Moving Contact Line Problem Research Articles

Numerical Simulation for Moving Contact Line with Continuous Finite Element Schemes

AbstractIn this paper, we compute a phase field (diffuse interface) model of Cahn-Hilliard type for moving contact line problems governing the motion of isothermal multiphase incompressible fluids. The generalized Navier boundary condition proposed by Qian et al. [1] is adopted here. We discretize model equations using a continuous finite element method in space and a modified midpoint scheme in time. We apply a penalty formulation to the continuity equation which may increase the stability in the pressure variable. Two kinds of immiscible fluids in a pipe and droplet displacement with a moving contact line under the effect of pressure driven shear flow are studied using a relatively coarse grid. We also derive the discrete energy law for the droplet displacement case, which is slightly different due to the boundary conditions. The accuracy and stability of the scheme are validated by examples, results and estimate order.

On the distinguished limits of the Navier slip model of the moving contact line problem

When a droplet spreads on a solid substrate, it is unclear what the correct boundary conditions are to impose at the moving contact line. The classical no-slip condition is generally acknowledged to lead to a non-integrable singularity at the moving contact line, which a slip condition, associated with a small slip parameter, ${\it\lambda}$, serves to alleviate. In this paper, we discuss what occurs as the slip parameter, ${\it\lambda}$, tends to zero. In particular, we explain how the zero-slip limit should be discussed in consideration of two distinguished limits: one where time is held constant, $t=O(1)$, and one where time tends to infinity at the rate $t=O(|\!\log {\it\lambda}|)$. The crucial result is that in the case where time is held constant, the ${\it\lambda}\rightarrow 0$ limit converges to the slip-free equation, and contact line slippage occurs as a regular perturbative effect. However, if ${\it\lambda}\rightarrow 0$ and $t\rightarrow \infty$, then contact line slippage is a leading-order singular effect.

The asymptotics of the moving contact line: cracking an old nut

AbstractFor contact line motion where the full Stokes flow equations hold, full matched asymptotic solutions using slip models have been obtained for droplet spreading and more general geometries. These solutions to the singular perturbation problem in the slip length, however, all involve matching through an intermediate region that is taken to be separate from the outer–inner regions. Here, we show that the intermediate region is in fact an overlap region representing extensions of both the outer and the inner region, allowing direct matching to proceed. In particular, we investigate in detail how a previously seen result of the matching of the cubes of the free surface slope is justified in the lubrication setting. We also extend this two-region direct matching to the more general Stokes flow case, offering a new perspective on the asymptotics of the moving contact line problem.

An efficient scheme for a phase field model for the moving contact line problem with variable density and viscosity

In this paper, we develop an efficient numerical method for the two phase moving contact line problem with variable density, viscosity, and slip length. The physical model is based on a phase field approach, which consists of a coupled system of the Cahn–Hilliard and Navier–Stokes equations with the generalized Navier boundary condition [1,2,5]. To overcome the difficulties due to large density and viscosity ratio, the Navier–Stokes equations are solved by a splitting method based on a pressure Poisson equation [11], while the Cahn–Hilliard equation is solved by a convex splitting method. We show that the method is stable under certain conditions. The linearized schemes are easy to implement and introduce only mild CFL time constraint. Numerical tests are carried out to verify the accuracy, stability and efficiency of the schemes. The method allows us to simulate the interface problems with extremely small interface thickness. Three dimensional simulations are included to validate the efficiency of the method.

Moving contact line problem: Advances and perspectives

The solid-liquid interface, which is ubiquitous in nature and our daily life, plays fundamental roles in a variety of physical-chemical-biological-mechanical phenomena, for example in lubrication, crystal growth, and many biological reactions that govern the building of human body and the functioning of brain. A surge of interests in the moving contact line (MCL) problem, which is still going on today, can be traced back to 1970s primarily because of the existence of the “Huh-Scriven paradox”. This paper, mainly from a solid mechanics perspective, describes very briefly the multidisciplinary nature of the MCL problem, then summarizes some major advances in this exciting research area, and some future directions are presented.

Numerical Simulations of Moving Contact Line Problems

Numerical Simulations of Moving Contact Line Problems

The contact line behaviour of solid-liquid-gas diffuse-interface models

A solid-liquid-gas moving contact line is considered through a diffuse-interface model with the classical boundary condition of no-slip at the solid surface. Examination of the asymptotic behaviour as the contact line is approached shows that the relaxation of the classical model of a sharp liquid-gas interface, whilst retaining the no-slip condition, resolves the stress, and pressure singularities associated with the moving contact line problem while the fluid velocity is well defined (not multi-valued). The moving contact line behaviour is analysed for a general problem relevant for any density dependent dynamic viscosity and volume viscosity, and for general microscopic contact angle and double well free-energy forms. Away from the contact line, analysis of the diffuse-interface model shows that the Navier–Stokes equations and classical interfacial boundary conditions are obtained at leading order in the sharp-interface limit, justifying the creeping flow problem imposed in an intermediate region in the seminal work of Seppecher [Int. J. Eng. Sci. 34, 977–992 (1996)]. Corrections to Seppecher's work are given, as an incorrect solution form was originally used.

On the moving contact line singularity: Asymptotics of a diffuse-interface model

The behaviour of a solid-liquid-gas system near the three-phase contact line is considered using a diffuse-interface model with no-slip at the solid and where the fluid phase is specified by a continuous density field. Relaxation of the classical approach of a sharp liquid-gas interface and careful examination of the asymptotic behaviour as the contact line is approached is shown to resolve the stress and pressure singularities associated with the moving contact line problem. Various features of the model are scrutinised, alongside extensions to incorporate slip, finite-time relaxation of the chemical potential, or a precursor film at the wall.

3D adaptive finite element method for a phase field model for the moving contact line problems

In this paper, we propose an adaptive finite element method for simulating the moving contact line problems in three dimensions. The model that we used is the coupled Cahn-Hilliard Navier-Stokes equations with the generalized Navier boundary condition(GNBC) proposed in [18]. In our algorithm, to improve the efficiency of the simulation, we use the residual type adaptive finite element algorithm. It is well known that the phase variable decays much faster away from the interface than the velocity variables. Therefore we use an adaptive strategy that will take into account of such difference. Numerical experiments show that our algorithm is both efficient and reliable.

A finite element method for the numerical solution of the coupled Cahn–Hilliard and Navier–Stokes system for moving contact line problems

In this paper, a semi-implicit finite element method is presented for the coupled Cahn–Hilliard and Navier–Stokes equations with the generalized Navier boundary condition for the moving contact line problems. In our method, the system is solved in a decoupled way. For the Cahn–Hilliard equations, a convex splitting scheme is used along with a P1-P1 finite element discretization. The scheme is unconditionally stable. A linearized semi-implicit P2-P0 mixed finite element method is employed to solve the Navier–Stokes equations. With our method, the generalized Navier boundary condition is extended to handle the moving contact line problems with complex boundary in a very natural way. The efficiency and capacity of the present method are well demonstrated with several numerical examples.

A gradient stable scheme for a phase field model for the moving contact line problem

In this paper, an efficient numerical scheme is designed for a phase field model for the moving contact line problem, which consists of a coupled system of the Cahn–Hilliard and Navier–Stokes equations with the generalized Navier boundary condition [1,2,4]. The nonlinear version of the scheme is semi-implicit in time and is based on a convex splitting of the Cahn–Hilliard free energy (including the boundary energy) together with a projection method for the Navier–Stokes equations. We show, under certain conditions, the scheme has the total energy decaying property and is unconditionally stable. The linearized scheme is easy to implement and introduces only mild CFL time constraint. Numerical tests are carried out to verify the accuracy and stability of the scheme. The behavior of the solution near the contact line is examined. It is verified that, when the interface intersects with the boundary, the consistent splitting scheme [21,22] for the Navier Stokes equations has the better accuracy for pressure.

Contact line moving on a solid

The moving contact line problem can be summarized as follows: consider the triple line where a solid, a liquid and its vapor meet. This contact line may be the perimeter of a liquid droplet standing on a solid. Suppose now that, because of gravity for instance, the droplet and so its perimeter slides on the solid surface. The boundary conditions for viscous fluids impose that the flow velocity on an immobile solid is zero. If one assumes that the liquid/vapor surface is a material surface, i.e. that it is convected by the fluid, the contact line cannot move with respect to the solid, contrary to what is observed. Over the years many suggestions have been made to solve this problem. I show that solutions relying on the introduction of microscopic length scales are not consistent within the general framework of continuum mechanics. To get consistent solutions, one needs to introduce evaporation/condensation near the moving line, in agreement with experimental findings.

Derivation of continuum models for the moving contact line problem based on thermodynamic principles

Abstract Contact line arises as the boundary of free boundaries in fluids. This problem is interesting and important, not only because it arises in many applications, but also because of the distinct mathematical and physical features it has, such as singularities, hysteresis, instabilities, competing scaling regimes, etc. For a long time, this area of study was plagued with conflicting theories and uncertainties regarding how the problem should be modeled. In the present paper we illustrate how continuum models for the moving contact line problem can be derived using very simple thermodynamic considerations. Both the sharp interface models and diffuse interface models are derived.

Continuum models for the contact line problem

Continuum models are derived for the moving contact line problem through a combination of macroscopic and microscopic considerations. Macroscopic thermodynamic argument is used to place constraints on the form of the boundary conditions at the solid surface and the contact line. This information is then used to set up molecular dynamics to measure the detailed functional dependence of the boundary conditions. Long range molecular forces are taken into account in the form of a surface potential. This allows us to handle the case of complete wetting as well as the case of partial wetting. In particular, we obtain a new continuum model for both cases in a unified form. Two main parameters and different spreading regimes are identified from the analysis of the energy dissipations for the continuum model. Scaling laws in these different regimes are derived. The new continuum model also allows us to derive boundary conditions for the lubrication approximation. Numerical results are presented for the thin film model and the effect of the boundary condition is investigated.

An augmented method for free boundary problems with moving contact lines

An augmented immersed interface method (IIM) is proposed for simulating one-phase moving contact line problems in which a liquid drop spreads or recoils on a solid substrate. While the present two-dimensional mathematical model is a free boundary problem, in our new numerical method, the fluid domain enclosed by the free boundary is embedded into a rectangular one so that the problem can be solved by a regular Cartesian grid method. We introduce an augmented variable along the free boundary so that the stress balancing boundary condition is satisfied. A hybrid time discretization is used in the projection method for better stability. The resultant Helmholtz/Poisson equations with interfaces then are solved by the IIM in an efficient way. Several numerical tests including an accuracy check, and the spreading and recoiling processes of a liquid drop are presented in detail.

Modelling and simulation of moving contact line problems with wetting effects

This paper presents a numerical scheme for computing moving contact line flows with wetting effects. The numerical scheme is based on Arbitrary Lagrangian Eulerian (ALE) finite elements on moving m...

Modelling and simulation of moving contact line problems with wetting effects

This paper presents a numerical scheme for computing moving contact line flows with wetting effects. The numerical scheme is based on Arbitrary Lagrangian Eulerian (ALE) finite elements on moving meshes. In the computations, the wetting effects are taken into account through a weak enforcement of the prescribed equilibrium contact angle into the model equations. The equilibrium contact angle is included in the variational form of the model by replacing the curvature with Laplace Beltrami operator and integration by parts. This weak implementation allows that the contact angle determined by the numerical scheme differs from the equilibrium value and develops a certain dynamics. The Laplace Beltrami operator technique with an interface/boundary resolved mesh is well-suited for describing the dynamic contact angle observed in experiments. We consider the spreading and the pendant liquid droplets to investigate this implementation of the contact angle. It is shown that the dynamic contact angle tends to the prescribed equilibrium contact angle when time goes to infinity. However, the dynamics of the contact angle is influenced by the slip at the moving contact line.

A combined BIE-FE method for the Stokes equations

A numerical algorithm for the biharmonic equation in domains with piecewise smooth boundaries is presented. It is intended for problems describing the Stokes flow in the situations where one has corners or cusps formed by parts of the domain boundary and, due to the nature of the boundary conditions on these parts of the boundary, these regions have a global effect on the shape of the whole domain and hence have to be resolved with sufficient accuracy. The algorithm combines the boundary integral equation method for the main part of the flow domain and the finite-element method which is used to resolve the corner/cusp regions. Two parts of the solution are matched along a numerical ‘internal interface’ or, as a variant, two interfaces, and they are determined simultaneously by inverting a combined matrix in the course of iterations. The algorithm is illustrated by considering the flow configuration of ‘curtain coating’, a flow where a sheet of liquid impinges onto a moving solid substrate, which is particularly sensitive to what happens in the corner region formed, physically, by the free surface and the solid boundary. The ‘moving contact line problem’ is addressed in the framework of an earlier developed interface formation model which treats the dynamic contact angle as part of the solution, as opposed to it being a prescribed function of the contact line speed, as in the so-called ‘slip models’. Keywords: Dynamic contact angle; finite elements; free surface flows; hybrid numerical technique; Stokes equations.

HYDRODYNAMIC BOUNDARY CONDITION AT THE FLUID-SOLID INTERFACE

While the no-slip boundary condition (no relative motion at the fluid-solid interface) has been universally accepted as a cornerstone of hydrodynamics, it was known for some time that it is incompatible with the moving contact line, defined as the motion of the line of intersection of the immiscible (two phase) fluid-fluid interface with the solid wall. By employing Onsager's principle of minimum energy dissipation, we show in this work that a continuum hydrodynamics, comprising the equations of motion as well as the relevant boundary conditions, can be obtained which resolves the moving contact line problem. Our derivation reveals that just as other dissipative system dynamics, the fluid-solid interfacial boundary condition should be consistent with the framework of linear response theory. Implications of our results are discussed.

Boundary conditions for the moving contact line problem

The physical processes near a moving contact line are investigated systematically using molecular dynamics and continuum mechanics. Constitutive relations for the friction force in the contact line region, the fluid-fluid interfacial force, and the stresses in the fluid-solid interfacial region are studied. Verification of force balance demonstrates the importance of the normal stress jump across the contact line region. Effective boundary conditions are derived using force balance. It is found that in the flow regime studied, the deviation of the wall contact angle from the equilibrium contact angle is proportional to the velocity of the contact line. The effective continuum model is solved numerically and the behavior of the apparent contact angle and the wall contact angle is studied. It is found that the fluid-fluid interface near the wall exhibits a universal behavior. The onset of the nonlinear response for the contact line motion is studied within the framework of Blake’s molecular kinetic theory.