Research ArticleWeak Nonlinear Double-Diffusive Magnetoconvection in aNewtonian Liquid under Temperature Modulation
B S Bhadauria and Palle Kiran
Department of Applied Mathematics School for Physical Sciences Babasaheb Bhimrao Ambedkar University Lucknow 226 025 India
Correspondence should be addressed to B S Bhadauria mathsbsbyahoocom
Received 22 February 2014 Accepted 15 June 2014 Published 6 July 2014
Academic Editor Yurong Liu
Copyright copy 2014 B S Bhadauria and P KiranThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The present paper deals with a weak nonlinear theory of double-diffusive magnetoconvection in an electrically conductingNewtonian liquid confined between two horizontal surfaces under a constant vertical magnetic field and subjected to imposedtime-periodic thermal boundariesThe temperature of bothwalls is varied time periodic in this caseThe disturbances are expandedin terms of power series of amplitude of convection which is assumed to be small Using nonautonomous Ginzburg-Landauequation the Nusselt and Sherwood numbers obtained analytically and studied heat and mass transport in the system Effectof various parameters on the heat and mass transport is discussed extensively It is found that the effect of magnetic field is tostabilize the system Further it is also notified that the heat and mass transport can be controlled by suitably adjusting the externalparameters of the system
1 Introduction
Double-diffusive convection is an important fluid dynamicsphenomenon that involves motions driven by two differentdensity gradients diffusing at different rates In double-diffusive convection the buoyancy force is affected not onlyby the difference of temperatures but also by the differenceof concentration of the fluid An example of double-diffusiveconvection is seen in oceanography lakes undergroundwater atmospheric pollution chemical processes laboratoryexperiments modeling of solar ponds [1] electrochemistrymagma chambers and sparks [2] Fernando and Brandt[3]formation of microstructure during the cooling of moltenmetals fluid flows around shrouded heat-dissipation finsmigration of moisture through air contained in fibrous insu-lations grain storage system the dispersion of contaminantsthrough water saturated soil crystal P growth solidifica-tion of binary mixtures and the underground disposal ofnuclear wastes Therefore much work has also been done ondouble-diffusive convection in an electrically conducted fluidlayer because of its natural occurrence as mentioned aboveapplications Convection in planetary cores stellar interiors
and Earthrsquos metallic core occurs in the presence of strongmagnetic field The study of double-diffusive magnetocon-vection has recently drawn the attention of astrophysicistsgeophysicists oceanographers engineers and a host of others[4 5]The study of magnetoconvection in an electrically con-ducting horizontal fluid layer was motivated by astrophysicaland geophysical applications relate in some or the otherway to the problems concerning the external constraintslike rotation or magnetic field operative on double-diffusivesystems in particular by observation of sunspots [6]
Thompson [7] and Chandrasekhar [8] were the first tostudy the magnetoconvection in horizontal fluid layer Lortz[9]was the first to study the effect ofmagnetic field ondouble-diffusive convection His object was to clarify some of themathematical aspects of stability criterion [10] but his analysisis silent about the detailed study of stability analysis Stommelet al [11] explained that the diffusion is generally a stabilizingfactor in a single-component fluid But in the case of two-component system it can act to release the potential energyin the component that is the heaviest at the top and make thesystem unstable Gotoh and Yamada [12] studied the problemof magnetoconvection in a horizontal layer of magnetic fluid
Hindawi Publishing CorporationInternational Journal of Engineering MathematicsVolume 2014 Article ID 296216 14 pageshttpdxdoiorg1011552014296216
2 International Journal of Engineering Mathematics
which is heated frombelow and cooled from above and foundcondition for onset of convection Oreper and Szekely [13]have found that the presence of a magnetic field can suppressnatural convection currents and that the strength of themagnetic field is one of the important factors in determiningthe quality of the crystal Rudraiah and Shivakumara [5]investigated both linear and nonlinear theory of this problemin detail They have shown that the magnetic field undercertain conditions makes the system unstable They havealso investigated the effect of magnetic field on the physicallypreferred cell pattern Rudraiah [14] investigated the interac-tion between double-diffusive convection and an externallyimposed vertical magnetic field in a Boussinesq fluid Sid-dheshwar and Pranesh [15] analyzed the role ofmagnetic fieldin the inhibition of natural convection driven by combinedbuoyancy and surface tension forces in a horizontal layer ofan electrically conducting Boussinesq fluid with suspendedparticles confined between an upper freeadiabatic and alower rigidisothermal boundary is considered in 1 g and 120583gsituations Bhadauria [16] also studied the effect of magneticfield on thermal modulated convection in the case of porousmedium Siddheshwar et al [17] performed a local nonlin-ear stability analysis of Rayleigh-Benard magnetoconvectionusing Ginzburg-Landau equation They showed that gravitymodulation can be used to enhance or diminish the heattransport in stationary magnetoconvection
The classical Rayleigh-Benard convection due to bottomheating is well known and highly explored phenomenongiven by Chandrasekhar [8] and Drazin and Reid [18]Many researchers under different physical models haveinvestigated thermal instability in a horizontal fluid layerwith temperature modulation in the absence of double-diffusive magnetoconvection Some of them are Venezian[19] who was the first to consider the effect of temperaturemodulation on thermal instability in a horizontal fluid layerRosenblat and Tanaka [20] studied the linear stability for afluid in a classical geometry of Benard by considering thetemperature modulation of rigid-rigid boundaries The firstnonlinear stability problem in a horizontal fluid layer undertemperature modulation of the boundaries was studied byRoppo et al [21] Bhadauria and Bhatia [22] studied theeffect of temperature modulation on thermal instability byconsidering rigid-rigid boundaries and different types oftemperature profiles Bhadauria [23] Malashetty and Swamy[24] and Bhadauria and Kiran [25] are the related problems
The problem of double-diffusive magnetoconvection iscalled thermohaline magnetoconvection when the two dif-fusive mechanisms are thermal and solute From the aboveliterature (second paragraph) it is observed that concerninglinear theory numerous studies on magnetoconvection ordouble-diffusive magnetoconvection have been made Butthe double-diffusive magnetoconvection is still in its infantstage in nonlinear case Liner theory gives information aboutonset of convection but fails at heat andmass transport Mostof the studies are made with uniform temperature gradientacross the boundaries In general there are many practicalsituations in which the temperature gradient is a function ofboth time and space One of the external effective mecha-nisms to regulate (or control) the convection by maintaining
Electrically conductingNewtonian fluid layer
O
HbkZ
T = T0 +ΔT
2[1 + 1205982120575cos(120596t)] S = S + S0
T = T0 +ΔT
2[1 minus 1205982120575cos(120596t)] S = S0 + ΔS 120577 = 0
X
120577 = d
Y
Figure 1 Physical configuration of the problem
a nonuniform temperature gradient across the fluid layer Wealso observe from the literature (third paragraph) that somework has been done under temperature modulation withoutmagnetic field and double-diffusive convection which isessential in thermal and engineering science as mentionedin the literature Therefore from the above understandingwe motivated to analyze a weak nonlinear thermohalinemagnetoconvection under temperature modulation whilederiving an amplitude of convection using Ginzburg-Landaumodel
2 Governing Equations
We consider an electrically conducting fluid layer of depth119889 confined between two infinite parallel horizontal planesat 119911 = 0 and 119911 = 119889 A Cartesian frame of reference hasbeen taken with the origin at the bottom and the 119911 axisvertically upward direction of the fluid layerThe surfaces aremaintained at a constant temperature gradient Δ119879119889 and aconstant magnetic field 119867119887 is applied across the fluid layer(as shown in Figure 1) Under the Boussinesq approximationthe dimensional governing equations for the study of double-diffusive magnetoconvection in an electrically conductingfluid layer are defined as
+ (q sdot nabla) H minus (H sdot nabla) q = ]119898nabla2H (6)
120588 = 1205880 [1 minus 120572 (119879 minus 1198790) + 120573 (119878 minus 1198780)] (7)
where q is velocity (119906 V 119908) 120583 is a viscosity 120583119898 is magneticpermeability ]119898 is magnetic viscosity 120581119879 is the thermaldiffusivity 120581119878 is the solutal diffusivity 119879 is temperature
International Journal of Engineering Mathematics 3
120572 is thermal expansion coefficient 120573 is solute expansioncoefficient and 120574 is the ration of heat capacity For simplicity120574 is taken to be unity in this paper 120588 is density g = (0 0 minus119892)
is gravitational acceleration while 1205880 is reference densityand H is the magnetic field The externally imposed thermalboundary conditions considered in this paper are given byVenezian [19]
119879 = 1198790 +Δ119879
2[1 + 120598
2120575 cos (120596119905)] at 119911 = 0
= 1198790 minusΔ119879
2[1 minus 120598
2120575 cos (120596119905 + 120579)] at 119911 = 119889
(8)
where 120575 is small amplitude of modulation Δ119879 is the tem-perature difference across the fluid layer 120596 is the modulationfrequency and 120579 is the phase difference Since we are not con-sidering cross diffusion terms the walls of the fluid layer areassumed to bemaintained at constant solute concentration asdefined below
119878 = 1198780 + Δ119878 at 119911 = 0
= 1198780 at 119911 = 119889
(9)
The basic state is assumed to be quiescent and the quantitiesin this state are given by
The basic state pressure field is not required here howeverthe basic temperature and solute fields are governed by thefollowing ordinary differential equations
120597119879119887
120597119905= 120581119879
1205972119879119887
1205971199112
1205972119878119887
1205971199112= 0 (11)
The solutions of (11) subjected to the boundary conditions(8)-(9) are given by
where 119879119904(119911) is the study temperature field and 1198791(119911 119905) is theoscillating part while Re stands for the real part We assumefinite amplitude perturbations on the basic state in the form
where the primes denote the quantities at the perturbationsSubstituting (13) into (1)ndash(6) and using the basic state resultswe obtain the following equations
nabla sdot q1015840 = 0
nabla sdot H1015840 = 0
120597q1015840
120597119905+ (q1015840 sdot nabla) q minus 120583119898
120597119905+ (q1015840 sdot nabla) H1015840 minus (H1015840 sdot nabla) q1015840 minus 119867119887
120597w1015840
120597119911= ]119898nabla
2H1015840
1205881015840= minus1205880 (120572119879
1015840minus 1205731198781015840)
(14)
Further we consider only two-dimensional disturbances inour study and hence the stream function 120595 and magneticpotential Φ are introduced as (u1015840 w1015840) = (120597120595120597119911 minus120597120595120597119909)
and (H1015840119909 H1015840119911) = (120597Φ120597119911 minus120597Φ120597119909) By eliminating density
and pressure terms from (14) and nondimensionalizing usingthe following transformations (1199091015840 1199101015840 1199111015840) = 119889(119909
where 1198770119888 is the critical value of the Rayleigh number atwhich the onset of convection takes place in the absenceof temperature modulation Now we solve the system fordifferent orders of 120598
At the lowest order we have
[[[[[[[[[[
[
minusnabla4
1198770119888
120597
120597119909minusRas 120597
120597119909minus119876Pm 120597
120597119911(nabla2)
120597
120597119909minusnabla2
0 0
120597
1205971199090 minus
1
Lenabla2
0
minus120597
1205971199110 0 minusPmnabla
2
]]]]]]]]]]
]
[[[
[
1205951
1198791
1198781
Φ1
]]]
]
=[[[
[
0
0
0
0
]]]
]
(24)
The solutions of the lowest order of the system subject to theboundary conditions equation (22) are
1205951 = 119860 (120591) sin (119896119888119909) sin (120587119911)
1198791 = minus119896119888
1205752119860 (120591) cos (119896119888119909) sin (120587119911)
1198781 = minus119896119888
1205752Le119860 (120591) cos (119896119888119909) sin (120587119911)
Φ1 =120587
Pm1205752119860 (120591) sin (119896119888119909) cos (120587119911)
(25)
where 1205752
= 1198962
119888+ 1205872 The critical value of the Rayleigh
number for the onset of magnetoconvection in the absenceof temperature modulation is
which is the same as Siddheshwar et al [17] when Ras = 0 andwe obtain classical results of Chandrasekhar [8] for withoutmagnetic field and single component fluid layer
International Journal of Engineering Mathematics 5
At the second order we have
[[[[[[[[[
[
minusnabla4
1198770119888
120597
120597119909minusRas 120597
120597119909minus119876Pm 120597
120597119911nabla2
120597
120597119909minusnabla2
0 0
120597
1205971199090 minus
1
Lenabla2
0
minus120597
1205971199110 0 minusPmnabla
2
]]]]]]]]]
]
[[[
[
1205952
1198792
1198782
Φ2
]]]
]
=[[[
[
11987721
11987722
11987723
11987724
]]]
]
11987721 = 0
11987722 =1205971205951
120597119909
1205971198791
120597119911minus1205971205951
120597119911
1205971198791
120597119909
11987723 =1205971205951
120597119909
1205971198781
120597119911minus1205971205951
120597119911
1205971198781
120597119909
11987724 =1205971205951
120597119909
120597Φ1
120597119911minus1205971205951
120597119911
120597Φ1
120597119909
(27)
The second order solutions subjected to the boundary condi-tions Equation (22) are obtained as follows
1205952 = 0
1198792 = minus1198962
119888
812058712057521198602(120591) sin (2120587119911)
1198782 = minus1198962
119888Le2
812058712057521198602(120591) sin (2120587119911)
Φ2 = minus1205872
8119896119888Pm212057521198602(120591) sin (2119896119888119909)
(28)
Thehorizontally averagedNusseltNu and Sherwood Shnum-bers for the stationary double-diffusive magnetoconvection(the mode considered in this problem) are given by
Nu (120591) = 1 +
[(1198961198882120587) int2120587119896119888
0(1205971198792120597119911) 119889119909]
119911=0
[(1198961198882120587) int2120587119896119888
0(120597119879119887120597119911 119889119909]
119911=0
Sh (120591) = 1 +
[(1198961198882120587) int2120587119896119888
0(1205971198782120597119911) 119889119909]
119911=0
[(1198961198882120587) int2120587119896119888
0(119889119878119887119889119911) 119889119909]
119911=0
(29)
One must note here that 1198912(119911 120591) is effective at 119874(1205982) andaffects Nu(120591) and Sh(120591) through 119860(120591) can be seen laterTherefore substituting 1198792 119879119887 1198782 respectively into (29) andsimplifying we obtain
Nu (120591) = 1 +1198962
119888
412057521198602(120591)
Sh (120591) = 1 +1198962
119888Le2
412057521198602(120591)
(30)
At the third order we have
[[[[[[[[[
[
minusnabla4
1198770119888
120597
120597119909minusRas 120597
120597119909minus119876Pm 120597
120597119911nabla2
120597
120597119909minusnabla2
0 0
120597
1205971199090 minus
1
Lenabla2
0
minus120597
1205971199110 0 minusPmnabla
2
]]]]]]]]]
]
[[[
[
1205953
1198793
1198783
Φ3
]]]
]
=[[[
[
11987731
11987732
11987733
11987734
]]]
]
(31)
where
11987731 = minus1
Pr120597
120597120591(nabla21205951) minus 1198772
1205971198791
120597119909minus Ras1205971198781
120597119909
minus 119876Pm(120597Φ1
120597119911
120597
120597119909(nabla2Φ2) minus
120597Φ2
120597119909
120597
120597119911(nabla2Φ1))
11987732 = minus1205971198791
120597120591+ 1205751198912 (119911 120591)
1205971205951
120597119909+1205971205951
120597119909
1205971198792
120597119911
11987733 = minus1205971198781
120597120591+1205971205951
120597119909
1205971198782
120597119911
11987734 = minus120597Φ1
120597120591minus1205971205951
120597119911
120597Φ2
120597119909
(32)
Substituting 1205951 1198791 and 1198792 into (32) we obtain the expres-sions for119877311198773211987733 and11987734 easily For the existence of thirdorder solution of the systemwe apply the solvability conditionwhich leads to arrive at the nonautonomous Ginzburg-Landau equation for stationary mode of convection withtime-periodic coefficients in the form
11986011198601015840(120591) minus 1198602119860 (120591) + 1198603119860(120591)
The Ginzburg-Landau equations given in (33) are Bernoulliequation and obtaining its analytical solution is difficultdue to its nonautonomous nature So that it has beensolved numerically using the in-built function NDSolve ofMathematica 8 subjected to the initial condition 119860(0) = 1198860where 1198860 is the chosen initial amplitude of convection In ourcalculations we may use 1198772 = 1198770119888 to keep the parameters tothe minimum
6 International Journal of Engineering Mathematics
4 Results and Discussion
External regulation of thermal instability is important tostudy the double-diffusive convection in a fluid layer Theobjective of this paper is to consider two such candidatesnamely vertical magnetic field and temperature modulationfor either enhancing or inhibiting convective heat and masstransport as is required by a real application The presentpaper deals with double-diffusive magnetoconvection undertemperature modulation by using Ginzburg-Landau equa-tion It is necessary to consider a nonlinear theory toanalyze heat and mass transfer which is not possible bythe linear theory We consider the direct mode (120581119878120581119879 lt
1 otherwise Hopf mode) in which the salt and heat makeopposing contributions (120581119879 = 120581119878) We also consider the effectof temperature modulation to be of order 119874(1205982) this leadsto small amplitude of modulation Such an assumption willhelp us in obtaining the amplitude equation of convectionin a rather simple and elegant manner and is much easierto obtain than in the case of the Lorenz model We give thefollowing features of the problem before our results
(1) the need for nonlinear stability analysis(2) the relation of the problem to a real application(3) the selection of all dimensionless parameters utilized
in computations
We consider the following three types of temperaturemodulation on the boundaries of the system
The parameters of the system are 119876 Pr Le Pm Ras120579 120575 120596 these parameters influence the convective heatand mass transfer The first five parameters are related tothe fluid layer and the last three parameters concern theexternal mechanisms of controlling convection The effect oftemperaturemodulation is represented by amplitude120575 whichlies around 03 due to the assumption The effect of electricalconductivity andmagnetic field comes through Pm119876 Thereis the property of the fluid coming into picture as well asthrough Prandtl number Pr Further the modulation of theboundary temperature is assumed to be of low frequency Atlow range of frequencies the effect of frequency on onset ofconvection as well as on heat and mass transport is minimalThis assumption is required in order to ensure that the systemdoes not pick up oscillatory convective mode at onset dueto modulation in a situation that is conductive otherwiseto stationary mode It is important at this stage to considerthe effect of 119876 Pr Le Pm Ras 120579 120575 120596 on the onset ofconvection The heat and mass transfer of the problem arequantified by the Nusselt and Sherwood numbers which aregiven in (30) Figures 2ndash7 show the individual effect of eachnondimensional parameter on heat and mass transfer
(1) Figures 2(a)ndash7(a) show that the effect of Chan-drasekhar number119876which is ratio of Lorentz force to viscousforce is to delay the onset of convection hence heat and mass
transfer The Nu and Sh start with one and for small values oftime 120591 increase and become constant for large values of time 120591in the case of [IPM] given in Figures 2(a) and 5(a) In the caseof [OPM LBMO] the effect of119876 shows oscillatory behaviourand increment in it decreases the magnitude of both Nu andSh Hence 119876 has stabilizing effect in all the three types ofmodulations given in Figures 3(a) 4(a) 6(a) and 7(a) so thatheat and mass transfer decrease with 119876
(2) The effect of Prandtl number Pr is to advance theconvection and hence heat and mass transfer which hassimilar behaviour of (1) given in Figures 2(b)ndash7(b)
(3) The effect of Lewis Le and Magnetic Prandtl Pmnumbers is to advance the convection and hence heat andmass transfer Hence both Le and Pmhave destabilizing effectof the system given in Figures 2(c)ndash7(c) and 2(d)ndash7(d) andhave similar behaviour of (1) these results are earlier obtainedby Siddheshwar et al [17]
(4)The effect of solutal Rayleigh numberRas is to increaseNu and Sh so that heat and mass transfer Hence it hasdestabilizing effect in all the three types ofmodulationswhichis given by the Figures 2(e)ndash7(e) and has similar behaviour of(1)Though the presence of a stabilizing gradient of solute willprevent the onset of convection the strong finite-amplitudemotions which exist for large Rayleigh numbers tend to mixthe solute and redistribute it so that the interior layers ofthe fluid are more neutrally stratified As a consequence theinhibiting effect of the solute gradient is greatly reduced andhence fluid will convect more and more heat and mass whenRas is increased
(5) In the case of [IPM] we observe no effect of amplitude120575 and frequency 120596 of modulation which is given by theFigures 2(f) and 5(f) But in the case of [OPM-LBMO] theincrement in 120575 leads to increment in magnitude of Nu andSh hence heat and mass transfer given in Figures 3(f) 4(f)6(f) and 7(f) the increment in120596 shortens thewavelength anddecreases in magnitude of Nu Sh and hence heat and masstransfer given in Figures 3(g) 4(g) 6(g) and 7(g) which arethe results obtained by Venezian [19] Siddheshwar et al [17]and Bhadauria and Kiran [25]
(6) From Figures 2(g) and 5(g) we observe that 119876 hasstrongly stabilizing effect [NuSh]119876=0 gt [NuSh]119876 = 0
(7) The comparison of three types of temperature mod-ulations is given in Figures 4(h) and 7(h) [NuSh]IPM lt
[NuSh]LBMOlt [NuSh]OPM
(8)The results of this work can be summarized as followsfrom Figures 2ndash7
The effect of temperature modulation on weak nonlineardouble-diffusive magnetoconvection has been analyzed byusing the nonautonomous Ginzburg-Landau equation Thefollowing conclusions are drawn from previous analysis
(1) The effect of IPM is negligible on heat and masstransport in the system
(2) In the case of IPM the effect of 120575 and 120596 is also foundto be negligible on heat and mass transport
(3) In the case of IPM the values of Nu and Sh increasesteadily for small values of time 120591 however Nu and Shbecome constant when 120591 is large
(4) The effect of increasing Pr Le Pm Ras is found toincrease in Nu and Sh thus increasing heat and masstransfer for all three types of modulations
(5) The effect of increasing 120575 is to increase the value of Nuand Sh for the case of OPM and LBMO hence heatand mass transfer
(6) The effect of increasing 120596 is to decrease the value ofNu and Sh for the case ofOPMand LBMO hence heatand mass transfer
(7) In the cases of OPM and LBMO the natures of Nuand Sh remain oscillatory
(8) Initially when 120591 is small the values of Nusselt andSherwood numbers start with 1 corresponding to theconduction state However as 120591 increases Nu andSh also increase thus increasing the heat and masstransfer
(9) The values of Nu and Sh for LBMO are greater thanthose in IPM but smaller than those in OPM
(10) The effect of magnetic field is to stabilize the system
Nomenclature
Latin Symbols
119860 Amplitude of convection119889 Depth of the fluid layerg Acceleration due to gravity119876 Chandrasekhar number
119876 = 1205831198981198672
11988711988921205880]]119898
119896119888 Critical wave numberNu(120591) Nusselt numberSh(120591) Sherwood number119901 Reduced pressurePr Prandtl number Pr = ]120581119879Pm Magnetic Prandtl number Pm = ]119898120581119879Le Lewis number Le = 120581119879120581119878
120596 Frequency of temperature modulation120575 Amplitude of temperature modulation120583 Dynamic coefficient of viscosity of the fluid120583119898 Magnetic permeability] Kinematic viscosity ] = 1205831205880
]119898 Magnetic viscosity120588 Fluid density120595 Stream function120595lowast Dimensionless stream function
Φ Magnetic potentialΦlowast Dimensionless magnetic potential
120591 Slow time 120591 = 1205982119905
120579 Phase angle1198791015840 Perturbed temperature
Other Symbols
nabla2 (12059721205971199092) + (120597
21205971199102) + (12059721205971199112)
14 International Journal of Engineering Mathematics
Subscripts
119887 Basic state119888 Critical0 Reference value
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was done during the lien sanctioned to theauthor by Banaras Hindu University Varanasi to work asprofessor of Mathematics at Department of Applied Math-ematics School for Physical Sciences Babasaheb BhimraoAmbedkar Central University Lucknow India The authorBS Bhadauria gratefully acknowledges Banaras Hindu Uni-versity Varanasi for the same Further the author Palle Kirangratefully acknowledges the financial assistance fromBabasa-heb Bhimrao Ambedkar Central University as a researchfellowship
References
[1] A Akbarzadeh and P Manins ldquoConvective layers generated byside walls in solar pondsrdquo Solar Energy vol 41 no 6 pp 521ndash529 1988
[2] H E Huppert and R S J Sparks ldquoDouble-diffusive convectiondue to crystallization in magmasrdquo Annual Review of Earth andPlanetary Sciences vol 12 pp 11ndash37 1984
[3] H J S Fernando and A Brandt ldquoRecent advances in doublediffusive convectionrdquoAppliedMechanics Reviews vol 47 pp c1ndashc7 1994
[4] J S Turner Buoyancy Effects in Fluids University Of Cam-bridge 1979
[5] N Rudraiah and I S Shivakumara ldquoDouble-diffusive convec-tion with an imposed magnetic fieldrdquo International Journal ofHeat and Mass Transfer vol 27 no 10 pp 1825ndash1836 1984
[6] J H Thomas and N O Weiss ldquoThe theory of sunspotsrdquo inSunspots Theory and Observations p 428 Kluwer AcademicPublishers Dordrecht The Netherlands 1992
[7] W B Thompson ldquoThermal convection in a magnetic fieldrdquoPhilosophical Magazine vol 42 pp 1417ndash1432 1951
[8] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityOxford University Press London UK 1961
[9] D Lortz ldquoA stability criterion for steady finite amplitudeconvection with an external magnetic fieldrdquo Journal of FluidMechanics vol 23 pp 113ndash128 1965
[10] W V R Malkus and G Veronis ldquoFinite amplitude cellularconvectionrdquo Journal of Fluid Mechanics vol 4 pp 225ndash2601958
[11] H Stommel A B Arons and D Blanchard ldquoAn oceanograph-ical curiosity the perpetual salt fountainrdquo Deep Sea Researchvol 3 no 2 pp 152ndash153 1956
[12] K Gotoh and M Yamada ldquoThermal convection of a horizontallayer of magnetic fluidsrdquo Journal of the Physical Society of Japanvol 51 no 9 pp 3042ndash3048 1982
[13] GMOreper and J Szekely ldquoThe effect of an externally imposedmagnetic field on buoyancy driven flow in a rectangular cavityrdquoJournal of Crystal Growth vol 64 no 3 pp 505ndash515 1983
[14] N Rudraiah ldquoDouble-diffusive magnetoconvectionrdquo Pramanavol 27 no 1-2 pp 233ndash266 1986
[15] P G Siddheshwar and S Pranesh ldquoMagnetoconvection in fluidswith suspended particles under 1g and 120583grdquo Aerospace Scienceand Technology vol 6 no 2 pp 105ndash114 2002
[16] B S Bhadauria ldquoCombined effect of temperature modulationand magnetic field on the onset of convection in an electricallyconducting-fluid-saturated porous mediumrdquo Journal of HeatTransfer vol 130 no 5 Article ID 052601 2008
[17] P G Siddheshwar B S Bhadauria P Mishra and A K Srivas-tava ldquoStudy of heat transport by stationarymagneto-convectionin aNewtonian liquid under temperature or gravitymodulationusing Ginzburg-Landau modelrdquo International Journal of Non-Linear Mechanics vol 47 no 5 pp 418ndash425 2012
[18] P G Drazin and W H Reid Hydrodynamic Stability Cam-bridge University Press Cambridge UK 2004
[19] G Venezian ldquoEffect of modulation on the onset of thermalconvectionrdquo Journal of Fluid Mechanics vol 35 no 2 pp 243ndash254 1969
[20] S Rosenblat and G A Tanaka ldquoModulation of thermal convec-tion instabilityrdquo Physics of Fluids vol 14 no 7 pp 1319ndash13221971
[21] M N Roppo S H Davis and S Rosenblat ldquoBenard convectionwith time-periodic heatingrdquoThe Physics of Fluids vol 27 no 4pp 796ndash803 1984
[22] B S Bhadauria and P K Bhatia ldquoTime-periodic heating ofRayleigh-Benard convectionrdquo Physica Scripta vol 66 no 1 pp59ndash65 2002
[23] B S Bhadauria ldquoTime-periodic heating of Rayleigh-Benardconvection in a vertical magnetic fieldrdquo Physica Scripta vol 73no 3 pp 296ndash302 2006
[24] M S Malashetty andM Swamy ldquoEffect of thermal modulationon the onset of convection in a rotating fluid layerrdquo InternationalJournal of Heat and Mass Transfer vol 51 no 11-12 pp 2814ndash2823 2008
[25] B S Bhadauria and P Kiran ldquoHeat transport in an anisotropicporous medium saturated with variable viscosity liquid undertemperature modulationrdquo Transport in Porous Media vol 100no 2 pp 279ndash295 2013
2 International Journal of Engineering Mathematics
which is heated frombelow and cooled from above and foundcondition for onset of convection Oreper and Szekely [13]have found that the presence of a magnetic field can suppressnatural convection currents and that the strength of themagnetic field is one of the important factors in determiningthe quality of the crystal Rudraiah and Shivakumara [5]investigated both linear and nonlinear theory of this problemin detail They have shown that the magnetic field undercertain conditions makes the system unstable They havealso investigated the effect of magnetic field on the physicallypreferred cell pattern Rudraiah [14] investigated the interac-tion between double-diffusive convection and an externallyimposed vertical magnetic field in a Boussinesq fluid Sid-dheshwar and Pranesh [15] analyzed the role ofmagnetic fieldin the inhibition of natural convection driven by combinedbuoyancy and surface tension forces in a horizontal layer ofan electrically conducting Boussinesq fluid with suspendedparticles confined between an upper freeadiabatic and alower rigidisothermal boundary is considered in 1 g and 120583gsituations Bhadauria [16] also studied the effect of magneticfield on thermal modulated convection in the case of porousmedium Siddheshwar et al [17] performed a local nonlin-ear stability analysis of Rayleigh-Benard magnetoconvectionusing Ginzburg-Landau equation They showed that gravitymodulation can be used to enhance or diminish the heattransport in stationary magnetoconvection
The classical Rayleigh-Benard convection due to bottomheating is well known and highly explored phenomenongiven by Chandrasekhar [8] and Drazin and Reid [18]Many researchers under different physical models haveinvestigated thermal instability in a horizontal fluid layerwith temperature modulation in the absence of double-diffusive magnetoconvection Some of them are Venezian[19] who was the first to consider the effect of temperaturemodulation on thermal instability in a horizontal fluid layerRosenblat and Tanaka [20] studied the linear stability for afluid in a classical geometry of Benard by considering thetemperature modulation of rigid-rigid boundaries The firstnonlinear stability problem in a horizontal fluid layer undertemperature modulation of the boundaries was studied byRoppo et al [21] Bhadauria and Bhatia [22] studied theeffect of temperature modulation on thermal instability byconsidering rigid-rigid boundaries and different types oftemperature profiles Bhadauria [23] Malashetty and Swamy[24] and Bhadauria and Kiran [25] are the related problems
The problem of double-diffusive magnetoconvection iscalled thermohaline magnetoconvection when the two dif-fusive mechanisms are thermal and solute From the aboveliterature (second paragraph) it is observed that concerninglinear theory numerous studies on magnetoconvection ordouble-diffusive magnetoconvection have been made Butthe double-diffusive magnetoconvection is still in its infantstage in nonlinear case Liner theory gives information aboutonset of convection but fails at heat andmass transport Mostof the studies are made with uniform temperature gradientacross the boundaries In general there are many practicalsituations in which the temperature gradient is a function ofboth time and space One of the external effective mecha-nisms to regulate (or control) the convection by maintaining
Electrically conductingNewtonian fluid layer
O
HbkZ
T = T0 +ΔT
2[1 + 1205982120575cos(120596t)] S = S + S0
T = T0 +ΔT
2[1 minus 1205982120575cos(120596t)] S = S0 + ΔS 120577 = 0
X
120577 = d
Y
Figure 1 Physical configuration of the problem
a nonuniform temperature gradient across the fluid layer Wealso observe from the literature (third paragraph) that somework has been done under temperature modulation withoutmagnetic field and double-diffusive convection which isessential in thermal and engineering science as mentionedin the literature Therefore from the above understandingwe motivated to analyze a weak nonlinear thermohalinemagnetoconvection under temperature modulation whilederiving an amplitude of convection using Ginzburg-Landaumodel
2 Governing Equations
We consider an electrically conducting fluid layer of depth119889 confined between two infinite parallel horizontal planesat 119911 = 0 and 119911 = 119889 A Cartesian frame of reference hasbeen taken with the origin at the bottom and the 119911 axisvertically upward direction of the fluid layerThe surfaces aremaintained at a constant temperature gradient Δ119879119889 and aconstant magnetic field 119867119887 is applied across the fluid layer(as shown in Figure 1) Under the Boussinesq approximationthe dimensional governing equations for the study of double-diffusive magnetoconvection in an electrically conductingfluid layer are defined as
+ (q sdot nabla) H minus (H sdot nabla) q = ]119898nabla2H (6)
120588 = 1205880 [1 minus 120572 (119879 minus 1198790) + 120573 (119878 minus 1198780)] (7)
where q is velocity (119906 V 119908) 120583 is a viscosity 120583119898 is magneticpermeability ]119898 is magnetic viscosity 120581119879 is the thermaldiffusivity 120581119878 is the solutal diffusivity 119879 is temperature
International Journal of Engineering Mathematics 3
120572 is thermal expansion coefficient 120573 is solute expansioncoefficient and 120574 is the ration of heat capacity For simplicity120574 is taken to be unity in this paper 120588 is density g = (0 0 minus119892)
is gravitational acceleration while 1205880 is reference densityand H is the magnetic field The externally imposed thermalboundary conditions considered in this paper are given byVenezian [19]
119879 = 1198790 +Δ119879
2[1 + 120598
2120575 cos (120596119905)] at 119911 = 0
= 1198790 minusΔ119879
2[1 minus 120598
2120575 cos (120596119905 + 120579)] at 119911 = 119889
(8)
where 120575 is small amplitude of modulation Δ119879 is the tem-perature difference across the fluid layer 120596 is the modulationfrequency and 120579 is the phase difference Since we are not con-sidering cross diffusion terms the walls of the fluid layer areassumed to bemaintained at constant solute concentration asdefined below
119878 = 1198780 + Δ119878 at 119911 = 0
= 1198780 at 119911 = 119889
(9)
The basic state is assumed to be quiescent and the quantitiesin this state are given by
The basic state pressure field is not required here howeverthe basic temperature and solute fields are governed by thefollowing ordinary differential equations
120597119879119887
120597119905= 120581119879
1205972119879119887
1205971199112
1205972119878119887
1205971199112= 0 (11)
The solutions of (11) subjected to the boundary conditions(8)-(9) are given by
where 119879119904(119911) is the study temperature field and 1198791(119911 119905) is theoscillating part while Re stands for the real part We assumefinite amplitude perturbations on the basic state in the form
where the primes denote the quantities at the perturbationsSubstituting (13) into (1)ndash(6) and using the basic state resultswe obtain the following equations
nabla sdot q1015840 = 0
nabla sdot H1015840 = 0
120597q1015840
120597119905+ (q1015840 sdot nabla) q minus 120583119898
120597119905+ (q1015840 sdot nabla) H1015840 minus (H1015840 sdot nabla) q1015840 minus 119867119887
120597w1015840
120597119911= ]119898nabla
2H1015840
1205881015840= minus1205880 (120572119879
1015840minus 1205731198781015840)
(14)
Further we consider only two-dimensional disturbances inour study and hence the stream function 120595 and magneticpotential Φ are introduced as (u1015840 w1015840) = (120597120595120597119911 minus120597120595120597119909)
and (H1015840119909 H1015840119911) = (120597Φ120597119911 minus120597Φ120597119909) By eliminating density
and pressure terms from (14) and nondimensionalizing usingthe following transformations (1199091015840 1199101015840 1199111015840) = 119889(119909
where 1198770119888 is the critical value of the Rayleigh number atwhich the onset of convection takes place in the absenceof temperature modulation Now we solve the system fordifferent orders of 120598
At the lowest order we have
[[[[[[[[[[
[
minusnabla4
1198770119888
120597
120597119909minusRas 120597
120597119909minus119876Pm 120597
120597119911(nabla2)
120597
120597119909minusnabla2
0 0
120597
1205971199090 minus
1
Lenabla2
0
minus120597
1205971199110 0 minusPmnabla
2
]]]]]]]]]]
]
[[[
[
1205951
1198791
1198781
Φ1
]]]
]
=[[[
[
0
0
0
0
]]]
]
(24)
The solutions of the lowest order of the system subject to theboundary conditions equation (22) are
1205951 = 119860 (120591) sin (119896119888119909) sin (120587119911)
1198791 = minus119896119888
1205752119860 (120591) cos (119896119888119909) sin (120587119911)
1198781 = minus119896119888
1205752Le119860 (120591) cos (119896119888119909) sin (120587119911)
Φ1 =120587
Pm1205752119860 (120591) sin (119896119888119909) cos (120587119911)
(25)
where 1205752
= 1198962
119888+ 1205872 The critical value of the Rayleigh
number for the onset of magnetoconvection in the absenceof temperature modulation is
which is the same as Siddheshwar et al [17] when Ras = 0 andwe obtain classical results of Chandrasekhar [8] for withoutmagnetic field and single component fluid layer
International Journal of Engineering Mathematics 5
At the second order we have
[[[[[[[[[
[
minusnabla4
1198770119888
120597
120597119909minusRas 120597
120597119909minus119876Pm 120597
120597119911nabla2
120597
120597119909minusnabla2
0 0
120597
1205971199090 minus
1
Lenabla2
0
minus120597
1205971199110 0 minusPmnabla
2
]]]]]]]]]
]
[[[
[
1205952
1198792
1198782
Φ2
]]]
]
=[[[
[
11987721
11987722
11987723
11987724
]]]
]
11987721 = 0
11987722 =1205971205951
120597119909
1205971198791
120597119911minus1205971205951
120597119911
1205971198791
120597119909
11987723 =1205971205951
120597119909
1205971198781
120597119911minus1205971205951
120597119911
1205971198781
120597119909
11987724 =1205971205951
120597119909
120597Φ1
120597119911minus1205971205951
120597119911
120597Φ1
120597119909
(27)
The second order solutions subjected to the boundary condi-tions Equation (22) are obtained as follows
1205952 = 0
1198792 = minus1198962
119888
812058712057521198602(120591) sin (2120587119911)
1198782 = minus1198962
119888Le2
812058712057521198602(120591) sin (2120587119911)
Φ2 = minus1205872
8119896119888Pm212057521198602(120591) sin (2119896119888119909)
(28)
Thehorizontally averagedNusseltNu and Sherwood Shnum-bers for the stationary double-diffusive magnetoconvection(the mode considered in this problem) are given by
Nu (120591) = 1 +
[(1198961198882120587) int2120587119896119888
0(1205971198792120597119911) 119889119909]
119911=0
[(1198961198882120587) int2120587119896119888
0(120597119879119887120597119911 119889119909]
119911=0
Sh (120591) = 1 +
[(1198961198882120587) int2120587119896119888
0(1205971198782120597119911) 119889119909]
119911=0
[(1198961198882120587) int2120587119896119888
0(119889119878119887119889119911) 119889119909]
119911=0
(29)
One must note here that 1198912(119911 120591) is effective at 119874(1205982) andaffects Nu(120591) and Sh(120591) through 119860(120591) can be seen laterTherefore substituting 1198792 119879119887 1198782 respectively into (29) andsimplifying we obtain
Nu (120591) = 1 +1198962
119888
412057521198602(120591)
Sh (120591) = 1 +1198962
119888Le2
412057521198602(120591)
(30)
At the third order we have
[[[[[[[[[
[
minusnabla4
1198770119888
120597
120597119909minusRas 120597
120597119909minus119876Pm 120597
120597119911nabla2
120597
120597119909minusnabla2
0 0
120597
1205971199090 minus
1
Lenabla2
0
minus120597
1205971199110 0 minusPmnabla
2
]]]]]]]]]
]
[[[
[
1205953
1198793
1198783
Φ3
]]]
]
=[[[
[
11987731
11987732
11987733
11987734
]]]
]
(31)
where
11987731 = minus1
Pr120597
120597120591(nabla21205951) minus 1198772
1205971198791
120597119909minus Ras1205971198781
120597119909
minus 119876Pm(120597Φ1
120597119911
120597
120597119909(nabla2Φ2) minus
120597Φ2
120597119909
120597
120597119911(nabla2Φ1))
11987732 = minus1205971198791
120597120591+ 1205751198912 (119911 120591)
1205971205951
120597119909+1205971205951
120597119909
1205971198792
120597119911
11987733 = minus1205971198781
120597120591+1205971205951
120597119909
1205971198782
120597119911
11987734 = minus120597Φ1
120597120591minus1205971205951
120597119911
120597Φ2
120597119909
(32)
Substituting 1205951 1198791 and 1198792 into (32) we obtain the expres-sions for119877311198773211987733 and11987734 easily For the existence of thirdorder solution of the systemwe apply the solvability conditionwhich leads to arrive at the nonautonomous Ginzburg-Landau equation for stationary mode of convection withtime-periodic coefficients in the form
11986011198601015840(120591) minus 1198602119860 (120591) + 1198603119860(120591)
The Ginzburg-Landau equations given in (33) are Bernoulliequation and obtaining its analytical solution is difficultdue to its nonautonomous nature So that it has beensolved numerically using the in-built function NDSolve ofMathematica 8 subjected to the initial condition 119860(0) = 1198860where 1198860 is the chosen initial amplitude of convection In ourcalculations we may use 1198772 = 1198770119888 to keep the parameters tothe minimum
6 International Journal of Engineering Mathematics
4 Results and Discussion
External regulation of thermal instability is important tostudy the double-diffusive convection in a fluid layer Theobjective of this paper is to consider two such candidatesnamely vertical magnetic field and temperature modulationfor either enhancing or inhibiting convective heat and masstransport as is required by a real application The presentpaper deals with double-diffusive magnetoconvection undertemperature modulation by using Ginzburg-Landau equa-tion It is necessary to consider a nonlinear theory toanalyze heat and mass transfer which is not possible bythe linear theory We consider the direct mode (120581119878120581119879 lt
1 otherwise Hopf mode) in which the salt and heat makeopposing contributions (120581119879 = 120581119878) We also consider the effectof temperature modulation to be of order 119874(1205982) this leadsto small amplitude of modulation Such an assumption willhelp us in obtaining the amplitude equation of convectionin a rather simple and elegant manner and is much easierto obtain than in the case of the Lorenz model We give thefollowing features of the problem before our results
(1) the need for nonlinear stability analysis(2) the relation of the problem to a real application(3) the selection of all dimensionless parameters utilized
in computations
We consider the following three types of temperaturemodulation on the boundaries of the system
The parameters of the system are 119876 Pr Le Pm Ras120579 120575 120596 these parameters influence the convective heatand mass transfer The first five parameters are related tothe fluid layer and the last three parameters concern theexternal mechanisms of controlling convection The effect oftemperaturemodulation is represented by amplitude120575 whichlies around 03 due to the assumption The effect of electricalconductivity andmagnetic field comes through Pm119876 Thereis the property of the fluid coming into picture as well asthrough Prandtl number Pr Further the modulation of theboundary temperature is assumed to be of low frequency Atlow range of frequencies the effect of frequency on onset ofconvection as well as on heat and mass transport is minimalThis assumption is required in order to ensure that the systemdoes not pick up oscillatory convective mode at onset dueto modulation in a situation that is conductive otherwiseto stationary mode It is important at this stage to considerthe effect of 119876 Pr Le Pm Ras 120579 120575 120596 on the onset ofconvection The heat and mass transfer of the problem arequantified by the Nusselt and Sherwood numbers which aregiven in (30) Figures 2ndash7 show the individual effect of eachnondimensional parameter on heat and mass transfer
(1) Figures 2(a)ndash7(a) show that the effect of Chan-drasekhar number119876which is ratio of Lorentz force to viscousforce is to delay the onset of convection hence heat and mass
transfer The Nu and Sh start with one and for small values oftime 120591 increase and become constant for large values of time 120591in the case of [IPM] given in Figures 2(a) and 5(a) In the caseof [OPM LBMO] the effect of119876 shows oscillatory behaviourand increment in it decreases the magnitude of both Nu andSh Hence 119876 has stabilizing effect in all the three types ofmodulations given in Figures 3(a) 4(a) 6(a) and 7(a) so thatheat and mass transfer decrease with 119876
(2) The effect of Prandtl number Pr is to advance theconvection and hence heat and mass transfer which hassimilar behaviour of (1) given in Figures 2(b)ndash7(b)
(3) The effect of Lewis Le and Magnetic Prandtl Pmnumbers is to advance the convection and hence heat andmass transfer Hence both Le and Pmhave destabilizing effectof the system given in Figures 2(c)ndash7(c) and 2(d)ndash7(d) andhave similar behaviour of (1) these results are earlier obtainedby Siddheshwar et al [17]
(4)The effect of solutal Rayleigh numberRas is to increaseNu and Sh so that heat and mass transfer Hence it hasdestabilizing effect in all the three types ofmodulationswhichis given by the Figures 2(e)ndash7(e) and has similar behaviour of(1)Though the presence of a stabilizing gradient of solute willprevent the onset of convection the strong finite-amplitudemotions which exist for large Rayleigh numbers tend to mixthe solute and redistribute it so that the interior layers ofthe fluid are more neutrally stratified As a consequence theinhibiting effect of the solute gradient is greatly reduced andhence fluid will convect more and more heat and mass whenRas is increased
(5) In the case of [IPM] we observe no effect of amplitude120575 and frequency 120596 of modulation which is given by theFigures 2(f) and 5(f) But in the case of [OPM-LBMO] theincrement in 120575 leads to increment in magnitude of Nu andSh hence heat and mass transfer given in Figures 3(f) 4(f)6(f) and 7(f) the increment in120596 shortens thewavelength anddecreases in magnitude of Nu Sh and hence heat and masstransfer given in Figures 3(g) 4(g) 6(g) and 7(g) which arethe results obtained by Venezian [19] Siddheshwar et al [17]and Bhadauria and Kiran [25]
(6) From Figures 2(g) and 5(g) we observe that 119876 hasstrongly stabilizing effect [NuSh]119876=0 gt [NuSh]119876 = 0
(7) The comparison of three types of temperature mod-ulations is given in Figures 4(h) and 7(h) [NuSh]IPM lt
[NuSh]LBMOlt [NuSh]OPM
(8)The results of this work can be summarized as followsfrom Figures 2ndash7
The effect of temperature modulation on weak nonlineardouble-diffusive magnetoconvection has been analyzed byusing the nonautonomous Ginzburg-Landau equation Thefollowing conclusions are drawn from previous analysis
(1) The effect of IPM is negligible on heat and masstransport in the system
(2) In the case of IPM the effect of 120575 and 120596 is also foundto be negligible on heat and mass transport
(3) In the case of IPM the values of Nu and Sh increasesteadily for small values of time 120591 however Nu and Shbecome constant when 120591 is large
(4) The effect of increasing Pr Le Pm Ras is found toincrease in Nu and Sh thus increasing heat and masstransfer for all three types of modulations
(5) The effect of increasing 120575 is to increase the value of Nuand Sh for the case of OPM and LBMO hence heatand mass transfer
(6) The effect of increasing 120596 is to decrease the value ofNu and Sh for the case ofOPMand LBMO hence heatand mass transfer
(7) In the cases of OPM and LBMO the natures of Nuand Sh remain oscillatory
(8) Initially when 120591 is small the values of Nusselt andSherwood numbers start with 1 corresponding to theconduction state However as 120591 increases Nu andSh also increase thus increasing the heat and masstransfer
(9) The values of Nu and Sh for LBMO are greater thanthose in IPM but smaller than those in OPM
(10) The effect of magnetic field is to stabilize the system
Nomenclature
Latin Symbols
119860 Amplitude of convection119889 Depth of the fluid layerg Acceleration due to gravity119876 Chandrasekhar number
119876 = 1205831198981198672
11988711988921205880]]119898
119896119888 Critical wave numberNu(120591) Nusselt numberSh(120591) Sherwood number119901 Reduced pressurePr Prandtl number Pr = ]120581119879Pm Magnetic Prandtl number Pm = ]119898120581119879Le Lewis number Le = 120581119879120581119878
120596 Frequency of temperature modulation120575 Amplitude of temperature modulation120583 Dynamic coefficient of viscosity of the fluid120583119898 Magnetic permeability] Kinematic viscosity ] = 1205831205880
]119898 Magnetic viscosity120588 Fluid density120595 Stream function120595lowast Dimensionless stream function
Φ Magnetic potentialΦlowast Dimensionless magnetic potential
120591 Slow time 120591 = 1205982119905
120579 Phase angle1198791015840 Perturbed temperature
Other Symbols
nabla2 (12059721205971199092) + (120597
21205971199102) + (12059721205971199112)
14 International Journal of Engineering Mathematics
Subscripts
119887 Basic state119888 Critical0 Reference value
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was done during the lien sanctioned to theauthor by Banaras Hindu University Varanasi to work asprofessor of Mathematics at Department of Applied Math-ematics School for Physical Sciences Babasaheb BhimraoAmbedkar Central University Lucknow India The authorBS Bhadauria gratefully acknowledges Banaras Hindu Uni-versity Varanasi for the same Further the author Palle Kirangratefully acknowledges the financial assistance fromBabasa-heb Bhimrao Ambedkar Central University as a researchfellowship
References
[1] A Akbarzadeh and P Manins ldquoConvective layers generated byside walls in solar pondsrdquo Solar Energy vol 41 no 6 pp 521ndash529 1988
[2] H E Huppert and R S J Sparks ldquoDouble-diffusive convectiondue to crystallization in magmasrdquo Annual Review of Earth andPlanetary Sciences vol 12 pp 11ndash37 1984
[3] H J S Fernando and A Brandt ldquoRecent advances in doublediffusive convectionrdquoAppliedMechanics Reviews vol 47 pp c1ndashc7 1994
[4] J S Turner Buoyancy Effects in Fluids University Of Cam-bridge 1979
[5] N Rudraiah and I S Shivakumara ldquoDouble-diffusive convec-tion with an imposed magnetic fieldrdquo International Journal ofHeat and Mass Transfer vol 27 no 10 pp 1825ndash1836 1984
[6] J H Thomas and N O Weiss ldquoThe theory of sunspotsrdquo inSunspots Theory and Observations p 428 Kluwer AcademicPublishers Dordrecht The Netherlands 1992
[7] W B Thompson ldquoThermal convection in a magnetic fieldrdquoPhilosophical Magazine vol 42 pp 1417ndash1432 1951
[8] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityOxford University Press London UK 1961
[9] D Lortz ldquoA stability criterion for steady finite amplitudeconvection with an external magnetic fieldrdquo Journal of FluidMechanics vol 23 pp 113ndash128 1965
[10] W V R Malkus and G Veronis ldquoFinite amplitude cellularconvectionrdquo Journal of Fluid Mechanics vol 4 pp 225ndash2601958
[11] H Stommel A B Arons and D Blanchard ldquoAn oceanograph-ical curiosity the perpetual salt fountainrdquo Deep Sea Researchvol 3 no 2 pp 152ndash153 1956
[12] K Gotoh and M Yamada ldquoThermal convection of a horizontallayer of magnetic fluidsrdquo Journal of the Physical Society of Japanvol 51 no 9 pp 3042ndash3048 1982
[13] GMOreper and J Szekely ldquoThe effect of an externally imposedmagnetic field on buoyancy driven flow in a rectangular cavityrdquoJournal of Crystal Growth vol 64 no 3 pp 505ndash515 1983
[14] N Rudraiah ldquoDouble-diffusive magnetoconvectionrdquo Pramanavol 27 no 1-2 pp 233ndash266 1986
[15] P G Siddheshwar and S Pranesh ldquoMagnetoconvection in fluidswith suspended particles under 1g and 120583grdquo Aerospace Scienceand Technology vol 6 no 2 pp 105ndash114 2002
[16] B S Bhadauria ldquoCombined effect of temperature modulationand magnetic field on the onset of convection in an electricallyconducting-fluid-saturated porous mediumrdquo Journal of HeatTransfer vol 130 no 5 Article ID 052601 2008
[17] P G Siddheshwar B S Bhadauria P Mishra and A K Srivas-tava ldquoStudy of heat transport by stationarymagneto-convectionin aNewtonian liquid under temperature or gravitymodulationusing Ginzburg-Landau modelrdquo International Journal of Non-Linear Mechanics vol 47 no 5 pp 418ndash425 2012
[18] P G Drazin and W H Reid Hydrodynamic Stability Cam-bridge University Press Cambridge UK 2004
[19] G Venezian ldquoEffect of modulation on the onset of thermalconvectionrdquo Journal of Fluid Mechanics vol 35 no 2 pp 243ndash254 1969
[20] S Rosenblat and G A Tanaka ldquoModulation of thermal convec-tion instabilityrdquo Physics of Fluids vol 14 no 7 pp 1319ndash13221971
[21] M N Roppo S H Davis and S Rosenblat ldquoBenard convectionwith time-periodic heatingrdquoThe Physics of Fluids vol 27 no 4pp 796ndash803 1984
[22] B S Bhadauria and P K Bhatia ldquoTime-periodic heating ofRayleigh-Benard convectionrdquo Physica Scripta vol 66 no 1 pp59ndash65 2002
[23] B S Bhadauria ldquoTime-periodic heating of Rayleigh-Benardconvection in a vertical magnetic fieldrdquo Physica Scripta vol 73no 3 pp 296ndash302 2006
[24] M S Malashetty andM Swamy ldquoEffect of thermal modulationon the onset of convection in a rotating fluid layerrdquo InternationalJournal of Heat and Mass Transfer vol 51 no 11-12 pp 2814ndash2823 2008
[25] B S Bhadauria and P Kiran ldquoHeat transport in an anisotropicporous medium saturated with variable viscosity liquid undertemperature modulationrdquo Transport in Porous Media vol 100no 2 pp 279ndash295 2013
International Journal of Engineering Mathematics 3
120572 is thermal expansion coefficient 120573 is solute expansioncoefficient and 120574 is the ration of heat capacity For simplicity120574 is taken to be unity in this paper 120588 is density g = (0 0 minus119892)
is gravitational acceleration while 1205880 is reference densityand H is the magnetic field The externally imposed thermalboundary conditions considered in this paper are given byVenezian [19]
119879 = 1198790 +Δ119879
2[1 + 120598
2120575 cos (120596119905)] at 119911 = 0
= 1198790 minusΔ119879
2[1 minus 120598
2120575 cos (120596119905 + 120579)] at 119911 = 119889
(8)
where 120575 is small amplitude of modulation Δ119879 is the tem-perature difference across the fluid layer 120596 is the modulationfrequency and 120579 is the phase difference Since we are not con-sidering cross diffusion terms the walls of the fluid layer areassumed to bemaintained at constant solute concentration asdefined below
119878 = 1198780 + Δ119878 at 119911 = 0
= 1198780 at 119911 = 119889
(9)
The basic state is assumed to be quiescent and the quantitiesin this state are given by
The basic state pressure field is not required here howeverthe basic temperature and solute fields are governed by thefollowing ordinary differential equations
120597119879119887
120597119905= 120581119879
1205972119879119887
1205971199112
1205972119878119887
1205971199112= 0 (11)
The solutions of (11) subjected to the boundary conditions(8)-(9) are given by
where 119879119904(119911) is the study temperature field and 1198791(119911 119905) is theoscillating part while Re stands for the real part We assumefinite amplitude perturbations on the basic state in the form
where the primes denote the quantities at the perturbationsSubstituting (13) into (1)ndash(6) and using the basic state resultswe obtain the following equations
nabla sdot q1015840 = 0
nabla sdot H1015840 = 0
120597q1015840
120597119905+ (q1015840 sdot nabla) q minus 120583119898
120597119905+ (q1015840 sdot nabla) H1015840 minus (H1015840 sdot nabla) q1015840 minus 119867119887
120597w1015840
120597119911= ]119898nabla
2H1015840
1205881015840= minus1205880 (120572119879
1015840minus 1205731198781015840)
(14)
Further we consider only two-dimensional disturbances inour study and hence the stream function 120595 and magneticpotential Φ are introduced as (u1015840 w1015840) = (120597120595120597119911 minus120597120595120597119909)
and (H1015840119909 H1015840119911) = (120597Φ120597119911 minus120597Φ120597119909) By eliminating density
and pressure terms from (14) and nondimensionalizing usingthe following transformations (1199091015840 1199101015840 1199111015840) = 119889(119909
where 1198770119888 is the critical value of the Rayleigh number atwhich the onset of convection takes place in the absenceof temperature modulation Now we solve the system fordifferent orders of 120598
At the lowest order we have
[[[[[[[[[[
[
minusnabla4
1198770119888
120597
120597119909minusRas 120597
120597119909minus119876Pm 120597
120597119911(nabla2)
120597
120597119909minusnabla2
0 0
120597
1205971199090 minus
1
Lenabla2
0
minus120597
1205971199110 0 minusPmnabla
2
]]]]]]]]]]
]
[[[
[
1205951
1198791
1198781
Φ1
]]]
]
=[[[
[
0
0
0
0
]]]
]
(24)
The solutions of the lowest order of the system subject to theboundary conditions equation (22) are
1205951 = 119860 (120591) sin (119896119888119909) sin (120587119911)
1198791 = minus119896119888
1205752119860 (120591) cos (119896119888119909) sin (120587119911)
1198781 = minus119896119888
1205752Le119860 (120591) cos (119896119888119909) sin (120587119911)
Φ1 =120587
Pm1205752119860 (120591) sin (119896119888119909) cos (120587119911)
(25)
where 1205752
= 1198962
119888+ 1205872 The critical value of the Rayleigh
number for the onset of magnetoconvection in the absenceof temperature modulation is
which is the same as Siddheshwar et al [17] when Ras = 0 andwe obtain classical results of Chandrasekhar [8] for withoutmagnetic field and single component fluid layer
International Journal of Engineering Mathematics 5
At the second order we have
[[[[[[[[[
[
minusnabla4
1198770119888
120597
120597119909minusRas 120597
120597119909minus119876Pm 120597
120597119911nabla2
120597
120597119909minusnabla2
0 0
120597
1205971199090 minus
1
Lenabla2
0
minus120597
1205971199110 0 minusPmnabla
2
]]]]]]]]]
]
[[[
[
1205952
1198792
1198782
Φ2
]]]
]
=[[[
[
11987721
11987722
11987723
11987724
]]]
]
11987721 = 0
11987722 =1205971205951
120597119909
1205971198791
120597119911minus1205971205951
120597119911
1205971198791
120597119909
11987723 =1205971205951
120597119909
1205971198781
120597119911minus1205971205951
120597119911
1205971198781
120597119909
11987724 =1205971205951
120597119909
120597Φ1
120597119911minus1205971205951
120597119911
120597Φ1
120597119909
(27)
The second order solutions subjected to the boundary condi-tions Equation (22) are obtained as follows
1205952 = 0
1198792 = minus1198962
119888
812058712057521198602(120591) sin (2120587119911)
1198782 = minus1198962
119888Le2
812058712057521198602(120591) sin (2120587119911)
Φ2 = minus1205872
8119896119888Pm212057521198602(120591) sin (2119896119888119909)
(28)
Thehorizontally averagedNusseltNu and Sherwood Shnum-bers for the stationary double-diffusive magnetoconvection(the mode considered in this problem) are given by
Nu (120591) = 1 +
[(1198961198882120587) int2120587119896119888
0(1205971198792120597119911) 119889119909]
119911=0
[(1198961198882120587) int2120587119896119888
0(120597119879119887120597119911 119889119909]
119911=0
Sh (120591) = 1 +
[(1198961198882120587) int2120587119896119888
0(1205971198782120597119911) 119889119909]
119911=0
[(1198961198882120587) int2120587119896119888
0(119889119878119887119889119911) 119889119909]
119911=0
(29)
One must note here that 1198912(119911 120591) is effective at 119874(1205982) andaffects Nu(120591) and Sh(120591) through 119860(120591) can be seen laterTherefore substituting 1198792 119879119887 1198782 respectively into (29) andsimplifying we obtain
Nu (120591) = 1 +1198962
119888
412057521198602(120591)
Sh (120591) = 1 +1198962
119888Le2
412057521198602(120591)
(30)
At the third order we have
[[[[[[[[[
[
minusnabla4
1198770119888
120597
120597119909minusRas 120597
120597119909minus119876Pm 120597
120597119911nabla2
120597
120597119909minusnabla2
0 0
120597
1205971199090 minus
1
Lenabla2
0
minus120597
1205971199110 0 minusPmnabla
2
]]]]]]]]]
]
[[[
[
1205953
1198793
1198783
Φ3
]]]
]
=[[[
[
11987731
11987732
11987733
11987734
]]]
]
(31)
where
11987731 = minus1
Pr120597
120597120591(nabla21205951) minus 1198772
1205971198791
120597119909minus Ras1205971198781
120597119909
minus 119876Pm(120597Φ1
120597119911
120597
120597119909(nabla2Φ2) minus
120597Φ2
120597119909
120597
120597119911(nabla2Φ1))
11987732 = minus1205971198791
120597120591+ 1205751198912 (119911 120591)
1205971205951
120597119909+1205971205951
120597119909
1205971198792
120597119911
11987733 = minus1205971198781
120597120591+1205971205951
120597119909
1205971198782
120597119911
11987734 = minus120597Φ1
120597120591minus1205971205951
120597119911
120597Φ2
120597119909
(32)
Substituting 1205951 1198791 and 1198792 into (32) we obtain the expres-sions for119877311198773211987733 and11987734 easily For the existence of thirdorder solution of the systemwe apply the solvability conditionwhich leads to arrive at the nonautonomous Ginzburg-Landau equation for stationary mode of convection withtime-periodic coefficients in the form
11986011198601015840(120591) minus 1198602119860 (120591) + 1198603119860(120591)
The Ginzburg-Landau equations given in (33) are Bernoulliequation and obtaining its analytical solution is difficultdue to its nonautonomous nature So that it has beensolved numerically using the in-built function NDSolve ofMathematica 8 subjected to the initial condition 119860(0) = 1198860where 1198860 is the chosen initial amplitude of convection In ourcalculations we may use 1198772 = 1198770119888 to keep the parameters tothe minimum
6 International Journal of Engineering Mathematics
4 Results and Discussion
External regulation of thermal instability is important tostudy the double-diffusive convection in a fluid layer Theobjective of this paper is to consider two such candidatesnamely vertical magnetic field and temperature modulationfor either enhancing or inhibiting convective heat and masstransport as is required by a real application The presentpaper deals with double-diffusive magnetoconvection undertemperature modulation by using Ginzburg-Landau equa-tion It is necessary to consider a nonlinear theory toanalyze heat and mass transfer which is not possible bythe linear theory We consider the direct mode (120581119878120581119879 lt
1 otherwise Hopf mode) in which the salt and heat makeopposing contributions (120581119879 = 120581119878) We also consider the effectof temperature modulation to be of order 119874(1205982) this leadsto small amplitude of modulation Such an assumption willhelp us in obtaining the amplitude equation of convectionin a rather simple and elegant manner and is much easierto obtain than in the case of the Lorenz model We give thefollowing features of the problem before our results
(1) the need for nonlinear stability analysis(2) the relation of the problem to a real application(3) the selection of all dimensionless parameters utilized
in computations
We consider the following three types of temperaturemodulation on the boundaries of the system
The parameters of the system are 119876 Pr Le Pm Ras120579 120575 120596 these parameters influence the convective heatand mass transfer The first five parameters are related tothe fluid layer and the last three parameters concern theexternal mechanisms of controlling convection The effect oftemperaturemodulation is represented by amplitude120575 whichlies around 03 due to the assumption The effect of electricalconductivity andmagnetic field comes through Pm119876 Thereis the property of the fluid coming into picture as well asthrough Prandtl number Pr Further the modulation of theboundary temperature is assumed to be of low frequency Atlow range of frequencies the effect of frequency on onset ofconvection as well as on heat and mass transport is minimalThis assumption is required in order to ensure that the systemdoes not pick up oscillatory convective mode at onset dueto modulation in a situation that is conductive otherwiseto stationary mode It is important at this stage to considerthe effect of 119876 Pr Le Pm Ras 120579 120575 120596 on the onset ofconvection The heat and mass transfer of the problem arequantified by the Nusselt and Sherwood numbers which aregiven in (30) Figures 2ndash7 show the individual effect of eachnondimensional parameter on heat and mass transfer
(1) Figures 2(a)ndash7(a) show that the effect of Chan-drasekhar number119876which is ratio of Lorentz force to viscousforce is to delay the onset of convection hence heat and mass
transfer The Nu and Sh start with one and for small values oftime 120591 increase and become constant for large values of time 120591in the case of [IPM] given in Figures 2(a) and 5(a) In the caseof [OPM LBMO] the effect of119876 shows oscillatory behaviourand increment in it decreases the magnitude of both Nu andSh Hence 119876 has stabilizing effect in all the three types ofmodulations given in Figures 3(a) 4(a) 6(a) and 7(a) so thatheat and mass transfer decrease with 119876
(2) The effect of Prandtl number Pr is to advance theconvection and hence heat and mass transfer which hassimilar behaviour of (1) given in Figures 2(b)ndash7(b)
(3) The effect of Lewis Le and Magnetic Prandtl Pmnumbers is to advance the convection and hence heat andmass transfer Hence both Le and Pmhave destabilizing effectof the system given in Figures 2(c)ndash7(c) and 2(d)ndash7(d) andhave similar behaviour of (1) these results are earlier obtainedby Siddheshwar et al [17]
(4)The effect of solutal Rayleigh numberRas is to increaseNu and Sh so that heat and mass transfer Hence it hasdestabilizing effect in all the three types ofmodulationswhichis given by the Figures 2(e)ndash7(e) and has similar behaviour of(1)Though the presence of a stabilizing gradient of solute willprevent the onset of convection the strong finite-amplitudemotions which exist for large Rayleigh numbers tend to mixthe solute and redistribute it so that the interior layers ofthe fluid are more neutrally stratified As a consequence theinhibiting effect of the solute gradient is greatly reduced andhence fluid will convect more and more heat and mass whenRas is increased
(5) In the case of [IPM] we observe no effect of amplitude120575 and frequency 120596 of modulation which is given by theFigures 2(f) and 5(f) But in the case of [OPM-LBMO] theincrement in 120575 leads to increment in magnitude of Nu andSh hence heat and mass transfer given in Figures 3(f) 4(f)6(f) and 7(f) the increment in120596 shortens thewavelength anddecreases in magnitude of Nu Sh and hence heat and masstransfer given in Figures 3(g) 4(g) 6(g) and 7(g) which arethe results obtained by Venezian [19] Siddheshwar et al [17]and Bhadauria and Kiran [25]
(6) From Figures 2(g) and 5(g) we observe that 119876 hasstrongly stabilizing effect [NuSh]119876=0 gt [NuSh]119876 = 0
(7) The comparison of three types of temperature mod-ulations is given in Figures 4(h) and 7(h) [NuSh]IPM lt
[NuSh]LBMOlt [NuSh]OPM
(8)The results of this work can be summarized as followsfrom Figures 2ndash7
The effect of temperature modulation on weak nonlineardouble-diffusive magnetoconvection has been analyzed byusing the nonautonomous Ginzburg-Landau equation Thefollowing conclusions are drawn from previous analysis
(1) The effect of IPM is negligible on heat and masstransport in the system
(2) In the case of IPM the effect of 120575 and 120596 is also foundto be negligible on heat and mass transport
(3) In the case of IPM the values of Nu and Sh increasesteadily for small values of time 120591 however Nu and Shbecome constant when 120591 is large
(4) The effect of increasing Pr Le Pm Ras is found toincrease in Nu and Sh thus increasing heat and masstransfer for all three types of modulations
(5) The effect of increasing 120575 is to increase the value of Nuand Sh for the case of OPM and LBMO hence heatand mass transfer
(6) The effect of increasing 120596 is to decrease the value ofNu and Sh for the case ofOPMand LBMO hence heatand mass transfer
(7) In the cases of OPM and LBMO the natures of Nuand Sh remain oscillatory
(8) Initially when 120591 is small the values of Nusselt andSherwood numbers start with 1 corresponding to theconduction state However as 120591 increases Nu andSh also increase thus increasing the heat and masstransfer
(9) The values of Nu and Sh for LBMO are greater thanthose in IPM but smaller than those in OPM
(10) The effect of magnetic field is to stabilize the system
Nomenclature
Latin Symbols
119860 Amplitude of convection119889 Depth of the fluid layerg Acceleration due to gravity119876 Chandrasekhar number
119876 = 1205831198981198672
11988711988921205880]]119898
119896119888 Critical wave numberNu(120591) Nusselt numberSh(120591) Sherwood number119901 Reduced pressurePr Prandtl number Pr = ]120581119879Pm Magnetic Prandtl number Pm = ]119898120581119879Le Lewis number Le = 120581119879120581119878
120596 Frequency of temperature modulation120575 Amplitude of temperature modulation120583 Dynamic coefficient of viscosity of the fluid120583119898 Magnetic permeability] Kinematic viscosity ] = 1205831205880
]119898 Magnetic viscosity120588 Fluid density120595 Stream function120595lowast Dimensionless stream function
Φ Magnetic potentialΦlowast Dimensionless magnetic potential
120591 Slow time 120591 = 1205982119905
120579 Phase angle1198791015840 Perturbed temperature
Other Symbols
nabla2 (12059721205971199092) + (120597
21205971199102) + (12059721205971199112)
14 International Journal of Engineering Mathematics
Subscripts
119887 Basic state119888 Critical0 Reference value
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was done during the lien sanctioned to theauthor by Banaras Hindu University Varanasi to work asprofessor of Mathematics at Department of Applied Math-ematics School for Physical Sciences Babasaheb BhimraoAmbedkar Central University Lucknow India The authorBS Bhadauria gratefully acknowledges Banaras Hindu Uni-versity Varanasi for the same Further the author Palle Kirangratefully acknowledges the financial assistance fromBabasa-heb Bhimrao Ambedkar Central University as a researchfellowship
References
[1] A Akbarzadeh and P Manins ldquoConvective layers generated byside walls in solar pondsrdquo Solar Energy vol 41 no 6 pp 521ndash529 1988
[2] H E Huppert and R S J Sparks ldquoDouble-diffusive convectiondue to crystallization in magmasrdquo Annual Review of Earth andPlanetary Sciences vol 12 pp 11ndash37 1984
[3] H J S Fernando and A Brandt ldquoRecent advances in doublediffusive convectionrdquoAppliedMechanics Reviews vol 47 pp c1ndashc7 1994
[4] J S Turner Buoyancy Effects in Fluids University Of Cam-bridge 1979
[5] N Rudraiah and I S Shivakumara ldquoDouble-diffusive convec-tion with an imposed magnetic fieldrdquo International Journal ofHeat and Mass Transfer vol 27 no 10 pp 1825ndash1836 1984
[6] J H Thomas and N O Weiss ldquoThe theory of sunspotsrdquo inSunspots Theory and Observations p 428 Kluwer AcademicPublishers Dordrecht The Netherlands 1992
[7] W B Thompson ldquoThermal convection in a magnetic fieldrdquoPhilosophical Magazine vol 42 pp 1417ndash1432 1951
[8] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityOxford University Press London UK 1961
[9] D Lortz ldquoA stability criterion for steady finite amplitudeconvection with an external magnetic fieldrdquo Journal of FluidMechanics vol 23 pp 113ndash128 1965
[10] W V R Malkus and G Veronis ldquoFinite amplitude cellularconvectionrdquo Journal of Fluid Mechanics vol 4 pp 225ndash2601958
[11] H Stommel A B Arons and D Blanchard ldquoAn oceanograph-ical curiosity the perpetual salt fountainrdquo Deep Sea Researchvol 3 no 2 pp 152ndash153 1956
[12] K Gotoh and M Yamada ldquoThermal convection of a horizontallayer of magnetic fluidsrdquo Journal of the Physical Society of Japanvol 51 no 9 pp 3042ndash3048 1982
[13] GMOreper and J Szekely ldquoThe effect of an externally imposedmagnetic field on buoyancy driven flow in a rectangular cavityrdquoJournal of Crystal Growth vol 64 no 3 pp 505ndash515 1983
[14] N Rudraiah ldquoDouble-diffusive magnetoconvectionrdquo Pramanavol 27 no 1-2 pp 233ndash266 1986
[15] P G Siddheshwar and S Pranesh ldquoMagnetoconvection in fluidswith suspended particles under 1g and 120583grdquo Aerospace Scienceand Technology vol 6 no 2 pp 105ndash114 2002
[16] B S Bhadauria ldquoCombined effect of temperature modulationand magnetic field on the onset of convection in an electricallyconducting-fluid-saturated porous mediumrdquo Journal of HeatTransfer vol 130 no 5 Article ID 052601 2008
[17] P G Siddheshwar B S Bhadauria P Mishra and A K Srivas-tava ldquoStudy of heat transport by stationarymagneto-convectionin aNewtonian liquid under temperature or gravitymodulationusing Ginzburg-Landau modelrdquo International Journal of Non-Linear Mechanics vol 47 no 5 pp 418ndash425 2012
[18] P G Drazin and W H Reid Hydrodynamic Stability Cam-bridge University Press Cambridge UK 2004
[19] G Venezian ldquoEffect of modulation on the onset of thermalconvectionrdquo Journal of Fluid Mechanics vol 35 no 2 pp 243ndash254 1969
[20] S Rosenblat and G A Tanaka ldquoModulation of thermal convec-tion instabilityrdquo Physics of Fluids vol 14 no 7 pp 1319ndash13221971
[21] M N Roppo S H Davis and S Rosenblat ldquoBenard convectionwith time-periodic heatingrdquoThe Physics of Fluids vol 27 no 4pp 796ndash803 1984
[22] B S Bhadauria and P K Bhatia ldquoTime-periodic heating ofRayleigh-Benard convectionrdquo Physica Scripta vol 66 no 1 pp59ndash65 2002
[23] B S Bhadauria ldquoTime-periodic heating of Rayleigh-Benardconvection in a vertical magnetic fieldrdquo Physica Scripta vol 73no 3 pp 296ndash302 2006
[24] M S Malashetty andM Swamy ldquoEffect of thermal modulationon the onset of convection in a rotating fluid layerrdquo InternationalJournal of Heat and Mass Transfer vol 51 no 11-12 pp 2814ndash2823 2008
[25] B S Bhadauria and P Kiran ldquoHeat transport in an anisotropicporous medium saturated with variable viscosity liquid undertemperature modulationrdquo Transport in Porous Media vol 100no 2 pp 279ndash295 2013
where 1198770119888 is the critical value of the Rayleigh number atwhich the onset of convection takes place in the absenceof temperature modulation Now we solve the system fordifferent orders of 120598
At the lowest order we have
[[[[[[[[[[
[
minusnabla4
1198770119888
120597
120597119909minusRas 120597
120597119909minus119876Pm 120597
120597119911(nabla2)
120597
120597119909minusnabla2
0 0
120597
1205971199090 minus
1
Lenabla2
0
minus120597
1205971199110 0 minusPmnabla
2
]]]]]]]]]]
]
[[[
[
1205951
1198791
1198781
Φ1
]]]
]
=[[[
[
0
0
0
0
]]]
]
(24)
The solutions of the lowest order of the system subject to theboundary conditions equation (22) are
1205951 = 119860 (120591) sin (119896119888119909) sin (120587119911)
1198791 = minus119896119888
1205752119860 (120591) cos (119896119888119909) sin (120587119911)
1198781 = minus119896119888
1205752Le119860 (120591) cos (119896119888119909) sin (120587119911)
Φ1 =120587
Pm1205752119860 (120591) sin (119896119888119909) cos (120587119911)
(25)
where 1205752
= 1198962
119888+ 1205872 The critical value of the Rayleigh
number for the onset of magnetoconvection in the absenceof temperature modulation is
which is the same as Siddheshwar et al [17] when Ras = 0 andwe obtain classical results of Chandrasekhar [8] for withoutmagnetic field and single component fluid layer
International Journal of Engineering Mathematics 5
At the second order we have
[[[[[[[[[
[
minusnabla4
1198770119888
120597
120597119909minusRas 120597
120597119909minus119876Pm 120597
120597119911nabla2
120597
120597119909minusnabla2
0 0
120597
1205971199090 minus
1
Lenabla2
0
minus120597
1205971199110 0 minusPmnabla
2
]]]]]]]]]
]
[[[
[
1205952
1198792
1198782
Φ2
]]]
]
=[[[
[
11987721
11987722
11987723
11987724
]]]
]
11987721 = 0
11987722 =1205971205951
120597119909
1205971198791
120597119911minus1205971205951
120597119911
1205971198791
120597119909
11987723 =1205971205951
120597119909
1205971198781
120597119911minus1205971205951
120597119911
1205971198781
120597119909
11987724 =1205971205951
120597119909
120597Φ1
120597119911minus1205971205951
120597119911
120597Φ1
120597119909
(27)
The second order solutions subjected to the boundary condi-tions Equation (22) are obtained as follows
1205952 = 0
1198792 = minus1198962
119888
812058712057521198602(120591) sin (2120587119911)
1198782 = minus1198962
119888Le2
812058712057521198602(120591) sin (2120587119911)
Φ2 = minus1205872
8119896119888Pm212057521198602(120591) sin (2119896119888119909)
(28)
Thehorizontally averagedNusseltNu and Sherwood Shnum-bers for the stationary double-diffusive magnetoconvection(the mode considered in this problem) are given by
Nu (120591) = 1 +
[(1198961198882120587) int2120587119896119888
0(1205971198792120597119911) 119889119909]
119911=0
[(1198961198882120587) int2120587119896119888
0(120597119879119887120597119911 119889119909]
119911=0
Sh (120591) = 1 +
[(1198961198882120587) int2120587119896119888
0(1205971198782120597119911) 119889119909]
119911=0
[(1198961198882120587) int2120587119896119888
0(119889119878119887119889119911) 119889119909]
119911=0
(29)
One must note here that 1198912(119911 120591) is effective at 119874(1205982) andaffects Nu(120591) and Sh(120591) through 119860(120591) can be seen laterTherefore substituting 1198792 119879119887 1198782 respectively into (29) andsimplifying we obtain
Nu (120591) = 1 +1198962
119888
412057521198602(120591)
Sh (120591) = 1 +1198962
119888Le2
412057521198602(120591)
(30)
At the third order we have
[[[[[[[[[
[
minusnabla4
1198770119888
120597
120597119909minusRas 120597
120597119909minus119876Pm 120597
120597119911nabla2
120597
120597119909minusnabla2
0 0
120597
1205971199090 minus
1
Lenabla2
0
minus120597
1205971199110 0 minusPmnabla
2
]]]]]]]]]
]
[[[
[
1205953
1198793
1198783
Φ3
]]]
]
=[[[
[
11987731
11987732
11987733
11987734
]]]
]
(31)
where
11987731 = minus1
Pr120597
120597120591(nabla21205951) minus 1198772
1205971198791
120597119909minus Ras1205971198781
120597119909
minus 119876Pm(120597Φ1
120597119911
120597
120597119909(nabla2Φ2) minus
120597Φ2
120597119909
120597
120597119911(nabla2Φ1))
11987732 = minus1205971198791
120597120591+ 1205751198912 (119911 120591)
1205971205951
120597119909+1205971205951
120597119909
1205971198792
120597119911
11987733 = minus1205971198781
120597120591+1205971205951
120597119909
1205971198782
120597119911
11987734 = minus120597Φ1
120597120591minus1205971205951
120597119911
120597Φ2
120597119909
(32)
Substituting 1205951 1198791 and 1198792 into (32) we obtain the expres-sions for119877311198773211987733 and11987734 easily For the existence of thirdorder solution of the systemwe apply the solvability conditionwhich leads to arrive at the nonautonomous Ginzburg-Landau equation for stationary mode of convection withtime-periodic coefficients in the form
11986011198601015840(120591) minus 1198602119860 (120591) + 1198603119860(120591)
The Ginzburg-Landau equations given in (33) are Bernoulliequation and obtaining its analytical solution is difficultdue to its nonautonomous nature So that it has beensolved numerically using the in-built function NDSolve ofMathematica 8 subjected to the initial condition 119860(0) = 1198860where 1198860 is the chosen initial amplitude of convection In ourcalculations we may use 1198772 = 1198770119888 to keep the parameters tothe minimum
6 International Journal of Engineering Mathematics
4 Results and Discussion
External regulation of thermal instability is important tostudy the double-diffusive convection in a fluid layer Theobjective of this paper is to consider two such candidatesnamely vertical magnetic field and temperature modulationfor either enhancing or inhibiting convective heat and masstransport as is required by a real application The presentpaper deals with double-diffusive magnetoconvection undertemperature modulation by using Ginzburg-Landau equa-tion It is necessary to consider a nonlinear theory toanalyze heat and mass transfer which is not possible bythe linear theory We consider the direct mode (120581119878120581119879 lt
1 otherwise Hopf mode) in which the salt and heat makeopposing contributions (120581119879 = 120581119878) We also consider the effectof temperature modulation to be of order 119874(1205982) this leadsto small amplitude of modulation Such an assumption willhelp us in obtaining the amplitude equation of convectionin a rather simple and elegant manner and is much easierto obtain than in the case of the Lorenz model We give thefollowing features of the problem before our results
(1) the need for nonlinear stability analysis(2) the relation of the problem to a real application(3) the selection of all dimensionless parameters utilized
in computations
We consider the following three types of temperaturemodulation on the boundaries of the system
The parameters of the system are 119876 Pr Le Pm Ras120579 120575 120596 these parameters influence the convective heatand mass transfer The first five parameters are related tothe fluid layer and the last three parameters concern theexternal mechanisms of controlling convection The effect oftemperaturemodulation is represented by amplitude120575 whichlies around 03 due to the assumption The effect of electricalconductivity andmagnetic field comes through Pm119876 Thereis the property of the fluid coming into picture as well asthrough Prandtl number Pr Further the modulation of theboundary temperature is assumed to be of low frequency Atlow range of frequencies the effect of frequency on onset ofconvection as well as on heat and mass transport is minimalThis assumption is required in order to ensure that the systemdoes not pick up oscillatory convective mode at onset dueto modulation in a situation that is conductive otherwiseto stationary mode It is important at this stage to considerthe effect of 119876 Pr Le Pm Ras 120579 120575 120596 on the onset ofconvection The heat and mass transfer of the problem arequantified by the Nusselt and Sherwood numbers which aregiven in (30) Figures 2ndash7 show the individual effect of eachnondimensional parameter on heat and mass transfer
(1) Figures 2(a)ndash7(a) show that the effect of Chan-drasekhar number119876which is ratio of Lorentz force to viscousforce is to delay the onset of convection hence heat and mass
transfer The Nu and Sh start with one and for small values oftime 120591 increase and become constant for large values of time 120591in the case of [IPM] given in Figures 2(a) and 5(a) In the caseof [OPM LBMO] the effect of119876 shows oscillatory behaviourand increment in it decreases the magnitude of both Nu andSh Hence 119876 has stabilizing effect in all the three types ofmodulations given in Figures 3(a) 4(a) 6(a) and 7(a) so thatheat and mass transfer decrease with 119876
(2) The effect of Prandtl number Pr is to advance theconvection and hence heat and mass transfer which hassimilar behaviour of (1) given in Figures 2(b)ndash7(b)
(3) The effect of Lewis Le and Magnetic Prandtl Pmnumbers is to advance the convection and hence heat andmass transfer Hence both Le and Pmhave destabilizing effectof the system given in Figures 2(c)ndash7(c) and 2(d)ndash7(d) andhave similar behaviour of (1) these results are earlier obtainedby Siddheshwar et al [17]
(4)The effect of solutal Rayleigh numberRas is to increaseNu and Sh so that heat and mass transfer Hence it hasdestabilizing effect in all the three types ofmodulationswhichis given by the Figures 2(e)ndash7(e) and has similar behaviour of(1)Though the presence of a stabilizing gradient of solute willprevent the onset of convection the strong finite-amplitudemotions which exist for large Rayleigh numbers tend to mixthe solute and redistribute it so that the interior layers ofthe fluid are more neutrally stratified As a consequence theinhibiting effect of the solute gradient is greatly reduced andhence fluid will convect more and more heat and mass whenRas is increased
(5) In the case of [IPM] we observe no effect of amplitude120575 and frequency 120596 of modulation which is given by theFigures 2(f) and 5(f) But in the case of [OPM-LBMO] theincrement in 120575 leads to increment in magnitude of Nu andSh hence heat and mass transfer given in Figures 3(f) 4(f)6(f) and 7(f) the increment in120596 shortens thewavelength anddecreases in magnitude of Nu Sh and hence heat and masstransfer given in Figures 3(g) 4(g) 6(g) and 7(g) which arethe results obtained by Venezian [19] Siddheshwar et al [17]and Bhadauria and Kiran [25]
(6) From Figures 2(g) and 5(g) we observe that 119876 hasstrongly stabilizing effect [NuSh]119876=0 gt [NuSh]119876 = 0
(7) The comparison of three types of temperature mod-ulations is given in Figures 4(h) and 7(h) [NuSh]IPM lt
[NuSh]LBMOlt [NuSh]OPM
(8)The results of this work can be summarized as followsfrom Figures 2ndash7
The effect of temperature modulation on weak nonlineardouble-diffusive magnetoconvection has been analyzed byusing the nonautonomous Ginzburg-Landau equation Thefollowing conclusions are drawn from previous analysis
(1) The effect of IPM is negligible on heat and masstransport in the system
(2) In the case of IPM the effect of 120575 and 120596 is also foundto be negligible on heat and mass transport
(3) In the case of IPM the values of Nu and Sh increasesteadily for small values of time 120591 however Nu and Shbecome constant when 120591 is large
(4) The effect of increasing Pr Le Pm Ras is found toincrease in Nu and Sh thus increasing heat and masstransfer for all three types of modulations
(5) The effect of increasing 120575 is to increase the value of Nuand Sh for the case of OPM and LBMO hence heatand mass transfer
(6) The effect of increasing 120596 is to decrease the value ofNu and Sh for the case ofOPMand LBMO hence heatand mass transfer
(7) In the cases of OPM and LBMO the natures of Nuand Sh remain oscillatory
(8) Initially when 120591 is small the values of Nusselt andSherwood numbers start with 1 corresponding to theconduction state However as 120591 increases Nu andSh also increase thus increasing the heat and masstransfer
(9) The values of Nu and Sh for LBMO are greater thanthose in IPM but smaller than those in OPM
(10) The effect of magnetic field is to stabilize the system
Nomenclature
Latin Symbols
119860 Amplitude of convection119889 Depth of the fluid layerg Acceleration due to gravity119876 Chandrasekhar number
119876 = 1205831198981198672
11988711988921205880]]119898
119896119888 Critical wave numberNu(120591) Nusselt numberSh(120591) Sherwood number119901 Reduced pressurePr Prandtl number Pr = ]120581119879Pm Magnetic Prandtl number Pm = ]119898120581119879Le Lewis number Le = 120581119879120581119878
120596 Frequency of temperature modulation120575 Amplitude of temperature modulation120583 Dynamic coefficient of viscosity of the fluid120583119898 Magnetic permeability] Kinematic viscosity ] = 1205831205880
]119898 Magnetic viscosity120588 Fluid density120595 Stream function120595lowast Dimensionless stream function
Φ Magnetic potentialΦlowast Dimensionless magnetic potential
120591 Slow time 120591 = 1205982119905
120579 Phase angle1198791015840 Perturbed temperature
Other Symbols
nabla2 (12059721205971199092) + (120597
21205971199102) + (12059721205971199112)
14 International Journal of Engineering Mathematics
Subscripts
119887 Basic state119888 Critical0 Reference value
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was done during the lien sanctioned to theauthor by Banaras Hindu University Varanasi to work asprofessor of Mathematics at Department of Applied Math-ematics School for Physical Sciences Babasaheb BhimraoAmbedkar Central University Lucknow India The authorBS Bhadauria gratefully acknowledges Banaras Hindu Uni-versity Varanasi for the same Further the author Palle Kirangratefully acknowledges the financial assistance fromBabasa-heb Bhimrao Ambedkar Central University as a researchfellowship
References
[1] A Akbarzadeh and P Manins ldquoConvective layers generated byside walls in solar pondsrdquo Solar Energy vol 41 no 6 pp 521ndash529 1988
[2] H E Huppert and R S J Sparks ldquoDouble-diffusive convectiondue to crystallization in magmasrdquo Annual Review of Earth andPlanetary Sciences vol 12 pp 11ndash37 1984
[3] H J S Fernando and A Brandt ldquoRecent advances in doublediffusive convectionrdquoAppliedMechanics Reviews vol 47 pp c1ndashc7 1994
[4] J S Turner Buoyancy Effects in Fluids University Of Cam-bridge 1979
[5] N Rudraiah and I S Shivakumara ldquoDouble-diffusive convec-tion with an imposed magnetic fieldrdquo International Journal ofHeat and Mass Transfer vol 27 no 10 pp 1825ndash1836 1984
[6] J H Thomas and N O Weiss ldquoThe theory of sunspotsrdquo inSunspots Theory and Observations p 428 Kluwer AcademicPublishers Dordrecht The Netherlands 1992
[7] W B Thompson ldquoThermal convection in a magnetic fieldrdquoPhilosophical Magazine vol 42 pp 1417ndash1432 1951
[8] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityOxford University Press London UK 1961
[9] D Lortz ldquoA stability criterion for steady finite amplitudeconvection with an external magnetic fieldrdquo Journal of FluidMechanics vol 23 pp 113ndash128 1965
[10] W V R Malkus and G Veronis ldquoFinite amplitude cellularconvectionrdquo Journal of Fluid Mechanics vol 4 pp 225ndash2601958
[11] H Stommel A B Arons and D Blanchard ldquoAn oceanograph-ical curiosity the perpetual salt fountainrdquo Deep Sea Researchvol 3 no 2 pp 152ndash153 1956
[12] K Gotoh and M Yamada ldquoThermal convection of a horizontallayer of magnetic fluidsrdquo Journal of the Physical Society of Japanvol 51 no 9 pp 3042ndash3048 1982
[13] GMOreper and J Szekely ldquoThe effect of an externally imposedmagnetic field on buoyancy driven flow in a rectangular cavityrdquoJournal of Crystal Growth vol 64 no 3 pp 505ndash515 1983
[14] N Rudraiah ldquoDouble-diffusive magnetoconvectionrdquo Pramanavol 27 no 1-2 pp 233ndash266 1986
[15] P G Siddheshwar and S Pranesh ldquoMagnetoconvection in fluidswith suspended particles under 1g and 120583grdquo Aerospace Scienceand Technology vol 6 no 2 pp 105ndash114 2002
[16] B S Bhadauria ldquoCombined effect of temperature modulationand magnetic field on the onset of convection in an electricallyconducting-fluid-saturated porous mediumrdquo Journal of HeatTransfer vol 130 no 5 Article ID 052601 2008
[17] P G Siddheshwar B S Bhadauria P Mishra and A K Srivas-tava ldquoStudy of heat transport by stationarymagneto-convectionin aNewtonian liquid under temperature or gravitymodulationusing Ginzburg-Landau modelrdquo International Journal of Non-Linear Mechanics vol 47 no 5 pp 418ndash425 2012
[18] P G Drazin and W H Reid Hydrodynamic Stability Cam-bridge University Press Cambridge UK 2004
[19] G Venezian ldquoEffect of modulation on the onset of thermalconvectionrdquo Journal of Fluid Mechanics vol 35 no 2 pp 243ndash254 1969
[20] S Rosenblat and G A Tanaka ldquoModulation of thermal convec-tion instabilityrdquo Physics of Fluids vol 14 no 7 pp 1319ndash13221971
[21] M N Roppo S H Davis and S Rosenblat ldquoBenard convectionwith time-periodic heatingrdquoThe Physics of Fluids vol 27 no 4pp 796ndash803 1984
[22] B S Bhadauria and P K Bhatia ldquoTime-periodic heating ofRayleigh-Benard convectionrdquo Physica Scripta vol 66 no 1 pp59ndash65 2002
[23] B S Bhadauria ldquoTime-periodic heating of Rayleigh-Benardconvection in a vertical magnetic fieldrdquo Physica Scripta vol 73no 3 pp 296ndash302 2006
[24] M S Malashetty andM Swamy ldquoEffect of thermal modulationon the onset of convection in a rotating fluid layerrdquo InternationalJournal of Heat and Mass Transfer vol 51 no 11-12 pp 2814ndash2823 2008
[25] B S Bhadauria and P Kiran ldquoHeat transport in an anisotropicporous medium saturated with variable viscosity liquid undertemperature modulationrdquo Transport in Porous Media vol 100no 2 pp 279ndash295 2013
International Journal of Engineering Mathematics 5
At the second order we have
[[[[[[[[[
[
minusnabla4
1198770119888
120597
120597119909minusRas 120597
120597119909minus119876Pm 120597
120597119911nabla2
120597
120597119909minusnabla2
0 0
120597
1205971199090 minus
1
Lenabla2
0
minus120597
1205971199110 0 minusPmnabla
2
]]]]]]]]]
]
[[[
[
1205952
1198792
1198782
Φ2
]]]
]
=[[[
[
11987721
11987722
11987723
11987724
]]]
]
11987721 = 0
11987722 =1205971205951
120597119909
1205971198791
120597119911minus1205971205951
120597119911
1205971198791
120597119909
11987723 =1205971205951
120597119909
1205971198781
120597119911minus1205971205951
120597119911
1205971198781
120597119909
11987724 =1205971205951
120597119909
120597Φ1
120597119911minus1205971205951
120597119911
120597Φ1
120597119909
(27)
The second order solutions subjected to the boundary condi-tions Equation (22) are obtained as follows
1205952 = 0
1198792 = minus1198962
119888
812058712057521198602(120591) sin (2120587119911)
1198782 = minus1198962
119888Le2
812058712057521198602(120591) sin (2120587119911)
Φ2 = minus1205872
8119896119888Pm212057521198602(120591) sin (2119896119888119909)
(28)
Thehorizontally averagedNusseltNu and Sherwood Shnum-bers for the stationary double-diffusive magnetoconvection(the mode considered in this problem) are given by
Nu (120591) = 1 +
[(1198961198882120587) int2120587119896119888
0(1205971198792120597119911) 119889119909]
119911=0
[(1198961198882120587) int2120587119896119888
0(120597119879119887120597119911 119889119909]
119911=0
Sh (120591) = 1 +
[(1198961198882120587) int2120587119896119888
0(1205971198782120597119911) 119889119909]
119911=0
[(1198961198882120587) int2120587119896119888
0(119889119878119887119889119911) 119889119909]
119911=0
(29)
One must note here that 1198912(119911 120591) is effective at 119874(1205982) andaffects Nu(120591) and Sh(120591) through 119860(120591) can be seen laterTherefore substituting 1198792 119879119887 1198782 respectively into (29) andsimplifying we obtain
Nu (120591) = 1 +1198962
119888
412057521198602(120591)
Sh (120591) = 1 +1198962
119888Le2
412057521198602(120591)
(30)
At the third order we have
[[[[[[[[[
[
minusnabla4
1198770119888
120597
120597119909minusRas 120597
120597119909minus119876Pm 120597
120597119911nabla2
120597
120597119909minusnabla2
0 0
120597
1205971199090 minus
1
Lenabla2
0
minus120597
1205971199110 0 minusPmnabla
2
]]]]]]]]]
]
[[[
[
1205953
1198793
1198783
Φ3
]]]
]
=[[[
[
11987731
11987732
11987733
11987734
]]]
]
(31)
where
11987731 = minus1
Pr120597
120597120591(nabla21205951) minus 1198772
1205971198791
120597119909minus Ras1205971198781
120597119909
minus 119876Pm(120597Φ1
120597119911
120597
120597119909(nabla2Φ2) minus
120597Φ2
120597119909
120597
120597119911(nabla2Φ1))
11987732 = minus1205971198791
120597120591+ 1205751198912 (119911 120591)
1205971205951
120597119909+1205971205951
120597119909
1205971198792
120597119911
11987733 = minus1205971198781
120597120591+1205971205951
120597119909
1205971198782
120597119911
11987734 = minus120597Φ1
120597120591minus1205971205951
120597119911
120597Φ2
120597119909
(32)
Substituting 1205951 1198791 and 1198792 into (32) we obtain the expres-sions for119877311198773211987733 and11987734 easily For the existence of thirdorder solution of the systemwe apply the solvability conditionwhich leads to arrive at the nonautonomous Ginzburg-Landau equation for stationary mode of convection withtime-periodic coefficients in the form
11986011198601015840(120591) minus 1198602119860 (120591) + 1198603119860(120591)
The Ginzburg-Landau equations given in (33) are Bernoulliequation and obtaining its analytical solution is difficultdue to its nonautonomous nature So that it has beensolved numerically using the in-built function NDSolve ofMathematica 8 subjected to the initial condition 119860(0) = 1198860where 1198860 is the chosen initial amplitude of convection In ourcalculations we may use 1198772 = 1198770119888 to keep the parameters tothe minimum
6 International Journal of Engineering Mathematics
4 Results and Discussion
External regulation of thermal instability is important tostudy the double-diffusive convection in a fluid layer Theobjective of this paper is to consider two such candidatesnamely vertical magnetic field and temperature modulationfor either enhancing or inhibiting convective heat and masstransport as is required by a real application The presentpaper deals with double-diffusive magnetoconvection undertemperature modulation by using Ginzburg-Landau equa-tion It is necessary to consider a nonlinear theory toanalyze heat and mass transfer which is not possible bythe linear theory We consider the direct mode (120581119878120581119879 lt
1 otherwise Hopf mode) in which the salt and heat makeopposing contributions (120581119879 = 120581119878) We also consider the effectof temperature modulation to be of order 119874(1205982) this leadsto small amplitude of modulation Such an assumption willhelp us in obtaining the amplitude equation of convectionin a rather simple and elegant manner and is much easierto obtain than in the case of the Lorenz model We give thefollowing features of the problem before our results
(1) the need for nonlinear stability analysis(2) the relation of the problem to a real application(3) the selection of all dimensionless parameters utilized
in computations
We consider the following three types of temperaturemodulation on the boundaries of the system
The parameters of the system are 119876 Pr Le Pm Ras120579 120575 120596 these parameters influence the convective heatand mass transfer The first five parameters are related tothe fluid layer and the last three parameters concern theexternal mechanisms of controlling convection The effect oftemperaturemodulation is represented by amplitude120575 whichlies around 03 due to the assumption The effect of electricalconductivity andmagnetic field comes through Pm119876 Thereis the property of the fluid coming into picture as well asthrough Prandtl number Pr Further the modulation of theboundary temperature is assumed to be of low frequency Atlow range of frequencies the effect of frequency on onset ofconvection as well as on heat and mass transport is minimalThis assumption is required in order to ensure that the systemdoes not pick up oscillatory convective mode at onset dueto modulation in a situation that is conductive otherwiseto stationary mode It is important at this stage to considerthe effect of 119876 Pr Le Pm Ras 120579 120575 120596 on the onset ofconvection The heat and mass transfer of the problem arequantified by the Nusselt and Sherwood numbers which aregiven in (30) Figures 2ndash7 show the individual effect of eachnondimensional parameter on heat and mass transfer
(1) Figures 2(a)ndash7(a) show that the effect of Chan-drasekhar number119876which is ratio of Lorentz force to viscousforce is to delay the onset of convection hence heat and mass
transfer The Nu and Sh start with one and for small values oftime 120591 increase and become constant for large values of time 120591in the case of [IPM] given in Figures 2(a) and 5(a) In the caseof [OPM LBMO] the effect of119876 shows oscillatory behaviourand increment in it decreases the magnitude of both Nu andSh Hence 119876 has stabilizing effect in all the three types ofmodulations given in Figures 3(a) 4(a) 6(a) and 7(a) so thatheat and mass transfer decrease with 119876
(2) The effect of Prandtl number Pr is to advance theconvection and hence heat and mass transfer which hassimilar behaviour of (1) given in Figures 2(b)ndash7(b)
(3) The effect of Lewis Le and Magnetic Prandtl Pmnumbers is to advance the convection and hence heat andmass transfer Hence both Le and Pmhave destabilizing effectof the system given in Figures 2(c)ndash7(c) and 2(d)ndash7(d) andhave similar behaviour of (1) these results are earlier obtainedby Siddheshwar et al [17]
(4)The effect of solutal Rayleigh numberRas is to increaseNu and Sh so that heat and mass transfer Hence it hasdestabilizing effect in all the three types ofmodulationswhichis given by the Figures 2(e)ndash7(e) and has similar behaviour of(1)Though the presence of a stabilizing gradient of solute willprevent the onset of convection the strong finite-amplitudemotions which exist for large Rayleigh numbers tend to mixthe solute and redistribute it so that the interior layers ofthe fluid are more neutrally stratified As a consequence theinhibiting effect of the solute gradient is greatly reduced andhence fluid will convect more and more heat and mass whenRas is increased
(5) In the case of [IPM] we observe no effect of amplitude120575 and frequency 120596 of modulation which is given by theFigures 2(f) and 5(f) But in the case of [OPM-LBMO] theincrement in 120575 leads to increment in magnitude of Nu andSh hence heat and mass transfer given in Figures 3(f) 4(f)6(f) and 7(f) the increment in120596 shortens thewavelength anddecreases in magnitude of Nu Sh and hence heat and masstransfer given in Figures 3(g) 4(g) 6(g) and 7(g) which arethe results obtained by Venezian [19] Siddheshwar et al [17]and Bhadauria and Kiran [25]
(6) From Figures 2(g) and 5(g) we observe that 119876 hasstrongly stabilizing effect [NuSh]119876=0 gt [NuSh]119876 = 0
(7) The comparison of three types of temperature mod-ulations is given in Figures 4(h) and 7(h) [NuSh]IPM lt
[NuSh]LBMOlt [NuSh]OPM
(8)The results of this work can be summarized as followsfrom Figures 2ndash7
The effect of temperature modulation on weak nonlineardouble-diffusive magnetoconvection has been analyzed byusing the nonautonomous Ginzburg-Landau equation Thefollowing conclusions are drawn from previous analysis
(1) The effect of IPM is negligible on heat and masstransport in the system
(2) In the case of IPM the effect of 120575 and 120596 is also foundto be negligible on heat and mass transport
(3) In the case of IPM the values of Nu and Sh increasesteadily for small values of time 120591 however Nu and Shbecome constant when 120591 is large
(4) The effect of increasing Pr Le Pm Ras is found toincrease in Nu and Sh thus increasing heat and masstransfer for all three types of modulations
(5) The effect of increasing 120575 is to increase the value of Nuand Sh for the case of OPM and LBMO hence heatand mass transfer
(6) The effect of increasing 120596 is to decrease the value ofNu and Sh for the case ofOPMand LBMO hence heatand mass transfer
(7) In the cases of OPM and LBMO the natures of Nuand Sh remain oscillatory
(8) Initially when 120591 is small the values of Nusselt andSherwood numbers start with 1 corresponding to theconduction state However as 120591 increases Nu andSh also increase thus increasing the heat and masstransfer
(9) The values of Nu and Sh for LBMO are greater thanthose in IPM but smaller than those in OPM
(10) The effect of magnetic field is to stabilize the system
Nomenclature
Latin Symbols
119860 Amplitude of convection119889 Depth of the fluid layerg Acceleration due to gravity119876 Chandrasekhar number
119876 = 1205831198981198672
11988711988921205880]]119898
119896119888 Critical wave numberNu(120591) Nusselt numberSh(120591) Sherwood number119901 Reduced pressurePr Prandtl number Pr = ]120581119879Pm Magnetic Prandtl number Pm = ]119898120581119879Le Lewis number Le = 120581119879120581119878
120596 Frequency of temperature modulation120575 Amplitude of temperature modulation120583 Dynamic coefficient of viscosity of the fluid120583119898 Magnetic permeability] Kinematic viscosity ] = 1205831205880
]119898 Magnetic viscosity120588 Fluid density120595 Stream function120595lowast Dimensionless stream function
Φ Magnetic potentialΦlowast Dimensionless magnetic potential
120591 Slow time 120591 = 1205982119905
120579 Phase angle1198791015840 Perturbed temperature
Other Symbols
nabla2 (12059721205971199092) + (120597
21205971199102) + (12059721205971199112)
14 International Journal of Engineering Mathematics
Subscripts
119887 Basic state119888 Critical0 Reference value
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was done during the lien sanctioned to theauthor by Banaras Hindu University Varanasi to work asprofessor of Mathematics at Department of Applied Math-ematics School for Physical Sciences Babasaheb BhimraoAmbedkar Central University Lucknow India The authorBS Bhadauria gratefully acknowledges Banaras Hindu Uni-versity Varanasi for the same Further the author Palle Kirangratefully acknowledges the financial assistance fromBabasa-heb Bhimrao Ambedkar Central University as a researchfellowship
References
[1] A Akbarzadeh and P Manins ldquoConvective layers generated byside walls in solar pondsrdquo Solar Energy vol 41 no 6 pp 521ndash529 1988
[2] H E Huppert and R S J Sparks ldquoDouble-diffusive convectiondue to crystallization in magmasrdquo Annual Review of Earth andPlanetary Sciences vol 12 pp 11ndash37 1984
[3] H J S Fernando and A Brandt ldquoRecent advances in doublediffusive convectionrdquoAppliedMechanics Reviews vol 47 pp c1ndashc7 1994
[4] J S Turner Buoyancy Effects in Fluids University Of Cam-bridge 1979
[5] N Rudraiah and I S Shivakumara ldquoDouble-diffusive convec-tion with an imposed magnetic fieldrdquo International Journal ofHeat and Mass Transfer vol 27 no 10 pp 1825ndash1836 1984
[6] J H Thomas and N O Weiss ldquoThe theory of sunspotsrdquo inSunspots Theory and Observations p 428 Kluwer AcademicPublishers Dordrecht The Netherlands 1992
[7] W B Thompson ldquoThermal convection in a magnetic fieldrdquoPhilosophical Magazine vol 42 pp 1417ndash1432 1951
[8] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityOxford University Press London UK 1961
[9] D Lortz ldquoA stability criterion for steady finite amplitudeconvection with an external magnetic fieldrdquo Journal of FluidMechanics vol 23 pp 113ndash128 1965
[10] W V R Malkus and G Veronis ldquoFinite amplitude cellularconvectionrdquo Journal of Fluid Mechanics vol 4 pp 225ndash2601958
[11] H Stommel A B Arons and D Blanchard ldquoAn oceanograph-ical curiosity the perpetual salt fountainrdquo Deep Sea Researchvol 3 no 2 pp 152ndash153 1956
[12] K Gotoh and M Yamada ldquoThermal convection of a horizontallayer of magnetic fluidsrdquo Journal of the Physical Society of Japanvol 51 no 9 pp 3042ndash3048 1982
[13] GMOreper and J Szekely ldquoThe effect of an externally imposedmagnetic field on buoyancy driven flow in a rectangular cavityrdquoJournal of Crystal Growth vol 64 no 3 pp 505ndash515 1983
[14] N Rudraiah ldquoDouble-diffusive magnetoconvectionrdquo Pramanavol 27 no 1-2 pp 233ndash266 1986
[15] P G Siddheshwar and S Pranesh ldquoMagnetoconvection in fluidswith suspended particles under 1g and 120583grdquo Aerospace Scienceand Technology vol 6 no 2 pp 105ndash114 2002
[16] B S Bhadauria ldquoCombined effect of temperature modulationand magnetic field on the onset of convection in an electricallyconducting-fluid-saturated porous mediumrdquo Journal of HeatTransfer vol 130 no 5 Article ID 052601 2008
[17] P G Siddheshwar B S Bhadauria P Mishra and A K Srivas-tava ldquoStudy of heat transport by stationarymagneto-convectionin aNewtonian liquid under temperature or gravitymodulationusing Ginzburg-Landau modelrdquo International Journal of Non-Linear Mechanics vol 47 no 5 pp 418ndash425 2012
[18] P G Drazin and W H Reid Hydrodynamic Stability Cam-bridge University Press Cambridge UK 2004
[19] G Venezian ldquoEffect of modulation on the onset of thermalconvectionrdquo Journal of Fluid Mechanics vol 35 no 2 pp 243ndash254 1969
[20] S Rosenblat and G A Tanaka ldquoModulation of thermal convec-tion instabilityrdquo Physics of Fluids vol 14 no 7 pp 1319ndash13221971
[21] M N Roppo S H Davis and S Rosenblat ldquoBenard convectionwith time-periodic heatingrdquoThe Physics of Fluids vol 27 no 4pp 796ndash803 1984
[22] B S Bhadauria and P K Bhatia ldquoTime-periodic heating ofRayleigh-Benard convectionrdquo Physica Scripta vol 66 no 1 pp59ndash65 2002
[23] B S Bhadauria ldquoTime-periodic heating of Rayleigh-Benardconvection in a vertical magnetic fieldrdquo Physica Scripta vol 73no 3 pp 296ndash302 2006
[24] M S Malashetty andM Swamy ldquoEffect of thermal modulationon the onset of convection in a rotating fluid layerrdquo InternationalJournal of Heat and Mass Transfer vol 51 no 11-12 pp 2814ndash2823 2008
[25] B S Bhadauria and P Kiran ldquoHeat transport in an anisotropicporous medium saturated with variable viscosity liquid undertemperature modulationrdquo Transport in Porous Media vol 100no 2 pp 279ndash295 2013
6 International Journal of Engineering Mathematics
4 Results and Discussion
External regulation of thermal instability is important tostudy the double-diffusive convection in a fluid layer Theobjective of this paper is to consider two such candidatesnamely vertical magnetic field and temperature modulationfor either enhancing or inhibiting convective heat and masstransport as is required by a real application The presentpaper deals with double-diffusive magnetoconvection undertemperature modulation by using Ginzburg-Landau equa-tion It is necessary to consider a nonlinear theory toanalyze heat and mass transfer which is not possible bythe linear theory We consider the direct mode (120581119878120581119879 lt
1 otherwise Hopf mode) in which the salt and heat makeopposing contributions (120581119879 = 120581119878) We also consider the effectof temperature modulation to be of order 119874(1205982) this leadsto small amplitude of modulation Such an assumption willhelp us in obtaining the amplitude equation of convectionin a rather simple and elegant manner and is much easierto obtain than in the case of the Lorenz model We give thefollowing features of the problem before our results
(1) the need for nonlinear stability analysis(2) the relation of the problem to a real application(3) the selection of all dimensionless parameters utilized
in computations
We consider the following three types of temperaturemodulation on the boundaries of the system
The parameters of the system are 119876 Pr Le Pm Ras120579 120575 120596 these parameters influence the convective heatand mass transfer The first five parameters are related tothe fluid layer and the last three parameters concern theexternal mechanisms of controlling convection The effect oftemperaturemodulation is represented by amplitude120575 whichlies around 03 due to the assumption The effect of electricalconductivity andmagnetic field comes through Pm119876 Thereis the property of the fluid coming into picture as well asthrough Prandtl number Pr Further the modulation of theboundary temperature is assumed to be of low frequency Atlow range of frequencies the effect of frequency on onset ofconvection as well as on heat and mass transport is minimalThis assumption is required in order to ensure that the systemdoes not pick up oscillatory convective mode at onset dueto modulation in a situation that is conductive otherwiseto stationary mode It is important at this stage to considerthe effect of 119876 Pr Le Pm Ras 120579 120575 120596 on the onset ofconvection The heat and mass transfer of the problem arequantified by the Nusselt and Sherwood numbers which aregiven in (30) Figures 2ndash7 show the individual effect of eachnondimensional parameter on heat and mass transfer
(1) Figures 2(a)ndash7(a) show that the effect of Chan-drasekhar number119876which is ratio of Lorentz force to viscousforce is to delay the onset of convection hence heat and mass
transfer The Nu and Sh start with one and for small values oftime 120591 increase and become constant for large values of time 120591in the case of [IPM] given in Figures 2(a) and 5(a) In the caseof [OPM LBMO] the effect of119876 shows oscillatory behaviourand increment in it decreases the magnitude of both Nu andSh Hence 119876 has stabilizing effect in all the three types ofmodulations given in Figures 3(a) 4(a) 6(a) and 7(a) so thatheat and mass transfer decrease with 119876
(2) The effect of Prandtl number Pr is to advance theconvection and hence heat and mass transfer which hassimilar behaviour of (1) given in Figures 2(b)ndash7(b)
(3) The effect of Lewis Le and Magnetic Prandtl Pmnumbers is to advance the convection and hence heat andmass transfer Hence both Le and Pmhave destabilizing effectof the system given in Figures 2(c)ndash7(c) and 2(d)ndash7(d) andhave similar behaviour of (1) these results are earlier obtainedby Siddheshwar et al [17]
(4)The effect of solutal Rayleigh numberRas is to increaseNu and Sh so that heat and mass transfer Hence it hasdestabilizing effect in all the three types ofmodulationswhichis given by the Figures 2(e)ndash7(e) and has similar behaviour of(1)Though the presence of a stabilizing gradient of solute willprevent the onset of convection the strong finite-amplitudemotions which exist for large Rayleigh numbers tend to mixthe solute and redistribute it so that the interior layers ofthe fluid are more neutrally stratified As a consequence theinhibiting effect of the solute gradient is greatly reduced andhence fluid will convect more and more heat and mass whenRas is increased
(5) In the case of [IPM] we observe no effect of amplitude120575 and frequency 120596 of modulation which is given by theFigures 2(f) and 5(f) But in the case of [OPM-LBMO] theincrement in 120575 leads to increment in magnitude of Nu andSh hence heat and mass transfer given in Figures 3(f) 4(f)6(f) and 7(f) the increment in120596 shortens thewavelength anddecreases in magnitude of Nu Sh and hence heat and masstransfer given in Figures 3(g) 4(g) 6(g) and 7(g) which arethe results obtained by Venezian [19] Siddheshwar et al [17]and Bhadauria and Kiran [25]
(6) From Figures 2(g) and 5(g) we observe that 119876 hasstrongly stabilizing effect [NuSh]119876=0 gt [NuSh]119876 = 0
(7) The comparison of three types of temperature mod-ulations is given in Figures 4(h) and 7(h) [NuSh]IPM lt
[NuSh]LBMOlt [NuSh]OPM
(8)The results of this work can be summarized as followsfrom Figures 2ndash7
The effect of temperature modulation on weak nonlineardouble-diffusive magnetoconvection has been analyzed byusing the nonautonomous Ginzburg-Landau equation Thefollowing conclusions are drawn from previous analysis
(1) The effect of IPM is negligible on heat and masstransport in the system
(2) In the case of IPM the effect of 120575 and 120596 is also foundto be negligible on heat and mass transport
(3) In the case of IPM the values of Nu and Sh increasesteadily for small values of time 120591 however Nu and Shbecome constant when 120591 is large
(4) The effect of increasing Pr Le Pm Ras is found toincrease in Nu and Sh thus increasing heat and masstransfer for all three types of modulations
(5) The effect of increasing 120575 is to increase the value of Nuand Sh for the case of OPM and LBMO hence heatand mass transfer
(6) The effect of increasing 120596 is to decrease the value ofNu and Sh for the case ofOPMand LBMO hence heatand mass transfer
(7) In the cases of OPM and LBMO the natures of Nuand Sh remain oscillatory
(8) Initially when 120591 is small the values of Nusselt andSherwood numbers start with 1 corresponding to theconduction state However as 120591 increases Nu andSh also increase thus increasing the heat and masstransfer
(9) The values of Nu and Sh for LBMO are greater thanthose in IPM but smaller than those in OPM
(10) The effect of magnetic field is to stabilize the system
Nomenclature
Latin Symbols
119860 Amplitude of convection119889 Depth of the fluid layerg Acceleration due to gravity119876 Chandrasekhar number
119876 = 1205831198981198672
11988711988921205880]]119898
119896119888 Critical wave numberNu(120591) Nusselt numberSh(120591) Sherwood number119901 Reduced pressurePr Prandtl number Pr = ]120581119879Pm Magnetic Prandtl number Pm = ]119898120581119879Le Lewis number Le = 120581119879120581119878
120596 Frequency of temperature modulation120575 Amplitude of temperature modulation120583 Dynamic coefficient of viscosity of the fluid120583119898 Magnetic permeability] Kinematic viscosity ] = 1205831205880
]119898 Magnetic viscosity120588 Fluid density120595 Stream function120595lowast Dimensionless stream function
Φ Magnetic potentialΦlowast Dimensionless magnetic potential
120591 Slow time 120591 = 1205982119905
120579 Phase angle1198791015840 Perturbed temperature
Other Symbols
nabla2 (12059721205971199092) + (120597
21205971199102) + (12059721205971199112)
14 International Journal of Engineering Mathematics
Subscripts
119887 Basic state119888 Critical0 Reference value
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was done during the lien sanctioned to theauthor by Banaras Hindu University Varanasi to work asprofessor of Mathematics at Department of Applied Math-ematics School for Physical Sciences Babasaheb BhimraoAmbedkar Central University Lucknow India The authorBS Bhadauria gratefully acknowledges Banaras Hindu Uni-versity Varanasi for the same Further the author Palle Kirangratefully acknowledges the financial assistance fromBabasa-heb Bhimrao Ambedkar Central University as a researchfellowship
References
[1] A Akbarzadeh and P Manins ldquoConvective layers generated byside walls in solar pondsrdquo Solar Energy vol 41 no 6 pp 521ndash529 1988
[2] H E Huppert and R S J Sparks ldquoDouble-diffusive convectiondue to crystallization in magmasrdquo Annual Review of Earth andPlanetary Sciences vol 12 pp 11ndash37 1984
[3] H J S Fernando and A Brandt ldquoRecent advances in doublediffusive convectionrdquoAppliedMechanics Reviews vol 47 pp c1ndashc7 1994
[4] J S Turner Buoyancy Effects in Fluids University Of Cam-bridge 1979
[5] N Rudraiah and I S Shivakumara ldquoDouble-diffusive convec-tion with an imposed magnetic fieldrdquo International Journal ofHeat and Mass Transfer vol 27 no 10 pp 1825ndash1836 1984
[6] J H Thomas and N O Weiss ldquoThe theory of sunspotsrdquo inSunspots Theory and Observations p 428 Kluwer AcademicPublishers Dordrecht The Netherlands 1992
[7] W B Thompson ldquoThermal convection in a magnetic fieldrdquoPhilosophical Magazine vol 42 pp 1417ndash1432 1951
[8] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityOxford University Press London UK 1961
[9] D Lortz ldquoA stability criterion for steady finite amplitudeconvection with an external magnetic fieldrdquo Journal of FluidMechanics vol 23 pp 113ndash128 1965
[10] W V R Malkus and G Veronis ldquoFinite amplitude cellularconvectionrdquo Journal of Fluid Mechanics vol 4 pp 225ndash2601958
[11] H Stommel A B Arons and D Blanchard ldquoAn oceanograph-ical curiosity the perpetual salt fountainrdquo Deep Sea Researchvol 3 no 2 pp 152ndash153 1956
[12] K Gotoh and M Yamada ldquoThermal convection of a horizontallayer of magnetic fluidsrdquo Journal of the Physical Society of Japanvol 51 no 9 pp 3042ndash3048 1982
[13] GMOreper and J Szekely ldquoThe effect of an externally imposedmagnetic field on buoyancy driven flow in a rectangular cavityrdquoJournal of Crystal Growth vol 64 no 3 pp 505ndash515 1983
[14] N Rudraiah ldquoDouble-diffusive magnetoconvectionrdquo Pramanavol 27 no 1-2 pp 233ndash266 1986
[15] P G Siddheshwar and S Pranesh ldquoMagnetoconvection in fluidswith suspended particles under 1g and 120583grdquo Aerospace Scienceand Technology vol 6 no 2 pp 105ndash114 2002
[16] B S Bhadauria ldquoCombined effect of temperature modulationand magnetic field on the onset of convection in an electricallyconducting-fluid-saturated porous mediumrdquo Journal of HeatTransfer vol 130 no 5 Article ID 052601 2008
[17] P G Siddheshwar B S Bhadauria P Mishra and A K Srivas-tava ldquoStudy of heat transport by stationarymagneto-convectionin aNewtonian liquid under temperature or gravitymodulationusing Ginzburg-Landau modelrdquo International Journal of Non-Linear Mechanics vol 47 no 5 pp 418ndash425 2012
[18] P G Drazin and W H Reid Hydrodynamic Stability Cam-bridge University Press Cambridge UK 2004
[19] G Venezian ldquoEffect of modulation on the onset of thermalconvectionrdquo Journal of Fluid Mechanics vol 35 no 2 pp 243ndash254 1969
[20] S Rosenblat and G A Tanaka ldquoModulation of thermal convec-tion instabilityrdquo Physics of Fluids vol 14 no 7 pp 1319ndash13221971
[21] M N Roppo S H Davis and S Rosenblat ldquoBenard convectionwith time-periodic heatingrdquoThe Physics of Fluids vol 27 no 4pp 796ndash803 1984
[22] B S Bhadauria and P K Bhatia ldquoTime-periodic heating ofRayleigh-Benard convectionrdquo Physica Scripta vol 66 no 1 pp59ndash65 2002
[23] B S Bhadauria ldquoTime-periodic heating of Rayleigh-Benardconvection in a vertical magnetic fieldrdquo Physica Scripta vol 73no 3 pp 296ndash302 2006
[24] M S Malashetty andM Swamy ldquoEffect of thermal modulationon the onset of convection in a rotating fluid layerrdquo InternationalJournal of Heat and Mass Transfer vol 51 no 11-12 pp 2814ndash2823 2008
[25] B S Bhadauria and P Kiran ldquoHeat transport in an anisotropicporous medium saturated with variable viscosity liquid undertemperature modulationrdquo Transport in Porous Media vol 100no 2 pp 279ndash295 2013
The effect of temperature modulation on weak nonlineardouble-diffusive magnetoconvection has been analyzed byusing the nonautonomous Ginzburg-Landau equation Thefollowing conclusions are drawn from previous analysis
(1) The effect of IPM is negligible on heat and masstransport in the system
(2) In the case of IPM the effect of 120575 and 120596 is also foundto be negligible on heat and mass transport
(3) In the case of IPM the values of Nu and Sh increasesteadily for small values of time 120591 however Nu and Shbecome constant when 120591 is large
(4) The effect of increasing Pr Le Pm Ras is found toincrease in Nu and Sh thus increasing heat and masstransfer for all three types of modulations
(5) The effect of increasing 120575 is to increase the value of Nuand Sh for the case of OPM and LBMO hence heatand mass transfer
(6) The effect of increasing 120596 is to decrease the value ofNu and Sh for the case ofOPMand LBMO hence heatand mass transfer
(7) In the cases of OPM and LBMO the natures of Nuand Sh remain oscillatory
(8) Initially when 120591 is small the values of Nusselt andSherwood numbers start with 1 corresponding to theconduction state However as 120591 increases Nu andSh also increase thus increasing the heat and masstransfer
(9) The values of Nu and Sh for LBMO are greater thanthose in IPM but smaller than those in OPM
(10) The effect of magnetic field is to stabilize the system
Nomenclature
Latin Symbols
119860 Amplitude of convection119889 Depth of the fluid layerg Acceleration due to gravity119876 Chandrasekhar number
119876 = 1205831198981198672
11988711988921205880]]119898
119896119888 Critical wave numberNu(120591) Nusselt numberSh(120591) Sherwood number119901 Reduced pressurePr Prandtl number Pr = ]120581119879Pm Magnetic Prandtl number Pm = ]119898120581119879Le Lewis number Le = 120581119879120581119878
120596 Frequency of temperature modulation120575 Amplitude of temperature modulation120583 Dynamic coefficient of viscosity of the fluid120583119898 Magnetic permeability] Kinematic viscosity ] = 1205831205880
]119898 Magnetic viscosity120588 Fluid density120595 Stream function120595lowast Dimensionless stream function
Φ Magnetic potentialΦlowast Dimensionless magnetic potential
120591 Slow time 120591 = 1205982119905
120579 Phase angle1198791015840 Perturbed temperature
Other Symbols
nabla2 (12059721205971199092) + (120597
21205971199102) + (12059721205971199112)
14 International Journal of Engineering Mathematics
Subscripts
119887 Basic state119888 Critical0 Reference value
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was done during the lien sanctioned to theauthor by Banaras Hindu University Varanasi to work asprofessor of Mathematics at Department of Applied Math-ematics School for Physical Sciences Babasaheb BhimraoAmbedkar Central University Lucknow India The authorBS Bhadauria gratefully acknowledges Banaras Hindu Uni-versity Varanasi for the same Further the author Palle Kirangratefully acknowledges the financial assistance fromBabasa-heb Bhimrao Ambedkar Central University as a researchfellowship
References
[1] A Akbarzadeh and P Manins ldquoConvective layers generated byside walls in solar pondsrdquo Solar Energy vol 41 no 6 pp 521ndash529 1988
[2] H E Huppert and R S J Sparks ldquoDouble-diffusive convectiondue to crystallization in magmasrdquo Annual Review of Earth andPlanetary Sciences vol 12 pp 11ndash37 1984
[3] H J S Fernando and A Brandt ldquoRecent advances in doublediffusive convectionrdquoAppliedMechanics Reviews vol 47 pp c1ndashc7 1994
[4] J S Turner Buoyancy Effects in Fluids University Of Cam-bridge 1979
[5] N Rudraiah and I S Shivakumara ldquoDouble-diffusive convec-tion with an imposed magnetic fieldrdquo International Journal ofHeat and Mass Transfer vol 27 no 10 pp 1825ndash1836 1984
[6] J H Thomas and N O Weiss ldquoThe theory of sunspotsrdquo inSunspots Theory and Observations p 428 Kluwer AcademicPublishers Dordrecht The Netherlands 1992
[7] W B Thompson ldquoThermal convection in a magnetic fieldrdquoPhilosophical Magazine vol 42 pp 1417ndash1432 1951
[8] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityOxford University Press London UK 1961
[9] D Lortz ldquoA stability criterion for steady finite amplitudeconvection with an external magnetic fieldrdquo Journal of FluidMechanics vol 23 pp 113ndash128 1965
[10] W V R Malkus and G Veronis ldquoFinite amplitude cellularconvectionrdquo Journal of Fluid Mechanics vol 4 pp 225ndash2601958
[11] H Stommel A B Arons and D Blanchard ldquoAn oceanograph-ical curiosity the perpetual salt fountainrdquo Deep Sea Researchvol 3 no 2 pp 152ndash153 1956
[12] K Gotoh and M Yamada ldquoThermal convection of a horizontallayer of magnetic fluidsrdquo Journal of the Physical Society of Japanvol 51 no 9 pp 3042ndash3048 1982
[13] GMOreper and J Szekely ldquoThe effect of an externally imposedmagnetic field on buoyancy driven flow in a rectangular cavityrdquoJournal of Crystal Growth vol 64 no 3 pp 505ndash515 1983
[14] N Rudraiah ldquoDouble-diffusive magnetoconvectionrdquo Pramanavol 27 no 1-2 pp 233ndash266 1986
[15] P G Siddheshwar and S Pranesh ldquoMagnetoconvection in fluidswith suspended particles under 1g and 120583grdquo Aerospace Scienceand Technology vol 6 no 2 pp 105ndash114 2002
[16] B S Bhadauria ldquoCombined effect of temperature modulationand magnetic field on the onset of convection in an electricallyconducting-fluid-saturated porous mediumrdquo Journal of HeatTransfer vol 130 no 5 Article ID 052601 2008
[17] P G Siddheshwar B S Bhadauria P Mishra and A K Srivas-tava ldquoStudy of heat transport by stationarymagneto-convectionin aNewtonian liquid under temperature or gravitymodulationusing Ginzburg-Landau modelrdquo International Journal of Non-Linear Mechanics vol 47 no 5 pp 418ndash425 2012
[18] P G Drazin and W H Reid Hydrodynamic Stability Cam-bridge University Press Cambridge UK 2004
[19] G Venezian ldquoEffect of modulation on the onset of thermalconvectionrdquo Journal of Fluid Mechanics vol 35 no 2 pp 243ndash254 1969
[20] S Rosenblat and G A Tanaka ldquoModulation of thermal convec-tion instabilityrdquo Physics of Fluids vol 14 no 7 pp 1319ndash13221971
[21] M N Roppo S H Davis and S Rosenblat ldquoBenard convectionwith time-periodic heatingrdquoThe Physics of Fluids vol 27 no 4pp 796ndash803 1984
[22] B S Bhadauria and P K Bhatia ldquoTime-periodic heating ofRayleigh-Benard convectionrdquo Physica Scripta vol 66 no 1 pp59ndash65 2002
[23] B S Bhadauria ldquoTime-periodic heating of Rayleigh-Benardconvection in a vertical magnetic fieldrdquo Physica Scripta vol 73no 3 pp 296ndash302 2006
[24] M S Malashetty andM Swamy ldquoEffect of thermal modulationon the onset of convection in a rotating fluid layerrdquo InternationalJournal of Heat and Mass Transfer vol 51 no 11-12 pp 2814ndash2823 2008
[25] B S Bhadauria and P Kiran ldquoHeat transport in an anisotropicporous medium saturated with variable viscosity liquid undertemperature modulationrdquo Transport in Porous Media vol 100no 2 pp 279ndash295 2013
The effect of temperature modulation on weak nonlineardouble-diffusive magnetoconvection has been analyzed byusing the nonautonomous Ginzburg-Landau equation Thefollowing conclusions are drawn from previous analysis
(1) The effect of IPM is negligible on heat and masstransport in the system
(2) In the case of IPM the effect of 120575 and 120596 is also foundto be negligible on heat and mass transport
(3) In the case of IPM the values of Nu and Sh increasesteadily for small values of time 120591 however Nu and Shbecome constant when 120591 is large
(4) The effect of increasing Pr Le Pm Ras is found toincrease in Nu and Sh thus increasing heat and masstransfer for all three types of modulations
(5) The effect of increasing 120575 is to increase the value of Nuand Sh for the case of OPM and LBMO hence heatand mass transfer
(6) The effect of increasing 120596 is to decrease the value ofNu and Sh for the case ofOPMand LBMO hence heatand mass transfer
(7) In the cases of OPM and LBMO the natures of Nuand Sh remain oscillatory
(8) Initially when 120591 is small the values of Nusselt andSherwood numbers start with 1 corresponding to theconduction state However as 120591 increases Nu andSh also increase thus increasing the heat and masstransfer
(9) The values of Nu and Sh for LBMO are greater thanthose in IPM but smaller than those in OPM
(10) The effect of magnetic field is to stabilize the system
Nomenclature
Latin Symbols
119860 Amplitude of convection119889 Depth of the fluid layerg Acceleration due to gravity119876 Chandrasekhar number
119876 = 1205831198981198672
11988711988921205880]]119898
119896119888 Critical wave numberNu(120591) Nusselt numberSh(120591) Sherwood number119901 Reduced pressurePr Prandtl number Pr = ]120581119879Pm Magnetic Prandtl number Pm = ]119898120581119879Le Lewis number Le = 120581119879120581119878
120596 Frequency of temperature modulation120575 Amplitude of temperature modulation120583 Dynamic coefficient of viscosity of the fluid120583119898 Magnetic permeability] Kinematic viscosity ] = 1205831205880
]119898 Magnetic viscosity120588 Fluid density120595 Stream function120595lowast Dimensionless stream function
Φ Magnetic potentialΦlowast Dimensionless magnetic potential
120591 Slow time 120591 = 1205982119905
120579 Phase angle1198791015840 Perturbed temperature
Other Symbols
nabla2 (12059721205971199092) + (120597
21205971199102) + (12059721205971199112)
14 International Journal of Engineering Mathematics
Subscripts
119887 Basic state119888 Critical0 Reference value
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was done during the lien sanctioned to theauthor by Banaras Hindu University Varanasi to work asprofessor of Mathematics at Department of Applied Math-ematics School for Physical Sciences Babasaheb BhimraoAmbedkar Central University Lucknow India The authorBS Bhadauria gratefully acknowledges Banaras Hindu Uni-versity Varanasi for the same Further the author Palle Kirangratefully acknowledges the financial assistance fromBabasa-heb Bhimrao Ambedkar Central University as a researchfellowship
References
[1] A Akbarzadeh and P Manins ldquoConvective layers generated byside walls in solar pondsrdquo Solar Energy vol 41 no 6 pp 521ndash529 1988
[2] H E Huppert and R S J Sparks ldquoDouble-diffusive convectiondue to crystallization in magmasrdquo Annual Review of Earth andPlanetary Sciences vol 12 pp 11ndash37 1984
[3] H J S Fernando and A Brandt ldquoRecent advances in doublediffusive convectionrdquoAppliedMechanics Reviews vol 47 pp c1ndashc7 1994
[4] J S Turner Buoyancy Effects in Fluids University Of Cam-bridge 1979
[5] N Rudraiah and I S Shivakumara ldquoDouble-diffusive convec-tion with an imposed magnetic fieldrdquo International Journal ofHeat and Mass Transfer vol 27 no 10 pp 1825ndash1836 1984
[6] J H Thomas and N O Weiss ldquoThe theory of sunspotsrdquo inSunspots Theory and Observations p 428 Kluwer AcademicPublishers Dordrecht The Netherlands 1992
[7] W B Thompson ldquoThermal convection in a magnetic fieldrdquoPhilosophical Magazine vol 42 pp 1417ndash1432 1951
[8] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityOxford University Press London UK 1961
[9] D Lortz ldquoA stability criterion for steady finite amplitudeconvection with an external magnetic fieldrdquo Journal of FluidMechanics vol 23 pp 113ndash128 1965
[10] W V R Malkus and G Veronis ldquoFinite amplitude cellularconvectionrdquo Journal of Fluid Mechanics vol 4 pp 225ndash2601958
[11] H Stommel A B Arons and D Blanchard ldquoAn oceanograph-ical curiosity the perpetual salt fountainrdquo Deep Sea Researchvol 3 no 2 pp 152ndash153 1956
[12] K Gotoh and M Yamada ldquoThermal convection of a horizontallayer of magnetic fluidsrdquo Journal of the Physical Society of Japanvol 51 no 9 pp 3042ndash3048 1982
[13] GMOreper and J Szekely ldquoThe effect of an externally imposedmagnetic field on buoyancy driven flow in a rectangular cavityrdquoJournal of Crystal Growth vol 64 no 3 pp 505ndash515 1983
[14] N Rudraiah ldquoDouble-diffusive magnetoconvectionrdquo Pramanavol 27 no 1-2 pp 233ndash266 1986
[15] P G Siddheshwar and S Pranesh ldquoMagnetoconvection in fluidswith suspended particles under 1g and 120583grdquo Aerospace Scienceand Technology vol 6 no 2 pp 105ndash114 2002
[16] B S Bhadauria ldquoCombined effect of temperature modulationand magnetic field on the onset of convection in an electricallyconducting-fluid-saturated porous mediumrdquo Journal of HeatTransfer vol 130 no 5 Article ID 052601 2008
[17] P G Siddheshwar B S Bhadauria P Mishra and A K Srivas-tava ldquoStudy of heat transport by stationarymagneto-convectionin aNewtonian liquid under temperature or gravitymodulationusing Ginzburg-Landau modelrdquo International Journal of Non-Linear Mechanics vol 47 no 5 pp 418ndash425 2012
[18] P G Drazin and W H Reid Hydrodynamic Stability Cam-bridge University Press Cambridge UK 2004
[19] G Venezian ldquoEffect of modulation on the onset of thermalconvectionrdquo Journal of Fluid Mechanics vol 35 no 2 pp 243ndash254 1969
[20] S Rosenblat and G A Tanaka ldquoModulation of thermal convec-tion instabilityrdquo Physics of Fluids vol 14 no 7 pp 1319ndash13221971
[21] M N Roppo S H Davis and S Rosenblat ldquoBenard convectionwith time-periodic heatingrdquoThe Physics of Fluids vol 27 no 4pp 796ndash803 1984
[22] B S Bhadauria and P K Bhatia ldquoTime-periodic heating ofRayleigh-Benard convectionrdquo Physica Scripta vol 66 no 1 pp59ndash65 2002
[23] B S Bhadauria ldquoTime-periodic heating of Rayleigh-Benardconvection in a vertical magnetic fieldrdquo Physica Scripta vol 73no 3 pp 296ndash302 2006
[24] M S Malashetty andM Swamy ldquoEffect of thermal modulationon the onset of convection in a rotating fluid layerrdquo InternationalJournal of Heat and Mass Transfer vol 51 no 11-12 pp 2814ndash2823 2008
[25] B S Bhadauria and P Kiran ldquoHeat transport in an anisotropicporous medium saturated with variable viscosity liquid undertemperature modulationrdquo Transport in Porous Media vol 100no 2 pp 279ndash295 2013
The effect of temperature modulation on weak nonlineardouble-diffusive magnetoconvection has been analyzed byusing the nonautonomous Ginzburg-Landau equation Thefollowing conclusions are drawn from previous analysis
(1) The effect of IPM is negligible on heat and masstransport in the system
(2) In the case of IPM the effect of 120575 and 120596 is also foundto be negligible on heat and mass transport
(3) In the case of IPM the values of Nu and Sh increasesteadily for small values of time 120591 however Nu and Shbecome constant when 120591 is large
(4) The effect of increasing Pr Le Pm Ras is found toincrease in Nu and Sh thus increasing heat and masstransfer for all three types of modulations
(5) The effect of increasing 120575 is to increase the value of Nuand Sh for the case of OPM and LBMO hence heatand mass transfer
(6) The effect of increasing 120596 is to decrease the value ofNu and Sh for the case ofOPMand LBMO hence heatand mass transfer
(7) In the cases of OPM and LBMO the natures of Nuand Sh remain oscillatory
(8) Initially when 120591 is small the values of Nusselt andSherwood numbers start with 1 corresponding to theconduction state However as 120591 increases Nu andSh also increase thus increasing the heat and masstransfer
(9) The values of Nu and Sh for LBMO are greater thanthose in IPM but smaller than those in OPM
(10) The effect of magnetic field is to stabilize the system
Nomenclature
Latin Symbols
119860 Amplitude of convection119889 Depth of the fluid layerg Acceleration due to gravity119876 Chandrasekhar number
119876 = 1205831198981198672
11988711988921205880]]119898
119896119888 Critical wave numberNu(120591) Nusselt numberSh(120591) Sherwood number119901 Reduced pressurePr Prandtl number Pr = ]120581119879Pm Magnetic Prandtl number Pm = ]119898120581119879Le Lewis number Le = 120581119879120581119878
120596 Frequency of temperature modulation120575 Amplitude of temperature modulation120583 Dynamic coefficient of viscosity of the fluid120583119898 Magnetic permeability] Kinematic viscosity ] = 1205831205880
]119898 Magnetic viscosity120588 Fluid density120595 Stream function120595lowast Dimensionless stream function
Φ Magnetic potentialΦlowast Dimensionless magnetic potential
120591 Slow time 120591 = 1205982119905
120579 Phase angle1198791015840 Perturbed temperature
Other Symbols
nabla2 (12059721205971199092) + (120597
21205971199102) + (12059721205971199112)
14 International Journal of Engineering Mathematics
Subscripts
119887 Basic state119888 Critical0 Reference value
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was done during the lien sanctioned to theauthor by Banaras Hindu University Varanasi to work asprofessor of Mathematics at Department of Applied Math-ematics School for Physical Sciences Babasaheb BhimraoAmbedkar Central University Lucknow India The authorBS Bhadauria gratefully acknowledges Banaras Hindu Uni-versity Varanasi for the same Further the author Palle Kirangratefully acknowledges the financial assistance fromBabasa-heb Bhimrao Ambedkar Central University as a researchfellowship
References
[1] A Akbarzadeh and P Manins ldquoConvective layers generated byside walls in solar pondsrdquo Solar Energy vol 41 no 6 pp 521ndash529 1988
[2] H E Huppert and R S J Sparks ldquoDouble-diffusive convectiondue to crystallization in magmasrdquo Annual Review of Earth andPlanetary Sciences vol 12 pp 11ndash37 1984
[3] H J S Fernando and A Brandt ldquoRecent advances in doublediffusive convectionrdquoAppliedMechanics Reviews vol 47 pp c1ndashc7 1994
[4] J S Turner Buoyancy Effects in Fluids University Of Cam-bridge 1979
[5] N Rudraiah and I S Shivakumara ldquoDouble-diffusive convec-tion with an imposed magnetic fieldrdquo International Journal ofHeat and Mass Transfer vol 27 no 10 pp 1825ndash1836 1984
[6] J H Thomas and N O Weiss ldquoThe theory of sunspotsrdquo inSunspots Theory and Observations p 428 Kluwer AcademicPublishers Dordrecht The Netherlands 1992
[7] W B Thompson ldquoThermal convection in a magnetic fieldrdquoPhilosophical Magazine vol 42 pp 1417ndash1432 1951
[8] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityOxford University Press London UK 1961
[9] D Lortz ldquoA stability criterion for steady finite amplitudeconvection with an external magnetic fieldrdquo Journal of FluidMechanics vol 23 pp 113ndash128 1965
[10] W V R Malkus and G Veronis ldquoFinite amplitude cellularconvectionrdquo Journal of Fluid Mechanics vol 4 pp 225ndash2601958
[11] H Stommel A B Arons and D Blanchard ldquoAn oceanograph-ical curiosity the perpetual salt fountainrdquo Deep Sea Researchvol 3 no 2 pp 152ndash153 1956
[12] K Gotoh and M Yamada ldquoThermal convection of a horizontallayer of magnetic fluidsrdquo Journal of the Physical Society of Japanvol 51 no 9 pp 3042ndash3048 1982
[13] GMOreper and J Szekely ldquoThe effect of an externally imposedmagnetic field on buoyancy driven flow in a rectangular cavityrdquoJournal of Crystal Growth vol 64 no 3 pp 505ndash515 1983
[14] N Rudraiah ldquoDouble-diffusive magnetoconvectionrdquo Pramanavol 27 no 1-2 pp 233ndash266 1986
[15] P G Siddheshwar and S Pranesh ldquoMagnetoconvection in fluidswith suspended particles under 1g and 120583grdquo Aerospace Scienceand Technology vol 6 no 2 pp 105ndash114 2002
[16] B S Bhadauria ldquoCombined effect of temperature modulationand magnetic field on the onset of convection in an electricallyconducting-fluid-saturated porous mediumrdquo Journal of HeatTransfer vol 130 no 5 Article ID 052601 2008
[17] P G Siddheshwar B S Bhadauria P Mishra and A K Srivas-tava ldquoStudy of heat transport by stationarymagneto-convectionin aNewtonian liquid under temperature or gravitymodulationusing Ginzburg-Landau modelrdquo International Journal of Non-Linear Mechanics vol 47 no 5 pp 418ndash425 2012
[18] P G Drazin and W H Reid Hydrodynamic Stability Cam-bridge University Press Cambridge UK 2004
[19] G Venezian ldquoEffect of modulation on the onset of thermalconvectionrdquo Journal of Fluid Mechanics vol 35 no 2 pp 243ndash254 1969
[20] S Rosenblat and G A Tanaka ldquoModulation of thermal convec-tion instabilityrdquo Physics of Fluids vol 14 no 7 pp 1319ndash13221971
[21] M N Roppo S H Davis and S Rosenblat ldquoBenard convectionwith time-periodic heatingrdquoThe Physics of Fluids vol 27 no 4pp 796ndash803 1984
[22] B S Bhadauria and P K Bhatia ldquoTime-periodic heating ofRayleigh-Benard convectionrdquo Physica Scripta vol 66 no 1 pp59ndash65 2002
[23] B S Bhadauria ldquoTime-periodic heating of Rayleigh-Benardconvection in a vertical magnetic fieldrdquo Physica Scripta vol 73no 3 pp 296ndash302 2006
[24] M S Malashetty andM Swamy ldquoEffect of thermal modulationon the onset of convection in a rotating fluid layerrdquo InternationalJournal of Heat and Mass Transfer vol 51 no 11-12 pp 2814ndash2823 2008
[25] B S Bhadauria and P Kiran ldquoHeat transport in an anisotropicporous medium saturated with variable viscosity liquid undertemperature modulationrdquo Transport in Porous Media vol 100no 2 pp 279ndash295 2013
The effect of temperature modulation on weak nonlineardouble-diffusive magnetoconvection has been analyzed byusing the nonautonomous Ginzburg-Landau equation Thefollowing conclusions are drawn from previous analysis
(1) The effect of IPM is negligible on heat and masstransport in the system
(2) In the case of IPM the effect of 120575 and 120596 is also foundto be negligible on heat and mass transport
(3) In the case of IPM the values of Nu and Sh increasesteadily for small values of time 120591 however Nu and Shbecome constant when 120591 is large
(4) The effect of increasing Pr Le Pm Ras is found toincrease in Nu and Sh thus increasing heat and masstransfer for all three types of modulations
(5) The effect of increasing 120575 is to increase the value of Nuand Sh for the case of OPM and LBMO hence heatand mass transfer
(6) The effect of increasing 120596 is to decrease the value ofNu and Sh for the case ofOPMand LBMO hence heatand mass transfer
(7) In the cases of OPM and LBMO the natures of Nuand Sh remain oscillatory
(8) Initially when 120591 is small the values of Nusselt andSherwood numbers start with 1 corresponding to theconduction state However as 120591 increases Nu andSh also increase thus increasing the heat and masstransfer
(9) The values of Nu and Sh for LBMO are greater thanthose in IPM but smaller than those in OPM
(10) The effect of magnetic field is to stabilize the system
Nomenclature
Latin Symbols
119860 Amplitude of convection119889 Depth of the fluid layerg Acceleration due to gravity119876 Chandrasekhar number
119876 = 1205831198981198672
11988711988921205880]]119898
119896119888 Critical wave numberNu(120591) Nusselt numberSh(120591) Sherwood number119901 Reduced pressurePr Prandtl number Pr = ]120581119879Pm Magnetic Prandtl number Pm = ]119898120581119879Le Lewis number Le = 120581119879120581119878
120596 Frequency of temperature modulation120575 Amplitude of temperature modulation120583 Dynamic coefficient of viscosity of the fluid120583119898 Magnetic permeability] Kinematic viscosity ] = 1205831205880
]119898 Magnetic viscosity120588 Fluid density120595 Stream function120595lowast Dimensionless stream function
Φ Magnetic potentialΦlowast Dimensionless magnetic potential
120591 Slow time 120591 = 1205982119905
120579 Phase angle1198791015840 Perturbed temperature
Other Symbols
nabla2 (12059721205971199092) + (120597
21205971199102) + (12059721205971199112)
14 International Journal of Engineering Mathematics
Subscripts
119887 Basic state119888 Critical0 Reference value
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was done during the lien sanctioned to theauthor by Banaras Hindu University Varanasi to work asprofessor of Mathematics at Department of Applied Math-ematics School for Physical Sciences Babasaheb BhimraoAmbedkar Central University Lucknow India The authorBS Bhadauria gratefully acknowledges Banaras Hindu Uni-versity Varanasi for the same Further the author Palle Kirangratefully acknowledges the financial assistance fromBabasa-heb Bhimrao Ambedkar Central University as a researchfellowship
References
[1] A Akbarzadeh and P Manins ldquoConvective layers generated byside walls in solar pondsrdquo Solar Energy vol 41 no 6 pp 521ndash529 1988
[2] H E Huppert and R S J Sparks ldquoDouble-diffusive convectiondue to crystallization in magmasrdquo Annual Review of Earth andPlanetary Sciences vol 12 pp 11ndash37 1984
[3] H J S Fernando and A Brandt ldquoRecent advances in doublediffusive convectionrdquoAppliedMechanics Reviews vol 47 pp c1ndashc7 1994
[4] J S Turner Buoyancy Effects in Fluids University Of Cam-bridge 1979
[5] N Rudraiah and I S Shivakumara ldquoDouble-diffusive convec-tion with an imposed magnetic fieldrdquo International Journal ofHeat and Mass Transfer vol 27 no 10 pp 1825ndash1836 1984
[6] J H Thomas and N O Weiss ldquoThe theory of sunspotsrdquo inSunspots Theory and Observations p 428 Kluwer AcademicPublishers Dordrecht The Netherlands 1992
[7] W B Thompson ldquoThermal convection in a magnetic fieldrdquoPhilosophical Magazine vol 42 pp 1417ndash1432 1951
[8] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityOxford University Press London UK 1961
[9] D Lortz ldquoA stability criterion for steady finite amplitudeconvection with an external magnetic fieldrdquo Journal of FluidMechanics vol 23 pp 113ndash128 1965
[10] W V R Malkus and G Veronis ldquoFinite amplitude cellularconvectionrdquo Journal of Fluid Mechanics vol 4 pp 225ndash2601958
[11] H Stommel A B Arons and D Blanchard ldquoAn oceanograph-ical curiosity the perpetual salt fountainrdquo Deep Sea Researchvol 3 no 2 pp 152ndash153 1956
[12] K Gotoh and M Yamada ldquoThermal convection of a horizontallayer of magnetic fluidsrdquo Journal of the Physical Society of Japanvol 51 no 9 pp 3042ndash3048 1982
[13] GMOreper and J Szekely ldquoThe effect of an externally imposedmagnetic field on buoyancy driven flow in a rectangular cavityrdquoJournal of Crystal Growth vol 64 no 3 pp 505ndash515 1983
[14] N Rudraiah ldquoDouble-diffusive magnetoconvectionrdquo Pramanavol 27 no 1-2 pp 233ndash266 1986
[15] P G Siddheshwar and S Pranesh ldquoMagnetoconvection in fluidswith suspended particles under 1g and 120583grdquo Aerospace Scienceand Technology vol 6 no 2 pp 105ndash114 2002
[16] B S Bhadauria ldquoCombined effect of temperature modulationand magnetic field on the onset of convection in an electricallyconducting-fluid-saturated porous mediumrdquo Journal of HeatTransfer vol 130 no 5 Article ID 052601 2008
[17] P G Siddheshwar B S Bhadauria P Mishra and A K Srivas-tava ldquoStudy of heat transport by stationarymagneto-convectionin aNewtonian liquid under temperature or gravitymodulationusing Ginzburg-Landau modelrdquo International Journal of Non-Linear Mechanics vol 47 no 5 pp 418ndash425 2012
[18] P G Drazin and W H Reid Hydrodynamic Stability Cam-bridge University Press Cambridge UK 2004
[19] G Venezian ldquoEffect of modulation on the onset of thermalconvectionrdquo Journal of Fluid Mechanics vol 35 no 2 pp 243ndash254 1969
[20] S Rosenblat and G A Tanaka ldquoModulation of thermal convec-tion instabilityrdquo Physics of Fluids vol 14 no 7 pp 1319ndash13221971
[21] M N Roppo S H Davis and S Rosenblat ldquoBenard convectionwith time-periodic heatingrdquoThe Physics of Fluids vol 27 no 4pp 796ndash803 1984
[22] B S Bhadauria and P K Bhatia ldquoTime-periodic heating ofRayleigh-Benard convectionrdquo Physica Scripta vol 66 no 1 pp59ndash65 2002
[23] B S Bhadauria ldquoTime-periodic heating of Rayleigh-Benardconvection in a vertical magnetic fieldrdquo Physica Scripta vol 73no 3 pp 296ndash302 2006
[24] M S Malashetty andM Swamy ldquoEffect of thermal modulationon the onset of convection in a rotating fluid layerrdquo InternationalJournal of Heat and Mass Transfer vol 51 no 11-12 pp 2814ndash2823 2008
[25] B S Bhadauria and P Kiran ldquoHeat transport in an anisotropicporous medium saturated with variable viscosity liquid undertemperature modulationrdquo Transport in Porous Media vol 100no 2 pp 279ndash295 2013
The effect of temperature modulation on weak nonlineardouble-diffusive magnetoconvection has been analyzed byusing the nonautonomous Ginzburg-Landau equation Thefollowing conclusions are drawn from previous analysis
(1) The effect of IPM is negligible on heat and masstransport in the system
(2) In the case of IPM the effect of 120575 and 120596 is also foundto be negligible on heat and mass transport
(3) In the case of IPM the values of Nu and Sh increasesteadily for small values of time 120591 however Nu and Shbecome constant when 120591 is large
(4) The effect of increasing Pr Le Pm Ras is found toincrease in Nu and Sh thus increasing heat and masstransfer for all three types of modulations
(5) The effect of increasing 120575 is to increase the value of Nuand Sh for the case of OPM and LBMO hence heatand mass transfer
(6) The effect of increasing 120596 is to decrease the value ofNu and Sh for the case ofOPMand LBMO hence heatand mass transfer
(7) In the cases of OPM and LBMO the natures of Nuand Sh remain oscillatory
(8) Initially when 120591 is small the values of Nusselt andSherwood numbers start with 1 corresponding to theconduction state However as 120591 increases Nu andSh also increase thus increasing the heat and masstransfer
(9) The values of Nu and Sh for LBMO are greater thanthose in IPM but smaller than those in OPM
(10) The effect of magnetic field is to stabilize the system
Nomenclature
Latin Symbols
119860 Amplitude of convection119889 Depth of the fluid layerg Acceleration due to gravity119876 Chandrasekhar number
119876 = 1205831198981198672
11988711988921205880]]119898
119896119888 Critical wave numberNu(120591) Nusselt numberSh(120591) Sherwood number119901 Reduced pressurePr Prandtl number Pr = ]120581119879Pm Magnetic Prandtl number Pm = ]119898120581119879Le Lewis number Le = 120581119879120581119878
120596 Frequency of temperature modulation120575 Amplitude of temperature modulation120583 Dynamic coefficient of viscosity of the fluid120583119898 Magnetic permeability] Kinematic viscosity ] = 1205831205880
]119898 Magnetic viscosity120588 Fluid density120595 Stream function120595lowast Dimensionless stream function
Φ Magnetic potentialΦlowast Dimensionless magnetic potential
120591 Slow time 120591 = 1205982119905
120579 Phase angle1198791015840 Perturbed temperature
Other Symbols
nabla2 (12059721205971199092) + (120597
21205971199102) + (12059721205971199112)
14 International Journal of Engineering Mathematics
Subscripts
119887 Basic state119888 Critical0 Reference value
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was done during the lien sanctioned to theauthor by Banaras Hindu University Varanasi to work asprofessor of Mathematics at Department of Applied Math-ematics School for Physical Sciences Babasaheb BhimraoAmbedkar Central University Lucknow India The authorBS Bhadauria gratefully acknowledges Banaras Hindu Uni-versity Varanasi for the same Further the author Palle Kirangratefully acknowledges the financial assistance fromBabasa-heb Bhimrao Ambedkar Central University as a researchfellowship
References
[1] A Akbarzadeh and P Manins ldquoConvective layers generated byside walls in solar pondsrdquo Solar Energy vol 41 no 6 pp 521ndash529 1988
[2] H E Huppert and R S J Sparks ldquoDouble-diffusive convectiondue to crystallization in magmasrdquo Annual Review of Earth andPlanetary Sciences vol 12 pp 11ndash37 1984
[3] H J S Fernando and A Brandt ldquoRecent advances in doublediffusive convectionrdquoAppliedMechanics Reviews vol 47 pp c1ndashc7 1994
[4] J S Turner Buoyancy Effects in Fluids University Of Cam-bridge 1979
[5] N Rudraiah and I S Shivakumara ldquoDouble-diffusive convec-tion with an imposed magnetic fieldrdquo International Journal ofHeat and Mass Transfer vol 27 no 10 pp 1825ndash1836 1984
[6] J H Thomas and N O Weiss ldquoThe theory of sunspotsrdquo inSunspots Theory and Observations p 428 Kluwer AcademicPublishers Dordrecht The Netherlands 1992
[7] W B Thompson ldquoThermal convection in a magnetic fieldrdquoPhilosophical Magazine vol 42 pp 1417ndash1432 1951
[8] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityOxford University Press London UK 1961
[9] D Lortz ldquoA stability criterion for steady finite amplitudeconvection with an external magnetic fieldrdquo Journal of FluidMechanics vol 23 pp 113ndash128 1965
[10] W V R Malkus and G Veronis ldquoFinite amplitude cellularconvectionrdquo Journal of Fluid Mechanics vol 4 pp 225ndash2601958
[11] H Stommel A B Arons and D Blanchard ldquoAn oceanograph-ical curiosity the perpetual salt fountainrdquo Deep Sea Researchvol 3 no 2 pp 152ndash153 1956
[12] K Gotoh and M Yamada ldquoThermal convection of a horizontallayer of magnetic fluidsrdquo Journal of the Physical Society of Japanvol 51 no 9 pp 3042ndash3048 1982
[13] GMOreper and J Szekely ldquoThe effect of an externally imposedmagnetic field on buoyancy driven flow in a rectangular cavityrdquoJournal of Crystal Growth vol 64 no 3 pp 505ndash515 1983
[14] N Rudraiah ldquoDouble-diffusive magnetoconvectionrdquo Pramanavol 27 no 1-2 pp 233ndash266 1986
[15] P G Siddheshwar and S Pranesh ldquoMagnetoconvection in fluidswith suspended particles under 1g and 120583grdquo Aerospace Scienceand Technology vol 6 no 2 pp 105ndash114 2002
[16] B S Bhadauria ldquoCombined effect of temperature modulationand magnetic field on the onset of convection in an electricallyconducting-fluid-saturated porous mediumrdquo Journal of HeatTransfer vol 130 no 5 Article ID 052601 2008
[17] P G Siddheshwar B S Bhadauria P Mishra and A K Srivas-tava ldquoStudy of heat transport by stationarymagneto-convectionin aNewtonian liquid under temperature or gravitymodulationusing Ginzburg-Landau modelrdquo International Journal of Non-Linear Mechanics vol 47 no 5 pp 418ndash425 2012
[18] P G Drazin and W H Reid Hydrodynamic Stability Cam-bridge University Press Cambridge UK 2004
[19] G Venezian ldquoEffect of modulation on the onset of thermalconvectionrdquo Journal of Fluid Mechanics vol 35 no 2 pp 243ndash254 1969
[20] S Rosenblat and G A Tanaka ldquoModulation of thermal convec-tion instabilityrdquo Physics of Fluids vol 14 no 7 pp 1319ndash13221971
[21] M N Roppo S H Davis and S Rosenblat ldquoBenard convectionwith time-periodic heatingrdquoThe Physics of Fluids vol 27 no 4pp 796ndash803 1984
[22] B S Bhadauria and P K Bhatia ldquoTime-periodic heating ofRayleigh-Benard convectionrdquo Physica Scripta vol 66 no 1 pp59ndash65 2002
[23] B S Bhadauria ldquoTime-periodic heating of Rayleigh-Benardconvection in a vertical magnetic fieldrdquo Physica Scripta vol 73no 3 pp 296ndash302 2006
[24] M S Malashetty andM Swamy ldquoEffect of thermal modulationon the onset of convection in a rotating fluid layerrdquo InternationalJournal of Heat and Mass Transfer vol 51 no 11-12 pp 2814ndash2823 2008
[25] B S Bhadauria and P Kiran ldquoHeat transport in an anisotropicporous medium saturated with variable viscosity liquid undertemperature modulationrdquo Transport in Porous Media vol 100no 2 pp 279ndash295 2013
The effect of temperature modulation on weak nonlineardouble-diffusive magnetoconvection has been analyzed byusing the nonautonomous Ginzburg-Landau equation Thefollowing conclusions are drawn from previous analysis
(1) The effect of IPM is negligible on heat and masstransport in the system
(2) In the case of IPM the effect of 120575 and 120596 is also foundto be negligible on heat and mass transport
(3) In the case of IPM the values of Nu and Sh increasesteadily for small values of time 120591 however Nu and Shbecome constant when 120591 is large
(4) The effect of increasing Pr Le Pm Ras is found toincrease in Nu and Sh thus increasing heat and masstransfer for all three types of modulations
(5) The effect of increasing 120575 is to increase the value of Nuand Sh for the case of OPM and LBMO hence heatand mass transfer
(6) The effect of increasing 120596 is to decrease the value ofNu and Sh for the case ofOPMand LBMO hence heatand mass transfer
(7) In the cases of OPM and LBMO the natures of Nuand Sh remain oscillatory
(8) Initially when 120591 is small the values of Nusselt andSherwood numbers start with 1 corresponding to theconduction state However as 120591 increases Nu andSh also increase thus increasing the heat and masstransfer
(9) The values of Nu and Sh for LBMO are greater thanthose in IPM but smaller than those in OPM
(10) The effect of magnetic field is to stabilize the system
Nomenclature
Latin Symbols
119860 Amplitude of convection119889 Depth of the fluid layerg Acceleration due to gravity119876 Chandrasekhar number
119876 = 1205831198981198672
11988711988921205880]]119898
119896119888 Critical wave numberNu(120591) Nusselt numberSh(120591) Sherwood number119901 Reduced pressurePr Prandtl number Pr = ]120581119879Pm Magnetic Prandtl number Pm = ]119898120581119879Le Lewis number Le = 120581119879120581119878
120596 Frequency of temperature modulation120575 Amplitude of temperature modulation120583 Dynamic coefficient of viscosity of the fluid120583119898 Magnetic permeability] Kinematic viscosity ] = 1205831205880
]119898 Magnetic viscosity120588 Fluid density120595 Stream function120595lowast Dimensionless stream function
Φ Magnetic potentialΦlowast Dimensionless magnetic potential
120591 Slow time 120591 = 1205982119905
120579 Phase angle1198791015840 Perturbed temperature
Other Symbols
nabla2 (12059721205971199092) + (120597
21205971199102) + (12059721205971199112)
14 International Journal of Engineering Mathematics
Subscripts
119887 Basic state119888 Critical0 Reference value
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was done during the lien sanctioned to theauthor by Banaras Hindu University Varanasi to work asprofessor of Mathematics at Department of Applied Math-ematics School for Physical Sciences Babasaheb BhimraoAmbedkar Central University Lucknow India The authorBS Bhadauria gratefully acknowledges Banaras Hindu Uni-versity Varanasi for the same Further the author Palle Kirangratefully acknowledges the financial assistance fromBabasa-heb Bhimrao Ambedkar Central University as a researchfellowship
References
[1] A Akbarzadeh and P Manins ldquoConvective layers generated byside walls in solar pondsrdquo Solar Energy vol 41 no 6 pp 521ndash529 1988
[2] H E Huppert and R S J Sparks ldquoDouble-diffusive convectiondue to crystallization in magmasrdquo Annual Review of Earth andPlanetary Sciences vol 12 pp 11ndash37 1984
[3] H J S Fernando and A Brandt ldquoRecent advances in doublediffusive convectionrdquoAppliedMechanics Reviews vol 47 pp c1ndashc7 1994
[4] J S Turner Buoyancy Effects in Fluids University Of Cam-bridge 1979
[5] N Rudraiah and I S Shivakumara ldquoDouble-diffusive convec-tion with an imposed magnetic fieldrdquo International Journal ofHeat and Mass Transfer vol 27 no 10 pp 1825ndash1836 1984
[6] J H Thomas and N O Weiss ldquoThe theory of sunspotsrdquo inSunspots Theory and Observations p 428 Kluwer AcademicPublishers Dordrecht The Netherlands 1992
[7] W B Thompson ldquoThermal convection in a magnetic fieldrdquoPhilosophical Magazine vol 42 pp 1417ndash1432 1951
[8] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityOxford University Press London UK 1961
[9] D Lortz ldquoA stability criterion for steady finite amplitudeconvection with an external magnetic fieldrdquo Journal of FluidMechanics vol 23 pp 113ndash128 1965
[10] W V R Malkus and G Veronis ldquoFinite amplitude cellularconvectionrdquo Journal of Fluid Mechanics vol 4 pp 225ndash2601958
[11] H Stommel A B Arons and D Blanchard ldquoAn oceanograph-ical curiosity the perpetual salt fountainrdquo Deep Sea Researchvol 3 no 2 pp 152ndash153 1956
[12] K Gotoh and M Yamada ldquoThermal convection of a horizontallayer of magnetic fluidsrdquo Journal of the Physical Society of Japanvol 51 no 9 pp 3042ndash3048 1982
[13] GMOreper and J Szekely ldquoThe effect of an externally imposedmagnetic field on buoyancy driven flow in a rectangular cavityrdquoJournal of Crystal Growth vol 64 no 3 pp 505ndash515 1983
[14] N Rudraiah ldquoDouble-diffusive magnetoconvectionrdquo Pramanavol 27 no 1-2 pp 233ndash266 1986
[15] P G Siddheshwar and S Pranesh ldquoMagnetoconvection in fluidswith suspended particles under 1g and 120583grdquo Aerospace Scienceand Technology vol 6 no 2 pp 105ndash114 2002
[16] B S Bhadauria ldquoCombined effect of temperature modulationand magnetic field on the onset of convection in an electricallyconducting-fluid-saturated porous mediumrdquo Journal of HeatTransfer vol 130 no 5 Article ID 052601 2008
[17] P G Siddheshwar B S Bhadauria P Mishra and A K Srivas-tava ldquoStudy of heat transport by stationarymagneto-convectionin aNewtonian liquid under temperature or gravitymodulationusing Ginzburg-Landau modelrdquo International Journal of Non-Linear Mechanics vol 47 no 5 pp 418ndash425 2012
[18] P G Drazin and W H Reid Hydrodynamic Stability Cam-bridge University Press Cambridge UK 2004
[19] G Venezian ldquoEffect of modulation on the onset of thermalconvectionrdquo Journal of Fluid Mechanics vol 35 no 2 pp 243ndash254 1969
[20] S Rosenblat and G A Tanaka ldquoModulation of thermal convec-tion instabilityrdquo Physics of Fluids vol 14 no 7 pp 1319ndash13221971
[21] M N Roppo S H Davis and S Rosenblat ldquoBenard convectionwith time-periodic heatingrdquoThe Physics of Fluids vol 27 no 4pp 796ndash803 1984
[22] B S Bhadauria and P K Bhatia ldquoTime-periodic heating ofRayleigh-Benard convectionrdquo Physica Scripta vol 66 no 1 pp59ndash65 2002
[23] B S Bhadauria ldquoTime-periodic heating of Rayleigh-Benardconvection in a vertical magnetic fieldrdquo Physica Scripta vol 73no 3 pp 296ndash302 2006
[24] M S Malashetty andM Swamy ldquoEffect of thermal modulationon the onset of convection in a rotating fluid layerrdquo InternationalJournal of Heat and Mass Transfer vol 51 no 11-12 pp 2814ndash2823 2008
[25] B S Bhadauria and P Kiran ldquoHeat transport in an anisotropicporous medium saturated with variable viscosity liquid undertemperature modulationrdquo Transport in Porous Media vol 100no 2 pp 279ndash295 2013
The effect of temperature modulation on weak nonlineardouble-diffusive magnetoconvection has been analyzed byusing the nonautonomous Ginzburg-Landau equation Thefollowing conclusions are drawn from previous analysis
(1) The effect of IPM is negligible on heat and masstransport in the system
(2) In the case of IPM the effect of 120575 and 120596 is also foundto be negligible on heat and mass transport
(3) In the case of IPM the values of Nu and Sh increasesteadily for small values of time 120591 however Nu and Shbecome constant when 120591 is large
(4) The effect of increasing Pr Le Pm Ras is found toincrease in Nu and Sh thus increasing heat and masstransfer for all three types of modulations
(5) The effect of increasing 120575 is to increase the value of Nuand Sh for the case of OPM and LBMO hence heatand mass transfer
(6) The effect of increasing 120596 is to decrease the value ofNu and Sh for the case ofOPMand LBMO hence heatand mass transfer
(7) In the cases of OPM and LBMO the natures of Nuand Sh remain oscillatory
(8) Initially when 120591 is small the values of Nusselt andSherwood numbers start with 1 corresponding to theconduction state However as 120591 increases Nu andSh also increase thus increasing the heat and masstransfer
(9) The values of Nu and Sh for LBMO are greater thanthose in IPM but smaller than those in OPM
(10) The effect of magnetic field is to stabilize the system
Nomenclature
Latin Symbols
119860 Amplitude of convection119889 Depth of the fluid layerg Acceleration due to gravity119876 Chandrasekhar number
119876 = 1205831198981198672
11988711988921205880]]119898
119896119888 Critical wave numberNu(120591) Nusselt numberSh(120591) Sherwood number119901 Reduced pressurePr Prandtl number Pr = ]120581119879Pm Magnetic Prandtl number Pm = ]119898120581119879Le Lewis number Le = 120581119879120581119878
120596 Frequency of temperature modulation120575 Amplitude of temperature modulation120583 Dynamic coefficient of viscosity of the fluid120583119898 Magnetic permeability] Kinematic viscosity ] = 1205831205880
]119898 Magnetic viscosity120588 Fluid density120595 Stream function120595lowast Dimensionless stream function
Φ Magnetic potentialΦlowast Dimensionless magnetic potential
120591 Slow time 120591 = 1205982119905
120579 Phase angle1198791015840 Perturbed temperature
Other Symbols
nabla2 (12059721205971199092) + (120597
21205971199102) + (12059721205971199112)
14 International Journal of Engineering Mathematics
Subscripts
119887 Basic state119888 Critical0 Reference value
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was done during the lien sanctioned to theauthor by Banaras Hindu University Varanasi to work asprofessor of Mathematics at Department of Applied Math-ematics School for Physical Sciences Babasaheb BhimraoAmbedkar Central University Lucknow India The authorBS Bhadauria gratefully acknowledges Banaras Hindu Uni-versity Varanasi for the same Further the author Palle Kirangratefully acknowledges the financial assistance fromBabasa-heb Bhimrao Ambedkar Central University as a researchfellowship
References
[1] A Akbarzadeh and P Manins ldquoConvective layers generated byside walls in solar pondsrdquo Solar Energy vol 41 no 6 pp 521ndash529 1988
[2] H E Huppert and R S J Sparks ldquoDouble-diffusive convectiondue to crystallization in magmasrdquo Annual Review of Earth andPlanetary Sciences vol 12 pp 11ndash37 1984
[3] H J S Fernando and A Brandt ldquoRecent advances in doublediffusive convectionrdquoAppliedMechanics Reviews vol 47 pp c1ndashc7 1994
[4] J S Turner Buoyancy Effects in Fluids University Of Cam-bridge 1979
[5] N Rudraiah and I S Shivakumara ldquoDouble-diffusive convec-tion with an imposed magnetic fieldrdquo International Journal ofHeat and Mass Transfer vol 27 no 10 pp 1825ndash1836 1984
[6] J H Thomas and N O Weiss ldquoThe theory of sunspotsrdquo inSunspots Theory and Observations p 428 Kluwer AcademicPublishers Dordrecht The Netherlands 1992
[7] W B Thompson ldquoThermal convection in a magnetic fieldrdquoPhilosophical Magazine vol 42 pp 1417ndash1432 1951
[8] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityOxford University Press London UK 1961
[9] D Lortz ldquoA stability criterion for steady finite amplitudeconvection with an external magnetic fieldrdquo Journal of FluidMechanics vol 23 pp 113ndash128 1965
[10] W V R Malkus and G Veronis ldquoFinite amplitude cellularconvectionrdquo Journal of Fluid Mechanics vol 4 pp 225ndash2601958
[11] H Stommel A B Arons and D Blanchard ldquoAn oceanograph-ical curiosity the perpetual salt fountainrdquo Deep Sea Researchvol 3 no 2 pp 152ndash153 1956
[12] K Gotoh and M Yamada ldquoThermal convection of a horizontallayer of magnetic fluidsrdquo Journal of the Physical Society of Japanvol 51 no 9 pp 3042ndash3048 1982
[13] GMOreper and J Szekely ldquoThe effect of an externally imposedmagnetic field on buoyancy driven flow in a rectangular cavityrdquoJournal of Crystal Growth vol 64 no 3 pp 505ndash515 1983
[14] N Rudraiah ldquoDouble-diffusive magnetoconvectionrdquo Pramanavol 27 no 1-2 pp 233ndash266 1986
[15] P G Siddheshwar and S Pranesh ldquoMagnetoconvection in fluidswith suspended particles under 1g and 120583grdquo Aerospace Scienceand Technology vol 6 no 2 pp 105ndash114 2002
[16] B S Bhadauria ldquoCombined effect of temperature modulationand magnetic field on the onset of convection in an electricallyconducting-fluid-saturated porous mediumrdquo Journal of HeatTransfer vol 130 no 5 Article ID 052601 2008
[17] P G Siddheshwar B S Bhadauria P Mishra and A K Srivas-tava ldquoStudy of heat transport by stationarymagneto-convectionin aNewtonian liquid under temperature or gravitymodulationusing Ginzburg-Landau modelrdquo International Journal of Non-Linear Mechanics vol 47 no 5 pp 418ndash425 2012
[18] P G Drazin and W H Reid Hydrodynamic Stability Cam-bridge University Press Cambridge UK 2004
[19] G Venezian ldquoEffect of modulation on the onset of thermalconvectionrdquo Journal of Fluid Mechanics vol 35 no 2 pp 243ndash254 1969
[20] S Rosenblat and G A Tanaka ldquoModulation of thermal convec-tion instabilityrdquo Physics of Fluids vol 14 no 7 pp 1319ndash13221971
[21] M N Roppo S H Davis and S Rosenblat ldquoBenard convectionwith time-periodic heatingrdquoThe Physics of Fluids vol 27 no 4pp 796ndash803 1984
[22] B S Bhadauria and P K Bhatia ldquoTime-periodic heating ofRayleigh-Benard convectionrdquo Physica Scripta vol 66 no 1 pp59ndash65 2002
[23] B S Bhadauria ldquoTime-periodic heating of Rayleigh-Benardconvection in a vertical magnetic fieldrdquo Physica Scripta vol 73no 3 pp 296ndash302 2006
[24] M S Malashetty andM Swamy ldquoEffect of thermal modulationon the onset of convection in a rotating fluid layerrdquo InternationalJournal of Heat and Mass Transfer vol 51 no 11-12 pp 2814ndash2823 2008
[25] B S Bhadauria and P Kiran ldquoHeat transport in an anisotropicporous medium saturated with variable viscosity liquid undertemperature modulationrdquo Transport in Porous Media vol 100no 2 pp 279ndash295 2013
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was done during the lien sanctioned to theauthor by Banaras Hindu University Varanasi to work asprofessor of Mathematics at Department of Applied Math-ematics School for Physical Sciences Babasaheb BhimraoAmbedkar Central University Lucknow India The authorBS Bhadauria gratefully acknowledges Banaras Hindu Uni-versity Varanasi for the same Further the author Palle Kirangratefully acknowledges the financial assistance fromBabasa-heb Bhimrao Ambedkar Central University as a researchfellowship
References
[1] A Akbarzadeh and P Manins ldquoConvective layers generated byside walls in solar pondsrdquo Solar Energy vol 41 no 6 pp 521ndash529 1988
[2] H E Huppert and R S J Sparks ldquoDouble-diffusive convectiondue to crystallization in magmasrdquo Annual Review of Earth andPlanetary Sciences vol 12 pp 11ndash37 1984
[3] H J S Fernando and A Brandt ldquoRecent advances in doublediffusive convectionrdquoAppliedMechanics Reviews vol 47 pp c1ndashc7 1994
[4] J S Turner Buoyancy Effects in Fluids University Of Cam-bridge 1979
[5] N Rudraiah and I S Shivakumara ldquoDouble-diffusive convec-tion with an imposed magnetic fieldrdquo International Journal ofHeat and Mass Transfer vol 27 no 10 pp 1825ndash1836 1984
[6] J H Thomas and N O Weiss ldquoThe theory of sunspotsrdquo inSunspots Theory and Observations p 428 Kluwer AcademicPublishers Dordrecht The Netherlands 1992
[7] W B Thompson ldquoThermal convection in a magnetic fieldrdquoPhilosophical Magazine vol 42 pp 1417ndash1432 1951
[8] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityOxford University Press London UK 1961
[9] D Lortz ldquoA stability criterion for steady finite amplitudeconvection with an external magnetic fieldrdquo Journal of FluidMechanics vol 23 pp 113ndash128 1965
[10] W V R Malkus and G Veronis ldquoFinite amplitude cellularconvectionrdquo Journal of Fluid Mechanics vol 4 pp 225ndash2601958
[11] H Stommel A B Arons and D Blanchard ldquoAn oceanograph-ical curiosity the perpetual salt fountainrdquo Deep Sea Researchvol 3 no 2 pp 152ndash153 1956
[12] K Gotoh and M Yamada ldquoThermal convection of a horizontallayer of magnetic fluidsrdquo Journal of the Physical Society of Japanvol 51 no 9 pp 3042ndash3048 1982
[13] GMOreper and J Szekely ldquoThe effect of an externally imposedmagnetic field on buoyancy driven flow in a rectangular cavityrdquoJournal of Crystal Growth vol 64 no 3 pp 505ndash515 1983
[14] N Rudraiah ldquoDouble-diffusive magnetoconvectionrdquo Pramanavol 27 no 1-2 pp 233ndash266 1986
[15] P G Siddheshwar and S Pranesh ldquoMagnetoconvection in fluidswith suspended particles under 1g and 120583grdquo Aerospace Scienceand Technology vol 6 no 2 pp 105ndash114 2002
[16] B S Bhadauria ldquoCombined effect of temperature modulationand magnetic field on the onset of convection in an electricallyconducting-fluid-saturated porous mediumrdquo Journal of HeatTransfer vol 130 no 5 Article ID 052601 2008
[17] P G Siddheshwar B S Bhadauria P Mishra and A K Srivas-tava ldquoStudy of heat transport by stationarymagneto-convectionin aNewtonian liquid under temperature or gravitymodulationusing Ginzburg-Landau modelrdquo International Journal of Non-Linear Mechanics vol 47 no 5 pp 418ndash425 2012
[18] P G Drazin and W H Reid Hydrodynamic Stability Cam-bridge University Press Cambridge UK 2004
[19] G Venezian ldquoEffect of modulation on the onset of thermalconvectionrdquo Journal of Fluid Mechanics vol 35 no 2 pp 243ndash254 1969
[20] S Rosenblat and G A Tanaka ldquoModulation of thermal convec-tion instabilityrdquo Physics of Fluids vol 14 no 7 pp 1319ndash13221971
[21] M N Roppo S H Davis and S Rosenblat ldquoBenard convectionwith time-periodic heatingrdquoThe Physics of Fluids vol 27 no 4pp 796ndash803 1984
[22] B S Bhadauria and P K Bhatia ldquoTime-periodic heating ofRayleigh-Benard convectionrdquo Physica Scripta vol 66 no 1 pp59ndash65 2002
[23] B S Bhadauria ldquoTime-periodic heating of Rayleigh-Benardconvection in a vertical magnetic fieldrdquo Physica Scripta vol 73no 3 pp 296ndash302 2006
[24] M S Malashetty andM Swamy ldquoEffect of thermal modulationon the onset of convection in a rotating fluid layerrdquo InternationalJournal of Heat and Mass Transfer vol 51 no 11-12 pp 2814ndash2823 2008
[25] B S Bhadauria and P Kiran ldquoHeat transport in an anisotropicporous medium saturated with variable viscosity liquid undertemperature modulationrdquo Transport in Porous Media vol 100no 2 pp 279ndash295 2013
