Open Access

ARTICLE

The Chebyshev Tau Spectral Method for the Solution of the Linear Stability Equations for Rayleigh-Bénard Convection with Melting

by Rubén Avila¹, Eduardo Ramos², S. N. Atluri³

1 Departamento de Termofluidos, Facultad de Ingeniería, Universidad Nacional Autónoma de México, Mexico D.F. C.P. 04510, ravila@servidor.unam.mx
2 Centro de Investigación en Energía, Universidad Nacional Autónoma de México, AP. P. 34, 62580, Temixco Mor. México, erm@mazatl.cie.unam.mx
3 Center for Aerospace Research & Education, University of California, Irvine

Computer Modeling in Engineering & Sciences 2009, 51(1), 73-92. https://doi.org/10.3970/cmes.2009.051.073

Abstract

A Chebyshev Tau numerical algorithm is presented to solve the perturbation equations that result from the linear stability analysis of the convective motion of a fluid layer that appears when an unconfined solid melts in the presence of gravity. The system of equations that describe the phenomenon constitute an eigenvalue problem whose accurate solution requires a robust method. We solve the equations with our method and briefly describe examples of the results. In the limit where the liquid-solid interface recedes at zero velocity the Rayleigh-Bénard solution is recovered. We show that the critical Rayleigh number Ra_c and the critical wave number a_c are monotonically decreasing functions of the rate of melting of the solid. We conclude that the parameters Ra_c and a_c are independent functions of the Prandtl number in the range 1 ≤ Pr ≤ 10,000. We also show that as the Pr number is reduced, Pr < 1, the critical parameters are nonmonotonic functions of the rate of melting.

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