Vol. 14 No. 3 (2015): Mapana Journal of Sciences
Research Articles

Gradient on Rayleigh–Bénard – Marangoni – Magnetoconvection in a Micropolar Fluid with Maxwell – Cattaneo Law

  • R V Kiran,
  • Attluri Kalyani
R V Kiran
Christ Junior College, Hosur Road, Bangalore 560 029, India;
Bio
Attluri Kalyani
Department of Mathematics, Christ University, Hosur Road, Bangalore 560 029, India
Bio
DOI: https://doi.org/10.12723/mjs.34.1

Published 2021-08-28

Keywords

  • Rayleigh-Bénard-Marangoni-Magneto-convection,
  • Maxwell-Cattaneo Law,
  • Micropolar fluid

Copyright (c) 2015

Creative Commons License

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Abstract

The effect of non-uniform temperature gradient on the onset of Rayleigh-Bénard-Marangoni- Magneto-convection in a Micropolar fluid with Maxwell-Cattaneo law is studied using the Galerkin technique. The eigen value is obtained for rigid-free velocity boundary combination with isothermal and adiabatic condition on the spin-vanishing boundaries. A linear stability analysis is performed. The influence of various parameters on the onset of convection has been analyzed. One linear and five non-linear temperature profiles are considered and their comparative influence on onset is discussed. The classical approach predicts an infinite speed for the propagation of heat.  The present non-classical theory involves a wave type heat transport (Second Sound) and does not suffer from the physically unacceptable drawback of infinite heat propagation speed.

References

  1. I. G. Currie, "The effect of heating rate on the stability of stationary fluids", Journal of Fluid Mechanics, vol. 29, pp. 337-347., 1967.
  2. D. A. Nield, “The onset of transient convective instability,” J Fluid Mech., vol. 71, 1975, pp. 3-11., 1975.
  3. Lebon. G., and Cloot. A., “Effects of non-uniform temperature gradients on Bénard-Marangoni instability,” J. Non-equin. Thermodyn., vol. 6, pp. 15-30, 1981.
  4. J. R. A. Pearson, “On convection cells induced by surface tension,” J. Fluid Mech., vol. 4, pp.489, 1958.
  5. M. Takashima, “Surface tension driven instability in a horizontal liquid layer with a deformable free surface Part I steady convection,” J. Phys. Soc. Japan, vol. 50, no. 8, pp. 27-45, 1981.
  6. M. Takashima, “Surface tension driven instability in a horizontal liquid layer with a deformable free surface Part II Overstability”. J. Phys. Soc. Japan, vol. 50, no. 8, pp. 27-51, 1981.
  7. K. A. Smith, “On convective instability induced by surface-tension gradients,” J. Fluid Mech. vol. 24, part 2, pp. 401, 1966.
  8. A.P. Garcia, and G. Carneiro, “Linear stability analysis of Bénard-Marangoni convection in fluids with a deformable free surface”. Phys. Fluids A, vol. 3, pp. 292, 1991.
  9. Rudraiah, N, “Surface tension driven convection subjected to rotation and non-uniform temperature gradient,” Mausam, vol. 37, pp. 39, 1986.
  10. T. Maekawa, and I. Tanasuwa, “Effect of magnetic field on onset of Marangoni convection”. Int. J. Heat Mass Transfer, vol. 31, no. 2, pp. 285, 1989.
  11. G. S. R. Sarma, “Marangoni convection in a fluid layer under the action of a transverse magnetic field,” Space Res. Vol. 19, pp. 575, 1979.
  12. G. S. R. Sarma, “Marangoni convection in a liquid layer under the simultaneous action of a transverse magnetic field and rotation”. Adv. Space Res. vol. 1, pp. 55, 1981.
  13. S. K. Wilson, “The effect of a uniform magnetic field on the onset of Marangoni convection in a layer of conducting fluid,” J. Mech. Appl. Math. vol. 46, pp. 211, 1993.
  14. S. K. Wilson, “ The effect of a uniform magnetic field on the onset of steady Benard-Marangoni convection in a layer of conducting fluid”. J. Engng Maths. vol. 27, , pp. 161, 1993.
  15. I. Hashim, I and S. K. Wilson, “The effect of a uniform vertical magnetic field on the onset of oscillatory Marangoni convection in a horizontal layer of conducting fluid,” Acta Mechanica. vol. 132, pp. 129, 1999.
  16. N. Rudraiah, and P.G. Siddheshwar, “Effects of non uniform temperature gradient on the onset of Marangoni convection in a fluid with suspended particles”. Journal of Aerospace Science and Technology, vol. 4, pp. 517, 2000.
  17. H. Power, “Bio-Fluid Mechanics,” Advances in fluid mechanics, W.I.T. Press, UK, vol. 3, pp. 336, 1995.
  18. Lukaszewicz, Micropolar fluid theory and applications, Birkhauser Boston, M. A., USA, 1998.
  19. A. C. Eringen, Micro continuum fluid theories, Springer Verlag, 1999.
  20. A. B. Datta, and V. U. D. Sastry, “Thermal instability of a horizontal layer of micropolar fluid heated from below,” Int. J. Engg. Sci., vol. 14, pp. 631-637, 1976.
  21. S. P. Bhattacharya, and S. K. Jena, “ Thermal instability of a horizontal layer of micropolar fluid with heat source” , Int. J. Engg. Sci., vol. 23, pp. 13-26, 1984.
  22. P. G. Siddheshwar, and S. Pranesh , “Magnetoconvection in a micropolar fluid,” Int.J.Engng. Sci. USA, vol. 36, pp. 1173-1181, 1998.
  23. P. G. Siddheshwar, and S. Pranesh , “Effect of temperature / gravity modulation on the onset of magneto-convection in weak electrically conducting fluids with internal angular momentum,” Journal of Magnetism and Magnetic Materials, USA, vol. 192, pp. 159-176, 1999.
  24. P. G. Siddheshwar, and S. Pranesh, “Magnetoconvection in fluids with suspended particles under 1g and g,” International Journal of Aerospace Science and Technology, France, vol. 6, pp. 105-114, 2001.
  25. P. G. Siddheshwar, and S. Pranesh, Linear and weakly non-Linear analyses of convection in a micropolar fluid, Hydrodynamics VI-Theory and Applications – Cheng & Yow (eds), pp. 487 – 493, 2005.
  26. P. G. Siddheshwar, and S. Pranesh, “Effects of non-uniform temperature gradients and magnetic field on the onset of convection in fluids with suspended particles under microgravity conditions,” Indian Journal of Engineering and Materials Sciences, vol. 8, pp. 77-83, 2001.
  27. P. G. Siddheshwar, and S. Pranesh, “Effects of a non-uniform basic temperature gradient on Rayleigh-Benard convection in a micropolar fluid,” International Journal of Engineering Science, USA, vol. 36, pp. 1183-1196, 1998.
  28. S. Pranesh, and Riya Baby, “Effect of non-uniform temperature gradient on the onset of Rayleigh-Bénard electro convection in a micropolar fluid”. Applied Mathematics, 3, issue 5, pp. 442, 2012.
  29. S. Pranesh and T. V.Joseph, “Effect of Non-Uniform Basic Temperature Gradient on the Onset of Rayleigh-Bénard-Marangoni Electro-Convection in a Micropolar Fluid,” Applied Mathematics, vol. 4, pp. 1180, 2013.
  30. J.C. Maxwell, “On the dynamical theory of gases”. Phil. Trans. R.Soc. London, vol. 157, pp. 49-88, 1867.
  31. C. Cattaneo, “Sulla condizione del Calore,” Atti del Semin. Matem.e Fis, Della Univ. Modena, vol. 3, pp. 83-101, 1948.
  32. Lindsay. and B. Stranghan, “Penetrative convection instability of a Micropolar fluid,” Int.J.Engng.Sci., vol. 12, pp. 1683, 1992.
  33. B. Straughan B, and F. Franchi, “Benard convection and the Cattaneo law of heat conduction,” Proc.R.Soc. Edin, vol. 96A, pp. 175-178, 1984.
  34. S. Pranesh and R.V. Kiran, “Study of Rayleigh-Bénard magneto convection in a Micropolar fluid with Maxwell – Cattaneo law,” Applied Mathematics, 1, pp. 470-480, 2010.
  35. S. Pranesh and R.V. Kiran, “Effect of non-uniform temperature gradient on the onset of Rayleigh–Bénard–Magnetoconvection in a Micropolar fluid with Maxwell–Cattaneo law,” Mapana Journal of sciences, vol. 23, pp.195, 2012.
  36. S. Pranesh and R.V. Kiran, “The study of effect of suction-injection-combination (SIC) on the onset of Rayleigh-Bénard-Magnetoconvection in a micropolar fluid with Maxwell-Cattaneo law,” American Journal of Pure Applied Mathematics, vol.2, no.1, , pp. 21, 2013.
  37. S. Pranesh and R.V. Kiran, “Rayleigh-Bénard Chandrasekhar Convection in An Electrically Conducting Fluid using Maxwell-Cattaneo Law with Temperature Modulation of the Boundaries,” International Journal of Engineering Research & Technology, vol. 4, issue. 10, pp. 174, 2015.
  38. P. Puri and P.M, “Jordan Stokes’s first problem for a dipolar fluid with a non classical heat conduction,” Journal of Engineering Mathematics, vol. 36, pp. 219-240, 1999.
  39. P. Puri and P. M. Jordan, “Wave structure in stokes second problem form dipolar fluid with nonclassical heat conduction,” Acta Mech., vol. 133, pp. 145-160, 1999.
  40. P. Puri and P. K. Kythe, “Nonclassical thermal effects in stokes second problem,” Acta Mech., vol. 112, pp. 1-9, 1995.
  41. P. Puri and P. K. Kythe, “Discontinuities in velocity gradients and temperature in the stokes first problem with nonclassical heat conduction,” Quart.Appl.Math., vol 55, pp. 167-176, 1997.
  42. B. Straughan,” Oscillatory convection and the Cattaneo law of heat conduction,” Ricerche mat., vol. 58, pp. 157-162, 2009.