Fundamental Solutions of the Creeping-Flow Equations

The Stokeslet A Fundamental Solution for the Creeping-Flow Equations 

We begin by considering the solution of the creeping-flow equations for a point force f in an unbounded fluid. This solution is a fundamental solution in the sense that it can be obtained formally from the governing equations 

This method is particularly simple to apply in the case of a point force f. The (disturbance) pressure must be a decaying harmonic ftmction that is linear in f and a true scalar. Hence, 

To this point, the solution is essentially identical to that obtained for uniform flow past a sphere in Section A. Beyond this, however, the two differ. To satisfy boundary conditions at the surface of a sphere, a nonzero value of a is necessary. However, for a point force, a = 0. The simplest way to see this is to note that c is dimensionless, whereas a must have dimensions of (length)2. But a point force has no characteristic length scale, so a = 0. Thus, 

To determine the scalar constant c, we make use of an integral relationship between the force f and the resultant force exerted on the fluid outside a control surface S. 

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We shall see that the stokeslet solution plays a fundamental role in creeping flow theory. We have already seen in Section E of Chap. 7 that it describes the disturbance velocity far away from a body of any shape that exerts a nonzero force on an unbounded fluid. Indeed, when nondimensionalized and expressed in spherical coordinates, it is identical to the velocity field, (7 151). In the next section we use the stokeslet solution to derive a general integral representation for solutions of the creeping-flow equations. 

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