Before revision:  We show that hydrodynamics tends to stabilize spherical droplets but that droplet division occurs for sufficiently strong chemical driving, sufficiently large droplet viscosity or sufficiently small surface tension.
After revision:  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 droplets but that droplet division occurs for sufficiently strong chemical driving, sufficiently large droplet viscosity or sufficiently small surface tension.