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 propose a time-advancing scheme for the discretization of the unsteady incompressible Navier-Stokes equations. At any time step, we are able to decouple velocity and pressure by solving some suitable elliptic problems. In particular, the problem related with the determination of the pressure does not require boundary conditions. The divergence free condition is imposed as a penalty term, according to an appropriate restatement of the original equations. Some experiments are carried out by approximating the space variables with the spectral Legendre collocation method.

A generalization of mesoscopic Lattice-Boltzmann models aimed at describing flows with solid/liquid phase transitions is presented. It exhibits lower computational costs with respect to the numerical schemes resulting from differential models, Moreover it is suitable to describe chaotic motions in the mushy zone.

A recent approach to generate a zero divergence velocity field by operating directly on the discretized Navier-Stokes equations is used to obtain the decoupling of the pressure from the velocity field. By following the methodology suggested by Amit, Hall, and Porsching the feasibility of treating three dimensional flows and multiply connected domains is analyzed. The present model keeps the main features of the classical vector potential method in that it generates a divergence-free velocity field through an algebraic manipulation of the discrete equations.

The non-inertial flow of a shear thinning fluid between intersecting planes is studied using a multi-parameter continuation technique. Unlike the classical linearly viscous fluid, it is found that boundary layers develop even in the case of non-inertial flows in both converging and diverging flow. The boundary layers develop due to the non-linearities in the equation which reflect the fact that the fluid can shear thin. © 1991.