This communication is devoted to the presentation of our recent results regarding the asymptotic analysis of a viscous flow in a tube with elastic walls. This study can be applied, for example, to the blood flow in an artery. With this aim, we consider the dynamic problem of the incompressible flow of a viscous fluid through a curved pipe with a smooth central curve. Our analysis leads to obtain an one dimensional model via singular perturbation of the Navier-Stokes system as $\varepsilon$, a non dimensional parameter related to the radius of cross-section of the tube, tends to zero. We allow the radius depend on tangential direction and time, so a coupling with an elastic or viscoelastic law on the wall of the pipe is possible.
To perform the asymptotic analysis, we do a change of variable to a reference domain where we assume the existence of asymptotic expansions on $\varepsilon$ for both velocity and pressure which, upon substitution on Navier-Stokes equations, leads to the characterization of various terms of the expansion. This allows us to obtain an approximation of the solution of the Navier-Stokes equations.
This paper is concerned with the application of finite element methods to obtain solutions for steady fully developed second-grade flows in a curved pipe of circular cross-section and arbitrary curvature ratio, under a given axial pressure gradient. The qualitative and quantitative behavior of the secondary flows is analyzed with respect to inertia and viscoelasticity.
We study the motion of an incompressible fluid in an $n+1$-dimensional infinite pipe $\,\La\,$ with an $L$-periodic shape in the $z=x_{n+1}$ direction. We set $\,x=(x_1,x_2,\cdots,x_{n})$, and $z=x_{n+1}$. We denote by $Σ_z$ the cross section of the pipe at the level $z\,,$ and by $v_z$ the $n+1$ component of the velocity. Fluid motion is described by the evolution Stokes or Navier-Stokes equations together with the non-slip boundary condition $\bv=\,0\,$. We look for solutions $\bv(x,z,t)$ with a given, arbitrary, $T-$time periodic total flux $\,\int_{Σ_z} \,v_z(x,z,t)\,dx=g(t)\,,$ which shou…
A modified Reynolds equation governing the steady flow of a fluid with low Reynolds number through a curvilinear, narrow tube, with its derivation from Stokes equations through asymptotic methods is presented. The channel considered may have large curvature and torsion. Approximations of the velocity and the pressure of the fluid inside the channel are constructed by artificially imposing appropriate boundary conditions at the inlet and the outlet. A justification for the approximations is provided along with a comparison with a simpler case.
We consider the Navier-Stokes equations in a three-dimensional curved thin domain around a given closed surface under Navier's slip boundary conditions. When the thickness of the thin domain is sufficiently small, we establish the global existence of a strong solution for large data. We also show several estimates for the strong solution with constants explicitly depending on the thickness of the thin domain. The proofs of these results are based on a standard energy method and a good product estimate for the convection and viscous terms following from a detailed study of average operators in …
We consider the three-dimensional incompressible Navier--Stokes equations in a curved thin domain with Navier's slip boundary conditions. The curved thin domain is defined as a region between two closed surfaces which are very close to each other and degenerates into a given closed surface as its width tends to zero. We establish the global-in-time existence and uniform estimates of a strong solution for large data when the width of the thin domain is very small. Moreover, we study a singular limit problem as the thickness of the thin domain tends to zero and rigorously derive limit equations …
We consider the Navier-Stokes equations with Navier's slip boundary conditions in a three-dimensional curved thin domain around a given closed surface. Under suitable assumptions we show that the average in the thin direction of a strong solution to the bulk Navier-Stokes equations converges weakly in appropriate function spaces on the limit surface as the thickness of the thin domain tends to zero. Moreover, we characterize the limit as a weak solution to limit equations, which are the damped and weighted Navier-Stokes equations on the limit surface. We also prove the strong convergence of th…
Brinkman's term for the steady-state Navier-Stokes equations with prescribed flux rate or pressure drop in perforated pipes
Fakultät für Mathematik, Universität Regensburg（雷根斯堡大学）；Institut de Mathématiques de Jussieu - Paris Rive Gauche, Université Paris Cité（巴黎西岱大学）；Facultad de Matemáticas, Pontificia Universidad Católica de Chile（智利天主教Pontificia大学）
Global weak solutions and incompressible limit to the isentropic compressible Navier-Stokes equations in 2D bounded domains with ripped density and large initial data
Fluid flow in pipes with discontinuous cross section or with kinks is described through balance laws with a non conservative product in the source. At jump discontinuities in the pipes' geometry, the physics of the problem suggests how to single out a solution. On this basis, we present a definition of solution for a general BV geometry and prove an existence result, consistent with a limiting procedure from piecewise constant geometries. In the case of a smoothly curved pipe we thus justify the appearance of the curvature in the source term of the linear momentum equation. These results are o…
