Effect of convective transport on droplet spinodal decomposition in fluids

The effect of convective transport on the late stage growth of droplets in the presence of sedimentation and shear flow is analyzed. The high Peclet number limit (UR/D)≫1 is considered, where U is the characteristic velocity, R is the radius of the droplet, and D is the diffusion coefficient. The growth of the droplet depends on the boundary condition for the fluid velocity at the droplet interface, and two types of boundary conditions are considered. For a rigid interface, which corresponds to the interface between a solid and a fluid, the tangential velocity is zero and the normal velocity is equal to the velocity of the surface. For a mobile interface, which corresponds to an interface between two fluids, the tangential and normal velocities are continuous. These results indicate that the scaling relations for the critical radius are Rc(t)∝t(1/2) for a sedimenting droplet with a rigid interface, Rc(t)∝t(2/3) for a sedimenting droplet with a mobile interface, Rc(t)∝t(3/7) for a droplet with a rigid interface in a simple shear flow, and Rc(t)∝t(1/2) for a droplet with a mobile interface in a simple shear flow. The rate of droplet growth is enhanced by a factor of Pe(1/3) for rigid interfaces and Pe(1/2) for mobile interfaces.