Method of regularised stokeslets: Flow analysis and improvement of convergence

TL;DR

This paper analyzes the flow induced by regularised stokeslets, deriving explicit formulas, examining flow properties, and proposing improved blob functions to reduce regularisation errors in low-Reynolds number flow simulations.

Contribution

It provides a detailed flow analysis of regularised stokeslets, derives explicit formulas, and introduces new blob functions to enhance computational accuracy.

Findings

Flow from regularised stokeslets can be expressed as a sum of singularities.

Spherically symmetric blobs simplify the flow analysis.

Negative force regions in blobs help reduce regularisation errors.

Abstract

Since their development in 2001, regularised stokeslets have become a popular numerical tool for low-Reynolds number flows since the replacement of a point force by a smoothed blob overcomes many computational difficulties associated with flow singularities (Cortez, 2001, \textit{SIAM J. Sci. Comput.} \textbf{23}, 1204). The physical changes to the flow resulting from this process are, however, unclear. In this paper, we analyse the flow induced by general regularised stokeslets. An explicit formula for the flow from any regularised stokeslet is first derived, which is shown to simplify for spherically symmetric blobs. Far from the centre of any regularised stokeslet we show that the flow can be written in terms of an infinite number of singularity solutions provided the blob decays sufficiently rapidly. This infinite number of singularities reduces to a point force and source dipole for spherically symmetric blobs. Slowly-decaying blobs induce additional flow resulting from the non-zero body forces acting on the fluid. We also show that near the centre of spherically symmetric regularised stokeslets the flow becomes isotropic, which contrasts with the flow anisotropy fundamental to viscous systems. The concepts developed are used to { identify blobs that reduce regularisation errors. These blobs contain regions of negative force in order to counter the flows produced in the regularisation process, but still retain a form convenient for computations.