On the Consistency of Dynamic Wetting Boundary Conditions for the Navier-Stokes-Cahn-Hilliard Equations

We investigate the limiting behavior of the Navier-Stokes-Cahn-Hilliard model for binary-fluid flows as the diffuse-interface thickness passes to zero, in the presence of fluid-fluid-solid contact lines. Allowing for motion of such contact lines relative to the solid substrate is required to adequately model multi-phase and multi-species fluid transport past and through solid media. Even though diffuse-interface models provide an inherent slip mechanism through the mobility-induced diffusion, this slip vanishes as the interface thickness and mobility parameter tend to zero in the so-called sharp-interface limit. The objective of this work is to present dynamic wetting and generalized Navier boundary conditions for diffuse-interface models that are consistent in the sharp-interface limit. We concentrate our analysis on the prototypical binary-fluid Couette-flow problems. To verify the consistency of the diffuse-interface model in the limit of vanishing interface thickness, we provide reference limit solutions of a corresponding sharp-interface model. For parameter values both at and away from the critical viscosity ratio, we present and compare the results of both the diffuse- and sharp-interface models. The close match between both model results indicates that the considered test case lends itself well as a benchmark for further research

Numerical investigation of the sharp-interface limit of the Navier-Stokes-Cahn-Hilliard equations

In this article, we study the behaviour of the Abels-Garcke-Grün Navier-Stokes-Cahn- Hilliard diffuse-interface model for binary-fluid flows, as the diffuse-interface thickness passes to zero. For the diffuse-interface model to approach a classical sharp-interface model in the limit, the so-called mobility parameter in the diffuse-interface model must scale appropriately with the interface-thickness parameter. In the literature various scaling relations in the range to have been proposed, but the optimal order to pass to the limit has not been explored previously. Our primary objective is to elucidate this optimal order of the - scaling relation in terms of the rate of convergence of the diffuse-interface solution to the sharp-interface solution. Additionally, we examine how the convergence rate is affected by a sub-optimal parameter scaling. We centre our investigation around the case of an oscillating droplet. To provide reference limit solutions, we derive new analytical expressions for small-amplitude oscillations of a viscous droplet in a viscous ambient fluid in two dimensions. For two distinct modes of oscillation, we probe the sharp-interface limit of the Navier-Stokes-Cahn-Hilliard equations by means of an adaptive finite-element method. The adaptive-refinement procedure enables us to consider diffuse-interface thicknesses that are significantly smaller than other relevant length scales in the droplet-oscillation problem, allowing an exploration of the asymptotic regime.</p

An adaptive isogeometric analysis approach to elasto-capillary fluid-solid interaction

We present an adaptive isogeometric-analysis approach to elasto-capillary fluid-solid interaction (FSI), based on a diffuse-interface model for the binary fluid and an Arbitrary-Lagrangian-Eulerian formulation for the FSI problem. We consider approximations constructed from adaptive high-regularity truncated hierarchical splines, as employed in the isogeometric analysis (IGA) paradigm. The considered adaptive strategy comprises a two-level hierarchical a posteriori error estimate. The hierarchical a posteriori error estimate directs a support-based refinement procedure. To attain robustness of the solution procedure for the aggregated binary-fluid-solid-interaction problem, we apply a fully monolithic solution procedure and we introduce a continuation process in which the diffuse interface of the binary fluid is artificially enlarged on the coarsest levels of the adaptive-refinement procedure. To assess the capability of the presented adaptive IGA method for elasto-capillary FSI, we conduct numerical computations for a configuration pertaining to a sessile droplet on a soft solid substrate

A robust and accurate adaptive approximation method for a diffuse-interface model of binary-fluid flows

We present an adaptive simulation framework for binary-fluid flows, based on the Abels–Garcke–Grün Navier–Stokes–Cahn–Hilliard (AGG NSCH) diffuse-interface model. The adaptive-refinement procedure is guided by a two-level hierarchical a-posteriori error estimate, and it effectively resolves the spatial multiscale behavior of the diffuse-interface model. To improve the robustness of the solution procedure and avoid severe time-step restrictions for small-interface thicknesses, we introduce an ɛ-continuation procedure, in which the diffuse interface thickness (ɛ) are enlarged on coarse meshes, and the mobility is scaled accordingly. To further accelerate the computations and improve robustness, we apply a modified Backward Euler scheme in the initial stages of the adaptive-refinement procedure in each time step, and a Crank–Nicolson scheme in the final stages of the refinement procedure. To enhance the robustness of the nonlinear solution procedure, we introduce a partitioned solution procedure for the linear tangent problems in Newton's method, based on a decomposition of the NSCH system into its NS and CH subsystems. We conduct a systematic investigation of the conditioning of the monolithic NSCH tangent matrix and of its NS and CH subsystems for a representative 2D model problem. To illustrate the properties of the presented adaptive simulation framework, we present numerical results for a 2D oscillating water droplet suspended in air, and we validate the obtained results versus those of a corresponding sharp-interface model