A compatible high-order meshless method for the Stokes equations with applications to suspension flows

TL;DR

This paper introduces a stable, high-order meshless method for solving the steady Stokes equations, effectively handling complex geometries and dense suspensions with implicit coupling to particle dynamics.

Contribution

It presents a novel meshless discretization that ensures compatibility between velocity and pressure, enabling high-order accuracy and efficient multigrid solutions for complex suspension flow problems.

Findings

Achieves equal-order convergence for velocity and pressure.

Effectively models dense suspensions with implicit particle coupling.

Resolves lubrication forces directly from Stokes solutions.

Abstract

A stable numerical solution of the steady Stokes problem requires compatibility between the choice of velocity and pressure approximation that has traditionally proven problematic for meshless methods. In this work, we present a discretization that couples a staggered scheme for pressure approximation with a divergence-free velocity reconstruction to obtain an adaptive, high-order, finite difference-like discretization that can be efficiently solved with conventional algebraic multigrid techniques. We use analytic benchmarks to demonstrate equal-order convergence for both velocity and pressure when solving problems with curvilinear geometries. In order to study problems in dense suspensions, we couple the solution for the flow to the equations of motion for freely suspended particles in an implicit monolithic scheme. The combination of high-order accuracy with fully-implicit schemes allows the accurate resolution of stiff lubrication forces directly from the solution of the Stokes problem without the need to introduce sub-grid lubrication models.