Stokes Flow in Container with Concave Bottom

γ

0.25

0.5

0.75

1.

1.25

1.5

2

α in degree

30

maximum cell measure

0.0025

zoom

Consider the Stokes flow in a lid-driven container with a concave bottom. Stokes flow occurs in a fluid with a low Reynolds number, Re<<1, when inertial forces are small compared to viscous forces. The container has straight sidewalls of height

W=L/γ

and width

L=1

, and a concave bottom with radius

R=1/(2cosα)

, centered at the mid-plane of the container;

α

is the angle between the straight sidewall and the curved bottom. The geometry of the container is outlined in red.

This problem has been solved numerically by Shankar [1], using an eigenfunction expansion.

This Demonstration uses the finite element method to solve the flow problem. We plot the streamlines for user-set values of the aspect ratio

γ

, the angle

α

, and the discretization refinement (i.e. the maximum cell measure value). The calculation time is larger for smaller values of "maximum cell measure". It is of interest is see how the primary eddy structure evolves as the geometry of the container is changed. The corner eddies can be resolved in the stream plot if you reduce "maximum cell measure".