Solution of three-dimensional steady state Stokes flow problems with exit boundary conditions by the boundary element method

This paper explores the Boundary Element formulation of “exit” boundary conditions for three-dimensional steady state Stokes flows. A previous investigation had formulated it for two- dimensional situations. The current study is a natural extension of that research. Specifically, the boundaries that have the normal-velocity gradient specified as zero or that have the pressure specified as any value can now be tackled in three dimensions. This would open up the Boundary Element Method (BEM) to a wide variety of Stokes flow problems. The procedure for both the boundary conditions begins with the creation of a replica of the exit plane at a small distance inside the domain. A discretization of the relevant mathematical quantities within the integration terms would result in the generation of the BEM equations that would include the above boundary conditions. This paper articulates all the mathematical intricacies of the procedure, especially for the more important specified-pressure boundary condition. The remainder of this paper demonstrates the new boundary conditions for a couple of relatively simple Stokes flow problems that were hitherto unsolvable by the standard BEM procedure. The BEM predictions matched excellently with the available analytical solutions. The article rounds off with concluding statements and possible research directions in the future.