# Effect of Non-Uniform Temperature Gradient on the Onset of Rayleigh–Bénard–Magnetoconvection in Micropolar Fluid with Maxwell–Cattaneo Law

### Abstract

The effect of non-uniform temperature gradient on the onset of Rayleigh-Bénard magnetoconvection in a Micropolar fluid with Maxwell-Cattaneo law is studied using the Galerkin technique. The eigenvalue is obtained for free-free, rigid-free and rigid-rigid velocity boundary combinations with isothermal 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 of convection 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

I G Currie, The effect of heating rate on the stability of stationary fluids, J. Fluid Mechanics, vol. 29, pp. 337-347, 1967.

D A Nield, The onset of transient convective instability, J Fluid Mech., vol. 71, pp. 3-11, 1975.

G Lebon and A Cloot, Effects of non-uniform temperature gradients on Bénard-Marangoni instability, J. Non-Equin. Thermodyn., vol. 6, pp. 15-30, 1981.

H Power, Bio-Fluid Mechanics, Advances in fluid mechanics, W.I.T. Press, UK, 1995.

Lukaszewicz, Micropolar fluid theory and applications, Birkhauser Boston, M. A., USA, 1998.

A C Eringen, Microcontinuum fluid theories, Springer Verlag, 1999.

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.

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, 1976.

P G Siddheshwar and S Pranesh, Magnetoconvection in a micropolar fluid, Int. J. Engng. Sci., vol. 36, pp. 1173-1181, 1998.

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 J. Engineering and Materials Sciences, vol. 8, pp. 77-83, 2001.

P G Siddheshwar and S Pranesh, Effects of a non-uniform basic temperature gradient on Rayleigh-Bénard convection in a micropolar fluid, International Journal of Engineering Science, vol. 36, pp. 1183-1196, 1998.

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, vol. 192, pp. 159-176, 1999.

P G Siddheshwar and S Pranesh, Magnetoconvection in fluids with suspended particles under 1g and g, Int. J. Aerospace Science and Technology, vol. 6, pp. 105-114, 2001.

P G Siddheshwar and S Pranesh, Linear and weakly non-Linear analyses of convection in a micropolar fluid, Hydrodynamics VI-Theory and Applications–Cheng and Yow (Eds.), pp. 487-493, 2005.

S Pranesh and Riya Baby, Effect of non-uniform temperature gradient on the onset of Rayleigh-Bénard Electroconvection in a micropolar Fluid, Applied Mathematics, vol.3, pp.442-450, 2012.

J C Maxwell, On the dynamical theory of gases, Phil. Trans. R. Soc. London, vol. 157, pp. 49-88, 1867.

C Cattaneo, Sulla condizione del Calore, Atti del Semin.Matem.e Fis, Della Univ. Modena, vol. 3, pp. 83-101, 1948.

Lindsay and B Stranghan, Penetrative convection instability of a micropolar fluid, Int. J. Engng. Sci., vol. 12, p. 1683, 1992.

B Straughan and F Franchi, Bénard convection and the Cattaneo law of heat conduction, Proc. R. Soc. Edin., vol. 96A, pp. 175-178, 1984.

S Pranesh and R V Kiran, Study of Rayleigh-Bénard magneto convection in a micropolar fluid with Maxwell–Cattaneo law, Applied Mathematics, vol. 1, pp. 470-480, 2010.

S Pranesh and S N Smita, Rayleigh-Bénard convection in a second-order fluid with Maxwell-Cattaneo law, Bulletin of Society for mathematical services and standard, vol. 1, No. 2, pp. 33-48, 2012.

P Puri and P M Jordan, Stokes’s first problem for a dipolar fluid with nonclassical heat conduction, J. Engineering Mathematics, vol. 36, pp. 219-240, 1999.

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.

P Puri and P K Kythe, Nonclassical thermal effects in stokes second problem, Acta Mech., vol. 112, pp. 1-9, 1995.

P Puri and P K Kyth, Discontinuities in velocity gradients and temperature in the Stokes first problem with nonclassical heat conduction, Quart. Appl. Math., vol. 55, pp. 167-176, 1997.

B Straughan, Oscillatory convection and the Cattaneo law of heat conduction, Ricerche mat., vol. 58, pp. 157-162, 2009.