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

Power and Miranda (1987) explained how integral equations of the secon d kind can be obtained for general exterior three-dimensional Stokes f lows. They observed that, although the double layer representation tha t leads to an integral equation of the second kind coming from the jum p property of its velocity field across the density carrying surface c an represent only those flow fields that correspond to a force and tor que free surface, the representation may be completed by adding terms that give arbitrary total force and torque in suitable linear combinat ions, precisely a Stokeslet and a Rotlet located in the interior of th e three-dimensional particles. Karrila and Kim (1989) called Power and Miranda's new method the completed double layer boundary integral equ ation method, since it involves the idea of completing the deficient r ange of the double layer operator. The main objective of this paper is to extend Power and Miranda's completed method to the problem of mult iple 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.