THE COMPLETED DOUBLE-LAYER BOUNDARY INTEGRAL-EQUATION METHOD FOR 2-DIMENSIONAL STOKES-FLOW

Power and Miranda (1987) explained how integral equations of the second kind can be obtained for general exterior three-dimensional Stokes flows. They observed that, although the double layer representation that leads to an integral equation of the second kind coming from the jump property of its velocity field across the density carrying surface can represent only those flow fields that correspond to a force and torque free surface, the representation may be completed by adding terms that give arbitrary total force and torque in suitable linear combinations, precisely a Stokeslet and a Rotlet located in the interior of the three-dimensional particles. Karrila and Kim (1989) called Power and Miranda's new method the completed double layer boundary integral equation method, since it involves the idea of completing the deficient range of the double layer operator. The main objective of this paper is to extend Power and Miranda's completed method to the problem of multiple cylinders in two-dimensional bounded and unbounded domains. This extension is not trivial, owing to the unbounded behaviour at infinity of the fundamental solution of the Stokes equation in two dimensions and the associated paradoxes arising from this unbounded behaviour.