Finite Difference Formulation for the Model of a Compressible Viscous and Heat-Conducting Micropolar Fluid with Spherical Symmetry

Abstract

We are dealing with the nonstationary 3D flow of a compressible viscous heat-conducting micropolar fluid, which is in the thermodynamical sense perfect and polytropic. It is assumed that the domain is a subset of R3 and that the fluid is bounded with two concentric spheres. The homogeneous boundary conditions for velocity, microrotation, heat flux, and spherical symmetry of the initial data are proposed. By using the assumption of the spherical symmetry, the problem reduces to the one-dimensional problem. The finite difference formulation of the considered problem is obtained by defining the finite difference approximate equation system. The corresponding approximate solutions converge to the generalized solution of our problem globally in time, which means that the defined numerical scheme is convergent. Numerical experiments are performed by applying the proposed finite difference formulation. We compare the numerical results obtained by using the finite difference and the Faedo–Galerkin approach and analyze the properties of the numerical solutions.