A NUMERICAL-METHOD FOR STUDY OF THE UNSTEADY VISCOUS-FLOW BETWEEN 2 CONCENTRIC ROTATING SPHERES

A simple but very efficient numerical method based on the finite diffe rence technique has been developed for solving time-dependent non-line ar flow problems. The governing equations of motion are discretized by a backward-time central-space scheme, whereby only the variables in t he non-linear terms other than the main variable of the transport equa tion are replaced with the corresponding values calculated previously. The resulting algebraic equations for each main variable are solved b y directly applying the Gaussian algorithm altered by considering the sparse structure of the diagonal-banded coefficient matrix adequately. The method is applied here to study the unsteady axisymmetric isother m flow of an incompressible viscous fluid in a spherical shell with a stationary inner sphere and a rotating outer sphere. The description g iven in literature of the flow under consideration concentrates analyt ically on the asymptotic behaviour for very large Reynolds number Re s tarting from the almost rigid rotation. The case of small or moderate Reynolds numbers could be studied numerically only for Re less-than-or -equal-to 3000 because of certain numerical difficulties, which alread y lead to discrepancies for Re > 1000. Therefore, no data are availabl e for the large intermediate region at high Reynolds numbers. In contr ast to literature, consistent solutions for a large range of Reynolds number from 10 to 20000 are obtained with the method described here. A comparison of the results with those in literature shows a good agree ment up to Re = 1000. At high Re the flow field confirms certain featu res such as the Stewartson shear layers as predicted by the asymptotic theory. With the results presented, a contribution is made for fillin g the gap between the asymptotic theory and numerical results in liter ature.