Low Prandtl number fluid convection modelled using symbolic algebra (REDUCE) and Matlab

Abstract

Using the Boussinesq approximation for a fluid of low Prandtl number, a low dimensional model of the onset of Rayleigh-Benard convection is developed. The initial roll mode instability is considered for a fluid, heated from below, between parallel, horizontal, non-slip, constant-temperature boundaries. Centre manifold theory provides a way of constructing a low dimensional model of the resulting two dimensional flow. Computer algebra implemented in reduce is used to symbolically expand the centre manifold as an asymptotic series in the convective amplitude. The spatial structure functions in this expansion are then found numerically in Matlab. A feature of this approach is that code output from reduce is used, with only minor syntactic editing, as the Matlab code to perform the numerical iteration. Thus a coding task which would have been difficult by hand is easily automated. The technique is generally applicable to perturbation expansions and its computational advantages over more formal Galerkin type expansions are discussed.