Shallow water viscous flows for arbitrary topopgraphy

Boutounet, Marc; Chupin, Laurent; Noble, Pascal; Vila, Jean Paul

doi:10.4310/cms.2008.v6.n1.a2

Cited by 26 publications

(30 citation statements)

References 11 publications

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“…The latter is a simplified form drawn from a family of models derived by Boutonet, Chupin, Noble, and Vila [2] where we omit, in particular, surface tension and some high-order terms. The present steady shallowwater model is one-dimensional (primes denote derivatives with respect to x 1 ) and expresses conservation of mass and momentum in the form…”

Section: Comparison With a Shallow-water Modelmentioning

confidence: 99%

“…Important results in this direction have been achieved by Wierschem,2 Aksel, and Scholle [13,14] using an analytical approach, similar in spirit to that of Yih [15] for a flat plane, based on an expansion of the free-surface Navier-Stokes equations in which the wavelength parameter (essentially the ratio of the film thickness to the wavelength of the bottom undulations) serves as the perturbation parameter. The main result is that the critical Reynolds number for the onset of surface waves is higher than that for a flat bottom.…”

Section: Introductionmentioning

confidence: 99%

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Numerical study of a thin liquid film flowing down an inclined wavy plane

Ern

Joubaud

Lelièvre

2011

Physica D: Nonlinear Phenomena

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To cite this version:Alexandre Ern, Rémi Joubaud, Tony Lelievre. Numerical study of a thin liquid film flowing down an inclined wavy plane. Physica D: Nonlinear Phenomena, Elsevier, 2011, 240 (21) AbstractWe investigate the stability of a thin liquid film flowing down an inclined wavy plane using a direct numerical solver based on a finite element/arbitrary Lagrangian Eulerian approximation of the free-surface Navier-Stokes equations. We study the dependence of the critical Reynolds number for the onset of surface wave instabilities on the inclination angle, the waviness parameter, and the wavelength parameter, focusing in particular on mild inclinations and relatively large waviness so that the bottom does not fall monotonously.In the present parameter range, shorter wavelengths and higher amplitude for the bottom undulation stabilize the flow. The dependence of the critical Reynolds number evaluated with the Nusselt flow rate on the inclination angle is more complex than the classical relation (5/6 times the cotangent of the inclination angle), but this dependence can be recovered if the actual flow rate at critical conditions is used instead.

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Section: Comparison With a Shallow-water Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical study of a thin liquid film flowing down an inclined wavy plane

Ern

Joubaud

Lelièvre

2011

Physica D: Nonlinear Phenomena

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“…Find the velocity field u and the mass density field ρ such that conditions (3), (6), (10) and (11) hold.…”

Section: Statement Of the 3d Problemmentioning

confidence: 99%

“…More general constitutive relations may be studied such as those included in [13]. Remark also that depending on the basal boundary condition (slip boundary condition or non-slip boundary condition), various shallow water type equations may be obtained, see for instance [20] and [10] for models coming from incompressible Navier-Stokes equations and [18] for models coming from Bingham type equations with Dirichlet boundary condition at the bottom. Here, we consider boundary conditions in the spirit of [20], namely Navier boundary conditions at the bottom.…”

Section: Introductionmentioning

confidence: 99%

Augmented Lagrangian Method and Compressible Visco-plastic Flows: Applications to Shallow Dense Avalanches

Bresch

Fernández-Nieto

Ionescu

et al. 2009

New Directions in Mathematical Fluid Mechanics

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In this paper we propose a well-balanced finite volume / augmented Lagrangian method for compressible viscoplastic models focusing on a compressible Bingham type system with applications to dense avalanches. For the sake of completeness we also present a method showing that such system may be derived for a shallow flow of a rigid-viscoplastic incompressible fluid, namely for incompressible Bingham type fluid with free surface. When the fluid is relatively shallow and spread slowly, lubrication-style asymptotic approximations can be used to build reduced models for the spreading dynamics, see for instance [N.J. Balmforth et al., J. Fluid Mech. (2002)] . When the motion is a little bit quicker, shallow water theory for non Newtonian flows may be tried to handle with for instance assuming Navier type boundary condition at the bottom. We start from the variational inequality for incompressible Bingham fluid and derive a shallow water type system. In the case where Bingham number and viscosity are set to zero we obtain the classical Shal- 1It combines well-balanced finite volume schemes for the spacial discretization with the augmented Lagrangian method to treat the associated optimization problem. Finally, we present various numerical tests.

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“…We derive thin film models generalizing results by Gerbeau, Perthame, see [15] and F.Marche, see [28] which consider Newtonian fluids with slip boundary condition at the bottom and appropriate range of adimensionnalized numbers. It could be interesting to perform similar asymptotic in Vila's framework, see [7,11] namely with no-slip boundary condition and infinitely large Reynolds number in the limit of infinitely thin layers. See [14] for such generalization but in the Bingham framework.…”

Section: Introductionmentioning

confidence: 99%

Two shallow-water type models for viscoelastic flows from kinetic theory for polymers solutions

Narbona-Reina

Bresch

2013

ESAIM: M2AN

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Abstract. In this work, depending on the relation between the Deborah, the Reynolds and the aspect ratio numbers, we formally derived shallow-water type systems starting from a micro-macro description for non-Newtonian fluids in a thin domain governed by an elastic dumbbell type model with a slip boundary condition at the bottom. The result has been announced by the authors in [G. Narbona-Reina, D. Bresch, Numer. Math. and Advanced Appl. Springer Verlag (2010)] and in the present paper, we provide a self-contained description, complete formal derivations and various numerical computations. In particular, we extend to FENE type systems the derivation of shallowwater models for Newtonian fluids that we can find for instance in [J. Mathematics Subject Classification.

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Shallow water viscous flows for arbitrary topopgraphy

Cited by 26 publications

References 11 publications

Numerical study of a thin liquid film flowing down an inclined wavy plane

Numerical study of a thin liquid film flowing down an inclined wavy plane

Augmented Lagrangian Method and Compressible Visco-plastic Flows: Applications to Shallow Dense Avalanches

Two shallow-water type models for viscoelastic flows from kinetic theory for polymers solutions

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