160 research outputs found
Mathematical derivation of viscous shallow-water equations with zero surface tension
The purpose of this paper is to derive rigorously the so called viscous
shallow water equations given for instance page 958-959 in [A. Oron, S.H.
Davis, S.G. Bankoff, Rev. Mod. Phys, 69 (1997), 931?980]. Such a system of
equations is similar to compressible Navier-Stokes equations for a barotropic
fluid with a non-constant viscosity. To do that, we consider a layer of
incompressible and Newtonian fluid which is relatively thin, assuming no
surface tension at the free surface. The motion of the fluid is described by 3d
Navier-Stokes equations with constant viscosity and free surface. We prove that
for a set of suitable initial data (asymptotically close to "shallow water
initial data"), the Cauchy problem for these equations is well-posed, and the
solution converges to the solution of viscous shallow water equations. More
precisely, we build the solution of the full problem as a perturbation of the
strong solution to the viscous shallow water equations. The method of proof is
based on a Lagrangian change of variable that fixes the fluid domain and we
have to prove the well-posedness in thin domains: we have to pay a special
attention to constants in classical Sobolev inequalities and regularity in
Stokes problem
Relative entropy for compressible Navier-Stokes equations with density dependent viscosities and applications
Recently, A. Vasseur and C. Yu have proved the existence of global
entropy-weak solutions to the compressible Navier-Stokes equations with
viscosities and and a pressure
law under the form with and
constants. In this note, we propose a non-trivial relative entropy for such
system in a periodic box and give some applications. This extends, in some
sense, results with constant viscosities initiated by E. Feiersl, B.J. Jin and
A. Novotny.
We present some mathematical results related to the weak-strong uniqueness,
convergence to a dissipative solution of compressible or incompressible Euler
equations. As a by-product, this mathematically justifies the convergence of
solutions of a viscous shallow water system to solutions of the inviscid
shall-water system
A compressible multifluid system with new physical relaxation terms
International audienceIn this paper, we rigorously derive a new compressible multifluid system from compressible Navier-Stokes equations with density-dependent viscosity in the one-dimensional in space setting. More precisely, we propose and mathematically derive a generalization of the usual one velocity Baer-Nunziato model with a new relaxation term in the PDE governing the volume fractions. This new relaxation term encodes the change of viscosity and pressure between the different fluids. For the reader's convenience, we first establish a formal derivation in the bifluid setting using a WKB decomposition and then we rigorously justify the multifluid homogenized system using a kinetic formulation via Young measures characterization
Two shallow-water type models for viscoelastic flows from kinetic theory for polymers solutions
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.-F. Gerbeau, B. Perthame, Discrete Contin. Dyn. Syst. (2001)] which assume an appropriate relation between the Reynolds number and the aspect ratio with slip boundary condition at the bottom. Under a radial hypothesis at the leading order, for small Deborah number, we find an interesting formulation where polymeric effect changes the drag term in the second order shallow-water formulation (obtained by J.-F. Gerbeau, B. Perthame). We also discuss intermediate Deborah number with a fixed Reynolds number where a strong coupling is found through a nonlinear time-dependent Fokker–Planck equation. This generalizes, at a formal level, the derivation in [L. Chupin, Meth. Appl. Anal. (2009)] including non-linear effects (shallow-water framework)
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