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

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 parameter M thermal diffusion effect as soret number S₁, permeability parameter K₁, Schimdt number S_c 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.

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 space 𝑦>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 𝜕𝑢𝜕𝑡=𝜈𝜕²𝑦𝜕𝑦²+𝐾⁰𝑁₀𝜕𝑣𝜌(𝑣−𝑢),(2.1)𝜕𝜕𝑇𝑡=𝐾𝑚⁰(𝑢−𝑣),(2.2)𝜕𝑡=𝜌𝐾𝐶^𝑇_𝑃𝜕²𝑇𝜕𝑦²𝜕𝐶,(2.3)𝜕𝑡=𝐷𝜕²𝐶𝜕𝑦²+𝐷_𝑇𝜕²𝑇𝜕𝑦²,(2.4) 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, 𝐾_𝑇 is the thermal conductivity, 𝐶_𝑃 is the specific heat at the constant pressure, 𝐷 is the molecular diffusivity, and 𝐷_𝑇 is the thermal diffusivity.

Applying the magnetic field, porous medium, free convection, mass transfer, and thermal diffusion along the 𝑥-axis the equation of motion (2.1) reduces to𝜕𝑢𝜕𝑡=𝜈𝜕²𝑢𝜕𝑦²+𝐾⁰𝑁₀𝜌(𝑣−𝑢)−𝜎𝐵²⁰𝑢𝜌−𝐾𝜈𝑢+𝑔𝛽𝜃+𝑔𝛽^𝜙,(2.5) where𝜃=𝑇−𝑇∞,𝜙=𝐶−𝐶∞(2.6)

The boundary conditions are𝜃A=t𝑦𝑣=𝑒0,^−𝜆2𝑡,𝜃=𝜙0=,𝑣∅𝑒=^−𝜆2𝑡0,as𝑢𝑦=⟶𝑣𝑒∞^−𝜆2𝑡.,(2.7)

Let the nondimensionals introduced be𝑦^∗=𝑦(𝛾𝑡)^1/2,𝑢^∗=𝑢𝜈,𝑣^∗=𝑣𝜈,𝑡^∗=𝑡𝜏=𝑚𝜏,𝐾₀,𝜃^∗=𝜃𝜈,𝜙^∗=𝜙𝜈.(2.8)

On applying nondimensionals the dimensionless forms of (2.5), (2.2), (2.3), and (2.4) are, respectively,𝜕𝑢𝜕𝑡=𝜕²𝑢𝜕𝑦²+𝑓(𝑣−𝑢)−𝑀𝑢−1𝐾₁𝜕𝑣𝑢+𝛽₁𝜃+𝛽₂𝜙,(2.9)𝜕𝜕𝑡𝜃=(𝑢−𝑣),(2.10)𝜕𝑡=𝑃1_𝑟𝜕²𝜃𝜕𝑦²𝜕𝜙,(2.11)𝜕𝑡=𝑆1_𝑐𝜕²𝜙𝜕𝑦²+1𝑆₁𝜕²𝜃𝜕𝑦²,(2.12) 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, 𝑆_𝑐 is the Schmidt number, 𝑃_𝑟 is the prandtl number, 𝐾₁ is the permeability parameter, and 𝑆₁ is the thermal diffusion parameter as soret number𝑓=𝑚𝑁⁰𝜌,𝑀=𝑚𝐾𝜎₀𝐵𝜌²⁰,𝛽¹𝑆=𝑔𝛽𝜏,𝛽₂=𝑔𝛽^′𝜏,_𝑐=𝜈𝐷,𝑃^𝑟=𝜌𝜈𝐶^𝑝𝐾𝜏,𝐾1₁=𝜈𝑇𝐾,𝑆¹=𝜈𝐷_𝑇.(2.13)

The boundary condition (2.7) is reduced to𝜃𝜃⟶=𝑒0,^−𝜆2𝑡,𝜙𝜙⟶=𝑒0,^−𝜆2𝑡,𝑢⟶𝑢=0𝑒a^−𝜆2𝑡s𝑦at⟶𝑦=∞0,.(2.14)

Let us choose the solutions of (2.9), (2.10), (2.11), and (2.12), respectively, as𝑣𝑢==𝐺𝐹((𝑦𝑦))𝑒𝑒^−𝜆2𝑡_−𝜆2𝑡,(2.15)𝜃=𝐻(𝑦)𝑒_−𝜆2𝑡,(2.16)𝜙=𝐼(𝑦)𝑒_−𝜆2𝑡,(2.17).(2.18)

Using the solution the boundary condition (2.14),𝐻⟶𝐻0=,1,𝐼⟶𝐼=01,,𝐹𝐹⟶=10ata𝑦t𝑦=⟶0,∞.(2.19)

By virtue of (2.15), (2.16), (2.17), and (2.18), (2.9), (2.10), (2.11), and (2.12) are, respectively, reduced to𝑑²𝐹𝑑𝑦²+𝑓⋅𝐺+𝐹𝜆²−𝑓−𝑀−1𝐾₁=−𝛽¹𝐺1−𝜆₂𝐻−𝛽₂𝐼,(2.20)𝑑=𝐹,(2.21)₂𝐻𝑑𝑦²𝑑+𝜆²𝐻𝑃_𝑟=0,(2.22)₂𝐼𝑑𝑦²+𝑑²𝐻𝑑𝑦²+𝜆²𝐼𝑆₁𝑆_𝑐=0.(2.23)

Eliminating 𝐺 from (2.20) and (2.21) we get𝑑²𝐹𝑑𝑦²+𝑓𝐹1−𝜆²+𝐹𝜆²−𝑓−𝑀−1𝐾=−𝛽¹𝑑𝐻−𝛽₂𝐼,(2.24)₂𝐹𝑑𝑦²+𝑛²𝐹=−𝛽₁𝐻−𝛽₂𝐼,(2.25) where 𝑛=[(𝜆⁴−𝜆²(1+𝑓+𝑀+𝐾¹⁻¹)+𝑀+𝐾¹⁻¹)/(𝜆²−1)]^1/2.

From  (2.22), we get𝐻=𝑒^{−𝑖𝑠𝑦},(2.26) where 𝑠=𝜆√𝑃_𝑟.

From  (2.23)𝐼=1−𝑠²𝑚²−𝑠²𝑒^{−𝑖𝑚𝑦}+𝑠²𝑚²−𝑠²𝑒^{−𝑖𝑠1𝑦},(2.27) where 𝑠₁=𝜆√𝑆_𝑐.

By the boundary condition (2.19) the solution of (2.25) is obtained as𝐹=𝑒^{−𝑖𝑛𝑦}+𝛽¹𝑛²−𝑠²𝑒^{−𝑖𝑛𝑦}−𝑒^{−𝑖𝑠𝑦}+𝛽²𝑛²−𝑠²¹𝑒^{−𝑖𝑛𝑦}−𝑒^{−𝑖𝑠1𝑦}.(2.28) Then from (2.15) we get the velocity of dusty gas as𝑢=𝑒^{−𝑖𝑛𝑦}+𝛽¹𝑛²−𝑠²𝑒^{−𝑖𝑛𝑦}−𝑒^{−𝑖𝑠𝑦}+𝛽²𝑛²−𝑠²¹𝑒^{−𝑖𝑛𝑦}−𝑒^{−𝑖𝑠1𝑦}𝑒^−𝜆2𝑡.(2.29) Real part of 𝑢 is given by 𝑢=𝑒^−𝜆2𝑡cos𝑛𝑦+𝛽¹𝑛²−𝑠²𝑒^−𝜆2𝑡(cos𝑛𝑦−cos𝑠𝑦)+𝛽²𝑛²−𝑠²¹𝑒^−𝜆2𝑡cos𝑛𝑦−cos𝑠₁𝑦.(2.30) Using (2.16), (2.21), and (2.28) the real part of velocity of dust particle 𝑣 is obtained as𝑣=11−𝜆²𝑒^−𝜆2𝑡cos𝑛𝑦+𝛽¹𝑛²−𝑠²𝑒^−𝜆2𝑡(cos𝑛𝑦−cos𝑠𝑦)+𝛽²𝑛²−𝑠²¹𝑒^−𝜆2𝑡cos𝑛𝑦−cos𝑠₁𝑦.(2.31) Using (2.26) temperature distribution is given by𝜃=𝑒^{−𝑖𝑠𝑦}𝑒^−𝜆2𝑡.(2.32)

Real part of 𝜃𝜃=𝑒^−𝜆2𝑡cos𝑠𝑦.(2.33)

Using (2.27), concentration is given by𝜙=1−𝑠²𝑚²−𝑠²𝑒^{−𝑖𝑚𝑦}+𝑠²𝑚²−𝑠²𝑒^{−𝑖𝑠1𝑦}𝑒^−𝜆2𝑡.(2.34)

Real part of Φ is given by 𝜙=1−𝑠²𝑚²−𝑠²cos𝑚𝑦+𝑚²𝑠−²𝑠²cos𝑠₁𝑦𝑒^−𝜆2𝑡.(2.35)

3. Results and Discussion

From the solid and dotted graphs of Figure 1 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,  𝑃_𝑟=.71,  𝛽₁=5,  𝛽₂=2.

Figure 1 

Velocity profile for 𝜆=.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 profile 𝜆=.5 and 𝑆𝑐=.6 and the different values of 𝑆1 and 𝑡.

Figure 3 

Concentration and temperature profile 𝜆=.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 Table 1 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, 𝑃_𝑟=.71, 𝛽₁=5, 𝛽₂=2 and different values of 𝑀, 𝐾₁, 𝑆_𝑐, and 𝑡 are as in Table 4.

Table 1 

Velocity profile of dusty fluid and dust particle at 𝜆=.5,  𝑓=.2,  𝑃_𝑟=.71,  𝛽₁=5,  𝛽₂=2, 𝑆₁=.38,  and 𝑆_𝑐=.4 at the different values of 𝑀, 𝐾₁, 𝑆_𝑐, and 𝑡.

Table 2 

The concentration profile is tabulated in Table 2 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, 𝑆_𝑐=.6 and the different values of 𝑆₁ and 𝑡.

Table 3 

As taken for velocity, values of concentration profile 𝜆=.5, 𝑆_𝑐=.6 and different values of 𝑆₁ and 𝑡 are 𝑆₁=.3,.4,.5,.6.

Table 4 

Different values of 𝑀,𝐾₁,𝑆_𝑐, and 𝑡 taken for various graphs.

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 of 𝑦 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.

Acknowledgment

The authors are highly thankful to provide to valuable suggestions to improve the quality of research paper.