Applied Mathematics and Mechanics

, Volume 34, Issue 2, pp 139–152 | Cite as

Viscous and Ohmic heating effects in doubly stratified free convective flow over vertical plate with radiation and chemical reaction

  • P. GanesanEmail author
  • R. K. Suganthi
  • P. Loganathan


An analysis is carried out to study the combined effects of viscous and Ohmic heating in the transient, free convective flow of a viscous, incompressible, and doubly stratified fluid past an isothermal vertical plate with radiation and chemical reactions. The governing boundary layer equations are solved numerically by an implicit finite difference scheme of the Crank-Nicolson type. The influence of different parameters on the velocity, the temperature, the concentration, the skin friction, the Nusselt number, and the Sherwood number is discussed with graphical illustrations. It is observed that an increase in either the thermal stratification or the mass stratification parameter decreases the velocity. An increase in the thermal stratification increases the concentration and decreases the temperature while an opposite effect is observed for an increase in the mass stratification. An augmentation in viscous and Ohmic heating increases the velocity and temperature while decreases the concentration. The results are found to be in good agreement with the existing solutions in literature.

Key words

thermal stratification mass stratification Ohmic dissipation viscous dissipation chemical reaction radiation 



spatial coordinate along the plate


dimensionless spatial coordinate along the plate


spatial coordinate normal to the plate


dimensionless spatial coordinate normal to the plate


velocity component along the plate


dimensionless velocity component along the X-direction


velocity component normal to the plate


dimensionless velocity component along the Y -direction


characteristic length




dimensionless time


temperature of the fluid


dimensionless temperature


concentration of the fluid


dimensionless concentration


thermal conductivity




magnetic parameter


thermal stratification parameter


mass stratification parameter


mass diffusion coefficient


thermal Grashof number


acceleration due to gravity


radiation parameter


chemical reaction parameter


dimensionless chemical reaction parameter


buoyancy ratio parameter


Prandtl number


Schmidt number


radiative heat flux


local Nusselt number

\(\overline {Nu} _X\)

average Nusselt number


local Sherwood number

\(\overline {Sh} _X\)

average Sherwood number

Greek symbols


thermal diffusivity


volumetric coefficient of thermal expansion


volumetric coefficient of expansion with concentration


kinematic viscosity


local skin friction

\(\bar \tau _X\)

average skin friction

Chinese Library Classification


2010 Mathematics Subject Classification

76D55 76R10 76R50 


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  1. [1]
    Siegel, R. Transient free convection from a vertical flat plate. ASME Journal of Heat Transfer, 80, 347–359 (1958)Google Scholar
  2. [2]
    Hellums, J. D. and Churchill, S. W. Transient and steady state, free and natural convection, numerical solutions, part I, the isothermal vertical plate. American Institute of Chemical Engireers Journal, 8, 690–692 (1962)CrossRefGoogle Scholar
  3. [3]
    Callahan, G. D. and Marner, W. J. Transient free convection with mass transfer on an isothermal vertical flat plate. International Journal of Heat and Mass Transfer, 19, 165–174 (1976)zbMATHCrossRefGoogle Scholar
  4. [4]
    Gebhart, B. and Pera, L. The nature of vertical natural convection flows resulting from the combined buoyancy effects of thermal and mass diffusion. International Journal of Heat and Mass Transfer, 14, 2025–2050 (1971)zbMATHCrossRefGoogle Scholar
  5. [5]
    Soundalgekar, V. M. and Ganesan, P. Finite difference analysis of transient free convection with mass transfer on an isothermal vertical flat plate. International Journal of Engineering Science, 19, 757–770 (1981)zbMATHCrossRefGoogle Scholar
  6. [6]
    Barvinschi, P. Numerical simulation of Ohmic heating in idealized thin-layer electrodeposition cells. Journal of Optoelectronics and Advanced Materials, 8, 271–279 (2006)Google Scholar
  7. [7]
    Gebhart, B. Effects of viscous dissipation in natural convection. Journal of Fluid Mechanics, 14, 225–232 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Soundalgekar, V. M., Lahurikar, R. M., and Pohanerkar, S. G. Transient free convection flow of an incompressible viscous dissipative fluid. Heat and Mass Transfer, 32, 301–305 (1997)CrossRefGoogle Scholar
  9. [9]
    Tacken, R. A. and Janssen, L. J. J. Applications of magnetoelectrolysis. Journal of Applied Electrochemistry, 25, 1–5 (1995)CrossRefGoogle Scholar
  10. [10]
    Takami, I., Hisayoshi, M., and Yasuhiro, F. Water electrolysis under a magnetic field. Journal of the Electrochemical Society, 154, 112–115 (2007)CrossRefGoogle Scholar
  11. [11]
    Palani, G. and Srikanth, U. MHD flow past a semi-infinite vertical plate with mass transfer. Nonlinear Analysis: Modelling and Control, 14, 345–356 (2009)Google Scholar
  12. [12]
    Ogulu, A. and Makinde, O. D. Unsteady hydromagnetic free convection flow of a dissipative and radiating fluid past a vertical plate with constant heat flux. Chemical Engineering Communications, 196, 454–462 (2009)CrossRefGoogle Scholar
  13. [13]
    Dulal, P. and Babulal, T. Buoyancy and chemical reaction effects on MHD mixed convection heat and mass transfer in a porous medium with thermal radiation and Ohmic heating. Communications in Nonlinear Science and Numeric Simulation, 15, 2878–2893 (2010)zbMATHCrossRefGoogle Scholar
  14. [14]
    Chen, C. H. Combined effects of Joule heating and viscous dissipation on magnetohydrodynamic flow past a permeable, stretching surface with free convection and radiative heat transfer. Journal of Heat Transfer, 132, 064503 (2010)CrossRefGoogle Scholar
  15. [15]
    Zueco, J. and Ahmed, S. Combined heat and mass transfer by mixed convection MHD flow along a porous plate with chemical reaction in presence of heat source. Appl. Math. Mech. -Engl. Ed., 31(10), 1217–1230 (2010) DOI 10.1007/s10483-010-1355-6MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Bikash, S. Effects of slip, viscous dissipation and Joule heating on the MHD flow and heat transfer of a second grade fluid past a radially stretching sheet. Appl. Math. Mech. -Engl. Ed., 31(12), 159–173 (2010) DOI 10.1007/s10483-010-0204-7zbMATHGoogle Scholar
  17. [17]
    Rao, J. A. and Shivaiah, S. Chemical reaction effects on unsteady MHD flow past semi-infinite vertical porous plate with viscous dissipation. Appl. Math. Mech. -Engl. Ed., 32(8), 1065–1078 (2011) DOI 10.1007/s10483-011-1481-6MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Chen, C. C. and Eichhorn, R. Natural convection from a vertical surface to a thermally stratified fluid. ASME Journal of Heat Transfer, 98, 446–451 (1976)CrossRefGoogle Scholar
  19. [19]
    Srinivasan, J. and Angirasa, D. Numerical study of double-diffusive free convection from a vertical surface. International Journal of Heat and Mass Transfer, 31, 2033–2038 (1988)CrossRefGoogle Scholar
  20. [20]
    Angirasa, D. and Srinivasan, J. Natural convection flows due to the combined buoyancy of heat and mass diffusion in a thermally stratified medium. ASME Journal of Heat Transfer, 111, 657–663 (1989)CrossRefGoogle Scholar
  21. [21]
    Saha, S. C. and Hossain, M. A. Natural convection flow with combined buoyancy effects due to thermal and mass diffusion in thermally stratified media. Non-linear Analysis: Modelling and Control, 9, 89–102 (2004)zbMATHGoogle Scholar
  22. [22]
    Takhar, H. S., Chamkha, A. J., and Nath, G. Natural convection flow from a continuously moving vertical surface immersed in a thermally stratified medium. Heat and Mass transfer, 38, 17–24 (2001)CrossRefGoogle Scholar
  23. [23]
    Muhaimin, I., Kandasamy, R., and Khamis, A. Numerical investigation of variable viscosities and thermal stratification effects on MHD mixed convective heat and mass transfer past a porous wedge in the presence of a chemical reaction. Appl. Math. Mech. -Engl. Ed., 30(11), 1353–1364 (2009) DOI 10.1007/s10483-009-1102-6zbMATHCrossRefGoogle Scholar
  24. [24]
    Rathish Kumar, B. V. and Shalini, G. Combined influence of mass and thermal stratification on double-diffusion non-Darcian natural convection from a wavy vertical wall to porous media. Journal of Heat Transfer, 127, 637–647 (2005)CrossRefGoogle Scholar
  25. [25]
    Sparrow, E. M. and Cess, R. D. Radiation Heat Transfer, Hemisphere Publishing Corporation, Washington, D. C. (1978)Google Scholar
  26. [26]
    Herrmann Schlichting. Boundary Layer Theory, John Wiley and Sons, New York (1969)Google Scholar
  27. [27]
    Carnahan, B., Luther, H. A., and Wilkes, J. O. Applied Numerical Methods, John Wiley and Sons, New York (1969)zbMATHGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsAnna University ChennaiChennaiIndia

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