Force-dependence of the rigid-body motion for an arbitrarily shaped particle in a forced, incompressible Stokes flow

TL;DR

This paper derives a new linear functional representation for the motion of arbitrarily shaped particles in low Reynolds number incompressible flows, emphasizing the force-dependence and solenoidal components of force distributions.

Contribution

It introduces an explicit linear functional of the curl of force distributions to represent rigid body motion, complementing traditional force-based representations and addressing incompressibility constraints.

Findings

Derived the representation of particle motion as a linear functional of curl of force.

Showed the new representation simplifies handling incompressibility constraints.

Illustrated the approach's utility in avoiding ambiguities in swimmer simulations.

Abstract

When a particle moves in a Newtonian flow at low Reynolds number, inertia is irrelevant and a linear relationship exists between velocities and forces. For incompressible flows, any force distribution $\mathbf{f}(\mathbf{r})$ acting in the fluid bulk induces flow and motion only through its solenoidal component. For force distributions that are spatially localized (i.e., vanish sufficiently fast at infinity), we derive the representation of the rigid body motion as an explicit linear functional of $\nabla\times\mathbf{f}$, which complements the usual representation in terms of $\mathbf{f}$. We illustrate the utility of this alternative representation, which has the advantage of having the incompressibility constraint built-in, in avoiding certain ambiguities that arise, e.g., when implementing approximations for swimmers.