Nonnegative weak solutions of thin-film equations related to viscous flows in cylindrical geometries

Marzuola, Jeremy L.; Swygert, Sterling; Taranets, Roman M.

doi:10.1007/s00028-019-00553-1

Cited by 8 publications

(7 citation statements)

References 25 publications

(31 reference statements)

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“…It turns out that in the specific scaling limit and geometry considered in [11], the motion of the dominant fluid can be approximated by means of the classical one fluid Taylor-Couette flow and the dynamics of the interface which separates both fluids, can be approximated by means of a suitable thin film like equation. Similar results for viscous flows in cylindrical geometries can be found in [9].…”

Section: Introductionsupporting

confidence: 84%

On the dynamics of thin layers of viscous flows inside another viscous fluid

Pernas-Castaño¹,

Velázquez²

2020

Preprint

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In this work we will study the dynamics of a thin layer of a viscous fluid which is embedded in the interior of another viscous fluid. The resulting flow can be approximated by means of the solutions of a free boundary problem for the Stokes equation in which one of the unknowns is the shape of a curve which approximates the geometry of the thin layer of fluid. We also derive the equation yielding the thickness of this fluid. This model, that will be termed as the Geometric Free Boundary Problem, will be derived using matched asymptotic expansions. We will prove that the Geometric Free Boundary Problem is well posed and the solutions of the thickness equation are well defined (in particular they do not yield breaking of fluid layers) as long as the solutions of the Geometric Free Boundary Problem exist.

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Section: Introductionsupporting

confidence: 84%

On the dynamics of thin layers of viscous flows inside another viscous fluid

Pernas-Castaño¹,

Velázquez²

2020

Preprint

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“…This topic has additionally been pursued in Bertozzi et al (1994), where the authors also study numerically the existence of singularities in finite and infinite time. The existence of global in time weak solutions to (1.4) which allow for film rupture has been studied in Marzuola et al (2019) for a cylindrical geometry.…”

Section: Introductionmentioning

confidence: 99%

Analysis of a Two-Fluid Taylor–Couette Flow with One Non-Newtonian Fluid

2022

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We study the dynamic behaviour of two viscous fluid films confined between two concentric cylinders rotating at a small relative velocity. It is assumed that the fluids are immiscible and that the volume of the outer fluid film is large compared to the volume of the inner one. Moreover, while the outer fluid is considered to have constant viscosity, the rheological behaviour of the inner thin film is determined by a strain-dependent power-law. Starting from a Navier–Stokes system, we formally derive evolution equations for the interface separating the two fluids. Two competing effects drive the dynamics of the interface, namely the surface tension and the shear stresses induced by the rotation of the cylinders. When the two effects are comparable, the solutions behave, for large times, as in the Newtonian regime. We also study the regime in which the surface tension effects dominate the stresses induced by the rotation of the cylinders. In this case, we prove local existence of positive weak solutions both for shear-thinning and shear-thickening fluids. In the latter case, we show that interfaces which are initially close to a circle converge to a circle in finite time and keep that shape for later times.

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“…In the low Reynolds number limit, classical lubrication models have been developed for the dynamics of falling viscous liquid films along an axisymmetric cylindrical fibre. Under the long-wave approximation, the resultant evolution equations are a family of fourth-order degenerate parabolic PDEs for the fluid film thickness [5,10,11,15,19]. These models incorporate gravity and both stabilizing and destabilizing roles of surface tension by characterizing the axial and azimuthal curvature of the free surface.…”

Section: Introductionmentioning

confidence: 99%

On travelling waves of a control-volume model for liquid films flowing down a fibre

Taranets¹,

Ji²,

Chugunova³

2023

Preprint

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This paper analytically investigates the solutions to a control-volume model for liquid films flowing down a vertical fibre. The dynamic evolution of the free surface is governed by a coupled degenerate nonlinear PDE system for the fluid film radius and the axial velocity. We prove the existence of weak solutions to the coupled system based on the application of a priori estimates derived for energy-entropy functionals. Existence of travelling wave solutions to the system is also established. Numerical studies are presented to illustrate the derived analytical results for both the dynamic PDE solutions and the travelling wave structures.

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Nonnegative weak solutions of thin-film equations related to viscous flows in cylindrical geometries

Cited by 8 publications

References 25 publications

On the dynamics of thin layers of viscous flows inside another viscous fluid

On the dynamics of thin layers of viscous flows inside another viscous fluid

Analysis of a Two-Fluid Taylor–Couette Flow with One Non-Newtonian Fluid

On travelling waves of a control-volume model for liquid films flowing down a fibre

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