  We study the hydrodynamics and shape changes of chemically active droplets.
In non-spherical droplets, surface tension generates hydrodynamic flows that
drive liquid droplets into a spherical shape. Here we show that spherical
droplets that are maintained away from thermodynamic equilibrium by chemical
reactions may not remain spherical but can undergo a shape instability which
can lead to spontaneous droplet division. In this case chemical activity acts
against surface tension and tension-induced hydrodynamic flows. By combining
low Reynolds-number hydrodynamics with phase separation dynamics and chemical
reaction kinetics we determine stability diagrams of spherical droplets as a
function of dimensionless viscosity and reaction parameters. We determine
concentration and flow fields inside and outside the droplets during shape
changes and division. Our work shows that hydrodynamic flows tends to stabilize
spherical shapes but that droplet division occurs for sufficiently strong
chemical driving, sufficiently large droplet viscosity or sufficiently small
surface tension. Active droplets could provide simple models for prebiotic
protocells that are able to proliferate. Our work captures the key
hydrodynamics of droplet division that could be observable in chemically active
colloidal droplets.
