Hydrodynamic Forces and Torques for a Nonspherical Particle

Drag on an Axisymmetric Body

The stress tensor for an incompressible viscous fluid is given as

,   where

(1)

When an arbitrary shaped particle moves in a fluid, the resultant of the stresses acting on the surface of the body give rise to a hydrodynamic force   (drag, lift, and side force components and a hydrodynamic torque ). In general these are given by

,

(2)

where S is the surface of the body and  is the position vector.

 For an axisymmetric body in a uniform flow, the resultant of the stresses is only a drag force acting parallel to the direction of the flow. For the example shown in Figure 1,

(3)

When an intrinsic coordinate is used,

.

(4)

The expression for  as evaluated by Happel an Brenner is given by

(5)

Using (5) in (3) after some algebra, it follows that

(6)

This equation relates the Stokes stream function  to the drag for an axisymmetric body.