Nonlinear longwave stability of two liquid layers coating both sides of a thick wall in presence of gravity

Abstract

The linear and nonlinear thermocapillary instability of two liquid layers coating both sides of a horizontal solid wall in the presence of gravity was investigated in the absence of thermal buoyancy. A set of two coupled nonlinear evolution equations was calculated under the small wavenumber approximation. The two liquid layers were horizontal and had thermal interaction through the wall by means of a temperature gradient perpendicular to the two layers and the wall. The linear growth rate of the system instability was investigated with respect to a variety of parameters. In the same way as in a previous report in the absence of gravity, here the second stationary mode was always stable. In the presence of gravity it was possible to have stationary convection alone, stationary and oscillatory convection and oscillatory convection alone, depending on the magnitudes and signs of the parameters. It was also shown that the linear stability can be described by means of three parameters alone, different to those of a previous paper. Numerical analysis of the nonlinear coupled evolution equations of the free surfaces deformations was performed to find out which conditions lead to the sinuous or to the varicose mode of instability.