Self propulsion of droplets driven by an active permeating gel

TL;DR

This paper models active droplet propulsion driven by permeation forces in a gel, analyzing flow fields, velocities, and energy dissipation, revealing optimal gel fractions and limits where classical models apply or fail.

Contribution

It introduces a model for droplet propulsion driven by permeation forces in a gel using the Brinkman equation, highlighting optimal conditions and limitations of existing theories.

Findings

Optimal gel fractions maximize droplet velocities.

Flow fields depend on the gel's permeability length scale.

Classical Stokes flow is recovered in the dilute limit.

Abstract

We discuss the flow field and propulsion velocity of active droplets, which are driven by body forces residing on a rigid gel. The latter is modelled as a porous medium which gives rise to permeation forces. In the simplest model, the Brinkman equation, the porous medium is characterised by a single length scale $\ell$ --the square root of the permeability. We compute the flow fields inside and outside of the droplet as well as the energy dissipation as a function of $\ell$. We furthermore show that there are optimal gel fractions, giving rise to maximal linear and rotational velocities. In the limit $\ell\to\infty$, corresponding to a very dilute gel, we recover Stokes flow. The opposite limit, $\ell\to 0$, corresponding to a space filling gel, is singular and not equivalent to Darcy's equation, which cannot account for self-propulsion.