A Rigid Body in an Unbounded Domain

if u( j is a constant, as in the present case, it can be shown that the integral of the double-layer potential is zero (the proof of this statement is left to the reader). Thus (8 119) 

generally, it is the motion of the particle that is specified, and the surface stress vector T( ,) n is unknown. However, if we evaluate (8-197) at points x,v on the boundary of the particle, it is converted into an integral equation from which the surface-stress distribution can be determined. Specifically, taking account of the condition (8-195), we obtain from (8-197) 

knowledge of the boundary velocity u(xs) and the form of the undisturbed flow evaluated at the body surface Uoo(xj) allows a direct calculation of the surface-force vector T n by means of a solution of the integral equation, (8-198). It is emphasized that we do not actually address the numerical problem of solving (8-198). We note, however, that it is an integral equation of the first kind, and it is known that there can be numerical difficulties with the solution of this class of integral equations. The reader who wishes to learn more about the details of numerical solution should consult one of the general reference books that were listed in the introduction to this section. 

The formulation (8-198) was used by Youngren and Acrivos16 to calculate the force on solid particles of different shapes translating through an unbounded stationary fluid, u, (xv ) = 0, in what was likely the first application of the boundary-integral method to creeping-flow problems. Many subsequent investigators have used it to calculate forces on bodies of complicated shape, in a variety of undisturbed flows.17 

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