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International Journal of Pure and Applied Researches; 2016 Vol. 1(1); ISSN: 2455-474X
Paper ID: A16104, Propagation of Characteristic Wave Front through a Two Phase Mixture of Gas and Dust Particles By Dr.
M. K. Shukla, pp. 18-30. Page 18
Propagation of Characteristic Wave Front through a Two Phase
Mixture of Gas and Dust Particles
By
Dr. M. K. ShuklaDepartment of Mathematics, Shia P.G. College
Lucknow, India.
Abstract
The aim of the present paper is to discuss weak-non-linear waves through a two-phase mixture of gas and
dust particles, when particle-volume-fraction appears as an additional variable. The solutions based on
an asymptotic procedure are obtained under the approximation that the characteristic length of the signal
is much shorter that the characteristic length of the medium.
Keywords: Propagation, Two phase, Gas particles.
1. Introduction
In the recent technological advancements in different branches of engineering
and science, the compressible flows of a dusty gas have been encountered. When a gas carries a lot of
solid particles, the two-phase relaxation phenomenon significantly affects the flow field. It has been
shown by Lick (1967) and Parker (1969) that a disturbance having a time-scale comparable to the
attenuation time may produce partially and fully dispersed shock in relaxing gases. However in some
physical situations a signal with characteristic time much shorter than the attenuation time may suffer
continual profile distortion leading to a shock formation. In some disturbances this catastrophic effect isdelayed by dispersion but in non-dispersive systems it arises due to non-linearity provided that the
propagation distances are sufficiently large (1975, 1971,1972). General discussion on small amplitude
waves with considerations of non-linear effects are typified in works of Light-hill (1949), Witham (1952)
and Lin (1955).
It is known that a gas flow with an appreciable amount of small solid particles
may exhibit significantly relaxation effects, as a result of the inability of the particles to follow the rapid
changes in velocity and temperature of the gas. Such effects are predominant in case of wave propagation
in dusty gases. The problem of acoustical damming in dusty gases has been dealt by Epstein with
linearized analysis (1955, 1941). Bhutani and Chandran (1977) have discussed the weak waves in dusty
gases using the characteristic coordinate system; they have analyzed the decay of the plane, cylindricaland spherical weak-waves in gas particle system.
The aim of the paper is to discuss weak-non-linear waves through a two-phase
mixture of gas and dust particles, when particle-volume fraction appears as an additional variable. The
solutions based on an asymptotic simple wave procedure, are obtained under the approximation that the
characteristic length of the signal is much shorter that the characteristic length of the medium.
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International Journal of Pure and Applied Researches; 2016 Vol. 1(1); ISSN: 2455-474X
Paper ID:A16104, Propagation of Characteristic Wave Front through a Two Phase Mixture of Gas and Dust Particles By Dr. M.
K. Shukla, pp. 18-30, Email: [email protected] Page 19
2. Basic equation and boundary conditions
The mathematical analysis of two-phase flows is considerably more difficult than
that of pure gas flows, and one of usual simplifying assumptions is that the volume occupied by the
particle can be neglected. At high gas densities (high pressure) or at high particle mass fraction, the
particle-volume fraction may become sufficiently large so that it formulated under the following
assumptions-
1.
The gas obeys the perfect gas law and specific heats are constant.
2. The particles are spherical, of uniform size and are uniformly distributed initially. The specific
heat is constant and temperature is uniform within each particle.
3. Particles do not interact on each other and their motion is negligible.
4. The viscosity and heat conductivity of gas are neglected except for the interaction with solid
particles.
5. The particles do not contribute to the pressure.
6. No external forces (such as gravity) or heat exchange affects the mixture and no mass transfer
takes place between the gas and particle.
Equations governing the motion of gas particle mixture, under above assumptions are given by
(1969).
2
, , , , 0, (2.1)1 1 1
p
x x x x
v
u v RT Ru uu T
, , , , , 0, (2.2)
1 1t x x x x
u u v v u
2
1 11, 1 , , ,
1 1
1 1, , 0, (2.3)
1 1 1 1
t x x x
m p p
x x
v T v
u vv uT T Tu T
C T T T T v u u vv
C
, , 0, (2.4)1t x v
v u
v vv
, , , 0, (2.5)t x x
v v
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International Journal of Pure and Applied Researches; 2016 Vol. 1(1); ISSN: 2455-474X
Paper ID:A16104, Propagation of Characteristic Wave Front through a Two Phase Mixture of Gas and Dust Particles By Dr. M.
K. Shukla, pp. 18-30, Email: [email protected] Page 20
, ,
0, (2.6) p
p t p x
T
T T T vT
where ,, , , , , p v p T u C C be the pressure, density, temperature, velocity, specific heats and specific heat
ratio of the gas and , , , p mT v C be the volume-fraction, temperature, velocity and specific heat of the
dust particles respectively.v
is relaxation time for particle velocity andT
is relaxation time for heat
transfer. A Comma followed by an index denotes the partial differentiation with respect to index.
Let 1/ 2
0 0 0/ ,a p be the speed of sound in undisturbed gas, where subscript ‘0’ is used to
indicate a constant reference value. If the time ‘t’ is non dimensionalized by ' 'v
the distance ‘x’ by
0' ',
va the speed of sound 0' ' by ' 'a a the gas velocity ‘u’ and particle velocity ‘v’ by 0' ',a gas
temperature ‘T’ and particle temperature ' ' pT by 0' 'T equation (2.1) to (2.6) can be written in the
following matrix form :
, , 0, (2.7)t x
U AU B
where
,
p
u
T U
v
T
T 1
0 0 01 1
A=
u
u
0 01 1
1 11 1 1 1 0
1 1 1 1
v u
u vT v u T T v uT
0 0 0 v 0 0
0 0 0 v 0
0 0
0 0 0 v
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Paper ID:A16104, Propagation of Characteristic Wave Front through a Two Phase Mixture of Gas and Dust Particles By Dr. M.
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2
2
1
0
1
1 1
1
0
p
v
pv
p
u v
p u v
T T B
v u
T T
vand .The relationshi p between ' ' and ' ' is given by
3 P , where is Pr adtl number.2
v m
T
T p
mT r v r
v
C
C
C PC
The six families of characteristics of the equation (2.7) are given by:
,
dxu
dt representing the gas-particle trajectory, and
dxv
dt (repeated three times), represent the solid particle trajectory, and triple degeneracy corresponds
to case for non-equilibrium flows and is a consequence of neglecting the partial pressure of the particle.
The remaining two characteristics representing waves propagating in x direction are
. (2.8)dx
u adt
where
1 1 / 2 1 .u K u
1/ 22
1 4 1 . / 2 1a K u uK T
and 1 1 .K u v
In the particular case when 0, equation (2.8) reduces to
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Paper ID:A16104, Propagation of Characteristic Wave Front through a Two Phase Mixture of Gas and Dust Particles By Dr. M.
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1/ 2, where . (2.9)dx
u a a T dt
3. Characteristic Front
In studying the wave phenomenon governed by hyperbolic equations, it is usually
more natural and convenient to use the characteristics of governing system as the reference coordinate
system. Let us introduce the characteristic variables ' ' and ' ' such that;
, , 0, (3.1)t x
u
, + , 0, (3.2)t x
u a
The leading characteristic front can be represented by 0 and if a gas particle crosses this front at time
‘t’ it’s path will be represented by .t Keeping in view of the properties of and , it is obvious
that the function , and , x t satisfy the following partial differential equations;
,, , , + , x ut x u a t
The transformation from space time (x,t) to the plane of characteristic
parameters , will be one to one if and only if Jacobian
, , , , - t, , J x t x t u u a t
or
1, , ,
2 1
u v J a t t
does not vanish or does not become infinity anywhere.
Since , 0,t from physical considerations a breakdown of solution in terms of
characteristic parameters will arise if and only if , 0,at
In terms of characteristic coordinates equations given by (2.7) reduces to the following
form.
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Paper ID:A16104, Propagation of Characteristic Wave Front through a Two Phase Mixture of Gas and Dust Particles By Dr. M.
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2
1 1, , , , , ,
2 1 1
, , , ,
1
1, , 0, (3.5)
2 1 1
p
u va u t T t T t
T t p t
pu va u v t t
p
1, , , , , , , , , ,
2 1 1
, , , , 0, (3.6)1
u v v ua p t p u t u t t t
pv t v t
2
1 1, , , ,
1 2 1
1, , , ,
1
11 , , , , , , , ,
1
1 1, , , ,
1 1 1
1
1
a
p
u v u vT t a T t
T u vt t
T v uT u t u t t t
T u vv t v t a
u vT T
, , 0, (3.7)t t
1 2, , , ,
2 1
1, , 0, (3.8)
2 1 1
u vu v v t a v t
u v u va t t
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1 2, , , ,
2 1
, , , , 0, (3.9)
u v
u v v t a t
v t v t
, ,
1 2, ,
2 1
1, , 0. (3.10)
2 1
p p
p
u vu v T t a T t
u va T T t t
If we consider the case in which the wave front is an outgoing characteristic
propagating into a uniform region, boundary conditions are given by
0'0, 1, 1 and at 0 pu v T T
and '( ), ( ), , at 0.u F x F t
In order to solve the problem we consider , where ( 1),F t f t Characterizes the amplitude
of disturbance and 0 1 . f t We can now assume a solution of system (3.5) to (3.10) of the form
0 1 2, , , 0 (3.11)Q Q Q
where ‘Q’ is any one of the independent variables , , , etc. If 0u T x the piston is at rest and hence,
0 0 0 0 0
0 0 0
0
0
0, 1, 1,1
, , 0. (3.12)1
pu v T T
a u
Substituting the expression (3.11) in equations (3.5 to 3.10) and
collecting the terms of order 0
. we have:
0 0 0, , ,0, . (3.13) x x t
Thus boundary conditions are:
0 0
0
, 0, at 0,
at 0. (3.14)
t x
t
Equation (3.13) on integration subject to boundary conditions (3.14) yield,
0 0, . (3.15) x t
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The complete solution to 00 be given by (3.12) and (3.15) and to this order the characteristics in the
physical plane are
,
. (3.16)
x
t x
When the expansion (3.11) is inserted in the governing equations (3.5), to (3.10) and
(3.3), the collection of terms of order gives the following set of equations for the first order solutions,
01 1 1 1 1 13/ 2,,
0
1 0, (3.17)1
p
oT u T u v
1 1 1 1 10
,0
, , , 0 (3.18)1
u u v v
11
1 1 1, 0
00 0,
1 1
0
0
11 1, ,
(11 1
,0, (3.19)
1
p p
uuT v v
T T
1 11
0
( ), 0, (3.20)
1
u v
v
1 1 10 0, 1 , , 0, (3.21)v v
1 1 1, 0, (3.22) p pT T T
and
1 1 1 1 1 1, , ,, . (3.23) x u x u a t
The boundary conditions for the first order solutions can be obtained as
1 1 10, , ' at 0, (3.24)t x f u f
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1 1 1 1 1 10. 0, 0, 0 at 0. (3.25) p
u v T T t
Since the Parandtl number of many gases is close to 2/3 and / ,m pC C is of
order unity for many gas-particle combinations hence for such cases temperature and velocity relaxation
times are approximately equal, i.e., .v T
In such approximate case 1 and 1. The Laplace
transform of ,Q with respect to is dented by ˆ , , whereQ
0
ˆ , , ,Q Q e d
equation (3.17) to (3.23) assume the following form
1 1 1 1 1
0
0 1 1
0
1ˆ ˆˆ ˆˆ1 [ ][ ]
1
ˆ ˆ 0,1
o
p
d T u T
d
u v
(3.26)
1 11 1 10 0
0 0
̂ ˆˆ ̂ ˆ[ ] 0,
1 1
du dvu v
d d
(3.27)
1 1
01 1 10 0
0 1 1
0
1ˆ ˆˆˆ ˆ[ ] ,
1
ˆ ˆ( ) 0,1 1
p
p
du dv u T vd d
T T
(3.28)
1 1 10 ˆ ˆ ˆ1 [ ] 0,v u v (3.29)
11 1
0 0
0 0
ˆˆ ˆ[ 1 ] 0, (3.30)
1
dvv
d
1 1 1ˆ ˆ ˆ 0, p pT T T (3.31)
and
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1
1 1 1 1 1ˆˆˆ ˆ ˆ ˆ, [ ]
u d d x x u a t
d d (3.32)
respectively, similarly the transformed boundary conditions are:
1 1 1ˆ ˆˆ ˆ ˆ0, , , at =0,t x f u f (3.33)
11 1 1 1 1ˆ ˆ ˆˆ ˆ ˆ0, 0, at 0. pu v T T t (3.34)
In view of equations (3.27) to (3.31) equation (3.26) yields following second order linear differential
equation;
1 1 12
2
2
ˆ ˆ ˆ2 0,
1
d u du Au
d d
(3.35)
where ‘A’ is constant given by
0
0 0
1 ( / )
( / ) 1 1 1
pC D B A
B C D
and
0
0 0
0
00
0
0
1 ,(1 {1 (1 )}
( 1)1 ,
1 11
1 .1 1
p
B
C
D
Solution of equation (3.35) satisfying the condition that 1
û is bounded as is
1ˆ expcu K (3.36)
where
1/ 2
11
A
and constant K c is determined by the boundary conditions;
1 ˆˆ , at 0,u f
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which gives 1 ˆˆ exp - . (3.37)u f
Thus equations (3.26) to (3.32), give
1ˆ
( , ) ( ) exp (- ),u f (3.38)
1 0ˆˆ , exp - / 1 1 ,v f (3.39)
1 00
0
0
ˆˆ ( , )= f( ) exp - [1 1 11
(1 )],
1
(3.40)
10 0 0
0 0 0
1 1 1 1 1ˆ, exp - , (3.41)
1 1 1 1 pT f
0 0 01
0 0 0
1 1 [1 1 ]ˆˆ , exp - , (3.42)
1 1 1 1 p
p
T f
0 01
0
1ˆˆ , exp - , (3.43)
1 1 f
1 ˆˆ exp - , x f (3.44)
and
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Paper ID:A16104, Propagation of Characteristic Wave Front through a Two Phase Mixture of Gas and Dust Particles By Dr. M.
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1
0 0 0
0 0
3/ 2
0 0 00 0
0 0 0
0 0
0
ˆ exp - 1ˆ , [ {2(1 (1 )) (2 ( 1)(1 ))}
2 1 1 1
(1 ) ( 1)(1 ){1 (1 ) }2(1 ) {1 (1 )} { (3.45)
1 (1 )(1 )
(1 )}]
(1 )
p
f t
respectively.
On inversion, it follows that
2, exp - 0 ,2
S i
S iu a f d
i
(3.46)
2ˆ, exp - 0 ,2
S i
S i x f d
i
(3.47)
0 0
0 0 0
3/ 2
0 00 0
0 0 0
0 0 2
0
ˆ exp exp - 1,
4 1 1 1
[ {2(1 (1 )) (2 ( 1)(1 ))}
(1 ) ( 1)(1 )(1 )2 (1 ){1 (1 )} {
(1 ) (1 ){(1 )
1
}] 0 ,1
S i
S i
p
f t
i
d
(3.48)
where S is a positive number satisfying the condition that all the singularities of these integral are to the
left of line S in the complex plane.
4. Conclusions
Equations (3.46) to (3.48) are solutions up to the first order of non-linear
equations (3.3) and (3.5) to (3.10) along with prescribed boundary conditions (2.10). The trajectory of the
outgoing waves and particle paths in the ( - ) x t plane is described by equations (3.46) and (3.48), which
evidently show the convergence of characteristics. The formation of a shock wave is characterized by
, =0.t
References
[1]. Epstein, P. S., and Carhrat, R. R., 1955; J.Acoust. Soc. Am. 25, 555.
[2]. Epstein, P. S., 1941; Contributions to Applied Mechanics, Thedore Von Marman Anniversary
Volume, Calif. Inst. Technol., 162.
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Paper ID:A16104, Propagation of Characteristic Wave Front through a Two Phase Mixture of Gas and Dust Particles By Dr. M.
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[3]. Lick, W.J., 1967; In Advance of Applied Mechanics 10, Academic, New York.
[4]. Lighthill, M. J., 1949; Phil. Mag. 50, 1179.
[5]. Lin, C. C., 1955; J. Maths. Phys. 33, 117.
[6]. Parker, D .F., 1973; Physics Fluids, 15, 256.
[7]. Parker, D. F., 1971; J. Inst. Math. Applice 7, 92.
[8]. Parker, D. F., 1969; J. Fluid Mech. 39, 793.
[9]. Seymour. B. R. and Mortell, M. P., 1965; Non Linear Geometrical Acoustics Mechanic Today 2,
Pergamon Press.
[10]. Witham, G. B., 1952; Comm. On Pure and Appl. Maths.,301.
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