Creeping Flows

Point Force Solution

 Consider the response of the Stokes equation to a point force exerted at the origin. Equation (1) and (2) may be rewritten as

(5)

(6)

The general solution to (5) and (6) in an unbounded domain is given as

(7)

Here,  is referred to as the Oseen tensor. The solution in a two-dimensional flow is given by

(8)

Using the Green theorem, the velocity vector may be represented as

(9)

where

(10)

and

(11)

Equations (9) and (11) give the velocity and the pressure fields for arbitrary given distribution of stress and velocity on solid surface boundaries.

 For an unbounded flow and for a point force , from (9) and (11) it follows that

(12)

(13)

which is referred to as a Stokeslet.