Creeping Flow Past a Sphere

Substituting (12) and (13) into (9), the explicit expression for the stream function becomes

The velocity components are then given by

Figure 2a shows the streamline for the creeping flow around a sphere. Comparing the streamlines of the creeping flow conditions to the potential flow one given by

and is plotted in Figure 2 b, it appears that the stream lines are more dispersed.

a) Viscous Flow

b) Potential Flow

Figure 2. Comparison of the streamlines for creeping and potential flows.

For moving spheres, the stream function is given by

a) Viscous Flow

b) Potential Flow

 Figure 3. Comparison of the streamlines for creeping potential flows in a moving frame.

For the moving sphere coordinates the corresponding streamlines are shown in Figure 3. Figure 3a shows that the particle appears to be dragging the viscous fluid as it moves, while Figure 3b through suggests that the particle pushes the fluid in the potential flow regime.