On the theory of body motion in confined Stokesian fluids

TL;DR

This paper introduces a theoretical framework for analyzing Stokes flow around bodies in confined fluids by decomposing the problem into simpler parts, enabling better understanding of forces and torques in such systems.

Contribution

It presents a reflection-based method to decompose confined Stokes flow into unbounded and confined components, with detailed convergence and error analysis.

Findings

Derived explicit expressions for forces and torques on bodies in confined Stokes flow.

Applied the theory to spheres with slip boundary conditions near walls.

Analyzed the effects of slip length and proximity on hydrodynamic forces.

Abstract

We propose a theoretical method to decompose the solution of a Stokes flow past a body immersed in a confined fluid in two simpler problems, related separately to the two geometrical elements of these systems: (i) the body immersed in the unbounded fluid (represented by its Fax\'en operators), and (ii) the domain of the confinement (represented by its Stokesian multipoles). Specifically, by using a reflection method, and assuming linear and reciprocal boundary conditions \citep{procopio-giona_pof}, we provide the expression for the velocity field, the forces, torques and higher-order moments acting on the body in terms of: (i) the volume moments of the body in the unbounded ambient flow; (ii) the multipoles in the domain of the confinement; (iii) the collection of all the volumetric moments on the body immersed in all the regular parts of the multipoles considered as ambient flows. A detailed convergence analysis of the reflection method is developed. In the light of practical applications, we estimate the truncation error committed by considering only the lower order moments (thus truncating the matrices) and the errors associated with the approximated expressions available in the literature for force and torques. We apply the theoretical results to the archetypal hydrodynamic system of a sphere with Navier-slip boundary conditions near a plane wall with no-slip boundary conditions, to determine forces and torques on a translating and rotating sphere as a function of the slip length and of the distance of the sphere from the plane. The hydromechanics of a spheroid is also addressed.