770
120597119910
1205971205771
120597119910)
1199054= (
1205971205771
120597119909)
2
+ (1205971205771
120597119910)
2
+1205971205770
120597119909
1205971205772
120597119909+1205971205770
120597119910
1205971205772
120597119910
119860 = (1 + (1205971205770
120597119909)
2
+ (1205971205770
120597119910)
2
)
(28)
and 119868 is the identity operator The matching conditions as120578 rarr plusmninfin are given by
V0sim V0
10038161003816100381610038161199111=plusmn0
(29)
V1sim (
120597V0
1205971199111
100381610038161003816100381610038161003816100381610038161199111=plusmn0
)120578 + V1
10038161003816100381610038161199111=plusmn0
(30)
V2sim1
2(1205972V0
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0
)1205782+ (
120597V1
1205971199111
100381610038161003816100381610038161003816100381610038161199111=plusmn0
)120578
+ V2
10038161003816100381610038161199111=plusmn0
(31)
V3sim1
6(1205973V0
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0
)1205783+1
2(1205972V1
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0
)1205782
+ (120597V2
1205971199111
100381610038161003816100381610038161003816100381610038161199111=plusmn0
)120578 + V3
10038161003816100381610038161199111=plusmn0
(32)
Mathematical Problems in Engineering 5
As 120578 rarr +infin
1205791asymp 1205791
10038161003816100381610038161199111=+0
+ (1205971205790
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
)120578 0997888rarr 0 (33)
As 120578 rarr minusinfin
1205791sim 1205791
10038161003816100381610038161199111=minus0
0997888rarr 1 (34)
From (18) we obtain
1205972V0
1205971205782= 0 (35)
Consequently we can conclude that V0(120578) is a linear
function of 120578 and identically constant since the velocity isbounded Equation (29) becomes
V0
10038161003816100381610038161199111=+0
= V0
10038161003816100381610038161199111=minus0
(36)
120597V0
120597120578= 0 (37)
From (36)-(37) we deduce that the first term in theexpression of the velocity V
0is continuous at the front By
substituting (37) into (21) we find
1198751198601205972V1
1205971205782+ 1199050
1205971199010
120597120578= 0 (38)
120597V0119909
120597119909minus120597V1119909
120597120578
1205971205770
120597119909+120597V0119910
120597119910minus120597V1119910
120597120578
1205971205770
120597119910+120597V1119911
120597120578= 0 (39)
Differentiating (39) with respect to 120578 one obtains
1205972V1119909
1205971205782
1205971205770
120597119909+1205972V1119910
1205971205782
1205971205770
120597119910minus1205972V1119911
1205971205782= 0 (40)
As a result (38) is a vectorial equation with three com-ponents We multiply the first component by 120597120577
0120597119909 the
second by 1205971205770120597119910 and the third by minus1 and adding we have
119875119860(1205972V1119909
1205971205782
1205971205770
120597119909+1205972V1119910
1205971205782
1205971205770
120597119910minus1205972V1119911
1205971205782) + 119860
1205971199010
120597120578= 0 (41)
Considering the two equations (38) and (40) we obtain
1205971199010
120597120578= 0 (42)
1205972V1
1205971205782= 0 (43)
From the boundedness of the velocity and (30) the con-tinuity of the first order term of the outer expansion and ofthe first derivative of the zero-order term we can write
120597V0
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
=120597V0
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
V1
10038161003816100381610038161199111=+0
= V1
10038161003816100381610038161199111=minus0
(44)
Differentiating (23) once and (26) twice with respect to 120578and using the three equations (37) (42) and (43) we get
1199050
12059721199011
1205971205782+ 119875119860
1205973V2
1205971205783= 0 (45)
1205973V2119909
1205971205783
1205971205770
120597119909+1205973V2119910
1205971205783
1205971205770
120597119910minus1205973V2119911
1205971205783= 0 (46)
As above we multiply the three components of (45)respectively by 120597120577
0120597119909 120597120577
0120597119910 and minus1 and adding we get
119875119860(1205973V2119909
1205971205783
1205971205770
120597119909+1205973V2119910
1205971205783
1205971205770
120597119910minus1205973V2119911
1205971205783) + 119860
12059721199011
1205971205782= 0 (47)
From the previous equation and using (45)-(46) we have
12059721199011
1205971205782= 0
1205973V2
1205971205783= 0
(48)
Knowing that the velocity is bounded and considering(31) we obtain the jump conditions in the following forms
1205972V0
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=+0
=1205972V0
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
120597V1
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
=120597V1
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
V2
10038161003816100381610038161199111=+0
= V2
10038161003816100381610038161199111=minus0
(49)
Differentiating (25) twice and (27) three times withrespect to 120578 and taking into account (37) (42) (48) we obtain
1199050
12059731199012
1205971205783+ 119875119860
1205974V3
1205971205784
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120574
12059721205791
1205971205782= 0
1205974V3119909
1205971205784
1205971205770
120597119909+1205974V3119910
1205971205784
1205971205770
120597119910minus1205974V3119911
1205971205784= 0
(50)
119875119860(1205974V3119909
1205971205784
1205971205770
120597119909+1205974V3119910
1205971205784
1205971205770
120597119910minus1205974V3119911
1205971205784) + 119860
12059731199012
1205971205783
minus 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905))
12059721205791
1205971205782= 0
(51)
From the last equation and using (50) we can write
11986012059731199012
1205971205783minus 119876 (1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905))
12059721205791
1205971205782
= 0
(52)
1205740
12059721205791
1205971205782=1205974V3
1205971205784 (53)
6 Mathematical Problems in Engineering
where 1205740is defined by
1205740= (minus
1205971205770
120597119909
119877
1198602 minus1205971205770
120597119910
119877
1198602119877
1198602minus119877
119860)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905))
(54)
Now integrating (53) with respect to 120578 using the system(32)ndash(34) lead to
1205973V0
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
minus1205973V0
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=+0
= minus1205740
1205971205790
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
(55)
As a first conclusion the velocity jump conditions acrossthe reaction front are given by (36) (44) (49) and (55)
From (20) (22) (37) we conclude that 0is a monotonic
function satisfying 0 lt 0lt 1 Therefore the reaction is of
zero order and 120601(0) equiv 1 Multiplying (19) by 120597120579
1120597120578 and
integrating the result yield
(1205971205791
120597120578)
210038161003816100381610038161003816100381610038161003816100381610038161003816+infin
minus (1205971205791
120597120578)
210038161003816100381610038161003816100381610038161003816100381610038161003816minusinfin
= 2119860minus1int
1205791|1199111=minus0
minusinfin
exp( 120591
1 + 120591120575) 119889120591
(56)
Subtracting (19) from (20) and integrating the result oneobtains
1205971205791
120597120578
100381610038161003816100381610038161003816100381610038161003816+infin
minus1205971205791
120597120578
100381610038161003816100381610038161003816100381610038161003816minusinfin
= minus119860minus1(1205971205770
120597119905+ 119904) (57)
where 119904 is given by
119904 = V0119909
1205971205770
120597119909+ V0119910
1205971205770
120597119910minus V0119911 (58)
As a second conclusion the temperature jump conditionsacross the reaction front are given by the two last equations(56)-(57) By using the matching conditions above andtruncating the expansion as
120579 asymp 1205790
1205791
10038161003816100381610038161199111=minus0
asymp 119885120579|1199111=+0
120577 asymp 1205770
V asymp V0
(59)
the jump conditions can be written as
(120597120579
1205971199111
)
2100381610038161003816100381610038161003816100381610038161003816+0
minus (120597120579
1205971199111
)
2100381610038161003816100381610038161003816100381610038161003816minus0
= 2119885(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot int
1205791|1199111=minus0
minusinfin
exp ( 120591
119885minus1 + 120591120575) 119889120591
120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
minus120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
= minus(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot (120597120577
120597119905+ (V119909
120597120577
120597119909+ V119910
120597120577
120597119910minus V119911)
100381610038161003816100381610038161003816100381610038161199111=+0
)
V119911
10038161003816100381610038161199111=+0
= V119911
10038161003816100381610038161199111=minus0
120597V119911
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
=120597V119911
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
1205972V119911
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=+0
=120597V119911
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
1205973V119911
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
minus1205973V119911
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=+0
= minus119877(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot ((1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
minus 1)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905))
120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
(60)
4 The Interface Problem and Perturbation
To study the propagation of polymerization front with aliquid reactant and liquid product the equation of motionhas been considered before and after the reaction zone Thischanges the jump conditions and influences the stabilityconditions of the frontal polymerization process Notice thatin the case of liquid-solid polymerization front the equationof motion is considered after the reaction zone The originalsystem (2) (3) (8) (9) leads to the following interfaceproblem
In the liquid monomer 119911 gt 120577
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 (61)
120572 = 0 (62)
Mathematical Problems in Engineering 7
120597V120597119905
+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579
0) 120574
(63)
div (V) = 0 (64)
In the liquid polymer 119911 lt 120577
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 (65)
120572 = 1 (66)
120597V120597119905
+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579
0) 120574
(67)
div (V) = 0 (68)
At the interface 119911 = 120577
120579|120577minus0 = 120579|120577+0
120597120579
120597119911
10038161003816100381610038161003816100381610038161003816120577=minus0minus120597120579
120597119911
10038161003816100381610038161003816100381610038161003816120577=+0= (1 + (
120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot (120597120577
120597119905+ (V119909
120597120577
120597119909+ V119910
120597120577
120597119910minus V119911)
10038161003816100381610038161003816100381610038161003816120577
)
(120597120579
120597119911)
2100381610038161003816100381610038161003816100381610038161003816120577minus0
minus (120597120579
120597119911)
21003816100381610038161003816100381610038161003816100381610038161205771+0
= minus2119885(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot int
120579|120577
minusinfin
exp ( 120591
119885minus1 + 120575120591) 119889120591
V119911
1003816100381610038161003816120577=minus0 = V119911
1003816100381610038161003816120577=+0
120597V119911
120597119911
10038161003816100381610038161003816100381610038161003816120577=minus0=120597V119911
120597119911
10038161003816100381610038161003816100381610038161003816120577=+0
1205972V119911
1205971199112
100381610038161003816100381610038161003816100381610038161003816120577=minus0
=1205972V119911
1205971199112
100381610038161003816100381610038161003816100381610038161003816120577=+0
1205973V119911
1205971199113
100381610038161003816100381610038161003816100381610038161003816120577=minus0
minus1205973V119911
1205971199113
100381610038161003816100381610038161003816100381610038161003816120577=+0
= minus119877(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot ((1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
minus 1)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (
120597120579
120597119911)
10038161003816100381610038161003816100381610038161003816120577=+0
(69)
with the conditions at infinity
119911 = minusinfin 120579 = 0 V = 0
119911 = +infin 120579 = minus1 V = 0
(70)
5 Stability Analysis
The interface problem analyzed in Section 4 has a travellingwave solution in the following form
(120579 (119909 119910 119911 119905) 120572 (119909 119910 119911 119905) V)
= (120579119904(119911 minus 119906119905) 120572
119904(119911 minus 119906119905) 0)
(120579119904 (119911 minus 119906119905) 120572119904 (119911 minus 119906119905))
=
(0 1) 1199112lt 0
(exp (minus1199061199112) minus 1 0) 119911
2gt 0
(71)
where 1199112= 119911minus119906119905 and 119906 is the speed of the stationary reaction
frontThis solution is a basic and stationary solution of inter-face problem (61)ndash(70) written in the moving coordinatesThen (61) (63) (65) and (67) can be replaced by
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 + 119906
120597120579
1205971199112
120597V120597119905
+ (Vnabla) V
= minusnabla119901 + 119875ΔV + 119906120597120579
1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (120579 + 120579
0) 120574
(72)
The other equations remain unchanged To study thestability of reaction front we seek the solution of the problemas follows
120579 = 120579119904+ 120579
119901 = 119901119904+ 119901
V = V119904+ V
(73)
where 120579 119901 and V are respectively small perturbation oftemperature pressure and velocity
8 Mathematical Problems in Engineering
Subtituting (73) into (64) (72) one obtains the first-orderterms as follows
In the liquid monomer 1199112gt 120585
120597120579
120597119905= Δ120579 + 119906
120597120579
1205971199112
minus V1199111205791015840
119904 (74)
120597V120597119905
= minusnabla119901 + 119875ΔV + 119906120597120579
1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579120574
(75)
div (V) = 0 (76)
In the liquid polymer 1199112lt 120585
120597120579
120597119905= Δ120579 + 119906
120597120579
1205971199112
(77)
120597V120597119905
= minusnabla119901 + 119875ΔV + 119906120597V1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579120574
(78)
div (V) = 0 (79)
where 120585 is defined by 120585 = 120577 minus 119906119905We note
(120579 V119911) =
(1205791 V1199111) for 119911
2lt 120585
(1205792 V1199112) for 119911
2lt 120585
(80)
and consider the perturbation as follows
120579119894= 120579119894(1199112 119905) exp (119895 (119896
1119909 + 1198962119910))
V119911119894= V119911119894(1199112 119905) exp (119895 (119896
1119909 + 1198962119910))
120585 = 1205981(119905) exp (119895 (119896
1119909 + 1198962119910))
(81)
where 119896119894 119894 = 1 2 and 120598
1denote respectively the wave
numbers (in 119909 and 119910 directions) and the amplitude of theperturbation and 119895
2= minus1 Then jump conditions (69) are
linearized by taking into account the fact that
120579|120585=plusmn0 = 120579119904(plusmn0) + 120585120579
1015840
119904(plusmn0) + 120579 (plusmn0)
120597120579
1205971199112
10038161003816100381610038161003816100381610038161003816120585=plusmn0
= 1205791015840
119904(plusmn0) + 120585120579
10158401015840
119904(plusmn0) +
120597120579
1205971199112
100381610038161003816100381610038161003816100381610038161003816120585=plusmn0
(82)
Then the higher-order terms are given as
1205792
100381610038161003816100381610038161199112=0minus 1205791
100381610038161003816100381610038161199112=0= 119906120585 (83)
1205971205792
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
minus1205971205791
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
= minus1199062120585 minus
120597120585
120597119905+ V119911
10038161003816100381610038161199112=0 (84)
1199062120585 +
1205971205792
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
= minus119885
1199061205791
100381610038161003816100381610038161199112=0 (85)
V2119911
10038161003816100381610038161199112=0= V1119911
10038161003816100381610038161199112=0 (86)
120597V1199112
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
=120597V1119911
1205971199112
100381610038161003816100381610038161003816100381610038161199112=0
(87)
1205972V2119911
12059711991122
1003816100381610038161003816100381610038161003816100381610038161199112=0
=1205972V1119911
12059711991122
1003816100381610038161003816100381610038161003816100381610038161199112=0
(88)
1205973V2119911
12059711991132
1003816100381610038161003816100381610038161003816100381610038161199112=0
=1205973V1119911
12059711991132
1003816100381610038161003816100381610038161003816100381610038161199112=0
(89)
Substituting (81) into (83)ndash(89) leads to
1205792(0 119905) minus 120579
1(0 119905) = 119906120598
1(119905)
1205791015840
2(0 119905) minus 120579
1015840
1(0 119905) = minus120598
1 (119905) 1199062minus 1205981015840
1(119905) + V
1 (0 119905)
1205981 (119905) 119906
2+ 1205791015840
2(0 119905) = minus
119885
1199061205791 (0 119905)
V(119894)2(0 119905) = V(119894)
1(0 119905) 119894 = 0 1 2 3
(90)
where
V(119894) =120597119894V120597V119894
1205791015840=120597120579119894
120597119911
1205981015840
1(119905) =
1198891205981(119905)
119889119905
(91)
After simplification (75) and (78) become
120597
120597119905ΔV119911minus 119906
120597
1205971199112
ΔV119911= 119875ΔΔV
119911+ 119876(1 + 120582
1sin (120590
1119905)
+ 1205822sin (120590
2119905) (
1205972
1205971199092+
1205972
1205971199102))120579
(92)
Substituting (71) (87) (88) into (74) (77) (92) andintroducing the variable 119908 = V10158401015840 minus 1198962V where 119896 = radic1198962
1+ 11989622
one obtains two systems of equations as follows
1199112lt 120585
120597119908
120597119905minus 1199061199081015840minus 119875 (119908
10158401015840minus 1198962119908)
= minus1198761198962(1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579
119908 = V10158401015840 minus 1198962V
120597120579
120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 0
(93)
Mathematical Problems in Engineering 9
0 5 10 15 20 25 300
50
100
150
200
250
300
350
Rc
1205901
1205902 = radic212059011205902 = radic71205901
1205902 = 0
1205902 = radic371205901
Figure 1 Critical Rayleigh number as a function of 1205901for different
frequencies ratio and for 1205902= 0 120582
1= 1205822= 5 119875 = 05 119885 = 8 and
119896 = 07
1199112gt 120585
120597119908
120597119905minus 1199061199081015840minus 119875 (119908
10158401015840minus 1198962119908)
= minus1198761198962(1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579
119908 = V10158401015840 minus 1198962V
120597120579
120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911
2) V
(94)
with the following boundary conditions
1205792(0 119905) minus 120579
1(0 119905) = 119906120598
1(119905) (95)
1205791015840
2(0 119905) minus 120579
1015840
1(0 119905) = minus119906
21205761(119905) minus 120576
1015840
1(119905) + V
1(0 119905) (96)
1205791015840
2(0 119905) +
119885
1199061205791 (0 119905) = minus119906
21205761 (119905) (97)
V(119894)1(0 119905) = V(119894)
2(0 119905) 119894 = 0 1 2 3 (98)
Combining the three boundary conditions (95)ndash(97)yields
1205791015840
1(0 119905) +
119885
1199061205791(0 119905) =
1
119906(1205792(0 119905) minus 120579
1(0 119905))
1015840
119905
minus V1(0 119905)
(99)
6 Numerical Results
The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix
Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved
Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590
1for different frequencies ratio and for
0 25 50 75 100 125 1500
5
10
15
20
25
Rc
1205822
1205821 = 20
1205821 = 50
1205821 = 80
1205821 = 150
Figure 2 Critical Rayleigh number as a function of 1205822for different
values of 1205821 1205902= radic2120590
1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07
0 25 50 75 100 125 150 175 2000
50
100
150
200
250
Rc
1205822
1205902 = radic371205901
1205902 = radic1712059011205902 = radic71205901
1205902 = 1radic712059011205902 = 1radic171205901
1205902 = 1radic371205901
Figure 3 Critical Rayleigh number as a function of 1205822for different
frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590
1= 5
1205902= 0 In the absence of gravitational modulation (120590
1= 1205902=
0) we find the same result (119877119888asymp 27) as in [8 14] It can be
observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the
frequency 1205901 the frequencies ratio 120590
21205901has no effect on
the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590
2= radic37120590
1) comparing
to the periodic modulation case (1205902= 0)
The critical Rayleigh number as a function of the ampli-tude 120582
2is shown in Figure 2 for different values of 120582
1 We can
conclude that for a fixed value of 1205822and increased values of
1205821 the polymerization front loses its stability monotonically
revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation
Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582
2for different frequencies ratio
It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582
2 For
large values of 1205822(1205822ge 100) the effect of the frequencies
ratio on the convective instability boundary is insignificant
10 Mathematical Problems in Engineering
0 25 50 75 100 125 150
k = 15
k = 2
k = 1
0
50
100
150
200
250
300
350
Rc
1205822
Figure 4 Critical Rayleigh number as a function of 1205822for different
values of number wave 1205902= radic7120590
1 1205901= 10 119875 = 05 119885 = 8 and
1205821= 5
0 25 50 75 100 125 1500
255075
100125150175200
Rc
1205822
1205902 = radic21205901
1205902 = radic71205901
Figure 5 Critical Rayleigh number as a function of 1205822for two
different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and
1205901= 10
Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7
the influence of the wave number on the critical Rayleighnumber versus 120582
2 The plots indicate that for large values of
the wave number the reaction front gains stabilityThe influence of 120582
2on the stability of the reaction front is
given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect
7 Conclusion
In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590
1the
critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies
ratio and for a given amplitude 1205822an increase of the ampli-
tude1205821destabilizes the reaction front Also the reaction front
becomes more stable by increasing the wave number
Appendix
The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911
119894= 119894ℎ where ℎ is the
space step and 119894 is a positive integerThe discretization in timeis given by 119905
119894= 119894120591 where 120591 is sufficiently small time step The
variables of discretization are 119908119895119894= 119908(119911
119894 119905119895) V119895119894= V(119911
119894 119905119895)
and 120579119895119894= 120579(119911119894 119905119895) One obtains the following
In the liquid monomer 119911 lt 120577
(minus1
ℎ2) 120579119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ) 120579119895+1
119894
+ (minus1
ℎ2minus119906
ℎ) 120579119895+1
119894+1=1
120591120579119895
119894
(minus119875
ℎ2)119908119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119908119895+1
119894+1=1
120591119908119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 120579119895
119894
(1
ℎ2) V119895+1119894minus1
+ (minus2
ℎ2minus 1198962) V119895+1119894
+ (1
ℎ2) V119895+1119894+1
= 119908119895
119894
(A1)
In the liquid polymer 119911 gt 120577
(minus1
ℎ2)119879119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ)119879119895+1
119894
+ (minus1
ℎ2minus119906
ℎ)119879119895+1
119894+1=1
120591119879119895
119894+ 119906 exp (minus119906119911
119894) 119881119895
119894
(minus119875
ℎ2)119882119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119882119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
119894+1=1
120591119882119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 119879119895
119894
(1
ℎ2)119881119895+1
119894minus1+ (minus
2
ℎ2minus 1198962)119881119895+1
119894+ (
1
ℎ2)119881119895+1
119894+1= 119882119895
119894
(A2)
From the jump conditions we have also the followingsystems
(minus1
ℎ2) 120579119895+1
1
+ (1
120591+
2
ℎ2+ 1198962+119906
ℎ+ (
1198881
1198882
)(1
ℎ2+119906
ℎ)) 120579119895+1
0
Mathematical Problems in Engineering 11
= (minus1
120591) 120579119895
0minus (
1
1199061205911198882
)(1
ℎ2+119906
ℎ) (119879119895
0minus 120579119895
0)
minus (V0
1198882
)(1
ℎ2+119906
ℎ)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
1
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198882
1198881
)1198790minus
1
1199061205911198881
(119879119895
0minus 120579119895
0) minus
V0
1198881
)
(A3)
(1
120591+
2
ℎ2+ 1198962+119906
ℎ+
1198882
ℎ21198881
)119879119895+1
0+ (
minus1
ℎ2minus119906
ℎ)119879119895+1
1
= (minus1
120591)119879119895
0minus 119906 exp (minus119906ℎ) minus ( 1
119906120591ℎ21198881
) (119879119895
0minus 120579119895
0)
minusV0
ℎ21198881
(A4)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
0
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198881
1198882
)1205790minus
1
1199061205911198882
)(119879119895
0minus 120579119895
0minusV0
1198882
)
(A5)
with
1198881= (
1
ℎminus119885
119906minus
1
119906120591)
1198882= (minus
1
ℎ+
1
119906120591)
(A6)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974
[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011
[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014
[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994
[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997
[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996
[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001
[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010
[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012
[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014
[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969
[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961
[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938
[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998
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Mathematical Problems in Engineering 3
120575 and 1205790are given respectively by 120575 = 119877
0119879119887119864 and 120579
0= (119879119887minus
1198790)119902 and 120583 = 2120581120590119888
2Next we perform the linear stability analysis to tackle the
interface problem
3 Approximation of Infinitely NarrowReaction Zone
To study the interface problem analytically we use a singularperturbation analysis where the reaction zone is infinitelynarrow and the reaction term is neglected outside the zone[13] In this way the problem can be reduced to an interfaceproblem
To perform a formal asymptotic analysis 120598 = 119885minus1 is
considered as a small parameter The new independent var-iable is given by 119911
1= 119911 minus 120577(119909 119910 119905) where 120577(119909 119910 119905) defines the
location of the reaction zone Introducing new functions 1205791
1205721 V1 and 119901
1as
120579 (119909 119910 119911 119905) = 1205791(119909 119910 119911
1 119905)
120572 (119909 119910 119911 119905) = 1205721(119909 119910 119911
1 119905)
V (119909 119910 119911 119905) = V1(119909 119910 119911
1 119905)
119901 (119909 119910 119911 119905) = 1199011(119909 119910 119911
1 119905)
(13)
The system of (8)ndash(11) can be written in the form (index 1 forthe new function is omitted)120597120579
120597119905minus120597120579
1205971199111
120597120577
120597119905+ (V sdot nabla) 120579
= Δ120579 + 119885 exp( 120579
119885minus1 + 120575120579)120601 (120572)
120597120572
120597119905minus120597120572
1205971199111
120597120577
120597119905+ (V sdot nabla) 120572 = 119885 exp( 120579
119885minus1 + 120575120579)120601 (120572)
120597V120597119905
minus120597V1205971199111
120597120577
120597119905+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (120579 + 120579
0) 120574
120597V119909
120597119909minus120597V119909
1205971199111
120597120577
120597119909+120597V119910
120597119910minus120597V119910
1205971199111
120597120577
120597119910+120597V119911
1205971199111
= 0
(14)
where Δ nabla and 119876 are given by
Δ =1205972
1205971199092+
1205972
1205971199102+
1205972
12059711991121
minus 21205972
1205971199091205971199111
120597120577
120597119909minus 2
1205972
1205971199101205971199111
120597120577
120597119910
+1205972
12059711991121
((120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus120597
1205971199111
(1205972120577
1205971199092+1205972120577
1205971199102)
nabla = (120597
120597119909minus
120597
1205971199111
120597120577
120597119909120597
120597119910minus
120597
1205971199111
120597120577
120597119910120597
1205971199111
)
119876 = 119875119877
(15)
To approximate the jump conditions and resolve theinterface problem we use the matched asymptotic expansionby seeking the outer solution of problem (14) in the form
120579 = 1205790+ 1205981205791+ sdot sdot sdot
120572 = 1205720+ 1205981205721+ sdot sdot sdot
V = V0+ 120598V1+ sdot sdot sdot
119901 = 1199010+ 1205981199011+ sdot sdot sdot
(16)
Introducing the stretched coordinate 120578 = 1199111120598minus1 where 120598 =
119885minus1 the inner solution can be approximated in the following
form
120579 = 1205981205791+ sdot sdot sdot
120572 = 0+ 1205981+ sdot sdot sdot
V = V0+ 120598V1+ sdot sdot sdot
119901 = 1199010+ 1205981199011+ sdot sdot sdot
120577 = 1205770+ 1205981205771+ sdot sdot sdot
(17)
Substituting the inner and outer solutions in (14) leads to
Order 120598minus2
119875(1 + (1205971205770
120597119909)
2
+ (1205971205770
120597119910)
2
)1205972V0
1205971205782= 0 (18)
Order 120598minus1
(1 + (1205971205770
120597119909)
2
+ (1205971205770
120597119910)
2
)12059721205791
1205971205782
+ exp( 1205791
1 + 1205751205791
)120601 (0) = 0
(19)
minus1205970
120597120578
1205971205770
120597119905minus1205970
120597120578(V0119909
1205971205770
120597119909+ V0119910
1205971205770
120597119910minus V0119911)
= exp( 1205791
1 + 1205751205791
)120601 (0)
(20)
minus120597V0119909
120597120578
1205971205770
120597119905minus V0119909
120597V0
120597120578
1205971205770
120597119909minus V0119910
120597V0
120597120578
1205971205770
120597119910+ V0119911
120597V0
120597120578
= 1199050
1205971199010
120597120578+ 119875(119860
1205972V1
1205971205782+ 1198650
120597V0
120597120578)
(21)
minus120597V0119909
120597120578
1205971205770
120597119909minus120597V0119910
120597120578
1205971205770
120597119910+120597V0119911
120597120578= 0 (22)
4 Mathematical Problems in Engineering
Order 1205980
120597V0
120597119905minus120597V1
120597120578
1205971205770
120597119905minus120597V0
120597120578
1205971205771
120597119905+ V0119909(120597V0
120597119909minus120597V1
120597120578
1205971205770
120597119909
minus120597V0
120597120578
1205971205771
120597119909) + V1119909
120597V0
120597120578
1205971205770
120597119909+ V1119910
120597V0
120597120578
1205971205770
120597119910
+ V0119910(120597V0
120597119910minus120597V1
120597120578
1205971205770
120597119910minus120597V0
120597120578
1205971205771
120597119910) + V0119911
120597V1
120597120578+ V1119911
sdot120597V0
120597120578= minusnabla01199010+ 1199051
1205971199010
120597120578+ 1199050
1205971199011
120597120578+ 119875(119860
1205972V2
1205971205782
+ 1199053
1205972V1
1205971205782+ 1198650
120597V1
120597120578+ 1199054
1205972V0
1205971205782+ 1198651
120597V0
120597120578+ Δ1V0)
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120574120579
0
(23)
120597V0119909
120597119909minus120597V1119909
120597120578
1205971205770
120597119909minus120597V0119909
120597120578
1205971205771
120597119909+120597V0119910
120597119910minus120597V1119910
120597120578
1205971205770
120597119910
minus120597V0119910
120597120578
1205971205771
120597119910+120597V1119911
120597120578= 0
(24)
Order 1205981
120597V1
120597119905minus (
120597V2
120597120578
1205971205770
120597119905+120597V1
120597120578
1205971205771
120597119905+120597V0
120597120578
1205971205772
120597119905) + V0119909(120597V1
120597119909
minus120597V1
120597120578
1205971205771
120597119909minus120597V0
120597120578
1205971205772
120597119909minus120597V2
120597120578
1205971205770
120597119909) + V1119909(120597V0
120597119909
minus120597V0
120597120578
1205971205771
120597119909minus120597V1
120597120578
1205971205770
120597119909) minus V2119909
120597V0
120597120578
1205971205770
120597119909+ V0119910(120597V1
120597119910
minus120597V1
120597120578
1205971205771
120597119910minus120597V0
120597120578
1205971205772
120597119910minus120597V2
120597120578
1205971205770
120597119910) + V1119910(120597V0
120597119910
minus120597V0
120597120578
1205971205771
120597119910minus120597V1
120597120578
1205971205770
120597119910) minus V2119910
120597V0
120597120578
1205971205770
120597119910+ V0119911
120597V2
120597120578
+ V1119911
120597V1
120597120578+ V2119911
120597V0
120597120578= 1199050
1205971199012
120597120578+ 1199051
1205971199011
120597120578minus nabla01199011+ 1199052
sdot1205971199010
120597120578+ 119875(119860
1205972V3
1205971205782+ 1199053
1205972V2
1205971205782+ 1198650
120597V2
120597120578+ 1198651
120597V1
120597120578
+ Δ1V1+ 1198652
120597V0
120597120578) + 119876 (1 + 120582
1sin (120590
1119905)
+ 1205822sin (120590
2119905)) 1205790120574
(25)
120597V1119909
120597119909minus120597V2119909
120597120578
1205971205770
120597119909minus120597V1119909
120597120578
1205971205771
120597119909minus120597V0119909
120597120578
1205971205772
120597119909+120597V1119910
120597119910
minus120597V2119910
120597120578
1205971205770
120597119910minus120597V1119910
120597120578
1205971205771
120597119910minus120597V0119910
120597120578
1205971205772
120597119910+120597V2119911
120597120578= 0
(26)
Order 1205982
120597V2119909
120597119909minus120597V3119909
120597120578
1205971205770
120597119909minus120597V2119909
120597120578
1205971205771
120597119909minus120597V1119909
120597120578
1205971205772
120597119909minus120597V0119909
120597120578
1205971205773
120597119909
+120597V2119910
120597119910minus120597V3119910
120597120578
1205971205770
120597119910minus120597V2119910
120597120578
1205971205771
120597119910minus120597V1119910
120597120578
1205971205772
120597119910
minus120597V0119910
120597120578
1205971205773
120597119910+120597V3119911
120597120578= 0
(27)
where
Δ1=
1205972
1205971199092+
1205972
1205971199102
nabla0= (
120597
120597119909120597
120597119910 0)
119865119894= minus2(
120597120577119894
120597119909
120597
120597119909+120597120577119894
120597119910
120597
120597119910) minus (
1205972120577119894
1205971199092+1205972120577119894
1205971199102) 119868
119894 = 0 1 2
1199050= (
1205971205770
1205971199091205971205770
120597119910 minus1)
119905119894= (
120597120577119894
120597119909120597120577119894
120597119910 minus 1) 119894 = 1 2
1199053= 2(
1205971205771
120597119909
1205971205770
120597119909+1205971205770
120597119910
1205971205771
120597119910)
1199054= (
1205971205771
120597119909)
2
+ (1205971205771
120597119910)
2
+1205971205770
120597119909
1205971205772
120597119909+1205971205770
120597119910
1205971205772
120597119910
119860 = (1 + (1205971205770
120597119909)
2
+ (1205971205770
120597119910)
2
)
(28)
and 119868 is the identity operator The matching conditions as120578 rarr plusmninfin are given by
V0sim V0
10038161003816100381610038161199111=plusmn0
(29)
V1sim (
120597V0
1205971199111
100381610038161003816100381610038161003816100381610038161199111=plusmn0
)120578 + V1
10038161003816100381610038161199111=plusmn0
(30)
V2sim1
2(1205972V0
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0
)1205782+ (
120597V1
1205971199111
100381610038161003816100381610038161003816100381610038161199111=plusmn0
)120578
+ V2
10038161003816100381610038161199111=plusmn0
(31)
V3sim1
6(1205973V0
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0
)1205783+1
2(1205972V1
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0
)1205782
+ (120597V2
1205971199111
100381610038161003816100381610038161003816100381610038161199111=plusmn0
)120578 + V3
10038161003816100381610038161199111=plusmn0
(32)
Mathematical Problems in Engineering 5
As 120578 rarr +infin
1205791asymp 1205791
10038161003816100381610038161199111=+0
+ (1205971205790
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
)120578 0997888rarr 0 (33)
As 120578 rarr minusinfin
1205791sim 1205791
10038161003816100381610038161199111=minus0
0997888rarr 1 (34)
From (18) we obtain
1205972V0
1205971205782= 0 (35)
Consequently we can conclude that V0(120578) is a linear
function of 120578 and identically constant since the velocity isbounded Equation (29) becomes
V0
10038161003816100381610038161199111=+0
= V0
10038161003816100381610038161199111=minus0
(36)
120597V0
120597120578= 0 (37)
From (36)-(37) we deduce that the first term in theexpression of the velocity V
0is continuous at the front By
substituting (37) into (21) we find
1198751198601205972V1
1205971205782+ 1199050
1205971199010
120597120578= 0 (38)
120597V0119909
120597119909minus120597V1119909
120597120578
1205971205770
120597119909+120597V0119910
120597119910minus120597V1119910
120597120578
1205971205770
120597119910+120597V1119911
120597120578= 0 (39)
Differentiating (39) with respect to 120578 one obtains
1205972V1119909
1205971205782
1205971205770
120597119909+1205972V1119910
1205971205782
1205971205770
120597119910minus1205972V1119911
1205971205782= 0 (40)
As a result (38) is a vectorial equation with three com-ponents We multiply the first component by 120597120577
0120597119909 the
second by 1205971205770120597119910 and the third by minus1 and adding we have
119875119860(1205972V1119909
1205971205782
1205971205770
120597119909+1205972V1119910
1205971205782
1205971205770
120597119910minus1205972V1119911
1205971205782) + 119860
1205971199010
120597120578= 0 (41)
Considering the two equations (38) and (40) we obtain
1205971199010
120597120578= 0 (42)
1205972V1
1205971205782= 0 (43)
From the boundedness of the velocity and (30) the con-tinuity of the first order term of the outer expansion and ofthe first derivative of the zero-order term we can write
120597V0
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
=120597V0
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
V1
10038161003816100381610038161199111=+0
= V1
10038161003816100381610038161199111=minus0
(44)
Differentiating (23) once and (26) twice with respect to 120578and using the three equations (37) (42) and (43) we get
1199050
12059721199011
1205971205782+ 119875119860
1205973V2
1205971205783= 0 (45)
1205973V2119909
1205971205783
1205971205770
120597119909+1205973V2119910
1205971205783
1205971205770
120597119910minus1205973V2119911
1205971205783= 0 (46)
As above we multiply the three components of (45)respectively by 120597120577
0120597119909 120597120577
0120597119910 and minus1 and adding we get
119875119860(1205973V2119909
1205971205783
1205971205770
120597119909+1205973V2119910
1205971205783
1205971205770
120597119910minus1205973V2119911
1205971205783) + 119860
12059721199011
1205971205782= 0 (47)
From the previous equation and using (45)-(46) we have
12059721199011
1205971205782= 0
1205973V2
1205971205783= 0
(48)
Knowing that the velocity is bounded and considering(31) we obtain the jump conditions in the following forms
1205972V0
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=+0
=1205972V0
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
120597V1
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
=120597V1
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
V2
10038161003816100381610038161199111=+0
= V2
10038161003816100381610038161199111=minus0
(49)
Differentiating (25) twice and (27) three times withrespect to 120578 and taking into account (37) (42) (48) we obtain
1199050
12059731199012
1205971205783+ 119875119860
1205974V3
1205971205784
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120574
12059721205791
1205971205782= 0
1205974V3119909
1205971205784
1205971205770
120597119909+1205974V3119910
1205971205784
1205971205770
120597119910minus1205974V3119911
1205971205784= 0
(50)
119875119860(1205974V3119909
1205971205784
1205971205770
120597119909+1205974V3119910
1205971205784
1205971205770
120597119910minus1205974V3119911
1205971205784) + 119860
12059731199012
1205971205783
minus 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905))
12059721205791
1205971205782= 0
(51)
From the last equation and using (50) we can write
11986012059731199012
1205971205783minus 119876 (1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905))
12059721205791
1205971205782
= 0
(52)
1205740
12059721205791
1205971205782=1205974V3
1205971205784 (53)
6 Mathematical Problems in Engineering
where 1205740is defined by
1205740= (minus
1205971205770
120597119909
119877
1198602 minus1205971205770
120597119910
119877
1198602119877
1198602minus119877
119860)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905))
(54)
Now integrating (53) with respect to 120578 using the system(32)ndash(34) lead to
1205973V0
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
minus1205973V0
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=+0
= minus1205740
1205971205790
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
(55)
As a first conclusion the velocity jump conditions acrossthe reaction front are given by (36) (44) (49) and (55)
From (20) (22) (37) we conclude that 0is a monotonic
function satisfying 0 lt 0lt 1 Therefore the reaction is of
zero order and 120601(0) equiv 1 Multiplying (19) by 120597120579
1120597120578 and
integrating the result yield
(1205971205791
120597120578)
210038161003816100381610038161003816100381610038161003816100381610038161003816+infin
minus (1205971205791
120597120578)
210038161003816100381610038161003816100381610038161003816100381610038161003816minusinfin
= 2119860minus1int
1205791|1199111=minus0
minusinfin
exp( 120591
1 + 120591120575) 119889120591
(56)
Subtracting (19) from (20) and integrating the result oneobtains
1205971205791
120597120578
100381610038161003816100381610038161003816100381610038161003816+infin
minus1205971205791
120597120578
100381610038161003816100381610038161003816100381610038161003816minusinfin
= minus119860minus1(1205971205770
120597119905+ 119904) (57)
where 119904 is given by
119904 = V0119909
1205971205770
120597119909+ V0119910
1205971205770
120597119910minus V0119911 (58)
As a second conclusion the temperature jump conditionsacross the reaction front are given by the two last equations(56)-(57) By using the matching conditions above andtruncating the expansion as
120579 asymp 1205790
1205791
10038161003816100381610038161199111=minus0
asymp 119885120579|1199111=+0
120577 asymp 1205770
V asymp V0
(59)
the jump conditions can be written as
(120597120579
1205971199111
)
2100381610038161003816100381610038161003816100381610038161003816+0
minus (120597120579
1205971199111
)
2100381610038161003816100381610038161003816100381610038161003816minus0
= 2119885(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot int
1205791|1199111=minus0
minusinfin
exp ( 120591
119885minus1 + 120591120575) 119889120591
120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
minus120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
= minus(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot (120597120577
120597119905+ (V119909
120597120577
120597119909+ V119910
120597120577
120597119910minus V119911)
100381610038161003816100381610038161003816100381610038161199111=+0
)
V119911
10038161003816100381610038161199111=+0
= V119911
10038161003816100381610038161199111=minus0
120597V119911
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
=120597V119911
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
1205972V119911
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=+0
=120597V119911
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
1205973V119911
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
minus1205973V119911
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=+0
= minus119877(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot ((1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
minus 1)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905))
120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
(60)
4 The Interface Problem and Perturbation
To study the propagation of polymerization front with aliquid reactant and liquid product the equation of motionhas been considered before and after the reaction zone Thischanges the jump conditions and influences the stabilityconditions of the frontal polymerization process Notice thatin the case of liquid-solid polymerization front the equationof motion is considered after the reaction zone The originalsystem (2) (3) (8) (9) leads to the following interfaceproblem
In the liquid monomer 119911 gt 120577
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 (61)
120572 = 0 (62)
Mathematical Problems in Engineering 7
120597V120597119905
+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579
0) 120574
(63)
div (V) = 0 (64)
In the liquid polymer 119911 lt 120577
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 (65)
120572 = 1 (66)
120597V120597119905
+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579
0) 120574
(67)
div (V) = 0 (68)
At the interface 119911 = 120577
120579|120577minus0 = 120579|120577+0
120597120579
120597119911
10038161003816100381610038161003816100381610038161003816120577=minus0minus120597120579
120597119911
10038161003816100381610038161003816100381610038161003816120577=+0= (1 + (
120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot (120597120577
120597119905+ (V119909
120597120577
120597119909+ V119910
120597120577
120597119910minus V119911)
10038161003816100381610038161003816100381610038161003816120577
)
(120597120579
120597119911)
2100381610038161003816100381610038161003816100381610038161003816120577minus0
minus (120597120579
120597119911)
21003816100381610038161003816100381610038161003816100381610038161205771+0
= minus2119885(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot int
120579|120577
minusinfin
exp ( 120591
119885minus1 + 120575120591) 119889120591
V119911
1003816100381610038161003816120577=minus0 = V119911
1003816100381610038161003816120577=+0
120597V119911
120597119911
10038161003816100381610038161003816100381610038161003816120577=minus0=120597V119911
120597119911
10038161003816100381610038161003816100381610038161003816120577=+0
1205972V119911
1205971199112
100381610038161003816100381610038161003816100381610038161003816120577=minus0
=1205972V119911
1205971199112
100381610038161003816100381610038161003816100381610038161003816120577=+0
1205973V119911
1205971199113
100381610038161003816100381610038161003816100381610038161003816120577=minus0
minus1205973V119911
1205971199113
100381610038161003816100381610038161003816100381610038161003816120577=+0
= minus119877(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot ((1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
minus 1)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (
120597120579
120597119911)
10038161003816100381610038161003816100381610038161003816120577=+0
(69)
with the conditions at infinity
119911 = minusinfin 120579 = 0 V = 0
119911 = +infin 120579 = minus1 V = 0
(70)
5 Stability Analysis
The interface problem analyzed in Section 4 has a travellingwave solution in the following form
(120579 (119909 119910 119911 119905) 120572 (119909 119910 119911 119905) V)
= (120579119904(119911 minus 119906119905) 120572
119904(119911 minus 119906119905) 0)
(120579119904 (119911 minus 119906119905) 120572119904 (119911 minus 119906119905))
=
(0 1) 1199112lt 0
(exp (minus1199061199112) minus 1 0) 119911
2gt 0
(71)
where 1199112= 119911minus119906119905 and 119906 is the speed of the stationary reaction
frontThis solution is a basic and stationary solution of inter-face problem (61)ndash(70) written in the moving coordinatesThen (61) (63) (65) and (67) can be replaced by
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 + 119906
120597120579
1205971199112
120597V120597119905
+ (Vnabla) V
= minusnabla119901 + 119875ΔV + 119906120597120579
1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (120579 + 120579
0) 120574
(72)
The other equations remain unchanged To study thestability of reaction front we seek the solution of the problemas follows
120579 = 120579119904+ 120579
119901 = 119901119904+ 119901
V = V119904+ V
(73)
where 120579 119901 and V are respectively small perturbation oftemperature pressure and velocity
8 Mathematical Problems in Engineering
Subtituting (73) into (64) (72) one obtains the first-orderterms as follows
In the liquid monomer 1199112gt 120585
120597120579
120597119905= Δ120579 + 119906
120597120579
1205971199112
minus V1199111205791015840
119904 (74)
120597V120597119905
= minusnabla119901 + 119875ΔV + 119906120597120579
1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579120574
(75)
div (V) = 0 (76)
In the liquid polymer 1199112lt 120585
120597120579
120597119905= Δ120579 + 119906
120597120579
1205971199112
(77)
120597V120597119905
= minusnabla119901 + 119875ΔV + 119906120597V1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579120574
(78)
div (V) = 0 (79)
where 120585 is defined by 120585 = 120577 minus 119906119905We note
(120579 V119911) =
(1205791 V1199111) for 119911
2lt 120585
(1205792 V1199112) for 119911
2lt 120585
(80)
and consider the perturbation as follows
120579119894= 120579119894(1199112 119905) exp (119895 (119896
1119909 + 1198962119910))
V119911119894= V119911119894(1199112 119905) exp (119895 (119896
1119909 + 1198962119910))
120585 = 1205981(119905) exp (119895 (119896
1119909 + 1198962119910))
(81)
where 119896119894 119894 = 1 2 and 120598
1denote respectively the wave
numbers (in 119909 and 119910 directions) and the amplitude of theperturbation and 119895
2= minus1 Then jump conditions (69) are
linearized by taking into account the fact that
120579|120585=plusmn0 = 120579119904(plusmn0) + 120585120579
1015840
119904(plusmn0) + 120579 (plusmn0)
120597120579
1205971199112
10038161003816100381610038161003816100381610038161003816120585=plusmn0
= 1205791015840
119904(plusmn0) + 120585120579
10158401015840
119904(plusmn0) +
120597120579
1205971199112
100381610038161003816100381610038161003816100381610038161003816120585=plusmn0
(82)
Then the higher-order terms are given as
1205792
100381610038161003816100381610038161199112=0minus 1205791
100381610038161003816100381610038161199112=0= 119906120585 (83)
1205971205792
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
minus1205971205791
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
= minus1199062120585 minus
120597120585
120597119905+ V119911
10038161003816100381610038161199112=0 (84)
1199062120585 +
1205971205792
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
= minus119885
1199061205791
100381610038161003816100381610038161199112=0 (85)
V2119911
10038161003816100381610038161199112=0= V1119911
10038161003816100381610038161199112=0 (86)
120597V1199112
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
=120597V1119911
1205971199112
100381610038161003816100381610038161003816100381610038161199112=0
(87)
1205972V2119911
12059711991122
1003816100381610038161003816100381610038161003816100381610038161199112=0
=1205972V1119911
12059711991122
1003816100381610038161003816100381610038161003816100381610038161199112=0
(88)
1205973V2119911
12059711991132
1003816100381610038161003816100381610038161003816100381610038161199112=0
=1205973V1119911
12059711991132
1003816100381610038161003816100381610038161003816100381610038161199112=0
(89)
Substituting (81) into (83)ndash(89) leads to
1205792(0 119905) minus 120579
1(0 119905) = 119906120598
1(119905)
1205791015840
2(0 119905) minus 120579
1015840
1(0 119905) = minus120598
1 (119905) 1199062minus 1205981015840
1(119905) + V
1 (0 119905)
1205981 (119905) 119906
2+ 1205791015840
2(0 119905) = minus
119885
1199061205791 (0 119905)
V(119894)2(0 119905) = V(119894)
1(0 119905) 119894 = 0 1 2 3
(90)
where
V(119894) =120597119894V120597V119894
1205791015840=120597120579119894
120597119911
1205981015840
1(119905) =
1198891205981(119905)
119889119905
(91)
After simplification (75) and (78) become
120597
120597119905ΔV119911minus 119906
120597
1205971199112
ΔV119911= 119875ΔΔV
119911+ 119876(1 + 120582
1sin (120590
1119905)
+ 1205822sin (120590
2119905) (
1205972
1205971199092+
1205972
1205971199102))120579
(92)
Substituting (71) (87) (88) into (74) (77) (92) andintroducing the variable 119908 = V10158401015840 minus 1198962V where 119896 = radic1198962
1+ 11989622
one obtains two systems of equations as follows
1199112lt 120585
120597119908
120597119905minus 1199061199081015840minus 119875 (119908
10158401015840minus 1198962119908)
= minus1198761198962(1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579
119908 = V10158401015840 minus 1198962V
120597120579
120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 0
(93)
Mathematical Problems in Engineering 9
0 5 10 15 20 25 300
50
100
150
200
250
300
350
Rc
1205901
1205902 = radic212059011205902 = radic71205901
1205902 = 0
1205902 = radic371205901
Figure 1 Critical Rayleigh number as a function of 1205901for different
frequencies ratio and for 1205902= 0 120582
1= 1205822= 5 119875 = 05 119885 = 8 and
119896 = 07
1199112gt 120585
120597119908
120597119905minus 1199061199081015840minus 119875 (119908
10158401015840minus 1198962119908)
= minus1198761198962(1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579
119908 = V10158401015840 minus 1198962V
120597120579
120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911
2) V
(94)
with the following boundary conditions
1205792(0 119905) minus 120579
1(0 119905) = 119906120598
1(119905) (95)
1205791015840
2(0 119905) minus 120579
1015840
1(0 119905) = minus119906
21205761(119905) minus 120576
1015840
1(119905) + V
1(0 119905) (96)
1205791015840
2(0 119905) +
119885
1199061205791 (0 119905) = minus119906
21205761 (119905) (97)
V(119894)1(0 119905) = V(119894)
2(0 119905) 119894 = 0 1 2 3 (98)
Combining the three boundary conditions (95)ndash(97)yields
1205791015840
1(0 119905) +
119885
1199061205791(0 119905) =
1
119906(1205792(0 119905) minus 120579
1(0 119905))
1015840
119905
minus V1(0 119905)
(99)
6 Numerical Results
The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix
Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved
Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590
1for different frequencies ratio and for
0 25 50 75 100 125 1500
5
10
15
20
25
Rc
1205822
1205821 = 20
1205821 = 50
1205821 = 80
1205821 = 150
Figure 2 Critical Rayleigh number as a function of 1205822for different
values of 1205821 1205902= radic2120590
1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07
0 25 50 75 100 125 150 175 2000
50
100
150
200
250
Rc
1205822
1205902 = radic371205901
1205902 = radic1712059011205902 = radic71205901
1205902 = 1radic712059011205902 = 1radic171205901
1205902 = 1radic371205901
Figure 3 Critical Rayleigh number as a function of 1205822for different
frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590
1= 5
1205902= 0 In the absence of gravitational modulation (120590
1= 1205902=
0) we find the same result (119877119888asymp 27) as in [8 14] It can be
observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the
frequency 1205901 the frequencies ratio 120590
21205901has no effect on
the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590
2= radic37120590
1) comparing
to the periodic modulation case (1205902= 0)
The critical Rayleigh number as a function of the ampli-tude 120582
2is shown in Figure 2 for different values of 120582
1 We can
conclude that for a fixed value of 1205822and increased values of
1205821 the polymerization front loses its stability monotonically
revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation
Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582
2for different frequencies ratio
It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582
2 For
large values of 1205822(1205822ge 100) the effect of the frequencies
ratio on the convective instability boundary is insignificant
10 Mathematical Problems in Engineering
0 25 50 75 100 125 150
k = 15
k = 2
k = 1
0
50
100
150
200
250
300
350
Rc
1205822
Figure 4 Critical Rayleigh number as a function of 1205822for different
values of number wave 1205902= radic7120590
1 1205901= 10 119875 = 05 119885 = 8 and
1205821= 5
0 25 50 75 100 125 1500
255075
100125150175200
Rc
1205822
1205902 = radic21205901
1205902 = radic71205901
Figure 5 Critical Rayleigh number as a function of 1205822for two
different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and
1205901= 10
Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7
the influence of the wave number on the critical Rayleighnumber versus 120582
2 The plots indicate that for large values of
the wave number the reaction front gains stabilityThe influence of 120582
2on the stability of the reaction front is
given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect
7 Conclusion
In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590
1the
critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies
ratio and for a given amplitude 1205822an increase of the ampli-
tude1205821destabilizes the reaction front Also the reaction front
becomes more stable by increasing the wave number
Appendix
The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911
119894= 119894ℎ where ℎ is the
space step and 119894 is a positive integerThe discretization in timeis given by 119905
119894= 119894120591 where 120591 is sufficiently small time step The
variables of discretization are 119908119895119894= 119908(119911
119894 119905119895) V119895119894= V(119911
119894 119905119895)
and 120579119895119894= 120579(119911119894 119905119895) One obtains the following
In the liquid monomer 119911 lt 120577
(minus1
ℎ2) 120579119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ) 120579119895+1
119894
+ (minus1
ℎ2minus119906
ℎ) 120579119895+1
119894+1=1
120591120579119895
119894
(minus119875
ℎ2)119908119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119908119895+1
119894+1=1
120591119908119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 120579119895
119894
(1
ℎ2) V119895+1119894minus1
+ (minus2
ℎ2minus 1198962) V119895+1119894
+ (1
ℎ2) V119895+1119894+1
= 119908119895
119894
(A1)
In the liquid polymer 119911 gt 120577
(minus1
ℎ2)119879119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ)119879119895+1
119894
+ (minus1
ℎ2minus119906
ℎ)119879119895+1
119894+1=1
120591119879119895
119894+ 119906 exp (minus119906119911
119894) 119881119895
119894
(minus119875
ℎ2)119882119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119882119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
119894+1=1
120591119882119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 119879119895
119894
(1
ℎ2)119881119895+1
119894minus1+ (minus
2
ℎ2minus 1198962)119881119895+1
119894+ (
1
ℎ2)119881119895+1
119894+1= 119882119895
119894
(A2)
From the jump conditions we have also the followingsystems
(minus1
ℎ2) 120579119895+1
1
+ (1
120591+
2
ℎ2+ 1198962+119906
ℎ+ (
1198881
1198882
)(1
ℎ2+119906
ℎ)) 120579119895+1
0
Mathematical Problems in Engineering 11
= (minus1
120591) 120579119895
0minus (
1
1199061205911198882
)(1
ℎ2+119906
ℎ) (119879119895
0minus 120579119895
0)
minus (V0
1198882
)(1
ℎ2+119906
ℎ)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
1
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198882
1198881
)1198790minus
1
1199061205911198881
(119879119895
0minus 120579119895
0) minus
V0
1198881
)
(A3)
(1
120591+
2
ℎ2+ 1198962+119906
ℎ+
1198882
ℎ21198881
)119879119895+1
0+ (
minus1
ℎ2minus119906
ℎ)119879119895+1
1
= (minus1
120591)119879119895
0minus 119906 exp (minus119906ℎ) minus ( 1
119906120591ℎ21198881
) (119879119895
0minus 120579119895
0)
minusV0
ℎ21198881
(A4)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
0
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198881
1198882
)1205790minus
1
1199061205911198882
)(119879119895
0minus 120579119895
0minusV0
1198882
)
(A5)
with
1198881= (
1
ℎminus119885
119906minus
1
119906120591)
1198882= (minus
1
ℎ+
1
119906120591)
(A6)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974
[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011
[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014
[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994
[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997
[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996
[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001
[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010
[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012
[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014
[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969
[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961
[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938
[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998
Submit your manuscripts athttpwwwhindawicom
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Order 1205980
120597V0
120597119905minus120597V1
120597120578
1205971205770
120597119905minus120597V0
120597120578
1205971205771
120597119905+ V0119909(120597V0
120597119909minus120597V1
120597120578
1205971205770
120597119909
minus120597V0
120597120578
1205971205771
120597119909) + V1119909
120597V0
120597120578
1205971205770
120597119909+ V1119910
120597V0
120597120578
1205971205770
120597119910
+ V0119910(120597V0
120597119910minus120597V1
120597120578
1205971205770
120597119910minus120597V0
120597120578
1205971205771
120597119910) + V0119911
120597V1
120597120578+ V1119911
sdot120597V0
120597120578= minusnabla01199010+ 1199051
1205971199010
120597120578+ 1199050
1205971199011
120597120578+ 119875(119860
1205972V2
1205971205782
+ 1199053
1205972V1
1205971205782+ 1198650
120597V1
120597120578+ 1199054
1205972V0
1205971205782+ 1198651
120597V0
120597120578+ Δ1V0)
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120574120579
0
(23)
120597V0119909
120597119909minus120597V1119909
120597120578
1205971205770
120597119909minus120597V0119909
120597120578
1205971205771
120597119909+120597V0119910
120597119910minus120597V1119910
120597120578
1205971205770
120597119910
minus120597V0119910
120597120578
1205971205771
120597119910+120597V1119911
120597120578= 0
(24)
Order 1205981
120597V1
120597119905minus (
120597V2
120597120578
1205971205770
120597119905+120597V1
120597120578
1205971205771
120597119905+120597V0
120597120578
1205971205772
120597119905) + V0119909(120597V1
120597119909
minus120597V1
120597120578
1205971205771
120597119909minus120597V0
120597120578
1205971205772
120597119909minus120597V2
120597120578
1205971205770
120597119909) + V1119909(120597V0
120597119909
minus120597V0
120597120578
1205971205771
120597119909minus120597V1
120597120578
1205971205770
120597119909) minus V2119909
120597V0
120597120578
1205971205770
120597119909+ V0119910(120597V1
120597119910
minus120597V1
120597120578
1205971205771
120597119910minus120597V0
120597120578
1205971205772
120597119910minus120597V2
120597120578
1205971205770
120597119910) + V1119910(120597V0
120597119910
minus120597V0
120597120578
1205971205771
120597119910minus120597V1
120597120578
1205971205770
120597119910) minus V2119910
120597V0
120597120578
1205971205770
120597119910+ V0119911
120597V2
120597120578
+ V1119911
120597V1
120597120578+ V2119911
120597V0
120597120578= 1199050
1205971199012
120597120578+ 1199051
1205971199011
120597120578minus nabla01199011+ 1199052
sdot1205971199010
120597120578+ 119875(119860
1205972V3
1205971205782+ 1199053
1205972V2
1205971205782+ 1198650
120597V2
120597120578+ 1198651
120597V1
120597120578
+ Δ1V1+ 1198652
120597V0
120597120578) + 119876 (1 + 120582
1sin (120590
1119905)
+ 1205822sin (120590
2119905)) 1205790120574
(25)
120597V1119909
120597119909minus120597V2119909
120597120578
1205971205770
120597119909minus120597V1119909
120597120578
1205971205771
120597119909minus120597V0119909
120597120578
1205971205772
120597119909+120597V1119910
120597119910
minus120597V2119910
120597120578
1205971205770
120597119910minus120597V1119910
120597120578
1205971205771
120597119910minus120597V0119910
120597120578
1205971205772
120597119910+120597V2119911
120597120578= 0
(26)
Order 1205982
120597V2119909
120597119909minus120597V3119909
120597120578
1205971205770
120597119909minus120597V2119909
120597120578
1205971205771
120597119909minus120597V1119909
120597120578
1205971205772
120597119909minus120597V0119909
120597120578
1205971205773
120597119909
+120597V2119910
120597119910minus120597V3119910
120597120578
1205971205770
120597119910minus120597V2119910
120597120578
1205971205771
120597119910minus120597V1119910
120597120578
1205971205772
120597119910
minus120597V0119910
120597120578
1205971205773
120597119910+120597V3119911
120597120578= 0
(27)
where
Δ1=
1205972
1205971199092+
1205972
1205971199102
nabla0= (
120597
120597119909120597
120597119910 0)
119865119894= minus2(
120597120577119894
120597119909
120597
120597119909+120597120577119894
120597119910
120597
120597119910) minus (
1205972120577119894
1205971199092+1205972120577119894
1205971199102) 119868
119894 = 0 1 2
1199050= (
1205971205770
1205971199091205971205770
120597119910 minus1)
119905119894= (
120597120577119894
120597119909120597120577119894
120597119910 minus 1) 119894 = 1 2
1199053= 2(
1205971205771
120597119909
1205971205770
120597119909+1205971205770
120597119910
1205971205771
120597119910)
1199054= (
1205971205771
120597119909)
2
+ (1205971205771
120597119910)
2
+1205971205770
120597119909
1205971205772
120597119909+1205971205770
120597119910
1205971205772
120597119910
119860 = (1 + (1205971205770
120597119909)
2
+ (1205971205770
120597119910)
2
)
(28)
and 119868 is the identity operator The matching conditions as120578 rarr plusmninfin are given by
V0sim V0
10038161003816100381610038161199111=plusmn0
(29)
V1sim (
120597V0
1205971199111
100381610038161003816100381610038161003816100381610038161199111=plusmn0
)120578 + V1
10038161003816100381610038161199111=plusmn0
(30)
V2sim1
2(1205972V0
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0
)1205782+ (
120597V1
1205971199111
100381610038161003816100381610038161003816100381610038161199111=plusmn0
)120578
+ V2
10038161003816100381610038161199111=plusmn0
(31)
V3sim1
6(1205973V0
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0
)1205783+1
2(1205972V1
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0
)1205782
+ (120597V2
1205971199111
100381610038161003816100381610038161003816100381610038161199111=plusmn0
)120578 + V3
10038161003816100381610038161199111=plusmn0
(32)
Mathematical Problems in Engineering 5
As 120578 rarr +infin
1205791asymp 1205791
10038161003816100381610038161199111=+0
+ (1205971205790
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
)120578 0997888rarr 0 (33)
As 120578 rarr minusinfin
1205791sim 1205791
10038161003816100381610038161199111=minus0
0997888rarr 1 (34)
From (18) we obtain
1205972V0
1205971205782= 0 (35)
Consequently we can conclude that V0(120578) is a linear
function of 120578 and identically constant since the velocity isbounded Equation (29) becomes
V0
10038161003816100381610038161199111=+0
= V0
10038161003816100381610038161199111=minus0
(36)
120597V0
120597120578= 0 (37)
From (36)-(37) we deduce that the first term in theexpression of the velocity V
0is continuous at the front By
substituting (37) into (21) we find
1198751198601205972V1
1205971205782+ 1199050
1205971199010
120597120578= 0 (38)
120597V0119909
120597119909minus120597V1119909
120597120578
1205971205770
120597119909+120597V0119910
120597119910minus120597V1119910
120597120578
1205971205770
120597119910+120597V1119911
120597120578= 0 (39)
Differentiating (39) with respect to 120578 one obtains
1205972V1119909
1205971205782
1205971205770
120597119909+1205972V1119910
1205971205782
1205971205770
120597119910minus1205972V1119911
1205971205782= 0 (40)
As a result (38) is a vectorial equation with three com-ponents We multiply the first component by 120597120577
0120597119909 the
second by 1205971205770120597119910 and the third by minus1 and adding we have
119875119860(1205972V1119909
1205971205782
1205971205770
120597119909+1205972V1119910
1205971205782
1205971205770
120597119910minus1205972V1119911
1205971205782) + 119860
1205971199010
120597120578= 0 (41)
Considering the two equations (38) and (40) we obtain
1205971199010
120597120578= 0 (42)
1205972V1
1205971205782= 0 (43)
From the boundedness of the velocity and (30) the con-tinuity of the first order term of the outer expansion and ofthe first derivative of the zero-order term we can write
120597V0
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
=120597V0
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
V1
10038161003816100381610038161199111=+0
= V1
10038161003816100381610038161199111=minus0
(44)
Differentiating (23) once and (26) twice with respect to 120578and using the three equations (37) (42) and (43) we get
1199050
12059721199011
1205971205782+ 119875119860
1205973V2
1205971205783= 0 (45)
1205973V2119909
1205971205783
1205971205770
120597119909+1205973V2119910
1205971205783
1205971205770
120597119910minus1205973V2119911
1205971205783= 0 (46)
As above we multiply the three components of (45)respectively by 120597120577
0120597119909 120597120577
0120597119910 and minus1 and adding we get
119875119860(1205973V2119909
1205971205783
1205971205770
120597119909+1205973V2119910
1205971205783
1205971205770
120597119910minus1205973V2119911
1205971205783) + 119860
12059721199011
1205971205782= 0 (47)
From the previous equation and using (45)-(46) we have
12059721199011
1205971205782= 0
1205973V2
1205971205783= 0
(48)
Knowing that the velocity is bounded and considering(31) we obtain the jump conditions in the following forms
1205972V0
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=+0
=1205972V0
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
120597V1
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
=120597V1
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
V2
10038161003816100381610038161199111=+0
= V2
10038161003816100381610038161199111=minus0
(49)
Differentiating (25) twice and (27) three times withrespect to 120578 and taking into account (37) (42) (48) we obtain
1199050
12059731199012
1205971205783+ 119875119860
1205974V3
1205971205784
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120574
12059721205791
1205971205782= 0
1205974V3119909
1205971205784
1205971205770
120597119909+1205974V3119910
1205971205784
1205971205770
120597119910minus1205974V3119911
1205971205784= 0
(50)
119875119860(1205974V3119909
1205971205784
1205971205770
120597119909+1205974V3119910
1205971205784
1205971205770
120597119910minus1205974V3119911
1205971205784) + 119860
12059731199012
1205971205783
minus 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905))
12059721205791
1205971205782= 0
(51)
From the last equation and using (50) we can write
11986012059731199012
1205971205783minus 119876 (1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905))
12059721205791
1205971205782
= 0
(52)
1205740
12059721205791
1205971205782=1205974V3
1205971205784 (53)
6 Mathematical Problems in Engineering
where 1205740is defined by
1205740= (minus
1205971205770
120597119909
119877
1198602 minus1205971205770
120597119910
119877
1198602119877
1198602minus119877
119860)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905))
(54)
Now integrating (53) with respect to 120578 using the system(32)ndash(34) lead to
1205973V0
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
minus1205973V0
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=+0
= minus1205740
1205971205790
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
(55)
As a first conclusion the velocity jump conditions acrossthe reaction front are given by (36) (44) (49) and (55)
From (20) (22) (37) we conclude that 0is a monotonic
function satisfying 0 lt 0lt 1 Therefore the reaction is of
zero order and 120601(0) equiv 1 Multiplying (19) by 120597120579
1120597120578 and
integrating the result yield
(1205971205791
120597120578)
210038161003816100381610038161003816100381610038161003816100381610038161003816+infin
minus (1205971205791
120597120578)
210038161003816100381610038161003816100381610038161003816100381610038161003816minusinfin
= 2119860minus1int
1205791|1199111=minus0
minusinfin
exp( 120591
1 + 120591120575) 119889120591
(56)
Subtracting (19) from (20) and integrating the result oneobtains
1205971205791
120597120578
100381610038161003816100381610038161003816100381610038161003816+infin
minus1205971205791
120597120578
100381610038161003816100381610038161003816100381610038161003816minusinfin
= minus119860minus1(1205971205770
120597119905+ 119904) (57)
where 119904 is given by
119904 = V0119909
1205971205770
120597119909+ V0119910
1205971205770
120597119910minus V0119911 (58)
As a second conclusion the temperature jump conditionsacross the reaction front are given by the two last equations(56)-(57) By using the matching conditions above andtruncating the expansion as
120579 asymp 1205790
1205791
10038161003816100381610038161199111=minus0
asymp 119885120579|1199111=+0
120577 asymp 1205770
V asymp V0
(59)
the jump conditions can be written as
(120597120579
1205971199111
)
2100381610038161003816100381610038161003816100381610038161003816+0
minus (120597120579
1205971199111
)
2100381610038161003816100381610038161003816100381610038161003816minus0
= 2119885(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot int
1205791|1199111=minus0
minusinfin
exp ( 120591
119885minus1 + 120591120575) 119889120591
120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
minus120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
= minus(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot (120597120577
120597119905+ (V119909
120597120577
120597119909+ V119910
120597120577
120597119910minus V119911)
100381610038161003816100381610038161003816100381610038161199111=+0
)
V119911
10038161003816100381610038161199111=+0
= V119911
10038161003816100381610038161199111=minus0
120597V119911
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
=120597V119911
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
1205972V119911
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=+0
=120597V119911
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
1205973V119911
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
minus1205973V119911
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=+0
= minus119877(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot ((1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
minus 1)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905))
120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
(60)
4 The Interface Problem and Perturbation
To study the propagation of polymerization front with aliquid reactant and liquid product the equation of motionhas been considered before and after the reaction zone Thischanges the jump conditions and influences the stabilityconditions of the frontal polymerization process Notice thatin the case of liquid-solid polymerization front the equationof motion is considered after the reaction zone The originalsystem (2) (3) (8) (9) leads to the following interfaceproblem
In the liquid monomer 119911 gt 120577
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 (61)
120572 = 0 (62)
Mathematical Problems in Engineering 7
120597V120597119905
+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579
0) 120574
(63)
div (V) = 0 (64)
In the liquid polymer 119911 lt 120577
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 (65)
120572 = 1 (66)
120597V120597119905
+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579
0) 120574
(67)
div (V) = 0 (68)
At the interface 119911 = 120577
120579|120577minus0 = 120579|120577+0
120597120579
120597119911
10038161003816100381610038161003816100381610038161003816120577=minus0minus120597120579
120597119911
10038161003816100381610038161003816100381610038161003816120577=+0= (1 + (
120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot (120597120577
120597119905+ (V119909
120597120577
120597119909+ V119910
120597120577
120597119910minus V119911)
10038161003816100381610038161003816100381610038161003816120577
)
(120597120579
120597119911)
2100381610038161003816100381610038161003816100381610038161003816120577minus0
minus (120597120579
120597119911)
21003816100381610038161003816100381610038161003816100381610038161205771+0
= minus2119885(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot int
120579|120577
minusinfin
exp ( 120591
119885minus1 + 120575120591) 119889120591
V119911
1003816100381610038161003816120577=minus0 = V119911
1003816100381610038161003816120577=+0
120597V119911
120597119911
10038161003816100381610038161003816100381610038161003816120577=minus0=120597V119911
120597119911
10038161003816100381610038161003816100381610038161003816120577=+0
1205972V119911
1205971199112
100381610038161003816100381610038161003816100381610038161003816120577=minus0
=1205972V119911
1205971199112
100381610038161003816100381610038161003816100381610038161003816120577=+0
1205973V119911
1205971199113
100381610038161003816100381610038161003816100381610038161003816120577=minus0
minus1205973V119911
1205971199113
100381610038161003816100381610038161003816100381610038161003816120577=+0
= minus119877(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot ((1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
minus 1)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (
120597120579
120597119911)
10038161003816100381610038161003816100381610038161003816120577=+0
(69)
with the conditions at infinity
119911 = minusinfin 120579 = 0 V = 0
119911 = +infin 120579 = minus1 V = 0
(70)
5 Stability Analysis
The interface problem analyzed in Section 4 has a travellingwave solution in the following form
(120579 (119909 119910 119911 119905) 120572 (119909 119910 119911 119905) V)
= (120579119904(119911 minus 119906119905) 120572
119904(119911 minus 119906119905) 0)
(120579119904 (119911 minus 119906119905) 120572119904 (119911 minus 119906119905))
=
(0 1) 1199112lt 0
(exp (minus1199061199112) minus 1 0) 119911
2gt 0
(71)
where 1199112= 119911minus119906119905 and 119906 is the speed of the stationary reaction
frontThis solution is a basic and stationary solution of inter-face problem (61)ndash(70) written in the moving coordinatesThen (61) (63) (65) and (67) can be replaced by
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 + 119906
120597120579
1205971199112
120597V120597119905
+ (Vnabla) V
= minusnabla119901 + 119875ΔV + 119906120597120579
1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (120579 + 120579
0) 120574
(72)
The other equations remain unchanged To study thestability of reaction front we seek the solution of the problemas follows
120579 = 120579119904+ 120579
119901 = 119901119904+ 119901
V = V119904+ V
(73)
where 120579 119901 and V are respectively small perturbation oftemperature pressure and velocity
8 Mathematical Problems in Engineering
Subtituting (73) into (64) (72) one obtains the first-orderterms as follows
In the liquid monomer 1199112gt 120585
120597120579
120597119905= Δ120579 + 119906
120597120579
1205971199112
minus V1199111205791015840
119904 (74)
120597V120597119905
= minusnabla119901 + 119875ΔV + 119906120597120579
1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579120574
(75)
div (V) = 0 (76)
In the liquid polymer 1199112lt 120585
120597120579
120597119905= Δ120579 + 119906
120597120579
1205971199112
(77)
120597V120597119905
= minusnabla119901 + 119875ΔV + 119906120597V1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579120574
(78)
div (V) = 0 (79)
where 120585 is defined by 120585 = 120577 minus 119906119905We note
(120579 V119911) =
(1205791 V1199111) for 119911
2lt 120585
(1205792 V1199112) for 119911
2lt 120585
(80)
and consider the perturbation as follows
120579119894= 120579119894(1199112 119905) exp (119895 (119896
1119909 + 1198962119910))
V119911119894= V119911119894(1199112 119905) exp (119895 (119896
1119909 + 1198962119910))
120585 = 1205981(119905) exp (119895 (119896
1119909 + 1198962119910))
(81)
where 119896119894 119894 = 1 2 and 120598
1denote respectively the wave
numbers (in 119909 and 119910 directions) and the amplitude of theperturbation and 119895
2= minus1 Then jump conditions (69) are
linearized by taking into account the fact that
120579|120585=plusmn0 = 120579119904(plusmn0) + 120585120579
1015840
119904(plusmn0) + 120579 (plusmn0)
120597120579
1205971199112
10038161003816100381610038161003816100381610038161003816120585=plusmn0
= 1205791015840
119904(plusmn0) + 120585120579
10158401015840
119904(plusmn0) +
120597120579
1205971199112
100381610038161003816100381610038161003816100381610038161003816120585=plusmn0
(82)
Then the higher-order terms are given as
1205792
100381610038161003816100381610038161199112=0minus 1205791
100381610038161003816100381610038161199112=0= 119906120585 (83)
1205971205792
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
minus1205971205791
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
= minus1199062120585 minus
120597120585
120597119905+ V119911
10038161003816100381610038161199112=0 (84)
1199062120585 +
1205971205792
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
= minus119885
1199061205791
100381610038161003816100381610038161199112=0 (85)
V2119911
10038161003816100381610038161199112=0= V1119911
10038161003816100381610038161199112=0 (86)
120597V1199112
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
=120597V1119911
1205971199112
100381610038161003816100381610038161003816100381610038161199112=0
(87)
1205972V2119911
12059711991122
1003816100381610038161003816100381610038161003816100381610038161199112=0
=1205972V1119911
12059711991122
1003816100381610038161003816100381610038161003816100381610038161199112=0
(88)
1205973V2119911
12059711991132
1003816100381610038161003816100381610038161003816100381610038161199112=0
=1205973V1119911
12059711991132
1003816100381610038161003816100381610038161003816100381610038161199112=0
(89)
Substituting (81) into (83)ndash(89) leads to
1205792(0 119905) minus 120579
1(0 119905) = 119906120598
1(119905)
1205791015840
2(0 119905) minus 120579
1015840
1(0 119905) = minus120598
1 (119905) 1199062minus 1205981015840
1(119905) + V
1 (0 119905)
1205981 (119905) 119906
2+ 1205791015840
2(0 119905) = minus
119885
1199061205791 (0 119905)
V(119894)2(0 119905) = V(119894)
1(0 119905) 119894 = 0 1 2 3
(90)
where
V(119894) =120597119894V120597V119894
1205791015840=120597120579119894
120597119911
1205981015840
1(119905) =
1198891205981(119905)
119889119905
(91)
After simplification (75) and (78) become
120597
120597119905ΔV119911minus 119906
120597
1205971199112
ΔV119911= 119875ΔΔV
119911+ 119876(1 + 120582
1sin (120590
1119905)
+ 1205822sin (120590
2119905) (
1205972
1205971199092+
1205972
1205971199102))120579
(92)
Substituting (71) (87) (88) into (74) (77) (92) andintroducing the variable 119908 = V10158401015840 minus 1198962V where 119896 = radic1198962
1+ 11989622
one obtains two systems of equations as follows
1199112lt 120585
120597119908
120597119905minus 1199061199081015840minus 119875 (119908
10158401015840minus 1198962119908)
= minus1198761198962(1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579
119908 = V10158401015840 minus 1198962V
120597120579
120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 0
(93)
Mathematical Problems in Engineering 9
0 5 10 15 20 25 300
50
100
150
200
250
300
350
Rc
1205901
1205902 = radic212059011205902 = radic71205901
1205902 = 0
1205902 = radic371205901
Figure 1 Critical Rayleigh number as a function of 1205901for different
frequencies ratio and for 1205902= 0 120582
1= 1205822= 5 119875 = 05 119885 = 8 and
119896 = 07
1199112gt 120585
120597119908
120597119905minus 1199061199081015840minus 119875 (119908
10158401015840minus 1198962119908)
= minus1198761198962(1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579
119908 = V10158401015840 minus 1198962V
120597120579
120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911
2) V
(94)
with the following boundary conditions
1205792(0 119905) minus 120579
1(0 119905) = 119906120598
1(119905) (95)
1205791015840
2(0 119905) minus 120579
1015840
1(0 119905) = minus119906
21205761(119905) minus 120576
1015840
1(119905) + V
1(0 119905) (96)
1205791015840
2(0 119905) +
119885
1199061205791 (0 119905) = minus119906
21205761 (119905) (97)
V(119894)1(0 119905) = V(119894)
2(0 119905) 119894 = 0 1 2 3 (98)
Combining the three boundary conditions (95)ndash(97)yields
1205791015840
1(0 119905) +
119885
1199061205791(0 119905) =
1
119906(1205792(0 119905) minus 120579
1(0 119905))
1015840
119905
minus V1(0 119905)
(99)
6 Numerical Results
The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix
Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved
Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590
1for different frequencies ratio and for
0 25 50 75 100 125 1500
5
10
15
20
25
Rc
1205822
1205821 = 20
1205821 = 50
1205821 = 80
1205821 = 150
Figure 2 Critical Rayleigh number as a function of 1205822for different
values of 1205821 1205902= radic2120590
1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07
0 25 50 75 100 125 150 175 2000
50
100
150
200
250
Rc
1205822
1205902 = radic371205901
1205902 = radic1712059011205902 = radic71205901
1205902 = 1radic712059011205902 = 1radic171205901
1205902 = 1radic371205901
Figure 3 Critical Rayleigh number as a function of 1205822for different
frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590
1= 5
1205902= 0 In the absence of gravitational modulation (120590
1= 1205902=
0) we find the same result (119877119888asymp 27) as in [8 14] It can be
observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the
frequency 1205901 the frequencies ratio 120590
21205901has no effect on
the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590
2= radic37120590
1) comparing
to the periodic modulation case (1205902= 0)
The critical Rayleigh number as a function of the ampli-tude 120582
2is shown in Figure 2 for different values of 120582
1 We can
conclude that for a fixed value of 1205822and increased values of
1205821 the polymerization front loses its stability monotonically
revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation
Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582
2for different frequencies ratio
It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582
2 For
large values of 1205822(1205822ge 100) the effect of the frequencies
ratio on the convective instability boundary is insignificant
10 Mathematical Problems in Engineering
0 25 50 75 100 125 150
k = 15
k = 2
k = 1
0
50
100
150
200
250
300
350
Rc
1205822
Figure 4 Critical Rayleigh number as a function of 1205822for different
values of number wave 1205902= radic7120590
1 1205901= 10 119875 = 05 119885 = 8 and
1205821= 5
0 25 50 75 100 125 1500
255075
100125150175200
Rc
1205822
1205902 = radic21205901
1205902 = radic71205901
Figure 5 Critical Rayleigh number as a function of 1205822for two
different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and
1205901= 10
Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7
the influence of the wave number on the critical Rayleighnumber versus 120582
2 The plots indicate that for large values of
the wave number the reaction front gains stabilityThe influence of 120582
2on the stability of the reaction front is
given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect
7 Conclusion
In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590
1the
critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies
ratio and for a given amplitude 1205822an increase of the ampli-
tude1205821destabilizes the reaction front Also the reaction front
becomes more stable by increasing the wave number
Appendix
The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911
119894= 119894ℎ where ℎ is the
space step and 119894 is a positive integerThe discretization in timeis given by 119905
119894= 119894120591 where 120591 is sufficiently small time step The
variables of discretization are 119908119895119894= 119908(119911
119894 119905119895) V119895119894= V(119911
119894 119905119895)
and 120579119895119894= 120579(119911119894 119905119895) One obtains the following
In the liquid monomer 119911 lt 120577
(minus1
ℎ2) 120579119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ) 120579119895+1
119894
+ (minus1
ℎ2minus119906
ℎ) 120579119895+1
119894+1=1
120591120579119895
119894
(minus119875
ℎ2)119908119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119908119895+1
119894+1=1
120591119908119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 120579119895
119894
(1
ℎ2) V119895+1119894minus1
+ (minus2
ℎ2minus 1198962) V119895+1119894
+ (1
ℎ2) V119895+1119894+1
= 119908119895
119894
(A1)
In the liquid polymer 119911 gt 120577
(minus1
ℎ2)119879119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ)119879119895+1
119894
+ (minus1
ℎ2minus119906
ℎ)119879119895+1
119894+1=1
120591119879119895
119894+ 119906 exp (minus119906119911
119894) 119881119895
119894
(minus119875
ℎ2)119882119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119882119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
119894+1=1
120591119882119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 119879119895
119894
(1
ℎ2)119881119895+1
119894minus1+ (minus
2
ℎ2minus 1198962)119881119895+1
119894+ (
1
ℎ2)119881119895+1
119894+1= 119882119895
119894
(A2)
From the jump conditions we have also the followingsystems
(minus1
ℎ2) 120579119895+1
1
+ (1
120591+
2
ℎ2+ 1198962+119906
ℎ+ (
1198881
1198882
)(1
ℎ2+119906
ℎ)) 120579119895+1
0
Mathematical Problems in Engineering 11
= (minus1
120591) 120579119895
0minus (
1
1199061205911198882
)(1
ℎ2+119906
ℎ) (119879119895
0minus 120579119895
0)
minus (V0
1198882
)(1
ℎ2+119906
ℎ)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
1
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198882
1198881
)1198790minus
1
1199061205911198881
(119879119895
0minus 120579119895
0) minus
V0
1198881
)
(A3)
(1
120591+
2
ℎ2+ 1198962+119906
ℎ+
1198882
ℎ21198881
)119879119895+1
0+ (
minus1
ℎ2minus119906
ℎ)119879119895+1
1
= (minus1
120591)119879119895
0minus 119906 exp (minus119906ℎ) minus ( 1
119906120591ℎ21198881
) (119879119895
0minus 120579119895
0)
minusV0
ℎ21198881
(A4)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
0
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198881
1198882
)1205790minus
1
1199061205911198882
)(119879119895
0minus 120579119895
0minusV0
1198882
)
(A5)
with
1198881= (
1
ℎminus119885
119906minus
1
119906120591)
1198882= (minus
1
ℎ+
1
119906120591)
(A6)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974
[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011
[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014
[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994
[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997
[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996
[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001
[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010
[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012
[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014
[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969
[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961
[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938
[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
As 120578 rarr +infin
1205791asymp 1205791
10038161003816100381610038161199111=+0
+ (1205971205790
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
)120578 0997888rarr 0 (33)
As 120578 rarr minusinfin
1205791sim 1205791
10038161003816100381610038161199111=minus0
0997888rarr 1 (34)
From (18) we obtain
1205972V0
1205971205782= 0 (35)
Consequently we can conclude that V0(120578) is a linear
function of 120578 and identically constant since the velocity isbounded Equation (29) becomes
V0
10038161003816100381610038161199111=+0
= V0
10038161003816100381610038161199111=minus0
(36)
120597V0
120597120578= 0 (37)
From (36)-(37) we deduce that the first term in theexpression of the velocity V
0is continuous at the front By
substituting (37) into (21) we find
1198751198601205972V1
1205971205782+ 1199050
1205971199010
120597120578= 0 (38)
120597V0119909
120597119909minus120597V1119909
120597120578
1205971205770
120597119909+120597V0119910
120597119910minus120597V1119910
120597120578
1205971205770
120597119910+120597V1119911
120597120578= 0 (39)
Differentiating (39) with respect to 120578 one obtains
1205972V1119909
1205971205782
1205971205770
120597119909+1205972V1119910
1205971205782
1205971205770
120597119910minus1205972V1119911
1205971205782= 0 (40)
As a result (38) is a vectorial equation with three com-ponents We multiply the first component by 120597120577
0120597119909 the
second by 1205971205770120597119910 and the third by minus1 and adding we have
119875119860(1205972V1119909
1205971205782
1205971205770
120597119909+1205972V1119910
1205971205782
1205971205770
120597119910minus1205972V1119911
1205971205782) + 119860
1205971199010
120597120578= 0 (41)
Considering the two equations (38) and (40) we obtain
1205971199010
120597120578= 0 (42)
1205972V1
1205971205782= 0 (43)
From the boundedness of the velocity and (30) the con-tinuity of the first order term of the outer expansion and ofthe first derivative of the zero-order term we can write
120597V0
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
=120597V0
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
V1
10038161003816100381610038161199111=+0
= V1
10038161003816100381610038161199111=minus0
(44)
Differentiating (23) once and (26) twice with respect to 120578and using the three equations (37) (42) and (43) we get
1199050
12059721199011
1205971205782+ 119875119860
1205973V2
1205971205783= 0 (45)
1205973V2119909
1205971205783
1205971205770
120597119909+1205973V2119910
1205971205783
1205971205770
120597119910minus1205973V2119911
1205971205783= 0 (46)
As above we multiply the three components of (45)respectively by 120597120577
0120597119909 120597120577
0120597119910 and minus1 and adding we get
119875119860(1205973V2119909
1205971205783
1205971205770
120597119909+1205973V2119910
1205971205783
1205971205770
120597119910minus1205973V2119911
1205971205783) + 119860
12059721199011
1205971205782= 0 (47)
From the previous equation and using (45)-(46) we have
12059721199011
1205971205782= 0
1205973V2
1205971205783= 0
(48)
Knowing that the velocity is bounded and considering(31) we obtain the jump conditions in the following forms
1205972V0
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=+0
=1205972V0
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
120597V1
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
=120597V1
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
V2
10038161003816100381610038161199111=+0
= V2
10038161003816100381610038161199111=minus0
(49)
Differentiating (25) twice and (27) three times withrespect to 120578 and taking into account (37) (42) (48) we obtain
1199050
12059731199012
1205971205783+ 119875119860
1205974V3
1205971205784
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120574
12059721205791
1205971205782= 0
1205974V3119909
1205971205784
1205971205770
120597119909+1205974V3119910
1205971205784
1205971205770
120597119910minus1205974V3119911
1205971205784= 0
(50)
119875119860(1205974V3119909
1205971205784
1205971205770
120597119909+1205974V3119910
1205971205784
1205971205770
120597119910minus1205974V3119911
1205971205784) + 119860
12059731199012
1205971205783
minus 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905))
12059721205791
1205971205782= 0
(51)
From the last equation and using (50) we can write
11986012059731199012
1205971205783minus 119876 (1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905))
12059721205791
1205971205782
= 0
(52)
1205740
12059721205791
1205971205782=1205974V3
1205971205784 (53)
6 Mathematical Problems in Engineering
where 1205740is defined by
1205740= (minus
1205971205770
120597119909
119877
1198602 minus1205971205770
120597119910
119877
1198602119877
1198602minus119877
119860)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905))
(54)
Now integrating (53) with respect to 120578 using the system(32)ndash(34) lead to
1205973V0
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
minus1205973V0
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=+0
= minus1205740
1205971205790
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
(55)
As a first conclusion the velocity jump conditions acrossthe reaction front are given by (36) (44) (49) and (55)
From (20) (22) (37) we conclude that 0is a monotonic
function satisfying 0 lt 0lt 1 Therefore the reaction is of
zero order and 120601(0) equiv 1 Multiplying (19) by 120597120579
1120597120578 and
integrating the result yield
(1205971205791
120597120578)
210038161003816100381610038161003816100381610038161003816100381610038161003816+infin
minus (1205971205791
120597120578)
210038161003816100381610038161003816100381610038161003816100381610038161003816minusinfin
= 2119860minus1int
1205791|1199111=minus0
minusinfin
exp( 120591
1 + 120591120575) 119889120591
(56)
Subtracting (19) from (20) and integrating the result oneobtains
1205971205791
120597120578
100381610038161003816100381610038161003816100381610038161003816+infin
minus1205971205791
120597120578
100381610038161003816100381610038161003816100381610038161003816minusinfin
= minus119860minus1(1205971205770
120597119905+ 119904) (57)
where 119904 is given by
119904 = V0119909
1205971205770
120597119909+ V0119910
1205971205770
120597119910minus V0119911 (58)
As a second conclusion the temperature jump conditionsacross the reaction front are given by the two last equations(56)-(57) By using the matching conditions above andtruncating the expansion as
120579 asymp 1205790
1205791
10038161003816100381610038161199111=minus0
asymp 119885120579|1199111=+0
120577 asymp 1205770
V asymp V0
(59)
the jump conditions can be written as
(120597120579
1205971199111
)
2100381610038161003816100381610038161003816100381610038161003816+0
minus (120597120579
1205971199111
)
2100381610038161003816100381610038161003816100381610038161003816minus0
= 2119885(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot int
1205791|1199111=minus0
minusinfin
exp ( 120591
119885minus1 + 120591120575) 119889120591
120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
minus120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
= minus(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot (120597120577
120597119905+ (V119909
120597120577
120597119909+ V119910
120597120577
120597119910minus V119911)
100381610038161003816100381610038161003816100381610038161199111=+0
)
V119911
10038161003816100381610038161199111=+0
= V119911
10038161003816100381610038161199111=minus0
120597V119911
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
=120597V119911
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
1205972V119911
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=+0
=120597V119911
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
1205973V119911
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
minus1205973V119911
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=+0
= minus119877(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot ((1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
minus 1)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905))
120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
(60)
4 The Interface Problem and Perturbation
To study the propagation of polymerization front with aliquid reactant and liquid product the equation of motionhas been considered before and after the reaction zone Thischanges the jump conditions and influences the stabilityconditions of the frontal polymerization process Notice thatin the case of liquid-solid polymerization front the equationof motion is considered after the reaction zone The originalsystem (2) (3) (8) (9) leads to the following interfaceproblem
In the liquid monomer 119911 gt 120577
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 (61)
120572 = 0 (62)
Mathematical Problems in Engineering 7
120597V120597119905
+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579
0) 120574
(63)
div (V) = 0 (64)
In the liquid polymer 119911 lt 120577
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 (65)
120572 = 1 (66)
120597V120597119905
+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579
0) 120574
(67)
div (V) = 0 (68)
At the interface 119911 = 120577
120579|120577minus0 = 120579|120577+0
120597120579
120597119911
10038161003816100381610038161003816100381610038161003816120577=minus0minus120597120579
120597119911
10038161003816100381610038161003816100381610038161003816120577=+0= (1 + (
120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot (120597120577
120597119905+ (V119909
120597120577
120597119909+ V119910
120597120577
120597119910minus V119911)
10038161003816100381610038161003816100381610038161003816120577
)
(120597120579
120597119911)
2100381610038161003816100381610038161003816100381610038161003816120577minus0
minus (120597120579
120597119911)
21003816100381610038161003816100381610038161003816100381610038161205771+0
= minus2119885(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot int
120579|120577
minusinfin
exp ( 120591
119885minus1 + 120575120591) 119889120591
V119911
1003816100381610038161003816120577=minus0 = V119911
1003816100381610038161003816120577=+0
120597V119911
120597119911
10038161003816100381610038161003816100381610038161003816120577=minus0=120597V119911
120597119911
10038161003816100381610038161003816100381610038161003816120577=+0
1205972V119911
1205971199112
100381610038161003816100381610038161003816100381610038161003816120577=minus0
=1205972V119911
1205971199112
100381610038161003816100381610038161003816100381610038161003816120577=+0
1205973V119911
1205971199113
100381610038161003816100381610038161003816100381610038161003816120577=minus0
minus1205973V119911
1205971199113
100381610038161003816100381610038161003816100381610038161003816120577=+0
= minus119877(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot ((1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
minus 1)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (
120597120579
120597119911)
10038161003816100381610038161003816100381610038161003816120577=+0
(69)
with the conditions at infinity
119911 = minusinfin 120579 = 0 V = 0
119911 = +infin 120579 = minus1 V = 0
(70)
5 Stability Analysis
The interface problem analyzed in Section 4 has a travellingwave solution in the following form
(120579 (119909 119910 119911 119905) 120572 (119909 119910 119911 119905) V)
= (120579119904(119911 minus 119906119905) 120572
119904(119911 minus 119906119905) 0)
(120579119904 (119911 minus 119906119905) 120572119904 (119911 minus 119906119905))
=
(0 1) 1199112lt 0
(exp (minus1199061199112) minus 1 0) 119911
2gt 0
(71)
where 1199112= 119911minus119906119905 and 119906 is the speed of the stationary reaction
frontThis solution is a basic and stationary solution of inter-face problem (61)ndash(70) written in the moving coordinatesThen (61) (63) (65) and (67) can be replaced by
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 + 119906
120597120579
1205971199112
120597V120597119905
+ (Vnabla) V
= minusnabla119901 + 119875ΔV + 119906120597120579
1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (120579 + 120579
0) 120574
(72)
The other equations remain unchanged To study thestability of reaction front we seek the solution of the problemas follows
120579 = 120579119904+ 120579
119901 = 119901119904+ 119901
V = V119904+ V
(73)
where 120579 119901 and V are respectively small perturbation oftemperature pressure and velocity
8 Mathematical Problems in Engineering
Subtituting (73) into (64) (72) one obtains the first-orderterms as follows
In the liquid monomer 1199112gt 120585
120597120579
120597119905= Δ120579 + 119906
120597120579
1205971199112
minus V1199111205791015840
119904 (74)
120597V120597119905
= minusnabla119901 + 119875ΔV + 119906120597120579
1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579120574
(75)
div (V) = 0 (76)
In the liquid polymer 1199112lt 120585
120597120579
120597119905= Δ120579 + 119906
120597120579
1205971199112
(77)
120597V120597119905
= minusnabla119901 + 119875ΔV + 119906120597V1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579120574
(78)
div (V) = 0 (79)
where 120585 is defined by 120585 = 120577 minus 119906119905We note
(120579 V119911) =
(1205791 V1199111) for 119911
2lt 120585
(1205792 V1199112) for 119911
2lt 120585
(80)
and consider the perturbation as follows
120579119894= 120579119894(1199112 119905) exp (119895 (119896
1119909 + 1198962119910))
V119911119894= V119911119894(1199112 119905) exp (119895 (119896
1119909 + 1198962119910))
120585 = 1205981(119905) exp (119895 (119896
1119909 + 1198962119910))
(81)
where 119896119894 119894 = 1 2 and 120598
1denote respectively the wave
numbers (in 119909 and 119910 directions) and the amplitude of theperturbation and 119895
2= minus1 Then jump conditions (69) are
linearized by taking into account the fact that
120579|120585=plusmn0 = 120579119904(plusmn0) + 120585120579
1015840
119904(plusmn0) + 120579 (plusmn0)
120597120579
1205971199112
10038161003816100381610038161003816100381610038161003816120585=plusmn0
= 1205791015840
119904(plusmn0) + 120585120579
10158401015840
119904(plusmn0) +
120597120579
1205971199112
100381610038161003816100381610038161003816100381610038161003816120585=plusmn0
(82)
Then the higher-order terms are given as
1205792
100381610038161003816100381610038161199112=0minus 1205791
100381610038161003816100381610038161199112=0= 119906120585 (83)
1205971205792
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
minus1205971205791
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
= minus1199062120585 minus
120597120585
120597119905+ V119911
10038161003816100381610038161199112=0 (84)
1199062120585 +
1205971205792
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
= minus119885
1199061205791
100381610038161003816100381610038161199112=0 (85)
V2119911
10038161003816100381610038161199112=0= V1119911
10038161003816100381610038161199112=0 (86)
120597V1199112
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
=120597V1119911
1205971199112
100381610038161003816100381610038161003816100381610038161199112=0
(87)
1205972V2119911
12059711991122
1003816100381610038161003816100381610038161003816100381610038161199112=0
=1205972V1119911
12059711991122
1003816100381610038161003816100381610038161003816100381610038161199112=0
(88)
1205973V2119911
12059711991132
1003816100381610038161003816100381610038161003816100381610038161199112=0
=1205973V1119911
12059711991132
1003816100381610038161003816100381610038161003816100381610038161199112=0
(89)
Substituting (81) into (83)ndash(89) leads to
1205792(0 119905) minus 120579
1(0 119905) = 119906120598
1(119905)
1205791015840
2(0 119905) minus 120579
1015840
1(0 119905) = minus120598
1 (119905) 1199062minus 1205981015840
1(119905) + V
1 (0 119905)
1205981 (119905) 119906
2+ 1205791015840
2(0 119905) = minus
119885
1199061205791 (0 119905)
V(119894)2(0 119905) = V(119894)
1(0 119905) 119894 = 0 1 2 3
(90)
where
V(119894) =120597119894V120597V119894
1205791015840=120597120579119894
120597119911
1205981015840
1(119905) =
1198891205981(119905)
119889119905
(91)
After simplification (75) and (78) become
120597
120597119905ΔV119911minus 119906
120597
1205971199112
ΔV119911= 119875ΔΔV
119911+ 119876(1 + 120582
1sin (120590
1119905)
+ 1205822sin (120590
2119905) (
1205972
1205971199092+
1205972
1205971199102))120579
(92)
Substituting (71) (87) (88) into (74) (77) (92) andintroducing the variable 119908 = V10158401015840 minus 1198962V where 119896 = radic1198962
1+ 11989622
one obtains two systems of equations as follows
1199112lt 120585
120597119908
120597119905minus 1199061199081015840minus 119875 (119908
10158401015840minus 1198962119908)
= minus1198761198962(1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579
119908 = V10158401015840 minus 1198962V
120597120579
120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 0
(93)
Mathematical Problems in Engineering 9
0 5 10 15 20 25 300
50
100
150
200
250
300
350
Rc
1205901
1205902 = radic212059011205902 = radic71205901
1205902 = 0
1205902 = radic371205901
Figure 1 Critical Rayleigh number as a function of 1205901for different
frequencies ratio and for 1205902= 0 120582
1= 1205822= 5 119875 = 05 119885 = 8 and
119896 = 07
1199112gt 120585
120597119908
120597119905minus 1199061199081015840minus 119875 (119908
10158401015840minus 1198962119908)
= minus1198761198962(1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579
119908 = V10158401015840 minus 1198962V
120597120579
120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911
2) V
(94)
with the following boundary conditions
1205792(0 119905) minus 120579
1(0 119905) = 119906120598
1(119905) (95)
1205791015840
2(0 119905) minus 120579
1015840
1(0 119905) = minus119906
21205761(119905) minus 120576
1015840
1(119905) + V
1(0 119905) (96)
1205791015840
2(0 119905) +
119885
1199061205791 (0 119905) = minus119906
21205761 (119905) (97)
V(119894)1(0 119905) = V(119894)
2(0 119905) 119894 = 0 1 2 3 (98)
Combining the three boundary conditions (95)ndash(97)yields
1205791015840
1(0 119905) +
119885
1199061205791(0 119905) =
1
119906(1205792(0 119905) minus 120579
1(0 119905))
1015840
119905
minus V1(0 119905)
(99)
6 Numerical Results
The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix
Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved
Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590
1for different frequencies ratio and for
0 25 50 75 100 125 1500
5
10
15
20
25
Rc
1205822
1205821 = 20
1205821 = 50
1205821 = 80
1205821 = 150
Figure 2 Critical Rayleigh number as a function of 1205822for different
values of 1205821 1205902= radic2120590
1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07
0 25 50 75 100 125 150 175 2000
50
100
150
200
250
Rc
1205822
1205902 = radic371205901
1205902 = radic1712059011205902 = radic71205901
1205902 = 1radic712059011205902 = 1radic171205901
1205902 = 1radic371205901
Figure 3 Critical Rayleigh number as a function of 1205822for different
frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590
1= 5
1205902= 0 In the absence of gravitational modulation (120590
1= 1205902=
0) we find the same result (119877119888asymp 27) as in [8 14] It can be
observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the
frequency 1205901 the frequencies ratio 120590
21205901has no effect on
the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590
2= radic37120590
1) comparing
to the periodic modulation case (1205902= 0)
The critical Rayleigh number as a function of the ampli-tude 120582
2is shown in Figure 2 for different values of 120582
1 We can
conclude that for a fixed value of 1205822and increased values of
1205821 the polymerization front loses its stability monotonically
revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation
Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582
2for different frequencies ratio
It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582
2 For
large values of 1205822(1205822ge 100) the effect of the frequencies
ratio on the convective instability boundary is insignificant
10 Mathematical Problems in Engineering
0 25 50 75 100 125 150
k = 15
k = 2
k = 1
0
50
100
150
200
250
300
350
Rc
1205822
Figure 4 Critical Rayleigh number as a function of 1205822for different
values of number wave 1205902= radic7120590
1 1205901= 10 119875 = 05 119885 = 8 and
1205821= 5
0 25 50 75 100 125 1500
255075
100125150175200
Rc
1205822
1205902 = radic21205901
1205902 = radic71205901
Figure 5 Critical Rayleigh number as a function of 1205822for two
different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and
1205901= 10
Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7
the influence of the wave number on the critical Rayleighnumber versus 120582
2 The plots indicate that for large values of
the wave number the reaction front gains stabilityThe influence of 120582
2on the stability of the reaction front is
given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect
7 Conclusion
In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590
1the
critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies
ratio and for a given amplitude 1205822an increase of the ampli-
tude1205821destabilizes the reaction front Also the reaction front
becomes more stable by increasing the wave number
Appendix
The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911
119894= 119894ℎ where ℎ is the
space step and 119894 is a positive integerThe discretization in timeis given by 119905
119894= 119894120591 where 120591 is sufficiently small time step The
variables of discretization are 119908119895119894= 119908(119911
119894 119905119895) V119895119894= V(119911
119894 119905119895)
and 120579119895119894= 120579(119911119894 119905119895) One obtains the following
In the liquid monomer 119911 lt 120577
(minus1
ℎ2) 120579119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ) 120579119895+1
119894
+ (minus1
ℎ2minus119906
ℎ) 120579119895+1
119894+1=1
120591120579119895
119894
(minus119875
ℎ2)119908119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119908119895+1
119894+1=1
120591119908119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 120579119895
119894
(1
ℎ2) V119895+1119894minus1
+ (minus2
ℎ2minus 1198962) V119895+1119894
+ (1
ℎ2) V119895+1119894+1
= 119908119895
119894
(A1)
In the liquid polymer 119911 gt 120577
(minus1
ℎ2)119879119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ)119879119895+1
119894
+ (minus1
ℎ2minus119906
ℎ)119879119895+1
119894+1=1
120591119879119895
119894+ 119906 exp (minus119906119911
119894) 119881119895
119894
(minus119875
ℎ2)119882119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119882119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
119894+1=1
120591119882119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 119879119895
119894
(1
ℎ2)119881119895+1
119894minus1+ (minus
2
ℎ2minus 1198962)119881119895+1
119894+ (
1
ℎ2)119881119895+1
119894+1= 119882119895
119894
(A2)
From the jump conditions we have also the followingsystems
(minus1
ℎ2) 120579119895+1
1
+ (1
120591+
2
ℎ2+ 1198962+119906
ℎ+ (
1198881
1198882
)(1
ℎ2+119906
ℎ)) 120579119895+1
0
Mathematical Problems in Engineering 11
= (minus1
120591) 120579119895
0minus (
1
1199061205911198882
)(1
ℎ2+119906
ℎ) (119879119895
0minus 120579119895
0)
minus (V0
1198882
)(1
ℎ2+119906
ℎ)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
1
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198882
1198881
)1198790minus
1
1199061205911198881
(119879119895
0minus 120579119895
0) minus
V0
1198881
)
(A3)
(1
120591+
2
ℎ2+ 1198962+119906
ℎ+
1198882
ℎ21198881
)119879119895+1
0+ (
minus1
ℎ2minus119906
ℎ)119879119895+1
1
= (minus1
120591)119879119895
0minus 119906 exp (minus119906ℎ) minus ( 1
119906120591ℎ21198881
) (119879119895
0minus 120579119895
0)
minusV0
ℎ21198881
(A4)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
0
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198881
1198882
)1205790minus
1
1199061205911198882
)(119879119895
0minus 120579119895
0minusV0
1198882
)
(A5)
with
1198881= (
1
ℎminus119885
119906minus
1
119906120591)
1198882= (minus
1
ℎ+
1
119906120591)
(A6)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974
[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011
[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014
[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994
[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997
[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996
[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001
[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010
[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012
[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014
[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969
[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961
[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938
[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998
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Mathematical Problems in Engineering
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Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
where 1205740is defined by
1205740= (minus
1205971205770
120597119909
119877
1198602 minus1205971205770
120597119910
119877
1198602119877
1198602minus119877
119860)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905))
(54)
Now integrating (53) with respect to 120578 using the system(32)ndash(34) lead to
1205973V0
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
minus1205973V0
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=+0
= minus1205740
1205971205790
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
(55)
As a first conclusion the velocity jump conditions acrossthe reaction front are given by (36) (44) (49) and (55)
From (20) (22) (37) we conclude that 0is a monotonic
function satisfying 0 lt 0lt 1 Therefore the reaction is of
zero order and 120601(0) equiv 1 Multiplying (19) by 120597120579
1120597120578 and
integrating the result yield
(1205971205791
120597120578)
210038161003816100381610038161003816100381610038161003816100381610038161003816+infin
minus (1205971205791
120597120578)
210038161003816100381610038161003816100381610038161003816100381610038161003816minusinfin
= 2119860minus1int
1205791|1199111=minus0
minusinfin
exp( 120591
1 + 120591120575) 119889120591
(56)
Subtracting (19) from (20) and integrating the result oneobtains
1205971205791
120597120578
100381610038161003816100381610038161003816100381610038161003816+infin
minus1205971205791
120597120578
100381610038161003816100381610038161003816100381610038161003816minusinfin
= minus119860minus1(1205971205770
120597119905+ 119904) (57)
where 119904 is given by
119904 = V0119909
1205971205770
120597119909+ V0119910
1205971205770
120597119910minus V0119911 (58)
As a second conclusion the temperature jump conditionsacross the reaction front are given by the two last equations(56)-(57) By using the matching conditions above andtruncating the expansion as
120579 asymp 1205790
1205791
10038161003816100381610038161199111=minus0
asymp 119885120579|1199111=+0
120577 asymp 1205770
V asymp V0
(59)
the jump conditions can be written as
(120597120579
1205971199111
)
2100381610038161003816100381610038161003816100381610038161003816+0
minus (120597120579
1205971199111
)
2100381610038161003816100381610038161003816100381610038161003816minus0
= 2119885(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot int
1205791|1199111=minus0
minusinfin
exp ( 120591
119885minus1 + 120591120575) 119889120591
120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
minus120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
= minus(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot (120597120577
120597119905+ (V119909
120597120577
120597119909+ V119910
120597120577
120597119910minus V119911)
100381610038161003816100381610038161003816100381610038161199111=+0
)
V119911
10038161003816100381610038161199111=+0
= V119911
10038161003816100381610038161199111=minus0
120597V119911
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
=120597V119911
1205971199111
100381610038161003816100381610038161003816100381610038161199111=minus0
1205972V119911
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=+0
=120597V119911
12059711991121
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
1205973V119911
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=minus0
minus1205973V119911
12059711991131
1003816100381610038161003816100381610038161003816100381610038161199111=+0
= minus119877(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot ((1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
minus 1)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905))
120597120579
1205971199111
100381610038161003816100381610038161003816100381610038161199111=+0
(60)
4 The Interface Problem and Perturbation
To study the propagation of polymerization front with aliquid reactant and liquid product the equation of motionhas been considered before and after the reaction zone Thischanges the jump conditions and influences the stabilityconditions of the frontal polymerization process Notice thatin the case of liquid-solid polymerization front the equationof motion is considered after the reaction zone The originalsystem (2) (3) (8) (9) leads to the following interfaceproblem
In the liquid monomer 119911 gt 120577
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 (61)
120572 = 0 (62)
Mathematical Problems in Engineering 7
120597V120597119905
+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579
0) 120574
(63)
div (V) = 0 (64)
In the liquid polymer 119911 lt 120577
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 (65)
120572 = 1 (66)
120597V120597119905
+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579
0) 120574
(67)
div (V) = 0 (68)
At the interface 119911 = 120577
120579|120577minus0 = 120579|120577+0
120597120579
120597119911
10038161003816100381610038161003816100381610038161003816120577=minus0minus120597120579
120597119911
10038161003816100381610038161003816100381610038161003816120577=+0= (1 + (
120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot (120597120577
120597119905+ (V119909
120597120577
120597119909+ V119910
120597120577
120597119910minus V119911)
10038161003816100381610038161003816100381610038161003816120577
)
(120597120579
120597119911)
2100381610038161003816100381610038161003816100381610038161003816120577minus0
minus (120597120579
120597119911)
21003816100381610038161003816100381610038161003816100381610038161205771+0
= minus2119885(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot int
120579|120577
minusinfin
exp ( 120591
119885minus1 + 120575120591) 119889120591
V119911
1003816100381610038161003816120577=minus0 = V119911
1003816100381610038161003816120577=+0
120597V119911
120597119911
10038161003816100381610038161003816100381610038161003816120577=minus0=120597V119911
120597119911
10038161003816100381610038161003816100381610038161003816120577=+0
1205972V119911
1205971199112
100381610038161003816100381610038161003816100381610038161003816120577=minus0
=1205972V119911
1205971199112
100381610038161003816100381610038161003816100381610038161003816120577=+0
1205973V119911
1205971199113
100381610038161003816100381610038161003816100381610038161003816120577=minus0
minus1205973V119911
1205971199113
100381610038161003816100381610038161003816100381610038161003816120577=+0
= minus119877(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot ((1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
minus 1)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (
120597120579
120597119911)
10038161003816100381610038161003816100381610038161003816120577=+0
(69)
with the conditions at infinity
119911 = minusinfin 120579 = 0 V = 0
119911 = +infin 120579 = minus1 V = 0
(70)
5 Stability Analysis
The interface problem analyzed in Section 4 has a travellingwave solution in the following form
(120579 (119909 119910 119911 119905) 120572 (119909 119910 119911 119905) V)
= (120579119904(119911 minus 119906119905) 120572
119904(119911 minus 119906119905) 0)
(120579119904 (119911 minus 119906119905) 120572119904 (119911 minus 119906119905))
=
(0 1) 1199112lt 0
(exp (minus1199061199112) minus 1 0) 119911
2gt 0
(71)
where 1199112= 119911minus119906119905 and 119906 is the speed of the stationary reaction
frontThis solution is a basic and stationary solution of inter-face problem (61)ndash(70) written in the moving coordinatesThen (61) (63) (65) and (67) can be replaced by
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 + 119906
120597120579
1205971199112
120597V120597119905
+ (Vnabla) V
= minusnabla119901 + 119875ΔV + 119906120597120579
1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (120579 + 120579
0) 120574
(72)
The other equations remain unchanged To study thestability of reaction front we seek the solution of the problemas follows
120579 = 120579119904+ 120579
119901 = 119901119904+ 119901
V = V119904+ V
(73)
where 120579 119901 and V are respectively small perturbation oftemperature pressure and velocity
8 Mathematical Problems in Engineering
Subtituting (73) into (64) (72) one obtains the first-orderterms as follows
In the liquid monomer 1199112gt 120585
120597120579
120597119905= Δ120579 + 119906
120597120579
1205971199112
minus V1199111205791015840
119904 (74)
120597V120597119905
= minusnabla119901 + 119875ΔV + 119906120597120579
1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579120574
(75)
div (V) = 0 (76)
In the liquid polymer 1199112lt 120585
120597120579
120597119905= Δ120579 + 119906
120597120579
1205971199112
(77)
120597V120597119905
= minusnabla119901 + 119875ΔV + 119906120597V1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579120574
(78)
div (V) = 0 (79)
where 120585 is defined by 120585 = 120577 minus 119906119905We note
(120579 V119911) =
(1205791 V1199111) for 119911
2lt 120585
(1205792 V1199112) for 119911
2lt 120585
(80)
and consider the perturbation as follows
120579119894= 120579119894(1199112 119905) exp (119895 (119896
1119909 + 1198962119910))
V119911119894= V119911119894(1199112 119905) exp (119895 (119896
1119909 + 1198962119910))
120585 = 1205981(119905) exp (119895 (119896
1119909 + 1198962119910))
(81)
where 119896119894 119894 = 1 2 and 120598
1denote respectively the wave
numbers (in 119909 and 119910 directions) and the amplitude of theperturbation and 119895
2= minus1 Then jump conditions (69) are
linearized by taking into account the fact that
120579|120585=plusmn0 = 120579119904(plusmn0) + 120585120579
1015840
119904(plusmn0) + 120579 (plusmn0)
120597120579
1205971199112
10038161003816100381610038161003816100381610038161003816120585=plusmn0
= 1205791015840
119904(plusmn0) + 120585120579
10158401015840
119904(plusmn0) +
120597120579
1205971199112
100381610038161003816100381610038161003816100381610038161003816120585=plusmn0
(82)
Then the higher-order terms are given as
1205792
100381610038161003816100381610038161199112=0minus 1205791
100381610038161003816100381610038161199112=0= 119906120585 (83)
1205971205792
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
minus1205971205791
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
= minus1199062120585 minus
120597120585
120597119905+ V119911
10038161003816100381610038161199112=0 (84)
1199062120585 +
1205971205792
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
= minus119885
1199061205791
100381610038161003816100381610038161199112=0 (85)
V2119911
10038161003816100381610038161199112=0= V1119911
10038161003816100381610038161199112=0 (86)
120597V1199112
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
=120597V1119911
1205971199112
100381610038161003816100381610038161003816100381610038161199112=0
(87)
1205972V2119911
12059711991122
1003816100381610038161003816100381610038161003816100381610038161199112=0
=1205972V1119911
12059711991122
1003816100381610038161003816100381610038161003816100381610038161199112=0
(88)
1205973V2119911
12059711991132
1003816100381610038161003816100381610038161003816100381610038161199112=0
=1205973V1119911
12059711991132
1003816100381610038161003816100381610038161003816100381610038161199112=0
(89)
Substituting (81) into (83)ndash(89) leads to
1205792(0 119905) minus 120579
1(0 119905) = 119906120598
1(119905)
1205791015840
2(0 119905) minus 120579
1015840
1(0 119905) = minus120598
1 (119905) 1199062minus 1205981015840
1(119905) + V
1 (0 119905)
1205981 (119905) 119906
2+ 1205791015840
2(0 119905) = minus
119885
1199061205791 (0 119905)
V(119894)2(0 119905) = V(119894)
1(0 119905) 119894 = 0 1 2 3
(90)
where
V(119894) =120597119894V120597V119894
1205791015840=120597120579119894
120597119911
1205981015840
1(119905) =
1198891205981(119905)
119889119905
(91)
After simplification (75) and (78) become
120597
120597119905ΔV119911minus 119906
120597
1205971199112
ΔV119911= 119875ΔΔV
119911+ 119876(1 + 120582
1sin (120590
1119905)
+ 1205822sin (120590
2119905) (
1205972
1205971199092+
1205972
1205971199102))120579
(92)
Substituting (71) (87) (88) into (74) (77) (92) andintroducing the variable 119908 = V10158401015840 minus 1198962V where 119896 = radic1198962
1+ 11989622
one obtains two systems of equations as follows
1199112lt 120585
120597119908
120597119905minus 1199061199081015840minus 119875 (119908
10158401015840minus 1198962119908)
= minus1198761198962(1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579
119908 = V10158401015840 minus 1198962V
120597120579
120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 0
(93)
Mathematical Problems in Engineering 9
0 5 10 15 20 25 300
50
100
150
200
250
300
350
Rc
1205901
1205902 = radic212059011205902 = radic71205901
1205902 = 0
1205902 = radic371205901
Figure 1 Critical Rayleigh number as a function of 1205901for different
frequencies ratio and for 1205902= 0 120582
1= 1205822= 5 119875 = 05 119885 = 8 and
119896 = 07
1199112gt 120585
120597119908
120597119905minus 1199061199081015840minus 119875 (119908
10158401015840minus 1198962119908)
= minus1198761198962(1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579
119908 = V10158401015840 minus 1198962V
120597120579
120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911
2) V
(94)
with the following boundary conditions
1205792(0 119905) minus 120579
1(0 119905) = 119906120598
1(119905) (95)
1205791015840
2(0 119905) minus 120579
1015840
1(0 119905) = minus119906
21205761(119905) minus 120576
1015840
1(119905) + V
1(0 119905) (96)
1205791015840
2(0 119905) +
119885
1199061205791 (0 119905) = minus119906
21205761 (119905) (97)
V(119894)1(0 119905) = V(119894)
2(0 119905) 119894 = 0 1 2 3 (98)
Combining the three boundary conditions (95)ndash(97)yields
1205791015840
1(0 119905) +
119885
1199061205791(0 119905) =
1
119906(1205792(0 119905) minus 120579
1(0 119905))
1015840
119905
minus V1(0 119905)
(99)
6 Numerical Results
The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix
Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved
Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590
1for different frequencies ratio and for
0 25 50 75 100 125 1500
5
10
15
20
25
Rc
1205822
1205821 = 20
1205821 = 50
1205821 = 80
1205821 = 150
Figure 2 Critical Rayleigh number as a function of 1205822for different
values of 1205821 1205902= radic2120590
1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07
0 25 50 75 100 125 150 175 2000
50
100
150
200
250
Rc
1205822
1205902 = radic371205901
1205902 = radic1712059011205902 = radic71205901
1205902 = 1radic712059011205902 = 1radic171205901
1205902 = 1radic371205901
Figure 3 Critical Rayleigh number as a function of 1205822for different
frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590
1= 5
1205902= 0 In the absence of gravitational modulation (120590
1= 1205902=
0) we find the same result (119877119888asymp 27) as in [8 14] It can be
observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the
frequency 1205901 the frequencies ratio 120590
21205901has no effect on
the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590
2= radic37120590
1) comparing
to the periodic modulation case (1205902= 0)
The critical Rayleigh number as a function of the ampli-tude 120582
2is shown in Figure 2 for different values of 120582
1 We can
conclude that for a fixed value of 1205822and increased values of
1205821 the polymerization front loses its stability monotonically
revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation
Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582
2for different frequencies ratio
It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582
2 For
large values of 1205822(1205822ge 100) the effect of the frequencies
ratio on the convective instability boundary is insignificant
10 Mathematical Problems in Engineering
0 25 50 75 100 125 150
k = 15
k = 2
k = 1
0
50
100
150
200
250
300
350
Rc
1205822
Figure 4 Critical Rayleigh number as a function of 1205822for different
values of number wave 1205902= radic7120590
1 1205901= 10 119875 = 05 119885 = 8 and
1205821= 5
0 25 50 75 100 125 1500
255075
100125150175200
Rc
1205822
1205902 = radic21205901
1205902 = radic71205901
Figure 5 Critical Rayleigh number as a function of 1205822for two
different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and
1205901= 10
Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7
the influence of the wave number on the critical Rayleighnumber versus 120582
2 The plots indicate that for large values of
the wave number the reaction front gains stabilityThe influence of 120582
2on the stability of the reaction front is
given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect
7 Conclusion
In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590
1the
critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies
ratio and for a given amplitude 1205822an increase of the ampli-
tude1205821destabilizes the reaction front Also the reaction front
becomes more stable by increasing the wave number
Appendix
The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911
119894= 119894ℎ where ℎ is the
space step and 119894 is a positive integerThe discretization in timeis given by 119905
119894= 119894120591 where 120591 is sufficiently small time step The
variables of discretization are 119908119895119894= 119908(119911
119894 119905119895) V119895119894= V(119911
119894 119905119895)
and 120579119895119894= 120579(119911119894 119905119895) One obtains the following
In the liquid monomer 119911 lt 120577
(minus1
ℎ2) 120579119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ) 120579119895+1
119894
+ (minus1
ℎ2minus119906
ℎ) 120579119895+1
119894+1=1
120591120579119895
119894
(minus119875
ℎ2)119908119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119908119895+1
119894+1=1
120591119908119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 120579119895
119894
(1
ℎ2) V119895+1119894minus1
+ (minus2
ℎ2minus 1198962) V119895+1119894
+ (1
ℎ2) V119895+1119894+1
= 119908119895
119894
(A1)
In the liquid polymer 119911 gt 120577
(minus1
ℎ2)119879119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ)119879119895+1
119894
+ (minus1
ℎ2minus119906
ℎ)119879119895+1
119894+1=1
120591119879119895
119894+ 119906 exp (minus119906119911
119894) 119881119895
119894
(minus119875
ℎ2)119882119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119882119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
119894+1=1
120591119882119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 119879119895
119894
(1
ℎ2)119881119895+1
119894minus1+ (minus
2
ℎ2minus 1198962)119881119895+1
119894+ (
1
ℎ2)119881119895+1
119894+1= 119882119895
119894
(A2)
From the jump conditions we have also the followingsystems
(minus1
ℎ2) 120579119895+1
1
+ (1
120591+
2
ℎ2+ 1198962+119906
ℎ+ (
1198881
1198882
)(1
ℎ2+119906
ℎ)) 120579119895+1
0
Mathematical Problems in Engineering 11
= (minus1
120591) 120579119895
0minus (
1
1199061205911198882
)(1
ℎ2+119906
ℎ) (119879119895
0minus 120579119895
0)
minus (V0
1198882
)(1
ℎ2+119906
ℎ)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
1
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198882
1198881
)1198790minus
1
1199061205911198881
(119879119895
0minus 120579119895
0) minus
V0
1198881
)
(A3)
(1
120591+
2
ℎ2+ 1198962+119906
ℎ+
1198882
ℎ21198881
)119879119895+1
0+ (
minus1
ℎ2minus119906
ℎ)119879119895+1
1
= (minus1
120591)119879119895
0minus 119906 exp (minus119906ℎ) minus ( 1
119906120591ℎ21198881
) (119879119895
0minus 120579119895
0)
minusV0
ℎ21198881
(A4)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
0
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198881
1198882
)1205790minus
1
1199061205911198882
)(119879119895
0minus 120579119895
0minusV0
1198882
)
(A5)
with
1198881= (
1
ℎminus119885
119906minus
1
119906120591)
1198882= (minus
1
ℎ+
1
119906120591)
(A6)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974
[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011
[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014
[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994
[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997
[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996
[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001
[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010
[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012
[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014
[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969
[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961
[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938
[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
120597V120597119905
+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579
0) 120574
(63)
div (V) = 0 (64)
In the liquid polymer 119911 lt 120577
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 (65)
120572 = 1 (66)
120597V120597119905
+ (V sdot nabla) V
= minusnabla119901 + 119875ΔV
+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579
0) 120574
(67)
div (V) = 0 (68)
At the interface 119911 = 120577
120579|120577minus0 = 120579|120577+0
120597120579
120597119911
10038161003816100381610038161003816100381610038161003816120577=minus0minus120597120579
120597119911
10038161003816100381610038161003816100381610038161003816120577=+0= (1 + (
120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot (120597120577
120597119905+ (V119909
120597120577
120597119909+ V119910
120597120577
120597119910minus V119911)
10038161003816100381610038161003816100381610038161003816120577
)
(120597120579
120597119911)
2100381610038161003816100381610038161003816100381610038161003816120577minus0
minus (120597120579
120597119911)
21003816100381610038161003816100381610038161003816100381610038161205771+0
= minus2119885(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot int
120579|120577
minusinfin
exp ( 120591
119885minus1 + 120575120591) 119889120591
V119911
1003816100381610038161003816120577=minus0 = V119911
1003816100381610038161003816120577=+0
120597V119911
120597119911
10038161003816100381610038161003816100381610038161003816120577=minus0=120597V119911
120597119911
10038161003816100381610038161003816100381610038161003816120577=+0
1205972V119911
1205971199112
100381610038161003816100381610038161003816100381610038161003816120577=minus0
=1205972V119911
1205971199112
100381610038161003816100381610038161003816100381610038161003816120577=+0
1205973V119911
1205971199113
100381610038161003816100381610038161003816100381610038161003816120577=minus0
minus1205973V119911
1205971199113
100381610038161003816100381610038161003816100381610038161003816120577=+0
= minus119877(1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
sdot ((1 + (120597120577
120597119909)
2
+ (120597120577
120597119910)
2
)
minus1
minus 1)
sdot (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (
120597120579
120597119911)
10038161003816100381610038161003816100381610038161003816120577=+0
(69)
with the conditions at infinity
119911 = minusinfin 120579 = 0 V = 0
119911 = +infin 120579 = minus1 V = 0
(70)
5 Stability Analysis
The interface problem analyzed in Section 4 has a travellingwave solution in the following form
(120579 (119909 119910 119911 119905) 120572 (119909 119910 119911 119905) V)
= (120579119904(119911 minus 119906119905) 120572
119904(119911 minus 119906119905) 0)
(120579119904 (119911 minus 119906119905) 120572119904 (119911 minus 119906119905))
=
(0 1) 1199112lt 0
(exp (minus1199061199112) minus 1 0) 119911
2gt 0
(71)
where 1199112= 119911minus119906119905 and 119906 is the speed of the stationary reaction
frontThis solution is a basic and stationary solution of inter-face problem (61)ndash(70) written in the moving coordinatesThen (61) (63) (65) and (67) can be replaced by
120597120579
120597119905+ (V sdot nabla) 120579 = Δ120579 + 119906
120597120579
1205971199112
120597V120597119905
+ (Vnabla) V
= minusnabla119901 + 119875ΔV + 119906120597120579
1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) (120579 + 120579
0) 120574
(72)
The other equations remain unchanged To study thestability of reaction front we seek the solution of the problemas follows
120579 = 120579119904+ 120579
119901 = 119901119904+ 119901
V = V119904+ V
(73)
where 120579 119901 and V are respectively small perturbation oftemperature pressure and velocity
8 Mathematical Problems in Engineering
Subtituting (73) into (64) (72) one obtains the first-orderterms as follows
In the liquid monomer 1199112gt 120585
120597120579
120597119905= Δ120579 + 119906
120597120579
1205971199112
minus V1199111205791015840
119904 (74)
120597V120597119905
= minusnabla119901 + 119875ΔV + 119906120597120579
1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579120574
(75)
div (V) = 0 (76)
In the liquid polymer 1199112lt 120585
120597120579
120597119905= Δ120579 + 119906
120597120579
1205971199112
(77)
120597V120597119905
= minusnabla119901 + 119875ΔV + 119906120597V1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579120574
(78)
div (V) = 0 (79)
where 120585 is defined by 120585 = 120577 minus 119906119905We note
(120579 V119911) =
(1205791 V1199111) for 119911
2lt 120585
(1205792 V1199112) for 119911
2lt 120585
(80)
and consider the perturbation as follows
120579119894= 120579119894(1199112 119905) exp (119895 (119896
1119909 + 1198962119910))
V119911119894= V119911119894(1199112 119905) exp (119895 (119896
1119909 + 1198962119910))
120585 = 1205981(119905) exp (119895 (119896
1119909 + 1198962119910))
(81)
where 119896119894 119894 = 1 2 and 120598
1denote respectively the wave
numbers (in 119909 and 119910 directions) and the amplitude of theperturbation and 119895
2= minus1 Then jump conditions (69) are
linearized by taking into account the fact that
120579|120585=plusmn0 = 120579119904(plusmn0) + 120585120579
1015840
119904(plusmn0) + 120579 (plusmn0)
120597120579
1205971199112
10038161003816100381610038161003816100381610038161003816120585=plusmn0
= 1205791015840
119904(plusmn0) + 120585120579
10158401015840
119904(plusmn0) +
120597120579
1205971199112
100381610038161003816100381610038161003816100381610038161003816120585=plusmn0
(82)
Then the higher-order terms are given as
1205792
100381610038161003816100381610038161199112=0minus 1205791
100381610038161003816100381610038161199112=0= 119906120585 (83)
1205971205792
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
minus1205971205791
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
= minus1199062120585 minus
120597120585
120597119905+ V119911
10038161003816100381610038161199112=0 (84)
1199062120585 +
1205971205792
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
= minus119885
1199061205791
100381610038161003816100381610038161199112=0 (85)
V2119911
10038161003816100381610038161199112=0= V1119911
10038161003816100381610038161199112=0 (86)
120597V1199112
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
=120597V1119911
1205971199112
100381610038161003816100381610038161003816100381610038161199112=0
(87)
1205972V2119911
12059711991122
1003816100381610038161003816100381610038161003816100381610038161199112=0
=1205972V1119911
12059711991122
1003816100381610038161003816100381610038161003816100381610038161199112=0
(88)
1205973V2119911
12059711991132
1003816100381610038161003816100381610038161003816100381610038161199112=0
=1205973V1119911
12059711991132
1003816100381610038161003816100381610038161003816100381610038161199112=0
(89)
Substituting (81) into (83)ndash(89) leads to
1205792(0 119905) minus 120579
1(0 119905) = 119906120598
1(119905)
1205791015840
2(0 119905) minus 120579
1015840
1(0 119905) = minus120598
1 (119905) 1199062minus 1205981015840
1(119905) + V
1 (0 119905)
1205981 (119905) 119906
2+ 1205791015840
2(0 119905) = minus
119885
1199061205791 (0 119905)
V(119894)2(0 119905) = V(119894)
1(0 119905) 119894 = 0 1 2 3
(90)
where
V(119894) =120597119894V120597V119894
1205791015840=120597120579119894
120597119911
1205981015840
1(119905) =
1198891205981(119905)
119889119905
(91)
After simplification (75) and (78) become
120597
120597119905ΔV119911minus 119906
120597
1205971199112
ΔV119911= 119875ΔΔV
119911+ 119876(1 + 120582
1sin (120590
1119905)
+ 1205822sin (120590
2119905) (
1205972
1205971199092+
1205972
1205971199102))120579
(92)
Substituting (71) (87) (88) into (74) (77) (92) andintroducing the variable 119908 = V10158401015840 minus 1198962V where 119896 = radic1198962
1+ 11989622
one obtains two systems of equations as follows
1199112lt 120585
120597119908
120597119905minus 1199061199081015840minus 119875 (119908
10158401015840minus 1198962119908)
= minus1198761198962(1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579
119908 = V10158401015840 minus 1198962V
120597120579
120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 0
(93)
Mathematical Problems in Engineering 9
0 5 10 15 20 25 300
50
100
150
200
250
300
350
Rc
1205901
1205902 = radic212059011205902 = radic71205901
1205902 = 0
1205902 = radic371205901
Figure 1 Critical Rayleigh number as a function of 1205901for different
frequencies ratio and for 1205902= 0 120582
1= 1205822= 5 119875 = 05 119885 = 8 and
119896 = 07
1199112gt 120585
120597119908
120597119905minus 1199061199081015840minus 119875 (119908
10158401015840minus 1198962119908)
= minus1198761198962(1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579
119908 = V10158401015840 minus 1198962V
120597120579
120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911
2) V
(94)
with the following boundary conditions
1205792(0 119905) minus 120579
1(0 119905) = 119906120598
1(119905) (95)
1205791015840
2(0 119905) minus 120579
1015840
1(0 119905) = minus119906
21205761(119905) minus 120576
1015840
1(119905) + V
1(0 119905) (96)
1205791015840
2(0 119905) +
119885
1199061205791 (0 119905) = minus119906
21205761 (119905) (97)
V(119894)1(0 119905) = V(119894)
2(0 119905) 119894 = 0 1 2 3 (98)
Combining the three boundary conditions (95)ndash(97)yields
1205791015840
1(0 119905) +
119885
1199061205791(0 119905) =
1
119906(1205792(0 119905) minus 120579
1(0 119905))
1015840
119905
minus V1(0 119905)
(99)
6 Numerical Results
The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix
Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved
Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590
1for different frequencies ratio and for
0 25 50 75 100 125 1500
5
10
15
20
25
Rc
1205822
1205821 = 20
1205821 = 50
1205821 = 80
1205821 = 150
Figure 2 Critical Rayleigh number as a function of 1205822for different
values of 1205821 1205902= radic2120590
1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07
0 25 50 75 100 125 150 175 2000
50
100
150
200
250
Rc
1205822
1205902 = radic371205901
1205902 = radic1712059011205902 = radic71205901
1205902 = 1radic712059011205902 = 1radic171205901
1205902 = 1radic371205901
Figure 3 Critical Rayleigh number as a function of 1205822for different
frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590
1= 5
1205902= 0 In the absence of gravitational modulation (120590
1= 1205902=
0) we find the same result (119877119888asymp 27) as in [8 14] It can be
observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the
frequency 1205901 the frequencies ratio 120590
21205901has no effect on
the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590
2= radic37120590
1) comparing
to the periodic modulation case (1205902= 0)
The critical Rayleigh number as a function of the ampli-tude 120582
2is shown in Figure 2 for different values of 120582
1 We can
conclude that for a fixed value of 1205822and increased values of
1205821 the polymerization front loses its stability monotonically
revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation
Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582
2for different frequencies ratio
It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582
2 For
large values of 1205822(1205822ge 100) the effect of the frequencies
ratio on the convective instability boundary is insignificant
10 Mathematical Problems in Engineering
0 25 50 75 100 125 150
k = 15
k = 2
k = 1
0
50
100
150
200
250
300
350
Rc
1205822
Figure 4 Critical Rayleigh number as a function of 1205822for different
values of number wave 1205902= radic7120590
1 1205901= 10 119875 = 05 119885 = 8 and
1205821= 5
0 25 50 75 100 125 1500
255075
100125150175200
Rc
1205822
1205902 = radic21205901
1205902 = radic71205901
Figure 5 Critical Rayleigh number as a function of 1205822for two
different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and
1205901= 10
Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7
the influence of the wave number on the critical Rayleighnumber versus 120582
2 The plots indicate that for large values of
the wave number the reaction front gains stabilityThe influence of 120582
2on the stability of the reaction front is
given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect
7 Conclusion
In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590
1the
critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies
ratio and for a given amplitude 1205822an increase of the ampli-
tude1205821destabilizes the reaction front Also the reaction front
becomes more stable by increasing the wave number
Appendix
The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911
119894= 119894ℎ where ℎ is the
space step and 119894 is a positive integerThe discretization in timeis given by 119905
119894= 119894120591 where 120591 is sufficiently small time step The
variables of discretization are 119908119895119894= 119908(119911
119894 119905119895) V119895119894= V(119911
119894 119905119895)
and 120579119895119894= 120579(119911119894 119905119895) One obtains the following
In the liquid monomer 119911 lt 120577
(minus1
ℎ2) 120579119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ) 120579119895+1
119894
+ (minus1
ℎ2minus119906
ℎ) 120579119895+1
119894+1=1
120591120579119895
119894
(minus119875
ℎ2)119908119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119908119895+1
119894+1=1
120591119908119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 120579119895
119894
(1
ℎ2) V119895+1119894minus1
+ (minus2
ℎ2minus 1198962) V119895+1119894
+ (1
ℎ2) V119895+1119894+1
= 119908119895
119894
(A1)
In the liquid polymer 119911 gt 120577
(minus1
ℎ2)119879119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ)119879119895+1
119894
+ (minus1
ℎ2minus119906
ℎ)119879119895+1
119894+1=1
120591119879119895
119894+ 119906 exp (minus119906119911
119894) 119881119895
119894
(minus119875
ℎ2)119882119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119882119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
119894+1=1
120591119882119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 119879119895
119894
(1
ℎ2)119881119895+1
119894minus1+ (minus
2
ℎ2minus 1198962)119881119895+1
119894+ (
1
ℎ2)119881119895+1
119894+1= 119882119895
119894
(A2)
From the jump conditions we have also the followingsystems
(minus1
ℎ2) 120579119895+1
1
+ (1
120591+
2
ℎ2+ 1198962+119906
ℎ+ (
1198881
1198882
)(1
ℎ2+119906
ℎ)) 120579119895+1
0
Mathematical Problems in Engineering 11
= (minus1
120591) 120579119895
0minus (
1
1199061205911198882
)(1
ℎ2+119906
ℎ) (119879119895
0minus 120579119895
0)
minus (V0
1198882
)(1
ℎ2+119906
ℎ)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
1
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198882
1198881
)1198790minus
1
1199061205911198881
(119879119895
0minus 120579119895
0) minus
V0
1198881
)
(A3)
(1
120591+
2
ℎ2+ 1198962+119906
ℎ+
1198882
ℎ21198881
)119879119895+1
0+ (
minus1
ℎ2minus119906
ℎ)119879119895+1
1
= (minus1
120591)119879119895
0minus 119906 exp (minus119906ℎ) minus ( 1
119906120591ℎ21198881
) (119879119895
0minus 120579119895
0)
minusV0
ℎ21198881
(A4)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
0
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198881
1198882
)1205790minus
1
1199061205911198882
)(119879119895
0minus 120579119895
0minusV0
1198882
)
(A5)
with
1198881= (
1
ℎminus119885
119906minus
1
119906120591)
1198882= (minus
1
ℎ+
1
119906120591)
(A6)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974
[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011
[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014
[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994
[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997
[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996
[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001
[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010
[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012
[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014
[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969
[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961
[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938
[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Subtituting (73) into (64) (72) one obtains the first-orderterms as follows
In the liquid monomer 1199112gt 120585
120597120579
120597119905= Δ120579 + 119906
120597120579
1205971199112
minus V1199111205791015840
119904 (74)
120597V120597119905
= minusnabla119901 + 119875ΔV + 119906120597120579
1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579120574
(75)
div (V) = 0 (76)
In the liquid polymer 1199112lt 120585
120597120579
120597119905= Δ120579 + 119906
120597120579
1205971199112
(77)
120597V120597119905
= minusnabla119901 + 119875ΔV + 119906120597V1205971199112
+ 119876 (1 + 1205821sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579120574
(78)
div (V) = 0 (79)
where 120585 is defined by 120585 = 120577 minus 119906119905We note
(120579 V119911) =
(1205791 V1199111) for 119911
2lt 120585
(1205792 V1199112) for 119911
2lt 120585
(80)
and consider the perturbation as follows
120579119894= 120579119894(1199112 119905) exp (119895 (119896
1119909 + 1198962119910))
V119911119894= V119911119894(1199112 119905) exp (119895 (119896
1119909 + 1198962119910))
120585 = 1205981(119905) exp (119895 (119896
1119909 + 1198962119910))
(81)
where 119896119894 119894 = 1 2 and 120598
1denote respectively the wave
numbers (in 119909 and 119910 directions) and the amplitude of theperturbation and 119895
2= minus1 Then jump conditions (69) are
linearized by taking into account the fact that
120579|120585=plusmn0 = 120579119904(plusmn0) + 120585120579
1015840
119904(plusmn0) + 120579 (plusmn0)
120597120579
1205971199112
10038161003816100381610038161003816100381610038161003816120585=plusmn0
= 1205791015840
119904(plusmn0) + 120585120579
10158401015840
119904(plusmn0) +
120597120579
1205971199112
100381610038161003816100381610038161003816100381610038161003816120585=plusmn0
(82)
Then the higher-order terms are given as
1205792
100381610038161003816100381610038161199112=0minus 1205791
100381610038161003816100381610038161199112=0= 119906120585 (83)
1205971205792
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
minus1205971205791
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
= minus1199062120585 minus
120597120585
120597119905+ V119911
10038161003816100381610038161199112=0 (84)
1199062120585 +
1205971205792
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
= minus119885
1199061205791
100381610038161003816100381610038161199112=0 (85)
V2119911
10038161003816100381610038161199112=0= V1119911
10038161003816100381610038161199112=0 (86)
120597V1199112
1205971199112
1003816100381610038161003816100381610038161003816100381610038161199112=0
=120597V1119911
1205971199112
100381610038161003816100381610038161003816100381610038161199112=0
(87)
1205972V2119911
12059711991122
1003816100381610038161003816100381610038161003816100381610038161199112=0
=1205972V1119911
12059711991122
1003816100381610038161003816100381610038161003816100381610038161199112=0
(88)
1205973V2119911
12059711991132
1003816100381610038161003816100381610038161003816100381610038161199112=0
=1205973V1119911
12059711991132
1003816100381610038161003816100381610038161003816100381610038161199112=0
(89)
Substituting (81) into (83)ndash(89) leads to
1205792(0 119905) minus 120579
1(0 119905) = 119906120598
1(119905)
1205791015840
2(0 119905) minus 120579
1015840
1(0 119905) = minus120598
1 (119905) 1199062minus 1205981015840
1(119905) + V
1 (0 119905)
1205981 (119905) 119906
2+ 1205791015840
2(0 119905) = minus
119885
1199061205791 (0 119905)
V(119894)2(0 119905) = V(119894)
1(0 119905) 119894 = 0 1 2 3
(90)
where
V(119894) =120597119894V120597V119894
1205791015840=120597120579119894
120597119911
1205981015840
1(119905) =
1198891205981(119905)
119889119905
(91)
After simplification (75) and (78) become
120597
120597119905ΔV119911minus 119906
120597
1205971199112
ΔV119911= 119875ΔΔV
119911+ 119876(1 + 120582
1sin (120590
1119905)
+ 1205822sin (120590
2119905) (
1205972
1205971199092+
1205972
1205971199102))120579
(92)
Substituting (71) (87) (88) into (74) (77) (92) andintroducing the variable 119908 = V10158401015840 minus 1198962V where 119896 = radic1198962
1+ 11989622
one obtains two systems of equations as follows
1199112lt 120585
120597119908
120597119905minus 1199061199081015840minus 119875 (119908
10158401015840minus 1198962119908)
= minus1198761198962(1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579
119908 = V10158401015840 minus 1198962V
120597120579
120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 0
(93)
Mathematical Problems in Engineering 9
0 5 10 15 20 25 300
50
100
150
200
250
300
350
Rc
1205901
1205902 = radic212059011205902 = radic71205901
1205902 = 0
1205902 = radic371205901
Figure 1 Critical Rayleigh number as a function of 1205901for different
frequencies ratio and for 1205902= 0 120582
1= 1205822= 5 119875 = 05 119885 = 8 and
119896 = 07
1199112gt 120585
120597119908
120597119905minus 1199061199081015840minus 119875 (119908
10158401015840minus 1198962119908)
= minus1198761198962(1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579
119908 = V10158401015840 minus 1198962V
120597120579
120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911
2) V
(94)
with the following boundary conditions
1205792(0 119905) minus 120579
1(0 119905) = 119906120598
1(119905) (95)
1205791015840
2(0 119905) minus 120579
1015840
1(0 119905) = minus119906
21205761(119905) minus 120576
1015840
1(119905) + V
1(0 119905) (96)
1205791015840
2(0 119905) +
119885
1199061205791 (0 119905) = minus119906
21205761 (119905) (97)
V(119894)1(0 119905) = V(119894)
2(0 119905) 119894 = 0 1 2 3 (98)
Combining the three boundary conditions (95)ndash(97)yields
1205791015840
1(0 119905) +
119885
1199061205791(0 119905) =
1
119906(1205792(0 119905) minus 120579
1(0 119905))
1015840
119905
minus V1(0 119905)
(99)
6 Numerical Results
The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix
Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved
Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590
1for different frequencies ratio and for
0 25 50 75 100 125 1500
5
10
15
20
25
Rc
1205822
1205821 = 20
1205821 = 50
1205821 = 80
1205821 = 150
Figure 2 Critical Rayleigh number as a function of 1205822for different
values of 1205821 1205902= radic2120590
1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07
0 25 50 75 100 125 150 175 2000
50
100
150
200
250
Rc
1205822
1205902 = radic371205901
1205902 = radic1712059011205902 = radic71205901
1205902 = 1radic712059011205902 = 1radic171205901
1205902 = 1radic371205901
Figure 3 Critical Rayleigh number as a function of 1205822for different
frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590
1= 5
1205902= 0 In the absence of gravitational modulation (120590
1= 1205902=
0) we find the same result (119877119888asymp 27) as in [8 14] It can be
observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the
frequency 1205901 the frequencies ratio 120590
21205901has no effect on
the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590
2= radic37120590
1) comparing
to the periodic modulation case (1205902= 0)
The critical Rayleigh number as a function of the ampli-tude 120582
2is shown in Figure 2 for different values of 120582
1 We can
conclude that for a fixed value of 1205822and increased values of
1205821 the polymerization front loses its stability monotonically
revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation
Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582
2for different frequencies ratio
It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582
2 For
large values of 1205822(1205822ge 100) the effect of the frequencies
ratio on the convective instability boundary is insignificant
10 Mathematical Problems in Engineering
0 25 50 75 100 125 150
k = 15
k = 2
k = 1
0
50
100
150
200
250
300
350
Rc
1205822
Figure 4 Critical Rayleigh number as a function of 1205822for different
values of number wave 1205902= radic7120590
1 1205901= 10 119875 = 05 119885 = 8 and
1205821= 5
0 25 50 75 100 125 1500
255075
100125150175200
Rc
1205822
1205902 = radic21205901
1205902 = radic71205901
Figure 5 Critical Rayleigh number as a function of 1205822for two
different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and
1205901= 10
Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7
the influence of the wave number on the critical Rayleighnumber versus 120582
2 The plots indicate that for large values of
the wave number the reaction front gains stabilityThe influence of 120582
2on the stability of the reaction front is
given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect
7 Conclusion
In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590
1the
critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies
ratio and for a given amplitude 1205822an increase of the ampli-
tude1205821destabilizes the reaction front Also the reaction front
becomes more stable by increasing the wave number
Appendix
The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911
119894= 119894ℎ where ℎ is the
space step and 119894 is a positive integerThe discretization in timeis given by 119905
119894= 119894120591 where 120591 is sufficiently small time step The
variables of discretization are 119908119895119894= 119908(119911
119894 119905119895) V119895119894= V(119911
119894 119905119895)
and 120579119895119894= 120579(119911119894 119905119895) One obtains the following
In the liquid monomer 119911 lt 120577
(minus1
ℎ2) 120579119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ) 120579119895+1
119894
+ (minus1
ℎ2minus119906
ℎ) 120579119895+1
119894+1=1
120591120579119895
119894
(minus119875
ℎ2)119908119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119908119895+1
119894+1=1
120591119908119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 120579119895
119894
(1
ℎ2) V119895+1119894minus1
+ (minus2
ℎ2minus 1198962) V119895+1119894
+ (1
ℎ2) V119895+1119894+1
= 119908119895
119894
(A1)
In the liquid polymer 119911 gt 120577
(minus1
ℎ2)119879119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ)119879119895+1
119894
+ (minus1
ℎ2minus119906
ℎ)119879119895+1
119894+1=1
120591119879119895
119894+ 119906 exp (minus119906119911
119894) 119881119895
119894
(minus119875
ℎ2)119882119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119882119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
119894+1=1
120591119882119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 119879119895
119894
(1
ℎ2)119881119895+1
119894minus1+ (minus
2
ℎ2minus 1198962)119881119895+1
119894+ (
1
ℎ2)119881119895+1
119894+1= 119882119895
119894
(A2)
From the jump conditions we have also the followingsystems
(minus1
ℎ2) 120579119895+1
1
+ (1
120591+
2
ℎ2+ 1198962+119906
ℎ+ (
1198881
1198882
)(1
ℎ2+119906
ℎ)) 120579119895+1
0
Mathematical Problems in Engineering 11
= (minus1
120591) 120579119895
0minus (
1
1199061205911198882
)(1
ℎ2+119906
ℎ) (119879119895
0minus 120579119895
0)
minus (V0
1198882
)(1
ℎ2+119906
ℎ)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
1
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198882
1198881
)1198790minus
1
1199061205911198881
(119879119895
0minus 120579119895
0) minus
V0
1198881
)
(A3)
(1
120591+
2
ℎ2+ 1198962+119906
ℎ+
1198882
ℎ21198881
)119879119895+1
0+ (
minus1
ℎ2minus119906
ℎ)119879119895+1
1
= (minus1
120591)119879119895
0minus 119906 exp (minus119906ℎ) minus ( 1
119906120591ℎ21198881
) (119879119895
0minus 120579119895
0)
minusV0
ℎ21198881
(A4)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
0
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198881
1198882
)1205790minus
1
1199061205911198882
)(119879119895
0minus 120579119895
0minusV0
1198882
)
(A5)
with
1198881= (
1
ℎminus119885
119906minus
1
119906120591)
1198882= (minus
1
ℎ+
1
119906120591)
(A6)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974
[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011
[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014
[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994
[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997
[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996
[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001
[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010
[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012
[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014
[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969
[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961
[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938
[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
0 5 10 15 20 25 300
50
100
150
200
250
300
350
Rc
1205901
1205902 = radic212059011205902 = radic71205901
1205902 = 0
1205902 = radic371205901
Figure 1 Critical Rayleigh number as a function of 1205901for different
frequencies ratio and for 1205902= 0 120582
1= 1205822= 5 119875 = 05 119885 = 8 and
119896 = 07
1199112gt 120585
120597119908
120597119905minus 1199061199081015840minus 119875 (119908
10158401015840minus 1198962119908)
= minus1198761198962(1 + 120582
1sin (120590
1119905) + 120582
2sin (120590
2119905)) 120579
119908 = V10158401015840 minus 1198962V
120597120579
120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911
2) V
(94)
with the following boundary conditions
1205792(0 119905) minus 120579
1(0 119905) = 119906120598
1(119905) (95)
1205791015840
2(0 119905) minus 120579
1015840
1(0 119905) = minus119906
21205761(119905) minus 120576
1015840
1(119905) + V
1(0 119905) (96)
1205791015840
2(0 119905) +
119885
1199061205791 (0 119905) = minus119906
21205761 (119905) (97)
V(119894)1(0 119905) = V(119894)
2(0 119905) 119894 = 0 1 2 3 (98)
Combining the three boundary conditions (95)ndash(97)yields
1205791015840
1(0 119905) +
119885
1199061205791(0 119905) =
1
119906(1205792(0 119905) minus 120579
1(0 119905))
1015840
119905
minus V1(0 119905)
(99)
6 Numerical Results
The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix
Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved
Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590
1for different frequencies ratio and for
0 25 50 75 100 125 1500
5
10
15
20
25
Rc
1205822
1205821 = 20
1205821 = 50
1205821 = 80
1205821 = 150
Figure 2 Critical Rayleigh number as a function of 1205822for different
values of 1205821 1205902= radic2120590
1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07
0 25 50 75 100 125 150 175 2000
50
100
150
200
250
Rc
1205822
1205902 = radic371205901
1205902 = radic1712059011205902 = radic71205901
1205902 = 1radic712059011205902 = 1radic171205901
1205902 = 1radic371205901
Figure 3 Critical Rayleigh number as a function of 1205822for different
frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590
1= 5
1205902= 0 In the absence of gravitational modulation (120590
1= 1205902=
0) we find the same result (119877119888asymp 27) as in [8 14] It can be
observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the
frequency 1205901 the frequencies ratio 120590
21205901has no effect on
the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590
2= radic37120590
1) comparing
to the periodic modulation case (1205902= 0)
The critical Rayleigh number as a function of the ampli-tude 120582
2is shown in Figure 2 for different values of 120582
1 We can
conclude that for a fixed value of 1205822and increased values of
1205821 the polymerization front loses its stability monotonically
revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation
Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582
2for different frequencies ratio
It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582
2 For
large values of 1205822(1205822ge 100) the effect of the frequencies
ratio on the convective instability boundary is insignificant
10 Mathematical Problems in Engineering
0 25 50 75 100 125 150
k = 15
k = 2
k = 1
0
50
100
150
200
250
300
350
Rc
1205822
Figure 4 Critical Rayleigh number as a function of 1205822for different
values of number wave 1205902= radic7120590
1 1205901= 10 119875 = 05 119885 = 8 and
1205821= 5
0 25 50 75 100 125 1500
255075
100125150175200
Rc
1205822
1205902 = radic21205901
1205902 = radic71205901
Figure 5 Critical Rayleigh number as a function of 1205822for two
different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and
1205901= 10
Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7
the influence of the wave number on the critical Rayleighnumber versus 120582
2 The plots indicate that for large values of
the wave number the reaction front gains stabilityThe influence of 120582
2on the stability of the reaction front is
given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect
7 Conclusion
In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590
1the
critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies
ratio and for a given amplitude 1205822an increase of the ampli-
tude1205821destabilizes the reaction front Also the reaction front
becomes more stable by increasing the wave number
Appendix
The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911
119894= 119894ℎ where ℎ is the
space step and 119894 is a positive integerThe discretization in timeis given by 119905
119894= 119894120591 where 120591 is sufficiently small time step The
variables of discretization are 119908119895119894= 119908(119911
119894 119905119895) V119895119894= V(119911
119894 119905119895)
and 120579119895119894= 120579(119911119894 119905119895) One obtains the following
In the liquid monomer 119911 lt 120577
(minus1
ℎ2) 120579119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ) 120579119895+1
119894
+ (minus1
ℎ2minus119906
ℎ) 120579119895+1
119894+1=1
120591120579119895
119894
(minus119875
ℎ2)119908119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119908119895+1
119894+1=1
120591119908119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 120579119895
119894
(1
ℎ2) V119895+1119894minus1
+ (minus2
ℎ2minus 1198962) V119895+1119894
+ (1
ℎ2) V119895+1119894+1
= 119908119895
119894
(A1)
In the liquid polymer 119911 gt 120577
(minus1
ℎ2)119879119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ)119879119895+1
119894
+ (minus1
ℎ2minus119906
ℎ)119879119895+1
119894+1=1
120591119879119895
119894+ 119906 exp (minus119906119911
119894) 119881119895
119894
(minus119875
ℎ2)119882119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119882119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
119894+1=1
120591119882119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 119879119895
119894
(1
ℎ2)119881119895+1
119894minus1+ (minus
2
ℎ2minus 1198962)119881119895+1
119894+ (
1
ℎ2)119881119895+1
119894+1= 119882119895
119894
(A2)
From the jump conditions we have also the followingsystems
(minus1
ℎ2) 120579119895+1
1
+ (1
120591+
2
ℎ2+ 1198962+119906
ℎ+ (
1198881
1198882
)(1
ℎ2+119906
ℎ)) 120579119895+1
0
Mathematical Problems in Engineering 11
= (minus1
120591) 120579119895
0minus (
1
1199061205911198882
)(1
ℎ2+119906
ℎ) (119879119895
0minus 120579119895
0)
minus (V0
1198882
)(1
ℎ2+119906
ℎ)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
1
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198882
1198881
)1198790minus
1
1199061205911198881
(119879119895
0minus 120579119895
0) minus
V0
1198881
)
(A3)
(1
120591+
2
ℎ2+ 1198962+119906
ℎ+
1198882
ℎ21198881
)119879119895+1
0+ (
minus1
ℎ2minus119906
ℎ)119879119895+1
1
= (minus1
120591)119879119895
0minus 119906 exp (minus119906ℎ) minus ( 1
119906120591ℎ21198881
) (119879119895
0minus 120579119895
0)
minusV0
ℎ21198881
(A4)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
0
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198881
1198882
)1205790minus
1
1199061205911198882
)(119879119895
0minus 120579119895
0minusV0
1198882
)
(A5)
with
1198881= (
1
ℎminus119885
119906minus
1
119906120591)
1198882= (minus
1
ℎ+
1
119906120591)
(A6)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974
[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011
[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014
[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994
[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997
[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996
[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001
[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010
[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012
[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014
[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969
[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961
[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938
[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
0 25 50 75 100 125 150
k = 15
k = 2
k = 1
0
50
100
150
200
250
300
350
Rc
1205822
Figure 4 Critical Rayleigh number as a function of 1205822for different
values of number wave 1205902= radic7120590
1 1205901= 10 119875 = 05 119885 = 8 and
1205821= 5
0 25 50 75 100 125 1500
255075
100125150175200
Rc
1205822
1205902 = radic21205901
1205902 = radic71205901
Figure 5 Critical Rayleigh number as a function of 1205822for two
different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and
1205901= 10
Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7
the influence of the wave number on the critical Rayleighnumber versus 120582
2 The plots indicate that for large values of
the wave number the reaction front gains stabilityThe influence of 120582
2on the stability of the reaction front is
given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect
7 Conclusion
In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590
1the
critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies
ratio and for a given amplitude 1205822an increase of the ampli-
tude1205821destabilizes the reaction front Also the reaction front
becomes more stable by increasing the wave number
Appendix
The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911
119894= 119894ℎ where ℎ is the
space step and 119894 is a positive integerThe discretization in timeis given by 119905
119894= 119894120591 where 120591 is sufficiently small time step The
variables of discretization are 119908119895119894= 119908(119911
119894 119905119895) V119895119894= V(119911
119894 119905119895)
and 120579119895119894= 120579(119911119894 119905119895) One obtains the following
In the liquid monomer 119911 lt 120577
(minus1
ℎ2) 120579119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ) 120579119895+1
119894
+ (minus1
ℎ2minus119906
ℎ) 120579119895+1
119894+1=1
120591120579119895
119894
(minus119875
ℎ2)119908119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119908119895+1
119894+1=1
120591119908119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 120579119895
119894
(1
ℎ2) V119895+1119894minus1
+ (minus2
ℎ2minus 1198962) V119895+1119894
+ (1
ℎ2) V119895+1119894+1
= 119908119895
119894
(A1)
In the liquid polymer 119911 gt 120577
(minus1
ℎ2)119879119895+1
119894minus1+ (
1
120591+
2
ℎ2+ 1198962+119906
ℎ)119879119895+1
119894
+ (minus1
ℎ2minus119906
ℎ)119879119895+1
119894+1=1
120591119879119895
119894+ 119906 exp (minus119906119911
119894) 119881119895
119894
(minus119875
ℎ2)119882119895+1
119894minus1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119882119895+1
119894
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
119894+1=1
120591119882119895
119894
minus 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895)) 119879119895
119894
(1
ℎ2)119881119895+1
119894minus1+ (minus
2
ℎ2minus 1198962)119881119895+1
119894+ (
1
ℎ2)119881119895+1
119894+1= 119882119895
119894
(A2)
From the jump conditions we have also the followingsystems
(minus1
ℎ2) 120579119895+1
1
+ (1
120591+
2
ℎ2+ 1198962+119906
ℎ+ (
1198881
1198882
)(1
ℎ2+119906
ℎ)) 120579119895+1
0
Mathematical Problems in Engineering 11
= (minus1
120591) 120579119895
0minus (
1
1199061205911198882
)(1
ℎ2+119906
ℎ) (119879119895
0minus 120579119895
0)
minus (V0
1198882
)(1
ℎ2+119906
ℎ)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
1
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198882
1198881
)1198790minus
1
1199061205911198881
(119879119895
0minus 120579119895
0) minus
V0
1198881
)
(A3)
(1
120591+
2
ℎ2+ 1198962+119906
ℎ+
1198882
ℎ21198881
)119879119895+1
0+ (
minus1
ℎ2minus119906
ℎ)119879119895+1
1
= (minus1
120591)119879119895
0minus 119906 exp (minus119906ℎ) minus ( 1
119906120591ℎ21198881
) (119879119895
0minus 120579119895
0)
minusV0
ℎ21198881
(A4)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
0
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198881
1198882
)1205790minus
1
1199061205911198882
)(119879119895
0minus 120579119895
0minusV0
1198882
)
(A5)
with
1198881= (
1
ℎminus119885
119906minus
1
119906120591)
1198882= (minus
1
ℎ+
1
119906120591)
(A6)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974
[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011
[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014
[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994
[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997
[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996
[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001
[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010
[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012
[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014
[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969
[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961
[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938
[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
= (minus1
120591) 120579119895
0minus (
1
1199061205911198882
)(1
ℎ2+119906
ℎ) (119879119895
0minus 120579119895
0)
minus (V0
1198882
)(1
ℎ2+119906
ℎ)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
1
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198882
1198881
)1198790minus
1
1199061205911198881
(119879119895
0minus 120579119895
0) minus
V0
1198881
)
(A3)
(1
120591+
2
ℎ2+ 1198962+119906
ℎ+
1198882
ℎ21198881
)119879119895+1
0+ (
minus1
ℎ2minus119906
ℎ)119879119895+1
1
= (minus1
120591)119879119895
0minus 119906 exp (minus119906ℎ) minus ( 1
119906120591ℎ21198881
) (119879119895
0minus 120579119895
0)
minusV0
ℎ21198881
(A4)
(minus119875
ℎ2)119908119895+1
1+ (
1
120591+2119875
ℎ2+ 1198751198962+119906
ℎ)119908119895+1
0
+ (minus119875
ℎ2minus119906
ℎ)119882119895+1
0= (
minus1
120591)119908119895
0
+ 1198761198962(1 + 120582
1sin (120590
1119905119895) + 1205822sin (120590
2119905119895))
sdot ((minus1198881
1198882
)1205790minus
1
1199061205911198882
)(119879119895
0minus 120579119895
0minusV0
1198882
)
(A5)
with
1198881= (
1
ℎminus119885
119906minus
1
119906120591)
1198882= (minus
1
ℎ+
1
119906120591)
(A6)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974
[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011
[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014
[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994
[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997
[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996
[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001
[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010
[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012
[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014
[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969
[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961
[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938
[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of