ESTIMATES FOR THE HOMOGENIZATION OF STOKES PROBLEM IN A PERFORATED DOMAIN

2018 ◽  
Vol 19 (1) ◽  
pp. 231-258 ◽  
Author(s):  
Amina Mecherbet ◽  
Matthieu Hillairet
Keyword(s):  
Stokes Equations ◽  
Stokes Problem ◽  
Brinkman Equation ◽  
Perforated Domain ◽  
Stat Phys ◽  
Quantitative Estimates

In this paper, we consider the Stokes equations in a perforated domain. When the number of holes increases while their radius tends to 0, it is proven in Desvillettes et al. [J. Stat. Phys. 131 (2008) 941–967], under suitable dilution assumptions, that the solution is well approximated asymptotically by solving a Stokes–Brinkman equation. We provide here quantitative estimates in $L^{p}$-norms of this convergence.

scholarly journals A Framework for Approximation of the Stokes Equations in an Axisymmetric Domain

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Niklas Ericsson
Keyword(s):  
Fourier Expansion ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Three Dimensional ◽  
Countable Family ◽  
Angular Variable ◽  
Two Dimensional ◽  
Sobolev Norms ◽  
Well Posed ◽  
Variational Formulations

Abstract We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain. By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an equivalent, countable family of decoupled two-dimensional problems. By using decomposition of three-dimensional Sobolev norms, we derive natural variational spaces for the two-dimensional problems, and show that the variational formulations are well-posed. We analyze the error due to Fourier truncation and conclude that, for data that are sufficiently regular, it suffices to solve a small number of two-dimensional problems.

Density-driven unstable flows of miscible fluids in a Hele-Shaw cell

2002 ◽  
Vol 451 ◽  
pp. 239-260 ◽  
Author(s):  
J. FERNANDEZ ◽  
P. KUROWSKI ◽  
P. PETITJEANS ◽  
E. MEIBURG
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Brinkman Equation ◽  
Length Scale ◽  
Dominant Wavelength ◽  
Miscible Fluids ◽  
Nonlinear Simulations ◽  
Instability Growth ◽  
Gap Width ◽  
Hele Shaw Cell

Density-driven instabilities between miscible fluids in a vertical Hele-Shaw cell are investigated by means of experimental measurements, as well as two- and three-dimensional numerical simulations. The experiments focus on the early stages of the instability growth, and they provide detailed information regarding the growth rates and most amplified wavenumbers as a function of the governing Rayleigh number Ra. They identify two clearly distinct parameter regimes: a low-Ra, ‘Hele-Shaw’ regime in which the dominant wavelength scales as Ra−1, and a high-Ra ‘gap’ regime in which the length scale of the instability is 5±1 times the gap width. The experiments are compared to a recent linear stability analysis based on the Brinkman equation. The analytical dispersion relationship for a step-like density profile reproduces the experimentally observed trend across the entire Ra range. Nonlinear simulations based on the two- and three-dimensional Stokes equations indicate that the high-Ra regime is characterized by an instability across the gap, wheras in the low-Ra regime a spanwise Hele-Shaw mode dominates.

scholarly journals Analysis of estimators for Stokes problem using a mixed approximation

2021 ◽  
Vol 39 (6) ◽  
pp. 105-128
Author(s):  
El Akkad Abdeslam
Keyword(s):  
Variational Formulation ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Mixed Finite Element ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Error Indicator ◽  
Commercial Code ◽  
Global Error Estimates

In this work, we introduce the steady Stokes equations with a new boundary condition, generalizes the Dirichlet and the Neumann conditions. Then we derive an adequate variational formulation of Stokes equations. It includes algorithms for discretization by mixed finite element methods. We use a block diagonal preconditioners for Stokes problem. We obtain a faster convergence when applying the preconditioned MINRES method. Two types of a posteriori error indicator are introduced and are shown to give global error estimates that are equivalent to the true discretization error. In order to evaluate the performance of the method, the numerical results are compared with some previously published works or with others coming from commercial code like Adina system.

A Simplified Parallel Two-Level Iterative Method for Simulation of Incompressible Navier-Stokes Equations

2015 ◽  
Vol 7 (6) ◽  
pp. 715-735
Author(s):  
Yueqiang Shang ◽  
Jin Qin
Keyword(s):  
Stokes Equations ◽  
Stokes Problem ◽  
Coarse Grid ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Driven Cavity ◽  
Step Flow ◽  
Grid Solution ◽  
Oseen Problem ◽  
Parallel Iterative

AbstractBased on two-grid discretization, a simplified parallel iterative finite element method for the simulation of incompressible Navier-Stokes equations is developed and analyzed. The method is based on a fixed point iteration for the equations on a coarse grid, where a Stokes problem is solved at each iteration. Then, on overlapped local fine grids, corrections are calculated in parallel by solving an Oseen problem in which the fixed convection is given by the coarse grid solution. Error bounds of the approximate solution are derived. Numerical results on examples of known analytical solutions, lid-driven cavity flow and backward-facing step flow are also given to demonstrate the effectiveness of the method.

BOUNDARY INTEGRAL EQUATIONS FOR A THREE-DIMENSIONAL STOKES–BRINKMAN CELL MODEL

2008 ◽  
Vol 18 (12) ◽  
pp. 2055-2085 ◽  
Author(s):  
MIRELA KOHR ◽  
G. P. RAJA SEKHAR ◽  
WOLFGANG L. WENDLAND
Keyword(s):  
Stokes Flow ◽  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Boundary Integral Equations ◽  
Cell Model ◽  
Uniqueness Result ◽  
Boundary Integral ◽  
Porous Particle ◽  
Brinkman Equation ◽  
Special Cases

The purpose of this paper is to prove the existence and uniqueness of the solution in Sobolev or Hölder spaces for a cell model problem which describes the Stokes flow of a viscous incompressible fluid in a bounded region past a porous particle. The flow within the porous particle is described by the Brinkman equation. In order to obtain the desired existence and uniqueness result, we use an indirect boundary integral formulation and potential theory for both Brinkman and Stokes equations. Some special cases, which refer to the cell model for a porous particle with large permeability, or to the exterior Stokes flow past a porous particle, are also presented.

Viscous flow in a channel with periodic cross-bridging fibres: exact solutions and Brinkman approximation

1991 ◽  
Vol 226 ◽  
pp. 125-148 ◽  
Author(s):  
Ruey-Yug Tsay ◽  
Sheldon Weinbaum
Keyword(s):  
Aspect Ratio ◽  
Drag Coefficient ◽  
Viscous Flow ◽  
Velocity Component ◽  
Stokes Equations ◽  
Interpolation Formula ◽  
Numerical Results ◽  
Brinkman Equation ◽  
Capillary Endothelium ◽  
Channel Height

A general solution of the three-dimensional Stokes equations is developed for the viscous flow past a square array of circular cylindrical fibres confined between two parallel walls. This doubly periodic solution, which is an extension of the theory developed by Lee & Fung (1969) for flow around a single fibre, successfully describes the transition in behaviour from the Hele-Shaw potential flow limit (aspect ratio B [Lt ] 1) to the viscous two-dimensional limiting case (B [Gt ] 1, Sangani & Acrivos 1982) for the hydrodynamic interaction between the fibres. These results are also compared with the solution of the Brinkman equation for the flow through a porous medium in a channel. This comparison shows that the Brinkman approximation is very good when B > 5, but breaks down when B [les ] O(1). A new interpolation formula is proposed for this last regime. Numerical results for the detailed velocity profiles, the drag coefficient f, and the Darcy permeability Kp are presented. It is shown that the velocity component perpendicular to the parallel walls is only significant within the viscous layers surrounding the fibres, whose thickness is of the order of half the channel height B′. One finds that when the aspect ratio B > 5, the neglect of the vertical velocity component vz can lead to large errors in the satisfaction of the no-slip boundary conditions on the surfaces of the fibres and large deviations from the approximate solution in Lee (1969), in which vz and the normal pressure field are neglected. The numerical results show that the drag coefficient of the fibrous bed increases dramatically when the open gap between adjacent fibres Δ′ becomes smaller than B′. The predictions of the new theory are used to examine the possibility that a cross-bridging slender fibre matrix can exist in the intercellular cleft of capillary endothelium as proposed by Curry & Michel (1980).

STEADY STOKES FLOWS WITH THRESHOLD SLIP BOUNDARY CONDITIONS

2005 ◽  
Vol 15 (08) ◽  
pp. 1141-1168 ◽  
Author(s):  
C. LE ROUX
Keyword(s):  
Stokes Equations ◽  
Stokes Problem ◽  
Nonlinear Function ◽  
Wall Slip ◽  
Slip Boundary Condition ◽  
Dirichlet Condition ◽  
Steady Flows ◽  
Slip Boundary ◽  
Rotationally Symmetric ◽  
Tangential Traction

We prove the existence, uniqueness and continuous dependence on the data of weak solutions to boundary-value problems that model steady flows of incompressible Newtonian fluids with wall slip in bounded domains. The flows satisfy the Stokes equations and a nonlinear slip boundary condition: for slip to occur, the magnitude of the tangential traction must exceed a prescribed threshold, which is independent of the normal stress, and where slip occurs the tangential traction is equal to a prescribed, possibly nonlinear, function of the slip velocity. In addition, a Dirichlet condition is imposed on a component of the boundary if the domain is rotationally symmetric. The method of proof is based on a variational inequality formulation of the problem and fixed point arguments which utilize wellposedness results for the Stokes problem with a slip condition of the "friction type".

scholarly journals An Oseen Two-Level Stabilized Mixed Finite-Element Method for the 2D/3D Stationary Navier-Stokes Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Aiwen Wang ◽  
Xin Zhao ◽  
Peihua Qin ◽  
Dongxiu Xie
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Mesh Size ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
Element Method

We investigate an Oseen two-level stabilized finite-element method based on the local pressure projection for the 2D/3D steady Navier-Stokes equations by the lowest order conforming finite-element pairs (i.e.,Q1−P0andP1−P0). Firstly, in contrast to other stabilized methods, they are parameter free, no calculation of higher-order derivatives and edge-based data structures, implemented at the element level with minimal cost. In addition, the Oseen two-level stabilized method involves solving one small nonlinear Navier-Stokes problem on the coarse mesh with mesh sizeH, a large general Stokes equation on the fine mesh with mesh sizeh=O(H)2. The Oseen two-level stabilized finite-element method provides an approximate solution (uh,ph) with the convergence rate of the same order as the usual stabilized finite-element solutions, which involves solving a large Navier-Stokes problem on a fine mesh with mesh sizeh. Therefore, the method presented in this paper can save a large amount of computational time. Finally, numerical tests confirm the theoretical results. Conclusion can be drawn that the Oseen two-level stabilized finite-element method is simple and efficient for solving the 2D/3D steady Navier-Stokes equations.

MINIMAL STABILIZATIONS OF THE Pk+1-Pk APPROXIMATION OF THE STATIONARY STOKES EQUATIONS

1995 ◽  
Vol 05 (02) ◽  
pp. 213-224 ◽  
Author(s):  
D. BOFFI
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Stabilization Method ◽  
Triangular Finite Element ◽  
Piecewise Polynomials

The triangular finite element approximation of the Stokes problem is analyzed by means of piecewise polynomials of degree k+1 for the velocity and discontinuous of degree k for the pressure. The known results are recalled, with particular interest in the restrictions required for the triangulation and in the stabilization procedures. A new less expensive stabilization method is proposed.

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