A variational front-tracking method for multiphase flow with triple junctions

TL;DR

The paper develops a variational front-tracking approach for sharp-interface multiphase Navier–Stokes flow with triple junctions, coupling a parametric description of interfaces to an unfitted Eulerian bulk solver. It provides linear unconditionally stable and structure-preserving discretizations, the latter achieving exact volume conservation via time-weighted normals and XFEM. The methods maintain energy dissipation at the discrete level and deliver accurate pressure jumps and robust interface evolution in both 2D and 3D, including three- and four-phase scenarios. Numerical experiments validate stability, volume preservation, and mesh quality, highlighting practical applicability to complex multiphase flows with triple junctions.

Abstract

We present and analyze a variational front-tracking method for a sharp-interface model of multiphase flow. The fluid interfaces between different phases are represented by curve networks in two space dimensions (2d) or surface clusters in three space dimensions (3d) with triple junctions where three interfaces meet, and boundary points/lines where an interface meets a fixed planar boundary. The model is described by the incompressible Navier--Stokes equations in the bulk domains, with classical interface conditions on the fluid interfaces, and appropriate boundary conditions at the triple junctions and boundary points/lines. We propose a weak formulation for the model, which combines a parametric formulation for the evolving interfaces and an Eulerian formulation for the bulk equations. We employ an unfitted discretization of the coupled formulation to obtain a fully discrete finite element method, where the existence and uniqueness of solutions can be shown under weak assumptions. The constructed method admits an unconditional stability result in terms of the discrete energy. Furthermore, we adapt the introduced method so that an exact volume preservation for each phase can be achieved for the discrete solutions. Numerical examples for three-phase flow and four-phase flow are presented to show the robustness and accuracy of the introduced methods.

Figures (15)

• Figure 1: Sketch of the local orientation of $(\Gamma_{s^k_1},\Gamma_{s^k_2},\Gamma_{s^k_3})$ at the triple junction line $\mathcal{T}_k$ ( blue). Depicted above is a plane that is perpendicular to $\mathcal{T}_k$. Left panel: the orientation $o^k:=(o_1^k, o_2^k, o_3^k)$ can be chosen as $(1,1,1)$. Right panel: $o^k$ can be chosen as $(1,1, -1)$.
• Figure 2: (${\rm adapt_{5,5}}$) Upper panel: pressure plots for the standard double bubble using P2-P1 without or with XFEM. Lower left panel: the time history of the energy for the standard double bubble. Lower right panel: pressure plot for the standard triple bubble using P2-P1 with XFEM.
• Figure 3: (${\rm adapt_{5,5}}$) The fluid interfaces and corresponding pressure plots for the nonsymmetric double bubble (upper panel) and triple bubble (lower panel).
• Figure 4: (${\rm adapt_{9,4}}$) Evolution of the triple junction. On top we show the fluid interfaces at times $t=0,5,10,20,30,40$, together with a visualization of the computational mesh at time $T=40$. Below are the time history of the total energy as well as a plot of the pressure and of some streamlines of the velocity at time $T=40$.
• Figure 5: ($2{\rm adapt_{9,4}}$) Snapshots of the fluid interfaces at times $t=0,0.5,\ldots, 3$, together with a visualization of the computational mesh at time $T=3$. On the right is plot of the pressure and of some streamlines of the velocity at time $t=3$. Compare also with Uemura10ripples.

Theorems & Definitions (14)

• Theorem 2.1: energy law for the strong solution
• proof
• Theorem 2.2: volume preservation for the strong solution
• proof
• Remark 2.3: curvatures at triple junctions
• Theorem 4.1: existence and uniqueness
• proof
• Remark 4.2
• Remark 4.3
• Theorem 4.4: unconditional stability