(142v) Non-Singular Boundary Integral Methods for Fluid Mechanics

A formulation of the boundary integral method that is applicable to the Laplace equation for the potential problem, the Helmholtz equation, and equations associated with Stokes flow and linear elastic deformations have been developed [1]. The approach is based on removing the usual weakly singular integral and the Cauchy principal value integral in the boundary integral method by subtracting the solution of a specially constructed related problem. We demonstrate the details of our approach using the potential problem from which it is easy to see how the method can be extended to the more complicated cases of Stokes flow and linearly elastic deformations. 

This non-singular boundary integral formulation simplifies numerical implementations of this popular technique. Validation of the approach is obtained by comparing numerical results for problems in Stokes flow for which analytic results are known [2, 3]. The flexibility of this method in colloidal hydrodynamics is demonstrated with examples where there may be a mix of boundary conditions based on the surface velocity or the surface traction.

[1] E. Klaseboer, Q. Sun and D. Y. C. Chan Non-singular boundary integral methods for fluid mechanics applications J Fluid Mech 696 (2012) 468-478.

[2] E. Klaseboer, R. Manica, D. Y. C. Chan and B. C. Khoo BEM simulations of potential flow with viscous effects as applied to a rising bubble Engg Analysis Boundary Elements 35 (2011) 489-494.

[3] Q. Sun, E. Klaseboer, B. C. Khoo and D. Y. C. Chan Implementation of a non-singular boundary integral method for Stokes flow Engg Analysis Boundary Elements (to be published 2012)