Perturbation, approximation and optimal control of a system governed by the Navier-Stokes equations coupled to the heat equation

A Cauchy-Kowaleska system is used to approximate the Navier-Stokes equations coupled with the heat equation, giving variational formulations of the perturbed and nonperturbed problems. A finite difference approximation of these equations is also given. A specific numerical example is considered: the stability of a fluid in mechanical equilibrium but in thermodynamic nonequilibrium with attention given to the case of the two-dimensional laminar flow induced by a temperature gradient in a closed cavity. Cauchy-Kowaleska and finite difference approximations are then applied to the problem of the optimal control of the system of equations under consideration. The calculation of temperature distribution in the two-dimensional laminar flow of a fluid in a square cavity is given as a numerical example.