Boundary integral equation approach for stokes flow with non-linear slip boundary condition

The numerical simulation of microfluid flow through the solution of governing equations based on continuum models has to be done under the consideration of appropriate slip boundary conditions to account for the velocity jump at the solid-fluid interface. The linear model proposed by Navier states a relation between the tangential shear rate and the fluid-wall velocity differences and has been successfully used in reproducing the characteristics of many types of flows (e.g. slit flows, rotating curved mixers, microbearings, among others), where the shear rate at solid-fluid interfaces remains linear because of the geometry smoothness. Despite this, there are some situations for which this linear dependency fails leading to unrealistic behaviour and an expression for the slip condition at a solid-liquid interface establishing the variation in the slip length in terms of the square root of the tangential shear rate needs to be used. This work employs a boundary integral equation formulation for Stokes slip flow based on the normal and tangential projection of the Green's integral representational formulae for the Stokes velocity field, which directly incorporates into the integral equations the local tangential shear rate at the wall surfaces. The universal slip flow boundary condition is presented in terms of the tangential surface traction allowing its inclusion into the normal and tangential projections of boundary integral equation formulation for Stokes flow. The Boundary Element Method (BEM) is employed to solve the resulting projections of the integral equations and the equation system is evaluated iteratively turning the non-linear term into a non-homogeneous constant vector by using results from previous iteration. This formulation is used to simulate flow between parallel plates and concentric rotating mixer. The numerical results obtained for both problems are validated with the corresponding analytical solutions with non-linear boundary condition, showing excellent agreements. Results obtained in this work extend the use of BEM for the study of microfluid flow, allowing the developed of more geometrical complex microfluidic applications.