# Thermal Diffusion and Mass Transfer Effects on MHD Flow of a Dusty Gas through Porous Medium

Authors: Rajesh Kumar; Devendra Kumar; R. K. Shrivastav
Journal: ISRN Applied Mathematics
 (2012)

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## Abstract

The present problem is concerned with the thermal diffusion mass transfer effects on MHD free convective flow of dusty gas through a porous medium induced by the motion of a semi-infinite flat plate moving with velocity decreasing “exponentially with time”. The effects of various parameters like magnetic parameterM thermal diffusion effect as soret number S1, permeability parameter K1, Schimdt number Sc are taken into account. The velocity profile, temperature field, and concentration of incompressible dusty gas and dust particles for several parameters are discussed numerically and explained graphically.


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## Body

## 1. Introduction

The thermal diffusion (commonly known as soret effect) for instance, has been utilized for isotope separation, and in mixtures between gases with very light-molecular-weight [H2, He] and the medium-molecular-weight [N2, air] the diffusion thermoeffect was found to be of a magnitude such that it cannot be neglected, Eckert and Darke [1]. In view of the importance of this diffusion thermoeffect, recently Basant Kumar and Singh [2] studied the free convection and mass transfer flow in an infinite vertical plate moving impulsively in its own plane, taking into account the soret effect.The problems of fluid mechanics involving gas particles mixture arise in many processes of practical importance. One of the earliest problems is that of the heat and mass transfer in the mist-flow region of a boiler tube. The liquid rocket is another example, usually the oxidizer vaporizes much more rapidly that the fuel spray and combustion occurs. Heterogeneously around each droplet, the length of the combustion chamber and the stability of the flow of acoustic or shock waves are practical two-phase flow problems. The study of the flow of dusty gases, which has gained increased attention recently has wide applications in environmental sciences; one finds in the literature an amazing number of derivations of equations for the flow of a gas particle mixture. The equations have been developed by the several authors for various special problem. Under various assumptions a few derivations primarily for the gas particle mixture are listed here, Saffman [3], Marble [4], and Soo [5]. Using the formulation of Saffman [3] several authors have given exact solutions of various dusty gas problems, Michael and Norey [6], Rao [7] Verma and Mathur [8], Singh [9], Rukmangadachari [10], and Mitra [11] Studied the problem of circular cylinders under various condition. Gupta [12] considered the unsteady flow of a dusty gas in a channel whose cross-section is an annular sector regarding the plate problems. Liu [13], Michael and Miller [14], Liu [15], and Vimal [16] studied the problems of infinite flat plate under various conditions. Mitra [17] has studied the flow of a dusty gas induced by the motion of a semi-infinite flat plate moving with velocity decreasing exponentially with time. Singh [18] has studied MHD flow of a dusty gas through a porous medium induced by the motion or a semi-infinite flat plate moving with velocity decreasing exponentially with time. Singh and Gupta [19] have discussed MHD free convective flow of a dusty gas through a porous medium induced by the motion of a semi-infinite flat plate moving with velocity decreasing exponentially with time. Recently, Singh and Varshney have seen the mass transfer effects on study of Singh and Gupta. In the present section, we are considering the problem of Singh and Varshney [20] taking thermal diffusion into account under the same conditions taken by Singh and Vaershney.


## 2. Mathematical Formulation of the Problem and Its Solution

We assume the dusty gas to be confined in the spacey>o and the flow is produced by the motion of the semi-infinite flat plate moving with velocity νe-λ2t in x direction, x-axis taken along the plate and y-axis to be measured normal to it. Since the plate is semi-infinite, all the physical quantities will be functions of y and t only. According to Saffman [3] the equation of motion of the dusty gas and the dust particles along the x-axis are, respectively, given by
(2.1)∂u∂t=ν∂2y∂y2+K0N0ρ(v-u),(2.2)∂v∂t=K0m(u-v),(2.3)∂T∂t=KTρCP∂2T∂y2,(2.4)∂C∂t=D∂2C∂y2+DT∂2T∂y2,
where u and v denote the velocity of gas and dust particles, respectively, ν is the kinemetic coefficient of viscosity of the gas, K0 is the stokes resistance coefficient, N0 is the number density of the dust particles which is taken to be constant, ρ is the density of the gas, m is the mass of dust particle, KT is the thermal conductivity, CP is the specific heat at the constant pressure, D is the molecular diffusivity, and DT is the thermal diffusivity.Applying the magnetic field, porous medium, free convection, mass transfer, and thermal diffusion along thex-axis the equation of motion (2.1) reduces to(2.5)∂u∂t=ν∂2u∂y2+K0N0ρ(v-u)-σB02uρ-νKu+gβθ+gβ′ϕ,
where(2.6)θ=T-T∞,ϕ=C-C∞The boundary conditions are(2.7)θ=ve-λ2t,ϕ=ve-λ2t,u=ve-λ2t,Aty=0,θ=0,Ø=0asy⟶∞.Let the nondimensionals introduced be(2.8)y*=y(γt)1/2,u*=uν,v*=vν,t*=tτ,τ=mK0,θ*=θν,ϕ*=ϕν.On applying nondimensionals the dimensionless forms of (2.5), (2.2), (2.3), and (2.4) are, respectively,(2.9)∂u∂t=∂2u∂y2+f(v-u)-Mu-1K1u+β1θ+β2ϕ,(2.10)∂v∂t=(u-v),(2.11)∂θ∂t=1Pr∂2θ∂y2,(2.12)∂ϕ∂t=1Sc∂2ϕ∂y2+1S1∂2θ∂y2,
where f is the mass of concentration of dust particles, M is the magnetic parameter, β1 volumetric expansion parameter, β2 is the mass expansion parameter, Sc is the Schmidt number, Pr is the prandtl number, K1 is the permeability parameter, and S1 is the thermal diffusion parameter as soret number(2.13)f=mN0ρ,M=mσB02K0ρ,β1=gβτ,β2=gβ′τ,Sc=νD,Pr=ρνCpKτ,1K1=νTK,S1=νDT.The boundary condition (2.7) is reduced to(2.14)θ=e-λ2t,ϕ=e-λ2t,u=e-λ2taty=0,θ⟶0,ϕ⟶0,u⟶0asy⟶∞.Let us choose the solutions of (2.9), (2.10), (2.11), and (2.12), respectively, as(2.15)u=F(y)e-λ2t,(2.16)v=G(y)e-λ2t,(2.17)θ=H(y)e-λ2t,(2.18)ϕ=I(y)e-λ2t.Using the solution the boundary condition (2.14),(2.19)H=1,I=1,F=1aty=0,H⟶0,I⟶0,F⟶0aty⟶∞.By virtue of (2.15), (2.16), (2.17), and (2.18), (2.9), (2.10), (2.11), and (2.12) are, respectively, reduced to(2.20)d2Fdy2+f⋅G+F(λ2-f-M-1K1)=-β1H-β2I,(2.21)G(1-λ2)=F,(2.22)d2Hdy2+λ2HPr=0,(2.23)d2Idy2+d2Hdy2+λ2IS1Sc=0.EliminatingG from (2.20) and (2.21) we get(2.24)d2Fdy2+fF1-λ2+F(λ2-f-M-1K)=-β1H-β2I,(2.25)d2Fdy2+n2F=-β1H-β2I,
where n=[(λ4-λ2(1+f+M+K1-1)+M+K1-1)/(λ2-1)]1/2.From  (2.22), we get(2.26)H=e-isy,
where s=λPr.From  (2.23)(2.27)I=(1-s2m2-s2)e-imy+s2m2-s2e-is1y,
where s1=λSc.By the boundary condition (2.19) the solution of (2.25) is obtained as(2.28)F=e-iny+β1n2-s2(e-iny-e-isy)+β2n2-s12(e-iny-e-is1y).
Then from (2.15) we get the velocity of dusty gas as(2.29)u=[e-iny+β1n2-s2(e-iny-e-isy)+β2n2-s12(e-iny-e-is1y)]e-λ2t.
Real part of u is given by
(2.30)u=e-λ2tcos⁡⁡ny+β1n2-s2e-λ2t(cos⁡⁡ny-cos⁡⁡sy)+β2n2-s12e-λ2t(cos⁡⁡ny-cos⁡⁡s1y).
Using (2.16), (2.21), and (2.28) the real part of velocity of dust particle v is obtained as(2.31)v=11-λ2[e-λ2tcos⁡⁡ny+β1n2-s2e-λ2t(cos⁡⁡ny-cos⁡⁡sy)+β2n2-s12e-λ2t(cos⁡⁡ny-cos⁡⁡s1y)].
Using (2.26) temperature distribution is given by(2.32)θ=e-isye-λ2t.Real part ofθ(2.33)θ=e-λ2tcos⁡⁡sy.Using (2.27), concentration is given by(2.34)ϕ=[(1-s2m2-s2)e-imy+s2m2-s2e-is1y]e-λ2t.Real part ofΦ is given by (2.35)ϕ=[(1-s2m2-s2)cos⁡⁡my+s2m2-s2cos⁡⁡s1y]e-λ2t.


## 3. Results and Discussion

From the solid and dotted graphs of Figure1 it is clear that velocity for dusty gas decrease with the increasing values of y and increases with the increasing values of t. And for the increasing values of λ keeping y, t, M, S1 constant the velocity of dusty gas increases as well as the velocity of dust particles decreases. Increasing values of t increases the velocity of dusty gas as well as the velocity of dust particles. From the solid and dotted graphs of Figure 2 it is noted that the temperature profile θ decreases and concentration profile ϕ increases when time t is increases. From the solid and dotted graphs of Figure 3 it is noted that the temperature profile decreases and the concentration profile ϕ increases as thermal diffusion S1 parameter as soret number increases at λ=.5,  f=.2,  Pr=.71,  β1=5,  β2=2.Figure 1
Velocity profile forλ=.5, f=.2, Pr=.71, β1=5, β2=2, S1=.38, Sc=.4 at the different values of M, K1, Sc, and t.Figure 2
Concentration profileλ=.5 and Sc=.6 and the different values of S1 and t.Figure 3
Concentration and temperature profileλ=.5, Sc=.6 and different values of S1 and t are S1=.3,.4,.5,.6 (see Table 3).The velocity profile for dusty gas is tabulated in Table1 and plotted in Figure 1 having solid graphs 1 to 3 for t=1 and dotted graphs 4 to 6 at t=3 and λ=.5, f=.2, Pr=.71, β1=5, β2=2 and different values of M, K1, Sc, and t are as in Table 4.Table 1
Velocity profile of dusty fluid and dust particle atλ=.5,  f=.2,  Pr=.71,  β1=5,  β2=2, S1=.38,  and Sc=.4 at the different values of M, K1, Sc, and t.
YGraph 1Graph 2Graph 3Graph 4Graph 5Graph 600.77880.77880.77881.03841.03841.03841−1.93407−1.90368−1.91017−2.57876−2.53823−2.54692−9.39053−9.37132−8.81633−12.5207−12.4951−11.75513−19.6764−20.0147−16.8816−26.2352−26.6863−22.50884−30.0193−31.554−25.2733−40.0257−42.072−35.6977Table 2
The concentration profile is tabulated in Table2 plotted in Figure 2 having graphs 1 to 3 at λ=.5 and different values of S1 and t are taken for velocity values of concentration at λ=.5, Sc=.6 and the different values of S1 and t.
YGraph 1Graph 2Graph 3Graph 4Graph 5Graph 600.77880.77880.77880.77880.77880.778810.541520.632840.726120.717320.683460.6427720.000170.261570.54730.54260.420790.28223−0.46131−0.170840.205610.28220.055090.176954−0.74208−0.48685−0.29205−0.02274−0.32410.57428Table 3
As taken for velocity, values of concentration profileλ=.5, Sc=.6 and different values of  S1 and t are S1=.3,.4,.5,.6.
yGraph 1Graph 2Graph 3Graph 4Graph 5Graph 600.77880.77880.77880.77880.77880.778810.717320.683460.642770.838860.87160.9115820.54260.420790.28221.009111.126061.2613730.28220.05509−0.176951.261371.476291.699474−0.02274−0.3241−0.574281.553631.830562.06203Table 4
Different values ofM,K1,Sc, and t taken for various graphs.
Graph 1Graph 2Graph 3Graph 4Graph 5Graph 6M0.10.20.10.10.10.2K1101020101010Sc0.40.40.40.80.40.4t111133


## 4. Conclusion

We conclude our study on thermal diffusion as well as mass transfer effect on MHD free convective flow of a dusty gas through a porous medium induced by the motion of a semi-infinite flat plate moving with velocity decreasing exponentially with time as follows.(1)
Increasing the viscoelastic parameter(λ) increases the velocity of dusty gas and decreases the velocity of dust particles as well as decreases the temperature profile and increases the concentration profile.(2)
Increasing values ofy decreases the velocity of dusty gas while it increases the velocity of dust particles.(3)
Increasing values of thermal diffusion parameters as soret number(S1) decreases the temperature profile (θ) and increases the concentration profile ϕ.



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*Source: 568278-2012-02-08.xml*