Hydrodynamic Forces and Torques for a Nonspherical Particle

Prolate Spheroid Translating in a Quiescent Fluid

The motion of a rigid prolate spheroid parallel to its axis of revolution as shown in Figure 4 is studied in this section. The appropriate coordinates system for this problem is the prolate coordinate system  with

(42)

Figure 4. Schematic of a prolate spheroids in creeping flow motion.

For convenience, we let

(43)

The surface . () are prolate spheroids. Then  and z may be expressed as

(44)

and

(45)

 Similar to the method used for the oblate spheroid, one can solve the equation of creeping motion in prolate spheroidal coordinates subject to appropriate boundary conditions. The stream function for a prolate spheroid translating with velocity U in the positive z-direction parallel to its axis of revolution is given as

(46)

 Using this expression in Equation (8), one obtains the force acting on the prolate spheroid, that is

(47)

where

(48)

and .

Equation (47) may be restated as

(49)

where the shape factor k is given by

(50)