An Energetic Variational Formulation with Phase Field Methods for Interfacial Dynamics of Complex Fluids: Advantages and Challenges

Abstract

Abstract. The use of a phase field to describe interfacial phenomena has a long and fruitful tradition. There are two key ingredients to the method: the transformation of Lagrangian description of geometric motions to Eulerian description framework, and the employment of the energetic variational procedure to derive the coupled systems. Several groups have used this theoretical framework to approximate Navier-Stokes systems for two-phase flows. Recently, we have adapted the method to simulate interfacial dynamics in blends of microstructured complex fluids. This review has two objectives. The first is to give a more or less self-contained exposition of the method. We will briefly review the literature, present the governing equations and discuss a numerical scheme based on different numerical schemes, such as spectral methods. The second objective is to elucidate the subtleties of the model that need to be handled properly for certain applications. These points, rarely discussed in the literature, are essential for a realistic representation of the physics and a successful numerical implementation. The advantages and limitations of the method will be illustrated by numerical examples. We hope that this review will encourage readers whose applications may potentially benefit from a similar approach to explore it further.

3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids

“…Equations (7)(8)(9)(10) form the governing equations for our two-phase system. For discretization using second-order finite elements, the fourth-order Cahn-Hilliard equation is decomposed into two second-order equations [20,35].…”

“…Because of its energy-based formalism and the physical picture of the diffuse-interface model, it has some unique features among interface-capturing methods [8]: (i) The evolution of the interface is self-consistent and requires no ad hoc intervention such as the re-initialization in level set methods. (ii) The theory has an energy law that ensures well-posedness in numerical computation [9,10].…”

Modeling and Numerical Approximation of Two-Phase Incompressible Flows by a Phase-Field Approach

“…When the density ratio is small, a usual approach is to use the Boussinesq approximation, i.e., replacing (2.14) by 22) where ρ 0 = ρ1+ρ2 2 and g(ρ 1 , ρ 2 ) is an additional gravitational force to account for the density difference. However, when the density ratio is large, the above Boussinesq approximation is no longer valid.…”

“…The idea behind the pressure-stabilization scheme is to replace the divergence free condition by 20) with = (δt) 2 . A first-order pressure-stabilization scheme based on (3.19) reads: 22) where…”

“…While it is theoretically more accurate to choose η and γ as small as possible, but it comes with added numerical costs as decreasing η requires more refined mesh to resolve the thinner interfacial region, and decreasing γ leads to longer relaxation time scales. We refer to [22] for a more detailed account on these issues.…”