The chebyshev tau spectral method for the solution of the linear stability equations for rayleigh-bénard convection with melting

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 Rac and the critical wave number ac are monotonically decreasing functions of the rate of melting of the solid. We conclude that the parameters Rac and ac 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.