Well-posedness of a Two-phase Flow with Soluble Surfactant

Bothe, Dieter; Prüß, Jan; Simonett, Gieri

doi:10.1007/3-7643-7385-7_3

Cited by 34 publications

(59 citation statements)

References 29 publications

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“…The correspondingly extended model is well-known in the Engineering Sciences; cf., e.g., [11]. Mathematically, its well-posedness has recently been obtained in [2], locally-in-time.…”

Section: Introductionmentioning

confidence: 99%

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

Bothe

Prüß

2008

J. Math. Fluid Mech.

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Two-phase flows with interface modeled as a Boussinesq-Scriven surface fluid are analysed concerning their fundamental mathematical properties. This extended form of the common sharp-interface model for two-phase flows includes both surface tension and surface viscosity. For this system of partial differential equations with free interface it is shown that the energy serves as a strict Ljapunov functional, where the equilibria of the model without boundary contact consist of zero velocity and spheres for the dispersed phase. The linearizations of the problem are derived formally, showing that equilibria are linearly stable, but nonzero velocities may lead to problems which linearly are not well-posed. This phenomenon does not occur in absence of surface viscosity. The present paper aims at initiating a rigorous mathematical study of two-phase flows with surface viscosity. (2000). Primary 35R35; secondary 35Q30, 76D45, 76T10. Mathematics Subject Classification

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“…The correspondingly extended model is well-known in the Engineering Sciences; cf., e.g., [11]. Mathematically, its well-posedness has recently been obtained in [2], locally-in-time.…”

Section: Introductionmentioning

confidence: 99%

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

Bothe

Prüß

2008

J. Math. Fluid Mech.

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“…The intersection between the interface and the control volume Σ(t) ∩ V (t) is denoted as S(t), with the boundary curve ∂S(t) and the outer unit normal m ⊥ n Σ to ∂S(t); see figure 1. The integral balance of surfactant molar mass for a moving control volume V (t) in absence of chemical reactions (or any other source term) [6] reads…”

Section: 1mentioning

confidence: 99%

Computational analysis of single rising bubbles influenced by soluble surfactant

et al. 2018

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This paper presents novel insights about the influence of soluble surfactants on bubble flows obtained by Direct Numerical Simulation (DNS). Surfactants are amphiphilic compounds which accumulate at fluid interfaces and significantly modify the respective interfacial properties, influencing also the overall dynamics of the flow. With the aid of DNS local quantities like the surfactant distribution on the bubble surface can be accessed for a better understanding of the physical phenomena occurring close to the interface. The core part of the physical model consists in the description of the surfactant transport in the bulk and on the deformable interface. The solution procedure is based on an Arbitrary Lagrangian-Eulerian (ALE) Interface-Tracking method. The existing methodology was enhanced to describe a wider range of physical phenomena. A subgridscale (SGS) model is employed in the cases where a fully resolved DNS for the species transport is not feasible due to high mesh resolution requirements and, therefore, high computational costs. After an exhaustive validation of the latest numerical developments, the DNS of single rising bubbles in contaminated solutions is compared to experimental results. The full velocity transients of the rising bubbles, especially the contaminated ones, are correctly reproduced by the DNS. The simulation results are then studied to gain a better understanding of the local bubble dynamics under the effect of soluble surfactant. One of the main insights is that the quasi-steady state of the rise velocity is reached without ad-and desorption being necessarily in local equilibrium.

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“…This function models an exchange rate towards a concentration equilibrium given by c D Hc , where H is the Henry constant. i D 1 corresponds to the correct mass balance for a moving surface, while the expression for i D 2 is included due to the fact that such models can be found in the literature elsewhere; see, for example, [7]. v M is the tangential and V is the normal velocity of , and Ä denotes its mean curvature.…”

Section: Introductionmentioning

confidence: 99%

Homogenization of a diffusion‐reaction system with surface exchange and evolving hypersurface

Dobberschütz

2014

Math Methods in App Sciences

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This paper is concerned with the homogenization of a diffusion-reaction process in a domain undergoing an evolution of the microstructure. The main novelty is the consideration of a chemical process on a hypersurface, where the surface itself is evolving. After transformation to a description in a fixed domain, we employ methods of periodic homogenization to arrive at a limit problem in the bulk coupled with problems in a reference cell. Here, the structural evolution takes place only in the reference structure. Additional problems come from nonlinear reaction rates, which require the Kolmogoroff compactness criterion for the derivation of suitable a priori estimates. The results can be used as a modeling tool and for the investigation of complex biological and chemical systems in porous media, for example, carbon sequestration or biological enhanced oil recovery. 559Details of the derivation can be found in [8]. The main physical assumption contained in the nondimensionalization procedure is the fact that the diffusivity on the boundary is orders of magnitude smaller than in the bulk. Of course, other scalings are possible as well. With the same method, one could, for example, consider the diffusion of a second substance in the bulk whose diffusivity D 0 satisfies D 0 D " 2 . A systematic investigation of the dependence of homogenization limits in reaction-diffusion scenarios on different scaling factors can be found in the study of Peter [9]. However, this only applies to processes in various bulk phases. To the knowledge of the author, no methods are known to treat systems similar to (3) and (4), where the ratio D D is not of order " 2 . Concerning the curvature, the assumption given earlier implies that the characteristic curvature is constant during the evolution of the surface. This restricts changes of the underlying geometry. For example, a process that transforms a spherical shape to a finely fingered structure might not fulfill these assumptions. Related worksSeveral works apply homogenization methods and coordinate transformations to treat problems in evolving domains: Peter [4,10] considers models for chemical degradation of concrete structures, where the domain consists of a porous matrix, an air, and a water phase. Because of a carbonation reaction, water is produced and thus leads to the growth of the latter phase. A summary of the mathematical method can also be found in [11]. Meier [5] considers a similar situation but focuses on mathematical properties of the two-scale limit system. In the latter references, one can also find information on how to incorporate a reaction driven change of the 561 1. One transforms the equations formally by using appropriate transformation rules for functions and differential operators. This is used in most works on problems in noncylindrical domains, for instance in the references cited above. 2. One defines a weak formulation of the problem in natural coordinates and uses integral transformations to obtain a weak formulation in referential coordinates. This rigor...

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Well-posedness of a Two-phase Flow with Soluble Surfactant

Cited by 34 publications

References 29 publications

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

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

Computational analysis of single rising bubbles influenced by soluble surfactant

Homogenization of a diffusion‐reaction system with surface exchange and evolving hypersurface

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