A robust finite difference method for confined and free surface flows with slip at the wall

Abstract

This paper describes the implementation of slip boundary conditions in an in-house finite difference code used to numerically solve the governing equations of complex transient free surface flows. A projection method is used to couple pressure and velocity, and an iterative procedure is used to couple stress to the pressure–velocity variables. The slip boundary conditions are explicitly computed (considering both Newtonian and viscoelastic tangential stress vectors) and have been implemented for both confined and free surface flows. We show that with the proposed implementation, even for high slip coefficients κ, no relaxation of the slip velocity is required (which is often used in the finite volume method). To validate and test the robustness of the implementation, we compare the numerical velocity and stress profiles for fully developed slip flows with the corresponding analytical solutions, and we consider two different case studies, flow in a confined channel (steady-state) and extrudate swell (transient), emphasizing the evaluation of slip velocity at the geometric singularity located on the corner of the channel exit. This study provides a reference point for assessing extrudate swell rates, focusing on how various parameters in the sPTT model influence them. Additionally, it introduces a method for addressing slip velocity at sharp corners in the outflow of free-surface region. Furthermore, the paper includes an analytical solution for fully developed flows of sPTT fluids with a Newtonian solvent, accounting for wall slip, as well as a numerical approach to implicitly calculate pressure gradients for a known average velocity and a specific set of sPTT model parameters (particularly useful in convergence tests).