Phase field simulation of drop formation in a coflowing fluid

We numerically investigate the dynamics of drop formation when a Newtonian fluid is injected through a tube into another immiscible, co-flowing Newtonian fluid with different density and viscosity using the phase field method. The two phase system is modeled by a coupled three dimensional Cahn-Hilliard and Navier-Stokes equation in cylindrical coordinates. And the contribution from the chemical potential has been taken into account in the classical Navier-Stokes equation. The numerical method involves a convex splitting scheme for the Cahn-Hilliard equation and a projection type scheme for the momentum equation. Our study of the dynamics of the drop formation is motivated by the experimental work by Utada et al [Phys. Rev. Lett. 99(2007), 094502] on dripping and jetting transition. The simulation results demonstrate that the process of drop formation can be reasonably predicated by the phase field model we used. Our simulations also identify two classes of dripping to jetting transition, one controlled by the Capillary number of the outer fluid and another one controlled by the Weber number of the inner fluid. The results match well with the experimental results in Utada et al [A. S. Utada, A. Fernandez-Nieves, H. A. Stone, and D. A. Weitz, Phys. Rev. Lett. 99(2007), 094502] and Zhang [Chem. Eng. Sci. 54(1999), 1759-1774]. We also study how the dynamics of the drop formation depends on the various physical parameters of the system.