doi: 10.17586/2226-1494-2016-16-1-161-167


BENCHMARK SOLUTIONS FOR STOKES EQUATIONS WITH VARIABLE VISCOSITY IN CYLINDRICAL AND SPHERICAL COORDINATES

I. V. Makeev, I. Y. Popov, I. V. Blinova


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For citation: Makeev I.V., Popov I.Yu., Blinova I.V. Benchmark solutions for Stokes equations with variable viscosity in cylindrical and spherical coordinates for algorithm testing. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2016, vol. 16, no. 1, pp. 161–167.

Abstract

Stokes flows in cylindrical and spherical geometry are considered. Such flows are rather natural for geophysics. We derive some exact particular solutions of Stokes and continuity equations for particular dependence of viscosity and density on cylindrical coordinates. These solutions correspond to axisymmetric flows for the case when viscosity is a function of radius. We suggest exact particular solutions of Stokes and continuity equations with variable viscosity and density in spherical coordinates for the case of spherically symmetric viscosity and density distributions. We demonstrate how these solutions can be used for creation of test problems suitable for benchmarking numerical algorithms. Examples of such benchmarking are presented. The advantage of this benchmarking approach is the ability to test numerical algorithms for variable viscosity and density gradients. We suggest numerical scheme of multigrid algorithm for solving Stokes and continuity equations with variable viscosity in a spherical coordinate system. Calculations are performed on a sequence of orthogonal staggered grids. The quality of the numerical scheme was verified by comparing the numerical solution with the analytical solution of the test problem. 


Keywords: Stokes equations, variable viscosity, Gauss-Seidel method, benchmark solutions, multigrid methods

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