Surface Evolution Equations

We investigate the moving contact line problem for two-phase incompressible flows with a kinematic approach.The key idea is to derive an evolution equation for the contact angle in terms of the transporting velocity field. It turns out that the resulting equation has a simple structure and expresses the time derivative of the contact angle in terms of the velocity gradient at the solid wall. Together with the additionally imposed boundary conditions for the velocity, it yields a more specific form of the contact angle evolution. Thus, the kinematic evolution equation is a tool to analyze the evolution of the contact angle. Since the transporting velocity field is required only on the moving interface, the kinematic evolution equation also applies when the interface moves with its own velocity independent of the fluid velocity. We apply the developed tool to a class of moving contact line models which employ the Navier slip boundary condition. We derive an explicit form of the contact angle evolution for sufficiently regular solutions, showing that such solutions are unphysical. Within the simplest model, this rigorously shows that the contact angle can only relax to equilibrium if some kind of singularity is present at the contact line. Moreover, we analyze more general models including surface tension gradients at the contact line, slip at the fluid-fluid interface and mass transfer across the fluid-fluid interface.

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“…The following definition of a C 1,2 -family of moving hypersurfaces can also be found in [22], [27] and in a similar form in [17].…”

Section: Mathematical Modeling 21 Notation and Mathematical Settingmentioning

confidence: 99%

A kinematic evolution equation for the dynamic contact angle and some consequences

Fricke

Köhne

Bothe

2019

Physica D: Nonlinear Phenomena

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“…We briefly recall some basic mathematical definitions for (moving) hypersurfaces; see [13,43,44] for more details.…”

Section: A Preliminaries On (Moving) Interfacesmentioning

confidence: 99%

“…In the present work, we focus on the level set method due to Osher and Sethian [8][9][10][11][12][13]. It represents the interface as the zero contour of a smooth function φ called the "level set function"…”

Section: Introductionmentioning

confidence: 99%

A locally signed-distance preserving level set method (SDPLS) for moving interfaces

Fricke¹,

Marić²,

Vučković³

et al. 2022

Preprint

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It is well-known that the standard level set advection equation does not preserve the signed distance property, which is a desirable property for the level set function representing a moving interface. Therefore, reinitialization or redistancing methods are frequently applied to restore the signed distance property while keeping the zero-contour fixed. As an alternative approach to these methods, we introduce a novel level set advection equation that intrinsically preserves the norm of the gradient at the interface, i.e. the local signed distance property. Mathematically, this is achieved by introducing a source term that is proportional to the local rate of interfacial area generation. The introduction of the source term turns the problem into a non-linear one. However, we show that by discretizing the source term explicitly in time, it is sufficient to solve a linear equation in each time step. Notably, without adjustment, the method works naturally in the case of a moving contact line. This is a major advantage since redistancing is known to be an issue when contact lines are involved (see, e.g., Della Rocca and Blanquart, 2014). We provide a first implementation of the method in a simple first-order upwind scheme.

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“…The following definition of a C 1,2 -family of moving hypersurfaces can also be found in [57], [16] and in a similar form in [58].…”

Section: A Appendixmentioning

confidence: 99%

“…The following definition of a C 1,2 -family of moving hypersurfaces can also be found in [57], [16] and in a similar form in [58]. (i) Each Σ(t) is an orientable C 2 -hypersurface in R 3 with unit normal field denoted as n Σ (t, •).…”

Section: A Appendixmentioning

confidence: 99%

Contact line advection using the geometrical Volume-of-Fluid method

Fricke

Marić

Bothe

2020

Journal of Computational Physics

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We consider the geometrical problem of the passive transport of a hypersurface by a prescribed velocity field in the special case where the hypersurface intersects the domain boundary. This problem emerges from the discretization of continuum models for dynamic wetting. The kinematic evolution equation for the dynamic contact angle (Fricke et al., 2019) expresses the fundamental relationship between the rate of change of the contact angle and the structure of the transporting velocity field. In the present study, it serves as a reference to verify the numerical transport of the contact angle. We employ the geometrical Volume-of-Fluid (VOF) method on a structured Cartesian grid to solve the hyperbolic transport equation for the interface in two spatial dimensions. We introduce generalizations of the Youngs and ELVIRA methods to reconstruct the interface close to the domain boundary. Both methods deliver first-order convergent results for the motion of the contact line. However, the Boundary Youngs method shows strong oscillations in the numerical contact angle that do not converge with mesh refinement. In contrast to that, the Boundary ELVIRA method provides linear convergence of the numerical contact angle transport.

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Surface Evolution Equations

Cited by 6 publications

References 129 publications

A kinematic evolution equation for the dynamic contact angle and some consequences

A kinematic evolution equation for the dynamic contact angle and some consequences

A locally signed-distance preserving level set method (SDPLS) for moving interfaces

Contact line advection using the geometrical Volume-of-Fluid method

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