Two-phase Stokes flow by capillarity in the plane: The case of different viscosities

Matioc, Bogdan–Vasile; Prokert, G Georg

doi:10.1007/s00030-022-00785-0

Cited by 4 publications

(2 citation statements)

References 22 publications

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“…Proof The property (i) is established in [31, Lemma 3.1]. The claim (ii) is proven for

k=2

in [34, Lemma 4.3] and the case

k\ge 3

follows from this result via induction. Moreover, (iii) is established in [33, Appendix C] and (iv) in [29, Lemma 2.5].

\Box

…”

Section: Solvability Of Some Boundary Value Problemsmentioning

confidence: 84%

The Mullins–Sekerka problem via the method of potentials

Escher,

Matioc,

Matioc

2024

Mathematische Nachrichten

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“…Proof The property (i) is established in [31, Lemma 3.1]. The claim (ii) is proven for

k=2

in [34, Lemma 4.3] and the case

k\ge 3

follows from this result via induction. Moreover, (iii) is established in [33, Appendix C] and (iv) in [29, Lemma 2.5].

\Box

…”

Section: Solvability Of Some Boundary Value Problemsmentioning

confidence: 84%

The Mullins–Sekerka problem via the method of potentials

Escher,

Matioc,

Matioc

2024

Mathematische Nachrichten

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“…In this work, the authors prove a global existence and uniqueness result for small initial data in the space of Fourier transforms of bounded measures. This scenario has also been studied by Matioc & Prokert [37,38], with equal viscosities and viscosity jump, respectively. The authors prove well-posedness in sub-critical Sobolev spaces with arbitrary size initial data and a criterion for global existence, using the potential theory approach.…”

Section: Introductionmentioning

confidence: 97%

Long time interface dynamics for gravity Stokes flow

Gancedo¹,

Granero-Belinchón²,

Salguero³

2022

Preprint

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We study the dynamics of the interface given by two incompressible viscous fluids in the Stokes regime filling a 2D horizontally periodic strip. The fluids are subject to the gravity force and the density difference induces the dynamics of the interface. We derive the contour dynamics formulation for this problem through a x 1 -periodic version of the Stokeslet. Using this new system, we show local-in-time well-posedness when the initial interface is described by a curve with no selfintersections and C 1+γ Hölder regularity, 0 < γ < 1. Global-in-time stability to the flat stationary state is proved in the Rayleigh-Taylor stable regime for small initial data in Sobolev spaces.

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Feedback Stabilization of a Two-Fluid Surface Tension System Modeling the Motion of a Soap Bubble at Low Reynolds Number: The Two-Dimensional Case

Court

2023

J. Math. Fluid Mech.

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The aim of this paper is to design a feedback operator for stabilizing in infinite time horizon a system modeling the interactions between a viscous incompressible fluid and the deformation of a soap bubble. The latter is represented by an interface separating a bounded domain of $$\mathbb {R}^2$$ R 2 into two connected parts filled with viscous incompressible fluids. The interface is a smooth perturbation of the 1-sphere, and the surrounding fluids satisfy the incompressible Stokes equations in time-dependent domains. The mean curvature of the surface defines a surface tension force which induces a jump of the normal trace of the Cauchy stress tensor. The response of the fluids is a velocity trace on the interface, governing the time evolution of the latter, via the equality of velocities. The data are assumed to be sufficiently small, in particular the initial perturbation, that is the initial shape of the soap bubble is close enough to a circle. The control function is a surface tension type force on the interface. We design it as the sum of two feedback operators: one is explicit, the second one is finite-dimensional. They enable us to define a control operator that stabilizes locally the soap bubble to a circle with an arbitrary exponential decay rate, up to translations, and up to non-contact with the outer boundary.

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Two-phase Stokes flow by capillarity in the plane: The case of different viscosities

Cited by 4 publications

References 22 publications

The Mullins–Sekerka problem via the method of potentials

The Mullins–Sekerka problem via the method of potentials

Long time interface dynamics for gravity Stokes flow

Feedback Stabilization of a Two-Fluid Surface Tension System Modeling the Motion of a Soap Bubble at Low Reynolds Number: The Two-Dimensional Case

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