On the Two-Phase Navier–Stokes Equations with Boussinesq–Scriven Surface Fluid

Bothe, Dieter; Prüß, Jan

doi:10.1007/s00021-008-0278-x

Cited by 75 publications

(49 citation statements)

References 8 publications

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“…The functions representing the physical fields live in space (on Γ) and time, ie, in the time interval = [0, T]. Therefore, Equations (5), (6), and (13) have to be solved in the space-time domain Γ × . Herein, we restrict ourselves to spatially fixed manifolds Γ.…”

Section: Instationary Navier-stokes Flowmentioning

confidence: 99%

“…Herein, Stokes and incompressible Navier‐Stokes flows on curved two‐dimensional (2D) manifolds are considered. The governing equations for flows on moving surfaces were discussed in the works of Bothe and Prüss and Jankuhn et al based on fundamental surface continuum mechanics and conservation laws, and in the work of Koba et al, an energetic approach was presented. Earlier works in a similar context may be traced back to other works .…”

Section: Introductionmentioning

confidence: 99%

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Higher‐order surface FEM for incompressible Navier‐Stokes flows on manifolds

Fries

2018

Numerical Methods in Fluids

101

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Summary Stationary and instationary Stokes and Navier‐Stokes flows are considered on two‐dimensional manifolds, ie, on curved surfaces in three dimensions. The higher‐order surface FEM is used for the approximation of the geometry, velocities, pressure, and Lagrange multiplier to enforce tangential velocities. Individual element orders are employed for these various fields. Streamline‐upwind stabilization is employed for flows at high Reynolds numbers. Applications are presented, which extend classical benchmark test cases from flat domains to general manifolds. Highly accurate solutions are obtained, and higher‐order convergence rates are confirmed.

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Section: Instationary Navier-stokes Flowmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Higher‐order surface FEM for incompressible Navier‐Stokes flows on manifolds

Fries

2018

Numerical Methods in Fluids

101

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“…We follow the approach in Bothe & Pruess (2010) to introduce the mathematical model. We therefore consider two immiscible Newtonian fluids with constant densities.…”

Section: Mathematical Modelmentioning

confidence: 99%

A finite element approach to incompressible two-phase flow on manifolds

Nitschke¹,

Voigt²,

Wensch³

2012

J. Fluid Mech.

111

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A two-phase Newtonian surface fluid is modelled as a surface Cahn-HilliardNavier-Stokes equation using a stream function formulation. This allows one to circumvent the subtleties in describing vectorial second-order partial differential equations on curved surfaces and allows for an efficient numerical treatment using parametric finite elements. The approach is validated for various test cases, including a vortex-trapping surface demonstrating the strong interplay of the surface morphology and the flow. Finally the approach is applied to a Rayleigh-Taylor instability and coarsening scenarios on various surfaces.

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“…In [4] the Boussinesq-Scriven surface fluid model was considered to formulate a continuum model for fluid membranes in a bulk fluid, which contains equations for a viscous fluid on a curved moving surface, and study the effect of membrane viscosity in the dynamics of fluid membranes. It was also studied in the context of two-phase flows [5,6,25] in which equations for a surface fluid are considered as the boundary condition on a fluid interface.…”

Section: Introductionmentioning

confidence: 99%

On singular limit equations for incompressible fluids in moving thin domains

Miura

2017

Quart. Appl. Math.

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Abstract. We consider the incompressible Euler and Navier-Stokes equations in a three-dimensional moving thin domain. Under the assumption that the moving thin domain degenerates into a two-dimensional moving closed surface as the width of the thin domain goes to zero, we give a heuristic derivation of singular limit equations on the degenerate moving surface of the Euler and Navier-Stokes equations in the moving thin domain and investigate relations between their energy structures. We also compare the limit equations with the Euler and Navier-Stokes equations on a stationary manifold, which are described in terms of the Levi-Civita connection. IntroductionFluid flows in a thin domain appear in many problems of natural sciences, e.g. ocean dynamics, geophysical fluid dynamics, and fluid flows in cell membranes. In the study of the incompressible Navier-Stokes equations in a three-dimensional thin domain mathematical researchers are mainly interested in global existence of a strong solution for large data since a three-dimensional thin domain with sufficiently small width can be considered "almost two-dimensional." It is also important to investigate the behavior of a solution as the width of a thin domain goes to zero. We may naturally ask whether we can derive limit equations as a thin domain degenerates into a two-dimensional set and compare properties of solutions to the original three-dimensional equations and the corresponding two-dimensional limit equations. There are several works studying such problems with a three-dimensional flat thin domain [15,16,29,33] of the formfor small ε > 0, where ω is a two-dimensional domain and g 0 and g 1 are functions on ω, and a three-dimensional thin spherical domain [34] which is a region between two concentric spheres of near radii. (We also refer to [28] for the strategy of analysis of the Euler equations in a flat and spherical thin domain and its limit equations.) However, mathematical studies of an incompressible fluid in a thin domain have not been done in the case where a thin domain and its degenerate set have more complicated geometric structures. (See [27] for the mathematical analysis of a reaction-diffusion equation in a thin domain degenerating into a lower dimensional manifold.) In this paper we are concerned with the incompressible Euler and Navier-Stokes equations in a three-dimensional thin domain that moves in time. The purpose of this paper is to give a heuristic derivation of singular limits of these equations as a moving thin domain degenerates into a two-dimensional moving closed surface. We also investigate relations between the energy structures of the incompressible fluid 2010 Mathematics Subject Classification. Primary: 35Q35, 35R01, 76M45; Secondary: 76A20.

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On the Two-Phase Navier–Stokes Equations with Boussinesq–Scriven Surface Fluid

Cited by 75 publications

References 8 publications

Higher‐order surface FEM for incompressible Navier‐Stokes flows on manifolds

Higher‐order surface FEM for incompressible Navier‐Stokes flows on manifolds

A finite element approach to incompressible two-phase flow on manifolds

On singular limit equations for incompressible fluids in moving thin domains

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