Steady 3D flow configurations for the horizontal thermal convection with thermocapillary effects

A vast literature exists on the Benard flow, the vertical thermal convection flow, but almost no result is known on the horizontal counterpart. On account of the wide range of applications in geophysics, astrophysics, metereology, and material science; we think that the horizontal thermal convection flow deserves as much consideration as the Benard problem. The present study is the first step towards the description of the bifurcation pattern of the horizontal thermal convection flow. We present several flow configurations of an incompressible Navier-Stokes fluid contained in a 4 × 1 × 1 parallelepipedic box with open top and vertical transversal walls at different temperature. The flow induced by the buoyancy force and the thermocapillary effects at the fluid/air interface is numerically computed within the Boussinesq approximation. The velocity-vorticity formulation with a fully implicit finite difference method are implemented in a parallel computational code. The strong coupling of the discrete equations ensures the correct mass balance at each time step and provides true-transient numerical flows. For fluids at Prandtl number Pr = 0.015 (semiconductors and liquid metals), we describe the changes in shape of the main vortex and the intensity of the speed of the flow versus variations of the Grashof number. In the case of absence of thermocapillary effects, we observe that as Gr increases, the flow exhibits more and more three-dimensional effects. When we add the thermocapillary effects, we observe an increase of the speed of the flow with the formation of a steep boundary layer around the liquid/air interface.