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21
Diffusion in PoroElastic Media
 Jour. Math. Anal. Appl
, 1998
"... . Existence, uniqueness and regularity theory is developed for a general initialboundaryvalue problem for a system of partial differential equations which describes the Biot consolidation model in poroelasticity as well as a coupled quasistatic problem in thermoelasticity. Additional effects of se ..."
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. Existence, uniqueness and regularity theory is developed for a general initialboundaryvalue problem for a system of partial differential equations which describes the Biot consolidation model in poroelasticity as well as a coupled quasistatic problem in thermoelasticity. Additional effects of secondary consolidation and pore fluid exposure on the boundary are included. This quasistatic system is resolved as an application of the theory of linear degenerate evolution equations in Hilbert space, and this leads to a precise description of the dynamics of the system. 1. Introduction We shall consider a system modeling diffusion in an elastic medium in the case for which the inertia effects are negligible. This quasistatic assumption arises naturally in the classical Biot model of consolidation for a linearly elastic and porous solid which is saturated by a slightly compressible viscous fluid. The fluid pressure is denoted by p(x; t) and the displacement of the structure by u(x; t). ...
Mathematical Derivation Of The Power Law Describing Polymer Flow Through A Thin Slab
"... . We consider the polymer flow through a slab of thickness ffl. The flow is described by 3D incompressible NavierStokes system with a nonlinear viscosity, being a power of a norm of the shear rate (power law). We consider the limit when ffl ! 0 and prove that the limit averaged velocity, averaged o ..."
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Cited by 3 (2 self)
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. We consider the polymer flow through a slab of thickness ffl. The flow is described by 3D incompressible NavierStokes system with a nonlinear viscosity, being a power of a norm of the shear rate (power law). We consider the limit when ffl ! 0 and prove that the limit averaged velocity, averaged over the thickness, satisfies a nonlinear twodimensional Poiseuille's law, with nonlinear viscosity depending on the power of the length of the gradient of the pressure. It is found out that the powers in the starting law and in the limit law are conjugate. Furthermore, we prove a convergence theorem for velocity and pressure in appropriate functional spaces. 1 Equipe d'Analyse Numerique, Universit'e de SaintEtienne, 23 rue du Dr. P. Michelon, 42023 SaintEtienne Cedex, France 2 University of Zagreb, Croatia L.A.N., Bat 101, Universit'e Claude Bernard, 43 Bd. du 11 Novembre 1918, F69622 Villeurbanne Cedex, France 2 Andro Mikeli'c and Roland Tapi'ero 1. STATEMENT OF THE PROBLEM AND...
Steady compressible Oseen flow with slip boundary conditions
, 807
"... We prove the existence of solution in a class H 2 (Ω) × H 1 (Ω) to steady compressible Oseen system with slip boundary conditions in a two dimensional, convex domain with the boundary of class H 5/2. The method is to regularize a weak solution obtained via the Galerkin method. The problem of regula ..."
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We prove the existence of solution in a class H 2 (Ω) × H 1 (Ω) to steady compressible Oseen system with slip boundary conditions in a two dimensional, convex domain with the boundary of class H 5/2. The method is to regularize a weak solution obtained via the Galerkin method. The problem of regularization is reduced to a problem of solvability of a certain transport equation by application of the Helmholtz decomposition. The method works under additional assumption on the geometry of the boundary.
Implicit Vectorial Operator Splitting For Incompressible NavierStokes Equations In Primitive Variables
, 2001
"... this paper only reions with rectilinear boundaries in cartes an coordinate s The rid is as00 ed uniform and s a ered. The eneralization to nonuniform rid dis tribution in each direction is trivial. As a featurin example is taken the liddriven vis cous incompres s ible flow in as quare cavity. Dri ..."
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this paper only reions with rectilinear boundaries in cartes an coordinate s The rid is as00 ed uniform and s a ered. The eneralization to nonuniform rid dis tribution in each direction is trivial. As a featurin example is taken the liddriven vis cous incompres s ible flow in as quare cavity. Driven cavity flow has always been a choice cas es tudy for each new s heme for NS equations The advanta e of this tes t problem is that its eometry is thes imples t pos= ble. The dis advanta e is that there ares in ularities in the points where the lid touches the vertical walls in thes ens e that the horizontal velocity componentis doublevalued there. It turns out, however, that the dis continuous boundary condition for horizontal velocity component does not pos e real di#culties . One of the important properties of the 2D liddriven cavity flow is that the trans ition to turbulence appears to be delayed to very lar e Reynolds numbers . This makes it a very ood te sg n round for new s hemes and al orithm s The abundant literature provides for the neces s ary checks for any new technique. Re ardles s to its tes t nature the liddriven cavity is hard enou h to tackle numerically and its s imulation is s till a formidable tas k in the cas of hi her Reynolds numbers Thesg5 le for quantitatively preci s prediction for hi h Reynolds numbers is s0 ll unfinisfi d proces s in the literature. Thes econd aim of the pres ent paperis to provide accurate data for various flow characteri s ic84 C. I. Christov, R. S. Marinova 2. Posing the Probl em ider a clos ed 2D domain # with a piecewis es mooth boundary ##. The vis cous incompre se ble flow in # is governed by the 2D Navier  Stokes equations 1 [u]=0, (1) 1 [v]=0, (2) coupled by the continuity equation ...
The nonlinear HodgeNavierStokes equations in Lipschitz domains
 the Journal of Differential and Integral Equations
, 2008
"... Abstract. We investigate the NavierStokes equations in a suitable functional setting, in a threedimensional bounded Lipschitz domain Ω, equipped with “free boundary ” conditions. In this context, we employ the FujitaKato method and prove the existence of a local mild solution. Our approach makes ..."
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Abstract. We investigate the NavierStokes equations in a suitable functional setting, in a threedimensional bounded Lipschitz domain Ω, equipped with “free boundary ” conditions. In this context, we employ the FujitaKato method and prove the existence of a local mild solution. Our approach makes essential use of the properties of the HodgeLaplacian in Lipschitz domains. 1.
Steady compressible NavierStokes flow in a square
, 901
"... We investigate a steady flow of compressible fluid with inflow boundary condition on the density and slip boundary conditions on the velocity in a square domain Q ∈ R 2. We show existence if a solution (v, ρ) ∈ W 2 p (Q) × W 1 p (Q) that is a small perturbation of a constant flow (¯v ≡ [1, 0], ¯ρ ..."
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We investigate a steady flow of compressible fluid with inflow boundary condition on the density and slip boundary conditions on the velocity in a square domain Q ∈ R 2. We show existence if a solution (v, ρ) ∈ W 2 p (Q) × W 1 p (Q) that is a small perturbation of a constant flow (¯v ≡ [1, 0], ¯ρ ≡ 1). We also show that this solution is unique in a class of small perturbations of the constant flow (¯v, ¯ρ). In order show the existence of the solution we adapt the techniques know from the theory of weak solutions. We apply the method of elliptic regularization and a fixed point argument. MSC: 35Q30; 76N10 Keywords: NavierStokes equations, steady compressible flow, inflow boundary condition, slip boundary conditions, strong solutions 1 Introduction and main results The problems of steady compressible flows described by the NavierStokes equations are usually considered with the homogeneous Dirichlet boundary conditions on the velocity. It is worth from the mathematical point of view, as well as in the eye of applications, to investigate different
unknown title
, 2009
"... On an inhomogeneous slipinflow boundary value problem for a steady flow of a viscous compressible fluid in a cylindrical domain ..."
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On an inhomogeneous slipinflow boundary value problem for a steady flow of a viscous compressible fluid in a cylindrical domain
Academy of Sciences
, 2013
"... Effective slip law for general viscous flows over oscillating surface ..."
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