Instabilities in a laterally heated liquid layer

Abstract

We study a convection problem in a free-surface container with lateral walls heated at different temperatures. The effects of buoyancy and thermocapillarity are taken into account. A basic convective state appears as soon as a temperature gradient with nonzero horizontal component is applied. This state bifurcates to new convective solutions for further values on the imposed temperature gradient. Our main contribution is to consider this situation in a container finite not only in the vertical coordinate, but also in the direction of the gradient. The third dimension is kept infinite. We determine the basic state, compare it with the usual one of parallel flow approach, and study its stability. When the lateral heating walls are considered new results are found. The boundary conditions on the top surface are no longer restricted to those that allow analytical solutions for the basic state, and we have considered for the heat interchange with the atmosphere the Newton law with constant ambient temperature. Due to this boundary condition, two control parameters related to the temperature field appear. One is the temperature difference between lateral walls as in previous research, and the new one is the temperature difference between the atmosphere and the cold wall. After a stationary bifurcation a three-dimensional structure which along the infinite direction consists of longitudinal rolls grows. On the vertical plane along the gradient direction this structure is nonhomogeneous but located near the hot side. These features coincide with observations of recent experiments.