Instabilities in a laterally heated liquid layer

We study a convection problem in a free-surface container with lateral wall s heated at different temperatures. The effects of buoyancy and thermocapil larity are taken into account. A basic convective state appears as soon as a temperature gradient with nonzero horizontal component is applied. This s tate bifurcates to new convective solutions for further values on the impos ed temperature gradient. Our main contribution is to consider this situatio n in a container finite not only in the vertical coordinate, but also in th e direction of the gradient. The third dimension is kept infinite. We deter mine the basic state, compare it with the usual one of parallel flow approa ch, and study its stability. When the lateral heating walls are considered new results are found. The boundary conditions on the top surface are no lo nger restricted to those that allow analytical solutions for the basic stat e, and we have considered for the heat interchange with the atmosphere the Newton law with constant ambient temperature. Due to this boundary conditio n, two control parameters related to the temperature field appear. One is t he temperature difference between lateral walls as in previous research, an d the new one is the temperature difference between the atmosphere and the cold wall. After a stationary bifurcation a three-dimensional structure whi ch along the infinite direction consists of longitudinal rolls grows. On th e vertical plane along the gradient direction this structure is nonhomogene ous but located near the hot side. These features coincide with observation s of recent experiments. (C) 2000 American Institute of Physics. [S1070-663 1(00)01904-8].