Periodic modulation of an equilibrium temperature gradient in a fluid layer and a saturated porous medium layer

Abstract

A linear problem of equilibrium stability in a heated from below two-layer system of a pure fluid layer and a saturated porous medium layer in the presence of varying temperature gradient under gravity is investigated. The problem is solved in the framework of Floquet theory. Numerical calculations are carried out on the basis of shooting method with orthogonalization and Galerkin method. The rectangular periodic modulation of heat flux is considered. This study is limited by the low-frequency modulation when one can neglect the spatial inhomogeneity of temperature gradient. Originally we present neutral curves of equilibrium stability in the presence of constant temperature gradient. After that we describe the stability maps obtained in the conditions of its periodic modulation. Resonance regions of the parametric instability with respect to the harmonic (with the period equal to the period of modulation) and subharmonic (with the period twice as large as the period of modulation) perturbations of the equilibrium were determined for various values of the effective Rayleigh number. The region limiting the baseband of instability was found. It was shown that at certain values of the frequency and modulation amplitude the convective flow could arise in the system only due to the periodic oscillations of temperature at its boundaries when the average temperature gradient was zero. The effect of conditions at the interface between the fluid and porous layers on the onset of convection in the system was studied. It was determined that perturbations of the smaller wave length, generally located in the fluid layer, were most affected by conditions at the interface between layers in contrast to the perturbations of the larger wave length propagating inside the saturated porous medium.