Computation of a regularized Brinkmanlet near a plane wall

Abstract

In this study, we derive a solution for Brinkman flow induced by a regularized point force in the presence of a plane wall. The process involves solving two coupled subproblems: one for the forced Brinkman flow in free-space and one for the unforced Brinkman flow in a half-space that enforces the no-slip boundary condition on the plane wall. Both subproblems are solved in the Fourier domain and a specific regularization of the force is designed so that the transformed solutions have Gaussian decay properties in certain limits. Similar to the singularly forced case, however, and as shown previously by others, there is no analytic inverse transform and thus it must be approximated numerically. The Gaussian decay of our solutions make the numerical approximation of the inverse transform more efficient. We provide detailed methodology to compute velocities from collections of regularized point forces and to solve for the forces at collections of points where the Brinkman velocity is prescribed. Because the unsteady Stokes equations can be recast as the Brinkman equations, our technique is applicable when solving for either type of flow, in the presence of a plane wall.