Evaluation of Slip Boundary Conditionwith Radial Basis Function for Implementation in Boundary Element Methods

Abstract

The slip flow regime emerges as a consequence of the characteristic length reduction in Micro Electro-Mechanical Systems (MEMS) that work with fluids (e.g. medical sample testing devices, drug delivery systems, micro heat exchangers and mixer, among others). The boundary condition imposed to account for slip flow when solving continuum based governing equations relates the wall and fluid velocity difference with the local shear rate projection in the tangential direction at the boundary. Several works have evaluated slip boundary conditions with diverse methods and approximations, in some cases misusing expressions derived for planar infinite surfaces aligned with coordinate axis, to analyse curved surfaces or corner flows. In this work, the creeping flow of a Newtonian fluid under linear slip boundary conditions is simulated applying the Boundary Element Method (BEM). Radial Basis Functions (RBF) are used to approximate the tangential shear rate projection in slip boundary conditions. Two types of interpolations schemes were implemented: a local interpolation and global interpolation. The first one evaluates the tangential shear rate from a finite set of points near the boundary nodes, while the second interpolation approach employs all nodes in the fluid domain (boundary and internal nodes). Two fluid flow problems are used to test the performance of the solution achieved: Couette and slit flow, both having analytical solutions to which the results obtained are compared. This implementation is also tested using three different RBF’s, where the numerical results show that the Generalized Thin Plate Spline (GTPS) function combined with a global interpolation scheme has the least square norm error, below 1%, for both fluid flow problems tested and less computational effort than a similarly accurate local approximation.