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

Gladys Narbona-Reina, Didier Bresch (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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...

Two-Layer Flow with One Viscous Layer in Inclined Channels

O. K. Matar, G. M. Sisoev, C. J. Lawrence (2008)

Mathematical Modelling of Natural Phenomena

We study pressure-driven, two-layer flow in inclined channels with high density and viscosity contrasts. We use a combination of asymptotic reduction, boundary-layer theory and the Karman-Polhausen approximation to derive evolution equations that describe the interfacial dynamics. Two distinguished limits are considered: where the viscosity ratio is small with density ratios of order unity, and where both density and viscosity ratios are small. The evolution equations account for the presence of...

Un résultat de convergence d'ordre deux en temps pour l'approximation des équations de Navier–Stokes par une technique de projection incrémentale

Jean-Luc Guermond (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The Navier–Stokes equations are approximated by means of a fractional step, Chorin–Temam projection method; the time derivative is approximated by a three-level backward finite difference, whereas the approximation in space is performed by a Galerkin technique. It is shown that the proposed scheme yields an error of 𝒪 ( δ t 2 + h l + 1 ) for the velocity in the norm of l2(L2(Ω)d), where l ≥ 1 is the polynomial degree of the velocity approximation. It is also shown that the splitting error of projection schemes based...

Uniform regularity for an isentropic compressible MHD- P 1 approximate model arising in radiation hydrodynamics

Tong Tang, Jianzhu Sun (2021)

Czechoslovak Mathematical Journal

It is well known that people can derive the radiation MHD model from an MHD- P 1 approximate model. As pointed out by F. Xie and C. Klingenberg (2018), the uniform regularity estimates play an important role in the convergence from an MHD- P 1 approximate model to the radiation MHD model. The aim of this paper is to prove the uniform regularity of strong solutions to an isentropic compressible MHD- P 1 approximate model arising in radiation hydrodynamics. Here we use the bilinear commutator and product estimates...

Uniqueness of weak solutions of the Navier-Stokes equations

Sadek Gala (2008)

Applications of Mathematics

Consider the Navier-Stokes equation with the initial data a L σ 2 ( d ) . Let u and v be two weak solutions with the same initial value a . If u satisfies the usual energy inequality and if v L 2 ( ( 0 , T ) ; X ˙ 1 ( d ) d ) where X ˙ 1 ( d ) is the multiplier space, then we have u = v .

Uniqueness results for some PDEs

Nader Masmoudi (2003)

Journées équations aux dérivées partielles

Existence of solutions to many kinds of PDEs can be proved by using a fixed point argument or an iterative argument in some Banach space. This usually yields uniqueness in the same Banach space where the fixed point is performed. We give here two methods to prove uniqueness in a more natural class. The first one is based on proving some estimates in a less regular space. The second one is based on a duality argument. In this paper, we present some results obtained in collaboration with Pierre-Louis...

Very weak solutions of the stationary Stokes equations in unbounded domains of half space type

Reinhard Farwig, Jonas Sauer (2015)

Mathematica Bohemica

We consider the theory of very weak solutions of the stationary Stokes system with nonhomogeneous boundary data and divergence in domains of half space type, such as + n , bent half spaces whose boundary can be written as the graph of a Lipschitz function, perturbed half spaces as local but possibly large perturbations of + n , and in aperture domains. The proofs are based on duality arguments and corresponding results for strong solutions in these domains, which have to be constructed in homogeneous...

Vorticity dynamics and numerical resolution of Navier-Stokes equations

Matania Ben-Artzi, Dalia Fishelov, Shlomo Trachtenberg (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We present a new methodology for the numerical resolution of the hydrodynamics of incompressible viscid newtonian fluids. It is based on the Navier-Stokes equations and we refer to it as the vorticity projection method. The method is robust enough to handle complex and convoluted configurations typical to the motion of biological structures in viscous fluids. Although the method is applicable to three dimensions, we address here in detail only the two dimensional case. We provide numerical data...

Vorticity dynamics and numerical Resolution of Navier-Stokes Equations

Matania Ben-Artzi, Dalia Fishelov, Shlomo Trachtenberg (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present a new methodology for the numerical resolution of the hydrodynamics of incompressible viscid newtonian fluids. It is based on the Navier-Stokes equations and we refer to it as the vorticity projection method. The method is robust enough to handle complex and convoluted configurations typical to the motion of biological structures in viscous fluids. Although the method is applicable to three dimensions, we address here in detail only the two dimensional case. We provide numerical data...

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