Diffuse interface surface tension models in an expanding flow

Abstract

We consider a diusive interface surface tension model under compressible ow. The equation of interest is the Cahn-Hilliard or Allen-Cahn equation with advection by a non-divergence free velocity eld. We prove that both model problems are well-posed. We are especially interested in the behavior of solutions with respect to droplet breakup phenomena. Numerical simulations of 1,2 and 3D all illustrate that the Cahn-Hilliard model is much more eective for droplet breakup. Using asymptotic methods we correctly predict the breakup condition for the Cahn-Hilliard model. Moreover, we prove that the Allen-Cahn model will not break up under certain circumstances due to a maximum principle. BackgroundThere is a need to develop simple computational models for surface tension in the droplet breakup phenomena. As an example, consider a piece of material that expands under a sudden pulse of energy that comes from laser fusion [21] or heavy ion fusion [12]. The material will breakup, and surface tension plays an important role in the ensuing dynamics. There are many numerical methods that deal with surface tension in twophase uids. This problem is known for its computational stiness. It contains two dierent time scales, the small surface tension time scale and the convection time scale. Three main algorithms exist for two-phase uids. The sharp interface method tracks the interface explicitly, yet it requires extensive processing when the interface splits and merges. Since droplet breakup involves mainly merging and splitting of the interface, we * W. Liu and A. Bertozzi are with the

Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation

“…Now, we test the advective AC equation with a sheering flow given by [5] V = (0, −v 0 y), with v 0 = 100, and the initial condition is the square data The interface length is taken as = 0.01. As the expanding case, the uniform solutions are obtained on the uniform mesh with the mesh size ∆x = ∆y = 1/32, and for the adaptive case we use the same settings.…”

“…Interfacial dynamics has great importance in the modeling of multi-phase flow and it plays an important role in different scientific and industrial applications such as micro-structure evolution and grain growth in material science [1], binary fluids flow movement [2], and complex interfacial dynamics [3]. There have been various diffuse interface models for multi-phase flow [4,5]. In this study, we consider a specific model of diffuse interface for two phase flow; Allen-Cahn (AC) equation with advection.…”

“…In (1), the bi-stable cubic nonlinearity f (u) is such that f (u) = F (u) = 2u(1 − u)(1 − 2u) for a double-well potential F (u), and V = (V 1 , V 2 ) T is the prescribed velocity field. The velocity field is either given or computed from the Navier-Stokes equation [4,5]. In many cases the velocity field V is divergence-free [6,7] and satisfies incompressible Navier-Stokes equations.…”

“…In this study we consider non-divergence-free velocity fields ∇·V = 0, either expanding ∇·V > 0 or contracting ∇·V < 0 [5]. In other words we consider advective AC equation in compressible fluids.…”

“…Problems with surface tension in two-phase fluids are known as multiscale problems with two different time scales, the small surface tension, and the convection time scale, which results in computational stiffness. Actually, there exists three main algorithms: the sharp interface algorithm method, the level-set algorithm method and the diffuse interface method [5]. The numerical simulations are illustrated using finite elements method in space and semi-implicit schemes or semi-implicit schemes with splitting in time [10,11].…”