A Mixed Finite Element Method for EWOD That Directly Computes the Position of the Moving Interface

Falk, Richard S.; Walker, Shawn W.

doi:10.1137/12088567x

Cited by 9 publications

(14 citation statements)

References 27 publications

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“…2.8). A related analysis, concerning Hele-Shaw flow without contact line effects, can be found in [32].…”

Section: Well-posednessmentioning

confidence: 99%

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A mixed formulation of a sharp interface model of stokes flow with moving contact lines

Walker

2014

ESAIM: M2AN

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Two-phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddlepoint) formulation of a Stokesian fluid with surface tension that interacts with a rigid substrate. The model is derived by an Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking level) and allows for moving contact lines and contact angle hysteresis and pinning through a variational inequality. Moreover, the formulation can be extended to include non-linear contact line motion models. We prove the well-posedness of the time semi-discrete system and fully discrete method using appropriate choices of finite element spaces. A formal energy law is derived for the semi-discrete and fully discrete formulations and preliminary error estimates are also given. Simulation results are presented for a droplet in multiple configurations to illustrate the method.Mathematics Subject Classification. 65N30, 65M12, 76D45, 76M30.Article published by EDP Sciences c EDP Sciences, SMAI 2014 Contact line "Paradox"There are many types of contact lines that appear in different physical situations (see Fig. 1). The most familiar concerns the peeling of adhesive tape ( Fig. 1a) [16,21,68]. Here the contact line (a point in 2-D) separates the tape into two disjoint regions: the part that is still attached to the substrate (where no-slip applies) and the other which is free to deform as a thin flexible body. Typically, one models the free part of the tape as elastic and captures the motion of the contact point with an inequality constraint [16,68]. The main thing to note is the contact point is not a material point. So its velocity is not a material velocity. Furthermore, the velocity of the contact line is not necessarily related to the velocity of the material tape. Lastly, it may seem that the velocity of the tape is discontinuous at the contact point. In reality, there is a small (curved) transition region at the contact point from flat to making an angle; ergo, no discontinuity.The situation is more complicated in the case of two immiscible fluids on a solid substrate. When the fluids are displaced, there arises the classic contact line paradox described in the seminal paper by [41] and addressed by others [9,10,28,54,55,61]. In [41], they assumed the wedge-shaped geometry depicted in Figure 1b. By applying free surface boundary conditions on the liquid-gas interface and no-slip conditions on the liquid-solid interface, they obtained a solution to the Navier-Stokes equations that has a logarithmic singularity in the rate of viscous dissipation in a small neighborhood of the moving contact line; clearly, a nonphysical result. The reason is because assuming a wedge-shape geometry and no-slip gives a discontinuity in the velocity boundary condition at the wedge-tip (i.e. the contact line). The singularity is then...

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“…2.8). A related analysis, concerning Hele-Shaw flow without contact line effects, can be found in [32].…”

Section: Well-posednessmentioning

confidence: 99%

“…3.5). In particular, we update the domain boundary with X n+1 h := W n+1 h , then use a smooth domain deformation to update the interior vertices of the mesh (see [32] for a similar approach).…”

Section: Mixed Formulationmentioning

confidence: 99%

A mixed formulation of a sharp interface model of stokes flow with moving contact lines

Walker

2014

ESAIM: M2AN

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“…The linear systems are solved by MATLAB's "backslash" command. Alternatively, one can use an iterative procedure such as Uzawa's algorithm; see [22,Section 7] for an example in a related problem.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Clearly, the bulk domains˝l,˝s follow the interface . Given V i C1 on i , it can be extended to the entire domain˝by a harmonic extension [22,65]…”

Section: Domain Velocitymentioning

confidence: 99%

A mixed formulation of the Stefan problem with surface tension

Davis¹,

Walker²

2016

Interfaces Free Bound.

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A dual formulation and finite element method is proposed and analyzed for simulating the Stefan problem with surface tension. The method uses a mixed form of the heat equation in the solid and liquid (bulk) domains, and imposes a weak formulation of the interface motion law (on the solidliquid interface) as a constraint. The basic unknowns are the heat fluxes and temperatures in the bulk, and the velocity and temperature on the interface. The formulation, as well as its discretization, is viewed as a saddle point system. Well-posedness of the time semi-discrete and fully discrete formulations is proved in three dimensions, as well as an a priori (stability) bound and conservation law. Simulations of interface growth (in two dimensions) are presented to illustrate the method.Our paper presents a completely mixed formulation of the Stefan problem, including the bulk heat equations [8]. In other words, we formulate the problem in a saddle-point framework, where the heat equations are in mixed form, and the interface motion law appears as a constraint in the system of equations with a balancing Lagrange multiplier that represents the interface temperature. To the best of our knowledge, this is a new method for the Stefan problem with surface tension. Some highlights of our method are the following.We prove that both the time semi-discrete and fully discrete systems have a priori bounds (in time) that mimic the continuous model. This assumes the interface velocity is reasonably regular and that there are no topological changes. Moreover, we can prove that both the time semi-discrete and fully discrete systems maintain conservation of thermal energy. In [5], they only achieve this for their discrete in space, continuous in time, scheme. The interface is represented by a surface triangulation that conforms to the bulk mesh which deforms with the interface. Hence, occasional re-meshing is needed, which is done by the method in [63]. One advantage of this method is that all integrals in the finite element formulation can be computed exactly. In addition, we do not need to compute the intersection of meshes at adjacent time steps to transfer solution variables from one mesh to the next (e.g. for computing L 2 projections from one mesh to another). Our method can be modified to include anisotropic surface tension via [5], which is relevant to crystal growth. The well-posedness of the method remains unchanged, as well as the a priori bound and conservation law. Other variations of the Stefan problem (e.g. Mullins-Sekerka) can be formulated with our approach by straightforward modifications. One can even include moving contact line effects when the solid phase is attached to a rigid boundary [60, 64]. SummaryIn Section 2, we describe the governing equations. Section 3 describes the fully continuous weak formulation and derives a formal a priori bound and conservation law. Section 4 explains the timediscretization and how the interface motion is handled. A variational formulation of the time semidiscrete problem is given, its ...

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“…The work here aims to provide a first step in moving from the study of the static problem described in [5] to the dynamic situation described above, via an efficient solution method for the underlying potential problem. Related work on the simulation of electrified fluids has been carried out using a range of techniques including asymptotic approaches [5,6], level set methods [7], finite element approaches for coupled fluid flow and dynamic interface models [8], and boundary integral methods for coupled potential and dynamic interface problems [9].…”

Section: Introductionmentioning

confidence: 99%

Boundary integral solution of potential problems arising in the modelling of electrified oil films

Chappell¹

2015

J. Integral Equations Applications

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We consider a class of potential problems on a periodic half-space for the modelling of electrified oil films, which are used in the development of novel switchable liquid optical devices (diffraction gratings). A boundary integral formulation which reduces the problem to the study of the oil-air interface alone is derived and solved in a highly efficient manner using the Nyström method. The oil films encountered experimentally are typically very thin and thus an interface-only integral representation is important for avoiding the near-singularity problems associated with boundary integral methods for long slender domains. The super-algebraic convergence of the proposed methods is discussed and demonstrated via appropriate numerical experiments.

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A Mixed Finite Element Method for EWOD That Directly Computes the Position of the Moving Interface

Cited by 9 publications

References 27 publications

A mixed formulation of a sharp interface model of stokes flow with moving contact lines

A mixed formulation of a sharp interface model of stokes flow with moving contact lines

A mixed formulation of the Stefan problem with surface tension

Boundary integral solution of potential problems arising in the modelling of electrified oil films

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