Self-propulsion of droplets driven by an active permeating gel

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 [Formula: see text] --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 [Formula: see text]. We furthermore show that there are optimal gel fractions, giving rise to maximal linear and rotational velocities. In the limit [Formula: see text], corresponding to a very dilute gel, we recover Stokes flow. The opposite limit, [Formula: see text], corresponding to a space filling gel, is singular and not equivalent to Darcy's equation, which cannot account for self-propulsion.