Self-propulsion of droplets driven by an active permeating gel

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 lengthscale $\ell$ --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 $\ell\rightarrow\infty$, corresponding to a very dilute gel, we recover Stokes flow. The opposite limit, $\ell\rightarrow 0$, corresponding to a space filling gel, is singular and not equivalent to Darcy’s equation, which cannot account for self-propulsion.

Graphical abstract