On pressure boundary conditions for thermoconvective problems

Solving numerically hydrodynamical problems of incompressible fluids arises the question of handling first order derivatives (those of pressure) in a closed container and determining its boundary conditions. A way to avoid the first point is to derive a Poisson equation for pressure, although the problem of taking the right boundary conditions still remains. To remove this problem another formulation of the problem has been used consisting of projecting the master equations into the space of divergence free velocity fields, so pressure is eliminated from the equations. This technique raises the order of the differential equations and additional boundary conditions may be required. Besides, high order derivatives are sometimes troublesome, specially in cylindrical coordinates due to the singularity at the origin. For these problems a low order formulation such as the primitive one is very convenient. We researched several pressure boundary conditions for the original formulation of thermoconvective problems. In particular we study the Marangoni instability of an infinite fluid layer and we show that the numerical results with a Chebyshev collocation method are highly correspondent to the analytical ∗Present address: School of Mathematics, University of Bristol. University Walk, Bristol, BS8 1TW. United Kingdom.