Properties and solution techniques for a mixed type boundary integral equation arising in creeping flow problems

Abstract

We consider the properties of a creeping flow, boundary integral operator which is the sum of a single layer potential arising from a surface density, and two double layer potentials, one each for the tangential and normal components of the same density with different coefficient weights. Specific cases of this operator are encountered quite frequently in the creeping flow literature, especially in the solution of problems with slip at an interface. We examine, in particular, the eigenspectrum of this mixed-type operator. As expected, when the coefficient of either the tangential or normal component of the double layer term is either much smaller in magnitude than unity or identically equal to zero, the solution of this operator for the unknown density is rendered ill-posed, and thus, solution with the Richardson iteration technique requires a prohibitively large number of iterations. However, a surprising result is that the solution via inversion (e.g. Gaussian elimination, LU decomposition, etc.), for the component of the density whose coefficient in the double layer term of the operator is non-zero, can actually be a well-posed problem. We verify our theoretical developments by considering a spherical viscous drop with a slipping interface suspended in a viscous liquid undergoing a uniaxial extensional flow, the interfacial slip being proportional to the local tangential stress. In this case, the operator is equivalent to the mixed type operator with the normal coefficient set to zero. We demonstrate that the solution of this operator for the tangential stress is a well-posed problem, and we delineate the parametric space for a stable solution scheme. © 2012 Elsevier Ltd.