Analysis of a thin film approximation for two-fluid Taylor-Couette flows

Castaño, Tania Pernas; Velázquez, Juan J. L.

doi:10.1016/j.jde.2019.12.005

Cited by 8 publications

(15 citation statements)

References 36 publications

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“…Equation (1.4) is studied in Pernas-Castaño and Velázquez (2020), where the authors observe the same asymptotic behaviour for initial interfaces close to a circle. In particular, it is shown that in the Newtonian case the solution is globally defined and the interface approaches quickly a circle which is initially not concentric with the rotating cylinders.…”

Section: Introductionmentioning

confidence: 85%

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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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Section: Introductionmentioning

confidence: 85%

“…InPernas-Castaño and Velázquez (2020), the case in which the layer of fluid closer to the external cylinder is much thinner than the inner one, has also been considered in the Newtonian case. However, since the analysis is similar we restrict ourselves in this paper to the case in which the thin layer is close to the internal cylinder.…”

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confidence: 99%

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

2022

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“…Also, Escher, Matioc & Matioc [23] considered the flow in porous media (see also Escher & Matioc [25], Matioc [39], Escher, Laurençot & Matioc [21], Laurençot & Matioc [35][36][37][38] and Bruell & Granero-Belinchón [10]) while the Stokes flow was considered by Escher, Matioc & Matioc [24] (see also Escher & Matioc [26] and Bruell & Granero-Belinchón [10]). A more recent reference is Pernas-Castaño & Velázquez [40], where the authors study the evolution of the interface between two different fluids in two concentric cylinders when the velocity is given by the Navier-Stokes equation and one of the fluids is thin.…”

Section: Introductionmentioning

confidence: 99%

On a thin film model with insoluble surfactant

Bruell

Granero-Belinchón

2020

Journal of Differential Equations

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This paper studies the existence and asymptotic behavior of global weak solutions for a thin film equation with insoluble surfactant under the influence of gravitational, capillary and van der Waals forces. We prove the existence of global weak solutions for medium sized initial data in large function spaces. Moreover, exponential decay towards the flat equilibrium state is established, where an estimate on the decay rate can be computed explicitly.2010 Mathematics Subject Classification. 35D30, 35B40, 35K52, 35K65, 76A20 .

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“…In particular, the stability of the flows for different ranges of viscosities, densities and surface tensions of the fluids in the absence of gravity is considered. The particular setting in which one of the fluids is localised in a thin layer is considered in [30] for the case in which both fluids are Newtonian.…”

Section: Introductionmentioning

confidence: 99%

“…Two physical assumptions are crucial for the derivation of the model. First, as in [30], we assume that the dynamics of the two-fluid system is described by a small perturbation of the Taylor-Couette flow for one single fluid confined between two cylinders. Second, while the outer fluid is assumed to be Newtonian, the inner fluid is assumed to be a non-Newtonian fluid with a strain-dependent viscosity .…”

Section: Introductionmentioning

confidence: 99%

Analysis of a two-fluid Taylor-Couette flow with one non-Newtonian fluid

Lienstromberg¹,

Pernas-Castaño²,

Velázquez³

2020

Preprint

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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.

show abstract

Analysis of a thin film approximation for two-fluid Taylor-Couette flows

Cited by 8 publications

References 36 publications

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

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

On a thin film model with insoluble surfactant

Analysis of a two-fluid Taylor-Couette flow with one non-Newtonian fluid

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