A partitioned numerical scheme for fluid–structure interaction with slip

¹ Applied and Computational Mathematics and Statistics, University of Notre Dame, USA.
² Department of Mathematics, University of California, Berkeley, USA.

^* Corresponding author: canics@berkeley.edu

Received: 12 July 2019
Accepted: 26 November 2020

Abstract

We present a loosely coupled, partitioned scheme for solving fluid–structure interaction (FSI) problems with the Navier slip boundary condition. The fluid flow is modeled by the Navier–Stokes equations for an incompressible, viscous fluid, interacting with a thin elastic structure modeled by the membrane or Koiter shell type equations. The fluid and structure are coupled via two sets of coupling conditions: a dynamic coupling condition describing balance of forces, and a kinematic coupling condition describing fluid slipping tangentially to the moving fluid–structure interface, with no penetration in the normal direction. Problems of this type arise in, e.g., FSI with hydrophobic structures or surfaces treated with a no-stick coating, and in biologic FSI involving rough surfaces of elastic tissues or tissue scaffolds. We propose a novel, efficient partitioned scheme where the fluid sub-problem is solved separately from the structure sub-problem, and there is no need for sub-iterations at every time step to achieve stability, convergence, and its first-order accuracy. We derive energy estimates, which prove that the proposed scheme is unconditionally stable for the corresponding linear problem. Moreover, we present convergence analysis and show that under a time-step condition, the method is first-order accurate in time and optimally convergent in space for a Finite Element Method-based spatial discretization. The theoretical rates of convergence in time are confirmed numerically on an example with an explicit solution using the method of manufactured solutions, and on a benchmark problem describing propagation of a pressure pulse in a two-dimensional channel. The effects of the slip rate and fluid viscosity on the FSI solution are numerically investigated in two additional examples: a 2D cylindrical FSI example for which an exact Navier slip Poiseuille-type solution is found and used for comparison, and a squeezed ketchup bottle example with gravity enhanced flow. We show that the Navier-slip boundary condition increases the outflow mass flow rate by 21% for a bottle angled at 45 degrees pointing downward, in the direction of gravity.