The boundary integral formulation of Stokes flows includes slender-body theory

TL;DR

This paper derives a slender-body theory for Stokes flows directly from the boundary integral formulation, showing that the flow around slender bodies can be represented by singularities along their centerline, unifying two classical approaches.

Contribution

The paper provides a rigorous derivation of slender-body theory from the boundary integral approach using matched asymptotic expansions, clarifying the mathematical basis of the singularity method.

Findings

Derivation of slender-body theory from boundary integral formulation.

Equivalence of the derived theory to classical singularity methods.

Limitations of slender solutions depending on flow type.

Abstract

The incompressible Stokes equations can classically be recast in a boundary integral (BI) representation, which provides a general method to solve low-Reynolds number problems analytically and computationally. Alternatively, one can solve the Stokes equations by using an appropriate distribution of flow singularities of the right strength within the boundary, a method particularly useful to describe the dynamics of long slender objects for which the numerical implementation of the BI representation becomes cumbersome. While the BI approach is a mathematical consequence of the Stokes equations, the singularity method involves making judicious guesses that can only be justified a posteriori. In this paper we use matched asymptotic expansions to derive an algebraically accurate slender-body theory directly from the BI representation able to handle arbitrary surface velocities and surface tractions. This expansion procedure leads to sets of uncoupled linear equations and to a single one-dimensional integral equation identical to that derived by Keller and Rubinow (1976) and Johnson (1979) using the singularity method. Hence we show that it is a mathematical consequence of the BI approach that the leading-order flow around a slender body can be represented using a distribution of singularities along its centreline. Furthermore when derived from either the single-layer or double-layer modified BI representation, general slender solutions are only possible in certain types of flow, in accordance with the limitations of these representations.