A numerical study of the buoyancy driven natural convection of liquid materials is presented. There is experimental evidence that at low values of the Grashof number (Cr) the flow is laminar and steady while is periodic and generally time dependent as the value of Cr increases. This behaviour affects the products in many applications, for instance the quality of semi-conductor crystals grown by the Czochralski and Bridgman techniques. The bifurcation pattern of this flow has been almost completely identified in the two dimensional case whereas in three dimensions we cannot mention any published result due to the inadequacy of the computational resources. In this work we shall present the flow configurations occuring in the case of a shallow (4.1.1) box where the temperature gradient is applied at the smaller vertical faces. The fluid considered has Prandtl number equal to 0.015. The vorticity-velocity formulation of the Navier-Stokes equations has been integrated by means of a fully implicit finite difference scheme. By starting from a steady flow configuration as Cr increases we shall describe the transition to a periodic flow, and then from this one the transition to a non-periodic time-dependent flow.
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