Parametric finite element approximation of two-phase Navier--Stokes flow with viscoelasticity

TL;DR

This work tackles the numerical approximation of two-phase viscoelastic flow governed by the Oldroyd-B model with possible stress diffusion ($\alpha$) using a parametric unfitted finite element method. It fuses a moving-interface framework with an unfitted bulk discretisation and an XFEM-enhanced pressure space to achieve unconditional stability and volume conservation, underpinned by a rigorously derived energy identity. The authors establish stability and existence of discrete solutions, develop a semi-discrete scheme with equidistributed interface nodes, and demonstrate good volume conservation and energy decay in 2D numerical tests, including retraction of an elongated bubble and rising-bubble scenarios across varying viscoelastic parameters. They also extend the framework to a variable shear modulus $G(\cdot,t)$, outlining discretisation and analysis adjustments. Overall, the approach provides a robust, energy-faithful computational platform for simulating viscoelastic two-phase flows with potential extensions to 3D and more complex rheologies.

Abstract

In this work, we present a parametric finite element approximation of two-phase Navier-Stokes flow with viscoelasticity. The free boundary problem is given by the viscoelastic Navier-Stokes equations in the two fluid phases, connected by jump conditions across the interface. The elasticity in the fluids is characterised using the Oldroyd-B model with possible stress diffusion. The model was originally introduced to approximate fluid-structure interaction problems between an incompressible Newtonian fluid and a hyperelastic neo-Hookean solid, which are possible limit cases of the model. We approximate a variational formulation of the model with an unfitted finite element method that uses piecewise linear parametric finite elements. The two-phase Navier-Stokes-Oldroyd-B system in the bulk regions is discretised in a way that guarantees unconditional solvability and stability for the coupled bulk-interface system. Good volume conservation properties for the two phases are observed in the case where the pressure approximation space is enriched with the help of an XFEM function. We show the applicability of our method with some numerical results.

Figures (26)

• Figure 1: A schematic sketch of the domain $\Omega$ in two space dimensions.
• Figure 2: A uniform bulk mesh, successively refined close to the interface.
• Figure 3: Time evolution of the interface in the retraction experiment at times $t \in \{0, 0.21, 0.375, 0.55, 1\}$.
• Figure 4: Time evolution of the energy contributions for the retraction experiment. From left to right: kinetic energy, elastic energy and interface length.
• Figure 5: Time evolution of the full energy and the dissipations $\mathcal{D}^{n+1}$ defined in \ref{['eq:dissipation']}.

Theorems & Definitions (11)

• Remark 3.1
• Theorem 3.3
• Theorem 3.4
• Lemma 3.5
• Lemma 3.6
• Lemma 3.7
• Lemma 3.8
• Lemma 3.9
• Theorem 3.10
• Lemma 3.11