Buoyancy Driven Convection in a Liquid Layer with Insulating Permeable Boundaries


  • A. K. Gupta
  • S. K. Kalta


Buoyancy, Convection, Insulating, Linear stability, Permeable, Slip velocity


In this paper, we investigate the onset of buoyancy driven thermal convection in a horizontal layer of fluid with thermally insulating permeable boundaries, using the classical linear stability analysis. It is proved that the principle of exchange of stabilities is valid. The eigenvalue problem is solved by using the Galerkin method. Results are obtained and discussed  for a wide range of values of the boundary parameters characterizing the permeable nature of boundaries. Attention is focused on a situation where the value of the critical Rayleigh number is less than that for the case when one of the boundaries is rigid while the other one is free and the convection is not maintained in general. In the case, when permeability parameter of either one of the two boundaries varies inversely to that of the other, we discover that the critical Rayleigh number decreases and goes through a lowest minimum at a certain value of the permeability parameter and this situation pertains when the critical wave number is zero. In addition, existing results for various combinations of the boundary conditions namely, when both the bounding surfaces are either dynamically free or rigid and when either one of them is dynamically free while the other one is rigid, are obtained as limiting cases of the boundary parameters.


H. Bénard, “Les tourbillons cellulaires dans une nappe liquide, Rev. Gén. Sci. pures et Appl.â€, Vol.11, pp.1261-1271, 1900.

H. Bénard, “Les tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en régime permanent, Ann. Chim. Phys.â€, Vol. 23, 62-144, 1901.

L. Rayleigh, “On convection currents in a horizontal layer of fluid, when the higher temperature is on the undersideâ€, Phil. Mag. Vol. 32, pp. 529-546, 1916.

S. Chandrasekhar, “Hydrodynamic and hydromagnetic stabilityâ€, London, Oxford University Press, 1961.

D. A. Nield, “The effect of permeable boundaries in the Bénard convection problemâ€, J. Math. Phys. Sci., Vol. 26, pp. 341-343, 1992.

A. K. Gupta, R. G. Shandil and S. Kumar, “On Rayleigh Bénard convection with porous boundariesâ€, Proc. Natl. Acad. Sci. Sect. A, Phys. Sci. Vol. 83, pp. 365-369, 2013.

G. S. Beavers and D. D. Joseph “Boundary conditions at a naturally permeable wallâ€, J. Fluid Mech., Vol. 30, pp.197-207, 1967.

A. Pellew and R. V. Southwell, “On the maintained convective motion in a fluid heated from belowâ€, Proc. Roy. Soc. London Ser. A, Vol. 176, pp. 312-343, 1940.

B. A. Finlayson, “The method of weighted residuals and variational principlesâ€, (New York Academic Press), 1972.

E. M. Sparrow, R. J. Goldstein, and V. K. Jonsson “Thermal instability in horizontal fluid layer: effects of boundary conditions and non-linear temperature profileâ€, J. Fluid Mech., Vol. 18, pp. 513-528, 1964.




How to Cite

Gupta, A. K., & Kalta, S. K. (2017). Buoyancy Driven Convection in a Liquid Layer with Insulating Permeable Boundaries. Asian Journal of Fuzzy and Applied Mathematics, 5(1). Retrieved from https://ajouronline.com/index.php/AJFAM/article/view/4598