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In this paper, we study the contact angle hysteresis on a slowly moving rough boundary using a phase-field model with a relaxed boundary condition. In particular, we want to model the recent experi...

The paper studies an Allen–Cahn-type equation defined on a time-dependent surface as a model of phase separation with order–disorder transition in a thin material layer. By a formal inner–outer expansion, it is shown that the limiting behavior of the solution is a geodesic mean curvature type flow in reference coordinates. A geometrically unfitted finite element method, known as a trace FEM, is considered for the numerical solution of the equation. The paper provides full stability analysis and convergence analysis that accounts for interpolation errors and an approximate recovery of the geometry.

By using the Onsager variational principle as an approximation tool, we develop a new diffusion generated motion method for wetting problems. The method uses a signed distance function to represent the interface between the liquid and vapor surface. In each iteration, a linear diffusion equation with a linear boundary condition is solved for one time step in addition to a simple re-distance step and a volume correction step. The method has a first-order convergence rate with respect to the time step size even in the vicinity three-phase contact points. Its energy stability property is analyzed by careful studies for some geometric flows on substrates. Numerical examples show that the method can be used to simulate complicated wetting problems on inhomogeneous surfaces.

It is proved that if \(\phi \) is a finite Blaschke product with four zeros, then \(M_\phi \) is reducible on the Dirichlet space with norm \(\Vert \ \Vert \) if and only if \(\phi =\phi _1\circ \phi _2\), where \(\phi _1, \phi _2\) are Blaschke products and \(\phi _2\) is equivalent to \(z^2\). Also, the same reducibility of \(M_\phi \) with finite Blaschke product \(\phi \) on the Dirichlet space under the equivalent norms \(\Vert \ \Vert _1\) and \(\Vert \ \Vert _0\) is given.

The threshold dynamics method developed by Merriman, Bence and Osher (MBO) is an efficient method for simulating the motion by mean curvature flow when the interface is away from the solid boundary. Direct generalization of MBO-type methods to the wetting problem with interfaces intersecting the solid boundary is not easy because solving the heat equation in a general domain with a wetting boundary condition is not as efficient as it is with the original MBO method. The dynamics of the contact point also follows a different law compared with the dynamics of the interface away from the boundary. In this paper, we develop an efficient volume preserving threshold dynamics method for simulating wetting on rough surfaces. This method is based on minimization of the weighted surface area functional over an extended domain that includes the solid phase. The method is simple, stable with O ( N log ź N ) complexity per time step and is not sensitive to the inhomogeneity or roughness of the solid boundary.

We study the interface dynamics and contact angle hysteresis in a two dimensional, chemically patterned channel described by the Cahn-Hilliard equation with a relaxation boundary condition. A system for the dynamics of the contact angle and contact point is derived in the sharp interface limit. We then analyze the behavior of the solution using the phase plane analysis. We observe the stick-slip of the contact point and the contact angle hysteresis. As the size of the pattern decreases to zero, the stick-slip becomes weaker but the hysteresis becomes stronger in the sense that one observes either the advancing contact angle or the receding contact angle without any switching in between. Numerical examples are presented to verify our analysis.

In this paper, we propose an approach for solving PDEs on evolving surfaces using a combination of the trace finite element method and a fast marching method. The numerical approach is based on the Eulerian description of the surface problem and employs a time-independent background mesh that is not fitted to the surface. The surface and its evolution may be given implicitly, for example, by the level set method. Extension of the PDE off the surface is not required. The method introduced in this paper naturally allows a surface to undergo topological changes and experience local geometric singularities. In the simplest setting, the numerical method is second order accurate in space and time. Higher order variants are feasible, but not studied in this paper. We show results of several numerical experiments, which demonstrate the convergence properties of the method and its ability to handle the case of the surface with topological changes.

Summary In this paper we present a finite element method (FEM) for two-phase incompressible flows with moving contact lines. We use a sharp interface Navier-Stokes model for the bulk phase fluid dynamics. Surface tension forces, including Marangoni forces and viscous interfacial effects, are modeled. For describing the moving contact lines we consider a class of continuum models which contains several special cases known from the literature. For the whole model, describing bulk fluid dynamics, surface tension forces and contact line forces, we derive a variational formulation and a corresponding energy estimate. For handling the evolving interface numerically the level-set technique is applied. The discontinuous pressure is accurately approximated by using a stabilized extended finite element space (XFEM). We apply a Nitsche technique to weakly impose the Navier slip conditions on the solid wall. A unified approach for discretization of the (different types of) surface tension forces and contact line forces is introduced. Results of numerical experiments are presented which illustrate the performance of the solver. This article is protected by copyright. All rights reserved.

We present a numerical implementation of a model for void coalescence and fracture in nonlinear elasticity. The model is similar to the Ambrosio–Tortorelli regularization of the standard free-discontinuity variational model for quasistatic brittle fracture. The main change is the introduction of a nonlinear polyconvex energy that allows for cavitation. This change requires new analytic and numerical techniques. We propose a numerical method based on alternating directional minimization and a stabilized Crouzeix–Raviart finite element discretization. The method is used in several experiments, including void coalescence, void creation under tensile stress, failure in perfect materials and in materials with hard inclusions. The experimental results show the ability of the model and the numerical method to study different failure mechanisms in rubber-like materials.

We develop a new multiscale finite element method for the Laplace equation with oscillating Neumann boundary conditions on rough boundaries. The key point is the introduction of a new boundary condition that incorporates both the microscopically geometrical and physical information of the rough boundary. Our approach applies to problems posed on a domain with a rough boundary as well as oscillating boundary conditions. We prove the method has a linear convergence rate in the energy norm with a weak resonance term for periodic roughness. Numerical results are reported for both periodic and nonperiodic roughness.

The sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for binary fluids with moving contact lines are studied by asymptotic analysis and numerical simulations. The effects of the mobility number as well as a phenomenological relaxation parameter on the boundary condition are considered. In asymptotic analysis, we consider both the cases that the mobility number is proportional to the Cahn number and the square of the Cahn number, and derive the sharp-interface limits for several set-ups of the boundary relaxation parameter. It is shown that the sharp-interface limit of the phase-field model is the standard two-phase incompressible Navier–Stokes equations coupled with several different slip boundary conditions. Numerical results are consistent with the analysis results and also illustrate the different convergence rates of the sharp-interface limits for different scalings of the two parameters.

In this paper, we study a numerical method for the solution of partial differential equations on evolving surfaces. The numerical method is built on the stabilized trace finite element method (TraceFEM) for the spatial discretization and finite differences for the time discretization. The TraceFEM uses a stationary background mesh, which can be chosen independent of time and the position of the surface. The stabilization ensures well-conditioning of the algebraic systems and defines a regular extension of the solution from the surface to its volumetric neighborhood. Having such an extension is essential for the numerical method to be well-defined. The paper proves numerical stability and optimal order error estimates for the case of simplicial background meshes and finite element spaces of order $m\ge1$. For the algebraic condition numbers of the resulting systems we prove estimates, which are independent of the position of the interface. The method allows that the surface and its evolution are given implicitly with the help of an indicator function. Results of numerical experiments for a set of 2D evolving surfaces are provided., 27 pages, 3 figures, 6 tables

We study a stationary wetting problem on rough and inhomogeneous solid surfaces. We derive a new formula for the apparent contact angle by an asymptotic two-scale homogenization method. The formula reduces to a modified Wenzel equation for geometrically rough surfaces and a modified Cassie equation for chemically inhomogeneous surfaces. Unlike the classical Wenzel and Cassie equations, the modified equations correspond to local minimizers of the total interface energy in the solid-liquid-air system, so that they are consistent with experimental observations. The homogenization results are proved rigorously by a variational method.