A finite volume formulation for simulating drop motion in a viscoelastic two-phase system

We present a numerical method for simulating the motion of a drop in a viscoelastic two-phase system. The axisymmetric incompressible flow equations combined with viscoelastic constitutive equations are solved on an orthogonal curvilinear coordinate system using the finite volume method. The artificial compressibility factor is introduced in the continuity equation to effectively increase the convergence rate. A flux-difference splitting scheme is utilized in both the momentum and constitutive equations. This scheme has been used in high-speed aerodynamics to automatically handle the change of type of equations between elliptic and hyperbolic. The combination of artificial compressibility and the flux-difference splitting scheme is capable of overcoming the numerical instability caused by high elasticity of the viscoelastic phase. The differential constitutive equations of the FENE-CR model are employed to model the polymer stress terms in the momentum equations if either phase of the two-fluid system is viscoelastic. The free-surface boundary condition consists of the dynamic and kinematic conditions. These are used to update the drop shape during its time evolution. Conforming to the deforming drop shape, a boundary-fitted grid is generated at each time step. The grid is generated by solving elliptic partial differential equations.