Thermal instability of a viscosity stratified fluid layer heated from below

Abstract

T HE stability of a fluid layer, under Boussinesq's approximation with viscosity n ( z ) heated from below between two parallel planes (at z = 0 and 1) is investigated under linear theory by the normal mode technique. The theoretical foundations for the correct interpretation of the problem of the onset of thermal instability in horizontal layers of fluid heated from below were laid by Rayleigh in a fundamental paper. He proved that the principle of the exchange of stabilities is valid for this problem for the case of two nondeformable free boundaries. The proof for the general case is due to Pellew and South well. 2 They also proved the existence of a variational principle. Many authors, after that, have discussed the stability of a fluid layer heated from below. All the authors have considered the viscosity to be constant. For the nonheatconducting, density, and viscosity stratified fluid, Chandrasekhar, 7 Hide, and Fan have made an analytical treatment of its stability. Banerjee and Kalthia 10 obtained some bounds for the growth rate of arbitrary small disturbances, and a sufficient condition for the stability. Chandra has modified their results, by relaxing the condition on the sign of D^. In this Note, the viscosity of the fluid is taken to be stratified in the vertical direction. It is proved that the transition from stability to instability must occur via the stationary state, and the solution of the characteristic value problem at the marginal state can be expressed in terms of variational principles. Critical Rayleigh number is obtained from the first-order solution to the eigenvalue problem. Consider a steady thermally stratified viscous fluid layer between two horizontal boundaries at z = 0 and d, which are maintained at constant temperatures T0 and Tj, respectively. Following the usual procedure of linearization of perturbation equations, assuming the functional dependence of the perturbed quantities on time and space coordinates of the form