Numerical study of the basic stationary spherical couette flows at low Reynolds numbers

Previously developed iterative numerical methods with splitting of boundary conditions intended for solving an axisymmetric Dirichlet boundary value problem for the stationary Navier–Stokes system in spherical layers are used to study the basic spherical Couette flows (SCFs) of a viscous incompressible fluid in a wide range of outer-to-inner radius ratios $R/r$ ($1.1\le R/r\le100$) and to classify such SCFs. An important balance regime is found in the case of counter-rotating boundary spheres. The methods converge at low Reynolds numbers ($\mathrm{Re}$), but a comparison with experimental data for SCFs in thin spherical layers show that they converge for $\mathrm{Re}$ sufficiently close to $\mathrm{Re}_{\mathrm{cr}}$. The methods are second-order accurate in the max norm for both velocity and pressure and exhibit high convergence rates when applied to boundary value problems for Stokes systems arising at simple iterations with respect to nonlinearity. Numerical experiments show that the Richardson extrapolation procedure, combined with the methods as applied to solve the nonlinear problem, improves the accuracy up to the fourth and third orders for the stream function and velocity, respectively, while, for the pressure, the accuracy remains of the second order but the error is nevertheless noticeably reduced. This property is used to construct reliable patterns of stream-function level curves for large values of $R/r$. The possible configurations of fluid-particle trajectories are also discussed.