Simulation of the convection in a porous medium in polar coordinates on a non-uniform grid

The problem of convection in an incompressible, heat-conducting fluid that saturates a long horizontal cylinder filled with a porous medium is considered. Based on the Darcy model and the equations in cylindrical coordinates, a scheme with staggered grids, a non-uniform distribution of nodes, as well as a special approximation near the pole, is developed. To discretize the problem in natural variables, an integro-interpolation method is used. The discrete transition to equations in terms of the stream function and temperature is performed. To determine the critical values of the Rayleigh number, formulas based on the zeros of Bessel functions are derived. The results of calculating the critical Rayleigh numbers are presented, Experiments were performed for a nonuniform distribution of nodes. The convective regimes that branching off of mechanical equilibrium are calculated.