TABLE OF CONTENTS The velocity field due to a point force applied at a certain point within a particular domain of a flow plays an important role in the analysis and computation of a variety of Stokes flows, including those due to the motion of particles and the propulsion of microscopic organisms. Physically, the flow due to a point force may be identified with the flow generated by the slow motion of a small particle in an otherwise quiescent fluid, as discussed in Section 5.6 within the more general context of flow at finite Reynolds number.
The velocity and modified pressure fields due to a point force are found by solving the continuity equation ? u = 0 and the singularly forced Stokes equation
which is the linearized version of Eq. (5.6.1); x 0 is the location of the point force, the constant b represents the direction and magnitude of the point force, and ? is the three-dimensional delta function. The solution of Eq. (6.4.1) must be found subject to a boundary condition that requires that the velocity vanish over a designated stationary solid boundary denoted by S B, that is, u = 0 when x is on S B.
To expedite the solution, we introduce the Green's function tensor G defined by the equation
where the factor 1/(8 ? ?) has been introduced for future convenience. To satisfy the condition that the velocity vanishes on
