Three-dimensional boundary singularity method for partial-slip flows

This study extends a 3-D method of the fundamental solutions (MFS) to Stokes flows with Knudsen numbers corresponding to partial-slip flows interior and exterior to spherical boundaries. The study focuses on the distribution of the singularities outside the fluid flow domain. Local spherical coordinates systems are thus introduced to accommodate the application of the method to flows with partial-slip boundary conditions, where velocities tangent to solid walls are proportional to shear rates at surfaces. The singularities are subject to variation in location and number to investigate their impacts on numerical accuracy. For the flow about a single sphere, it is shown that the numerical accuracy improves when the singularities move towards the sphere center. When the singularities are located too far away from the observation points toward the center of sphere, the solution of the linear system fails. The reasons that cause the failure are explored and optimal location of singularities is found. The flow between two concentric spheres is used for further validation of the developed method for a combination of convex and concave surfaces. Finally the application of the method to flows about two separately spaced spheres is presented. Numerical results obtained compare favorably with analytical solutions for presented test cases. It is shown that a moderate number of singularities can be used in combination with a proper location of singularities to achieve a prescribed accuracy. (C) 2010 Elsevier Ltd. All rights reserved.