Efficient numerical methods for the Maxey-Riley-Gatignol equations with Basset history term

TL;DR

This paper introduces a finite difference numerical method for solving the Maxey-Riley-Gatignol equations with Basset history term, comparing its efficiency and convergence to existing approaches, and analyzing effects of initial conditions and particle properties.

Contribution

It develops a finite difference approach for the MRGE with Basset history term, offering an alternative to polynomial expansion methods and analyzing its performance.

Findings

All methods achieve theoretical convergence for neutrally buoyant particles with zero initial velocity.

Order reduction occurs when initial velocity is non-zero or particle density differs from fluid.

Finite difference schemes show competitive efficiency and convergence properties.

Abstract

The Maxey-Riley-Gatignol equations (MRGE) describe the motion of a finite-sized, spherical particle in a fluid. Because of wake effects, the force acting on a particle depends on its past trajectory. This is modelled by an integral term in the MRGE, also called Basset force, that makes its numerical solution challenging and memory intensive. A recent approach proposed by Prasath et al. exploits connections between the integral term and fractional derivatives to reformulate the MRGE as a time-dependent partial differential equation on a semi-infinite pseudo-space. They also propose a numerical algorithm based on polynomial expansions. This paper develops a numerical approach based on finite difference instead, by adopting techniques by Koleva et al. and Fazio et al. to cope with the issues of having an unbounded spatial domain. We compare convergence order and computational efficiency for particles of varying size and density of the polynomial expansion by Prasath et al., our finite difference schemes and a direct integrator for the MRGE based on multi-step methods proposed by Daitche. While all methods achieve their theoretical convergence order for neutrally buoyant particles with zero initial relative velocity, they suffer from various degrees of order reduction if the initial relative velocity is non-zero or the particle has a different density than the fluid.