Fig. 1: Shapes and velocity field of a motile contractile droplet.

a Initial configuration of an active droplet. The red arrows indicate the direction of the polarization field P. The droplet is placed within a microfluidic channel of size L_y = 500 and L_z = 170. Here only a portion of the lattice is shown. b Intermediate pre-motile state of the contractile suspension with ζ = −8 × 10⁻⁴. The droplet elongates perpendicularly to the direction of the polarization which remains essentially parallel to the y direction. c However, this value of ζ is sufficiently high to destabilize the polarization, which gives rise to a large splay deformation. Once this occurs, the drop acquires a unidirectional motion along the direction indicated by the green arrow. d Schematic view of the hydrodynamic flow produced by a contractile material, such as actomyosin. The myosin protein pulls two actin filaments along opposite directions (indicated by tick gray arrows), yielding a four-roll flow in their surroundings. Panel (d) is adapted from²⁰. e A minimal model of the force dipole produced by a contractile material. The thick black arrows indicate the direction of the force dipole while the circles represent the emerging four vortices of fluid. f and g Velocity field of the pre-motile (b) and motile (c) states. In the former, the fluid is pulled inward along the equator (parallel to the direction of P) and emitted axially (perpendicularly to P), giving rise to a macroscopic four-vortex structure. In the latter, a splay distortion fosters the formation of two counter-rotating vortices pushing the drop forward. The droplet radius at equilibrium is R = 45 lattice sites and the color map represents the value of the order parameter ϕ₁, ranging between 0 (black) and 2 (yellow).