Numerical studies of viscous incompressible flow between two rotating concentric spheres

Abstract

The problem of determining the induced steady axially symmetric motion of an incompressible viscous fluid confined between two concentric spheres, with the outer sphere rotating with constant angular velocity and the inner sphere fixed, is numerically investigated for large Reynolds number. The governing Navier-Stokes equations expressed in terms of a stream function-vorticity formulation are reduced to a set of nonlinear ordinary differential equations in the radial variable, one of second order and the other of fourth order, by expanding the flow variables as an infinite series of orthogonal Gegenbauer functions. The numerical investigation is based on a finite-difference technique which does not involve iterations and which is valid for arbitrary large Reynolds number. Present calculations are performed for Reynolds numbers as large as 5000. The resulting flow patterns are displayed in the form of level curves. The results show a stable configuration consistent with experimental results with no evidence of any disjoint closed curves.

Application of the Homotopy Analysis Method to Fluid Flow Problems

“…It seems hard to obtain analytic solutions of unsteady boundarylayer flows, which are valid for all time. Currently, an analytical method for strongly nonlinear problems, namely the homotopy analysis method [18], has been developed and successfully applied to many kinds of nonlinear problems in science and engineering [20,39,43,80,82,[114][115][116][117][118]. To the best of our knowledge, no one has reported any analytic solution of the unsteady boundary-layer flow over a semi-infinite flat plate, which is valid and accurate for all time.…”

Series Solutions of Unsteady Boundary‐Layer Flows over a Stretching Flat Plate

“…where the prime denotes the differentiation with respect to τ or ξ . According to the definitions (29) and (46) and the boundary conditions (48), it is reasonable to assume that F(z, ξ ) can be expressed by the following set of base functions…”

“…Thus, the homotopy analysis method is a kind of unification of the nonperturbation techniques, and therefore is more general, so that it is valid for more of highly nonlinear problems. The homotopy analysis method has been successfully applied to many nonlinear problems [29,[36][37][38][39][40][41][42][43][44][45][46]. It should be emphasized that a few new solutions have been found [47,48] by means of this analytic method, which were not discovered by other analytic techniques and even by numerical techniques.…”

Transitional vortices in wide‐gap spherical annulus flow

“…We hope to obtain stability limits for wide-gap spherical shells that will show regions of stability of the steady states, including level curves corresponding to the various flow regimes. This study involves the use of a numerical method developed recently in a previous study to investigate steady-state axisymmetric confined swirling flows [5]. An extension of the numerical method is presented in this paper to include parameter continuation.…”

“…5 shows the basic flow at Re = 100. It can be seen from the figure that the basic flow consists of a large-scale meridional circulation which is present on either side of the equator.…”