International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 568278 | 9 pages | https://doi.org/10.5402/2012/568278

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

Academic Editor: X. Wen
Received14 Sep 2011
Accepted27 Oct 2011
Published08 Feb 2012

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 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.

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 𝜈𝑒𝜆2𝑡 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=𝜈2𝑦𝜕𝑦2+𝐾0𝑁0𝜌(𝑣𝑢),(2.1)𝜕𝑣=𝐾𝜕𝑡0𝑚(𝑢𝑣),(2.2)𝜕𝑇=𝐾𝜕𝑡𝑇𝜌𝐶𝑃𝜕2𝑇𝜕𝑦2,(2.3)𝜕𝐶𝜕𝜕𝑡=𝐷2𝐶𝜕𝑦2+𝐷𝑇𝜕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, 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 the 𝑥-axis the equation of motion (2.1) reduces to𝜕𝑢𝜕𝜕𝑡=𝜈2𝑢𝜕𝑦2+𝐾0𝑁0𝜌(𝑣𝑢)𝜎𝐵20𝑢𝜌𝜈𝐾𝑢+𝑔𝛽𝜃+𝑔𝛽𝜙,(2.5) where𝜃=𝑇𝑇,𝜙=𝐶𝐶(2.6)

The boundary conditions are𝜃=𝑣𝑒𝜆2𝑡,𝜙=𝑣𝑒𝜆2𝑡,𝑢=𝑣𝑒𝜆2𝑡,At𝑦=0,𝜃=0,=0as𝑦.(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=𝜕𝜕𝑡2𝑢𝜕𝑦21+𝑓(𝑣𝑢)𝑀𝑢𝐾1𝑢+𝛽1𝜃+𝛽2𝜙,(2.9)𝜕𝑣𝜕𝑡=(𝑢𝑣),(2.10)𝜕𝜃=1𝜕𝑡𝑃𝑟𝜕2𝜃𝜕𝑦2,(2.11)𝜕𝜙=1𝜕𝑡𝑆𝑐𝜕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, 𝑆𝑐 is the Schmidt number, 𝑃𝑟 is the prandtl number, 𝐾1 is the permeability parameter, and 𝑆1 is the thermal diffusion parameter as soret number𝑓=𝑚𝑁0𝜌,𝑀=𝑚𝜎𝐵20𝐾0𝜌,𝛽1=𝑔𝛽𝜏,𝛽2=𝑔𝛽𝑆𝜏,𝑐=𝜈𝐷,𝑃𝑟=𝜌𝜈𝐶𝑝,1𝐾𝜏𝐾1=𝜈𝑇𝐾,𝑆1=𝜈𝐷𝑇.(2.13)

The boundary condition (2.7) is reduced to𝜃=𝑒𝜆2𝑡,𝜙=𝑒𝜆2𝑡,𝑢=𝑒𝜆2𝑡at𝑦=0,𝜃0,𝜙0,𝑢0as𝑦.(2.14)

Let us choose the solutions of (2.9), (2.10), (2.11), and (2.12), respectively, as𝑢=𝐹(𝑦)𝑒𝜆2𝑡,(2.15)𝑣=𝐺(𝑦)𝑒𝜆2𝑡,(2.16)𝜃=𝐻(𝑦)𝑒𝜆2𝑡,(2.17)𝜙=𝐼(𝑦)𝑒𝜆2𝑡.(2.18)

Using the solution the boundary condition (2.14),𝐻=1,𝐼=1,𝐹=1at𝑦=0,𝐻0,𝐼0,𝐹0at𝑦.(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 to𝑑2𝐹𝑑𝑦2𝜆+𝑓𝐺+𝐹21𝑓𝑀𝐾1=𝛽1𝐻𝛽2𝐺𝐼,(2.20)1𝜆2𝑑=𝐹,(2.21)2𝐻𝑑𝑦2+𝜆2𝐻𝑃𝑟𝑑=0,(2.22)2𝐼𝑑𝑦2+𝑑2𝐻𝑑𝑦2+𝜆2𝐼𝑆1𝑆𝑐=0.(2.23)

Eliminating 𝐺 from (2.20) and (2.21) we get𝑑2𝐹𝑑𝑦2+𝑓𝐹1𝜆2𝜆+𝐹21𝑓𝑀𝐾=𝛽1𝐻𝛽2𝐼𝑑,(2.24)2𝐹𝑑𝑦2+𝑛2𝐹=𝛽1𝐻𝛽2𝐼,(2.25) where 𝑛=[(𝜆4𝜆2(1+𝑓+𝑀+𝐾11)+𝑀+𝐾11)/(𝜆21)]1/2.

From  (2.22), we get𝐻=𝑒𝑖𝑠𝑦,(2.26) where 𝑠=𝜆𝑃𝑟.

From  (2.23)𝑠𝐼=12𝑚2𝑠2𝑒𝑖𝑚𝑦+𝑠2𝑚2𝑠2𝑒𝑖𝑠1𝑦,(2.27) where 𝑠1=𝜆𝑆𝑐.

By the boundary condition (2.19) the solution of (2.25) is obtained as𝐹=𝑒𝑖𝑛𝑦+𝛽1𝑛2𝑠2𝑒𝑖𝑛𝑦𝑒𝑖𝑠𝑦+𝛽2𝑛2𝑠21𝑒𝑖𝑛𝑦𝑒𝑖𝑠1𝑦.(2.28) Then from (2.15) we get the velocity of dusty gas as𝑒𝑢=𝑖𝑛𝑦+𝛽1𝑛2𝑠2𝑒𝑖𝑛𝑦𝑒𝑖𝑠𝑦+𝛽2𝑛2𝑠21𝑒𝑖𝑛𝑦𝑒𝑖𝑠1𝑦𝑒𝜆2𝑡.(2.29) Real part of 𝑢 is given by 𝑢=𝑒𝜆2𝑡𝛽cos𝑛𝑦+1𝑛2𝑠2𝑒𝜆2𝑡𝛽(cos𝑛𝑦cos𝑠𝑦)+2𝑛2𝑠21𝑒𝜆2𝑡cos𝑛𝑦cos𝑠1𝑦.(2.30) Using (2.16), (2.21), and (2.28) the real part of velocity of dust particle 𝑣 is obtained as1𝑣=1𝜆2𝑒𝜆2𝑡𝛽cos𝑛𝑦+1𝑛2𝑠2𝑒𝜆2𝑡𝛽(cos𝑛𝑦cos𝑠𝑦)+2𝑛2𝑠21𝑒𝜆2𝑡cos𝑛𝑦cos𝑠1𝑦.(2.31) Using (2.26) temperature distribution is given by𝜃=𝑒𝑖𝑠𝑦𝑒𝜆2𝑡.(2.32)

Real part of 𝜃𝜃=𝑒𝜆2𝑡cos𝑠𝑦.(2.33)

Using (2.27), concentration is given by𝑠𝜙=12𝑚2𝑠2𝑒𝑖𝑚𝑦+𝑠2𝑚2𝑠2𝑒𝑖𝑠1𝑦𝑒𝜆2𝑡.(2.34)

Real part of Φ is given by 𝑠𝜙=12𝑚2𝑠2𝑠cos𝑚𝑦+2𝑚2𝑠2cos𝑠1𝑦𝑒𝜆2𝑡.(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,  𝑃𝑟=.71,  𝛽1=5,  𝛽2=2.

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, 𝑃𝑟=.71, 𝛽1=5, 𝛽2=2 and different values of 𝑀, 𝐾1, 𝑆𝑐, and 𝑡 are as in Table 4.


𝑌 Graph 1Graph 2Graph 3Graph 4Graph 5Graph 6

00.77880.77880.77881.03841.03841.0384
1−1.93407−1.90368−1.91017−2.57876−2.53823−2.5469
2−9.39053−9.37132−8.81633−12.5207−12.4951−11.7551
3−19.6764−20.0147−16.8816−26.2352−26.6863−22.5088
4−30.0193−31.554−25.2733−40.0257−42.072−35.6977


𝑌 Graph 1Graph 2Graph 3Graph 4Graph 5Graph 6

00.77880.77880.77880.77880.77880.7788
10.541520.632840.726120.717320.683460.64277
20.000170.261570.54730.54260.420790.2822
3−0.46131−0.170840.205610.28220.055090.17695
4−0.74208−0.48685−0.29205−0.02274−0.32410.57428


yGraph 1Graph 2Graph 3Graph 4Graph 5Graph 6

00.77880.77880.77880.77880.77880.7788
10.717320.683460.642770.838860.87160.91158
20.54260.420790.28221.009111.126061.26137
30.28220.05509−0.176951.261371.476291.69947
4−0.02274−0.3241−0.574281.553631.830562.06203


Graph 1Graph 2Graph 3Graph 4Graph 5Graph 6

𝑀 0.10.20.10.10.10.2
𝐾 1 101020101010
𝑆 𝑐 0.40.40.40.80.40.4
𝑡 111133

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.

References

  1. R. M. Eckert and Darke, Analysis of Heat and Mass Transfer, McGraw Hill, New York, NY, USA, 1972.
  2. J. Basant Kumar and A. K. Singh, “Soret effects on free-convection and mass transfer flow in the stokes problem for a infinite vertical plate,” Astrophysics and Space Science, vol. 173, no. 2, pp. 251–255, 1990. View at: Publisher Site | Google Scholar
  3. P. G. Saffman, “On the stability of laminar flow of a dusty gas,” Journal of Fluid Mechanics, vol. 13, pp. 120–128, 1962. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. F. E. Marble, Annual Review of Fluid Mechanics, vol. 2, Annual Reviews, Inc., Palo Alto, Calif, USA, 1970.
  5. S. L. Soo, Fluid Dynamics of Multiphase Systems, Blaisdeel, Boston, Mass, USA, 1971.
  6. D. H. Michael and P. W. Norey, “The laminar flow of a dusty gas between two rotating cylinders,” Quarterly Journal of Mechanics and Applied Mathematics, vol. 21, no. 3, pp. 375–388, 1968. View at: Publisher Site | Google Scholar
  7. S. S. Rao, “Unsteady flow of dust viscous liquid through circular cylinder,” Def. Sci. Jour., vol. 19, p. 135, 1969. View at: Google Scholar
  8. P. D. Verma and A. K. Mathur, “A numerical solution of the flow of a dusty viscous liquid through of a circular cylinder,” Indian Journal of Pure and Applied Mathematics, vol. 4, p. 133, 1977. View at: Google Scholar
  9. D. Sing, Indian Journal of Physics, vol. 47, p. 341, 1972.
  10. E. Rukmangadachari, “Dusty viscous flow through a cylinder rectangular cross-section under time dependent pressure gradient,” Indian Journal of Pure and Applied Mathematics, vol. 9, no. 8, 1978. View at: Google Scholar
  11. P. J. Mitra, Math. Phys. Sci., vol. 13, no. 4, 1979.
  12. S. C. Gupta, “Unsteady MHD flow in a circular pipe under transverse magnetic field,” Indian Journal of Pure and Applied Mathematics, vol. 19, no. 6, 1979. View at: Google Scholar
  13. J. T. C. Liu, “Flow of a dusty gas through a channel arbitrary time varying pressure gradient,” Astronautica Actu., vol. 17, p. 851, 1972. View at: Google Scholar
  14. D. H. Michael and D. A. Miller, “Oscillatory of a plate in a binary mixture of incompressible Newtonian fluid,” Mathematika, vol. 13, p. 197, 1966. View at: Google Scholar
  15. J. T. C. Liu, “Flow induced by an oscillating infinite flat plate in a dusty gas,” Physics of Fluids, vol. 9, no. 9, pp. 1716–1720, 1966. View at: Google Scholar
  16. C. S. Vimal, Defence Science Journal, vol. 22, p. 231, 1972.
  17. P. Mitra, “Flow of non-Newtonian fluid,” Acta Ciencia Indica, vol. 6, no. 3, p. 144, 1980. View at: Google Scholar
  18. K. P. Singh, “Unsteady flow of a stratified viscous fluid through a porous medium between two parallel plates with variable magnetic induction,” Acta Ciencia India, vol. 25, no. 4, 1999. View at: Google Scholar
  19. P. Singh and C. B. Gupta, “Hall currents effects on MHD free convection flow with heat and mass transfer in the presence of radiation and heat source,” Jour PAS, vol. 8, pp. 193–204, 2002. View at: Google Scholar
  20. Singh and Varshney, Acta Ciencia Indica, vol. 32, no. 1, p. 301, 2005.

Copyright © 2012 Rajesh Kumar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

544 Views | 430 Downloads | 0 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.