Convective instability of a fluid layer in a modulated external force field: PMM vol. 36, n≗1, 1972, pp. 152–157

Abstract

The onset of convection in a layer of an incompressible fluid with free boundaries is considered. The temperature at the layer boundaries, the density of the internal heat sources and the strength of the gravity field are all assumed to be T-periodic. The existence of the critical Rayleigh number and the T-periodicity of the neutral perturbation are proved for the case when the unperturbed temperature gradient is negative throughout the layer. These results are obtained by reducing the linearized problem to an ordinary differential equation in certain Banach space and applying the theory of the linear positive operators [1]. The onset of convection under the action of time-periodic forces is dealt with in [2–9]. The stability of equilibrium of a horizontal layer with free and rigid boundaries was investigated and numerical methods were used to determine the limits of stability [2] under the assumption that the temperature gradient was independent of the vertical coordinate. The method of averaging over small oscillations was used in [3, 4] to study the influence of high frequency vertical oscillations on the onset of convection. Use of the method of averaging for the abstract parabolic equations, in particular for the convection problem, was substantiated in [5, 6]. Convection in a cavity of square cross section in the case when the fluid is heated from below and is acted upon by vibrational forces, is studied in [7] and a numerical solution given for the nonlinear convection equations. The stability of equilibrium in the case when the temperature gradient depends on the vertical coordinate is studied in [8, 9], namely the convection in a deep vessel the surface temperature in which varies periodically with time is dealt with in [8], and the convection in a horizontal layer with the temperature varying periodically at the free boundaries, the amplitude of the modulations being small, in [9].