"Second-Order Finite Difference Solutions for the Flow Between Rotating Concentric
Spheres"
by Steve Schwengels,
David H. Schultz and Willam Shay
This paper describes a second-order method to calculate approximate solutions
to flow of viscous incompressible fluid between rotating concentric spheres.
The governing partial differential equations are presented in the
stream-vorticity formulation and are written as a series of second-order
equations. The technique employed makes use of second-order approximations
for all terms in the governing equations and is dependent upon the direction
of flow at a given point. This upwind technique has allowed us to generate
approximate solutions with larger Reynolds numbers than has generally been
possible for second and higher-order techniques. Solutions have been obtained
with Reynolds numbers as large as 3000 and with grids as fine as a 40 x 40
mesh. Results are displayed in the form of level curves for both the stream
and vorticity functions. A dimensionless quantity related to the torque
acting on both spheres has been calculated from the approximate solution and
compared with other results. Results with smaller Reynolds numbers such as
100 and 1000 are in excellent agreement with other published results.