The completed double layer boundary integral equation method for two-dimensional Stokes flow

Power and Miranda (1987) explained how integral equations ofthe second kind can be obtained for general exterior three-dimensionalStokes flows. They observed that, although the double layerrepresentation that leads to an integral equation of the secondkind coming from the jump property of its velocity field acrossthe density carrying surface can represent only those flow fieldsthat correspond to a force and torque free surface, the representationmay be completed by adding terms that give arbitrary total forceand torque in suitable linear combinations, precisely a Stokesletand a Rotlet located in the interior of the three-dimensionalparticles. Karrila and Kim (1989) called Power and Miranda'snew method the completed double layer boundary integral equationmethod, since it involves the idea of completing the deficientrange of the double layer operator. The main objective of thispaper is to extend Power and Miranda's completed method to theproblem of multiple cylinders in twodimensional bounded andunbounded domains. This extension is not trivial, owing to theunbounded behaviour at infinity of the fundamental solutionof the Stokes equation in two dimensions and the associatedparadoxes arising from this unbounded behaviour.