Boundary integral method for Stokes flow with linear slip flow conditions in curved surfaces

Abstract

The no slip boundary condition is traditionally used to predict velocity fields in macro scale flows. When the scale of the problem is about the size of the mean free path of particles, it is necessary to consider that the flow slips over the solid surfaces and the boundary condition must be changed to improve the description of the flow behaviour with continuous governing fluid flow equations. Navier's slip boundary condition states that the relative velocity of the fluid respect to the wall is directly proportionally to the local tangential shear stress. The proportionally constant is called the slip length, which represent the hypothetical distance at the wall needed to satisfy the condition of no-slip flow. Some works have misused boundary conditions derived from Navier's work to model slip flow behaviour for example by employing expressions, for diagonal and curved surfaces, that were derived for flat infinite surfaces aligned with coordinate axes. In this work, the creeping flow of a Newtonian fluid under linear slip conditions is simulated for the cases of a Slit and a Couette mixer by means of the Boundary Element Method (BEM). In the evaluation of such flows, different magnitudes of slip length from 0 (no slip) to 1.0 are analysed in an effort to understand the effect of the slip boundary condition on the physical behaviour of the simulation system. Analytic solutions for both geometries under slip flow are used to estimate L2 norm error, which is below 0.25% for Couette flow and 1.25% for Slit flow, validating the approximation applied.