A Level-Set Method for Computing Solutions to Viscoelastic Two-Phase Flow

Abstract

A finite-element code based on the level-set method is developed for simulating the motion of viscoelastic two-phase flow problems. This method is a generalization of the finite-difference approach described in [1–4] for computing solutions to two-phase problems of inviscid and viscous fluids. The Marchuk–Yanenko operator-splitting technique is used to decouple the difficulties associated with the nonlinear convection term, the incompressibility constraint, the viscoelastic term, and the interface motion problem. The nonlinear convection problem is solved using a least-squares conjugate gradient algorithm, and the Stokes-like problem is solved using a conjugate gradient algorithm. The constitutive equation is solved using a scheme that guarantees the positive definiteness of the configuration tensor, while the convection term in the constitutive equation is discretized using a third-order upwinding scheme. The code is verified by performing a convergence study to show that the results are independent of the mesh and time-step sizes. Using our code we have studied the deformation of drops in simple shear and pressure-driven flows and of bubbles in gravity-driven flows over a wide range of dimensionless capillary (Ca) and Deborah numbers (De). For a Newtonian bubble rising in a quiescent viscoelastic liquid we find that there are limiting values of the parameters De and Ca, above which the bubble assumes a characteristic shape with a cusp-like trailing edge. The front of the bubble, however, remains round, as the local viscoelastic and viscous stresses act to round the bubble. In a pressure-driven flow the drop is stretched so that its front, which is closer to the channel center, remains round, and the trailing edge, which is closer to the channel wall, becomes sharp. These numerical results are in agreement with the experimental observations.