Fully coupled implicit finite-volume algorithm for viscoelastic interfacial flows

TL;DR

This paper introduces a fully coupled implicit finite-volume framework for simulating incompressible viscoelastic interfacial flows, solving pressure, velocity and the six independent polymer-stress components in a single linear system. It employs an upper-convected Maxwell constitutive model with limited extensibility and shear-thinning, and uses a front-tracking method to represent the fluid interface with surface tension effects. The solver advances all couplings implicitly, including stress–velocity and pressure–velocity interactions, without relying on log-conformation transformations, and demonstrates robust accuracy at very high Weissenberg numbers ($Wi$ up to $10^4$) across four challenging test cases. The results show good agreement with experimental and reference data, highlighting the framework’s capability to predict strongly elastic interfacial dynamics reliably, which is significant for practical viscoelastic multiphase flows in engineering and biology.

Abstract

A fully coupled implicit finite-volume algorithm for incompressible viscoelastic interfacial flows is proposed, whereby the viscoelasticity of the flow is described by an upper-convected Maxwell constitutive model, including limited extensibility and shear-thinning behaviour. The governing equations describing the conservation of continuity and momentum, as well as the constitutive model are discretized using standard finite-volume methods and are solved for pressure, velocity and the polymer stress tensor in a single linear system of equations. Treating all terms of the linearized and discretized governing equations implicit in velocity, pressure and/or the components of the polymer stress tensor, a tightly coupled system of equations is obtained. The interface separating the interacting bulk phases and the surface tension acting at the fluid interface are modelled using a state-of-the-art front-tracking method. We demonstrate the capabilities of the proposed numerical framework with four representative test cases, including the deformation of a viscoelastic droplet in shear flow at large Weissenberg numbers of up to Wi=10^4, and the jump discontinuity of the rise velocity of a bubble rising in a viscoelastic liquid as a result of a "negative wake". Contrary to previous studies using segregated algorithms, the proposed fully coupled implicit algorithm does not apply or require a log-conformation approach to predict these flows. Overall, the fully implicit coupled front-tracking formulation provides a robust framework to reliable numerical predictions of strongly elastic interfacial flows at large Weissenberg numbers.

Figures (12)

• Figure 1: Interpolation stencils of the velocity at face $f$, with the adjacent cells $P$ and $Q$, considered for the stress-velocity coupling.
• Figure 2: Flow chart of the solution procedure of the discretized and linearized system of governing equations, where $n$ is the nonlinear iteration counter, $\Gamma = \{u,v,w,p,\tau_{\text{p},xx},\tau_{\text{p},yy}, \tau_{\text{p},zz}, \tau_{\text{p},xy}, \tau_{\text{p},xz}, \tau_{\text{p},yz}\}$ are the solution variables and $F_f$ is the flux through mesh face $f$ (see Section \ref{['sec:mwi']}). The coefficient matrix $\mathbf{A}$ holds all coefficients for the implicitly sought solution variables $\Gamma^{(n+1)}$ of the discretized governing equations and $\boldsymbol{\zeta}$ is the solution vector. The right-hand side vector $\mathbf{b}$ holds the deferred contributions of the previous iteration ($\Gamma^{(n)}$, $\vartheta_f^{(n)}$) and the contributions of the previous time-levels ($\Gamma^{(t-\Delta t_1)}$, $\Gamma^{(t-\Delta t_1-\Delta t_2)}$, $\vartheta_f^{(t-\Delta t_1)}$).
• Figure 3: Velocity contours of the considered LPTT fluid in a lid-driven cavity at steady state, for $\mathrm{Wi} \in \{1,5\}$, on an equidistant Cartesian mesh with $160 \times 160$ cells.
• Figure 4: Velocity profiles along the respective centerlines of the lid-driven cavity, obtained on the reference mesh with $160 \times 160$ cells, and convergence of the error $\varepsilon_{\tau_{\mathrm{p},xx}}$ in normal polymer stress component $\tau_{\mathrm{p},xx}$ relative to the reference mesh, for $\mathrm{Wi} = 1$. The results of Yapici2012 are shown for reference.
• Figure 5: Velocity profiles along the respective centerlines of the lid-driven cavity, obtained on the reference mesh with $160 \times 160$ cells, and covergence of the error $\varepsilon_{\tau_{\mathrm{p},xx}}$ in normal polymer stress component $\tau_{\mathrm{p},xx}$ relative to the reference mesh, for $\mathrm{Wi} = 5$. The results of Yapici2012 are shown for reference.