A decoupling method with different subdomain time steps for the nonstationary stokes–darcy model

This report analyzes a multirate, decoupling algorithm, which allows different time steps in the fluid region and the porous region for the nonstationary Stokes–Darcy problem. The method presented requires only one, uncoupled Stokes and Darcy subphysics and subdomain solve per time step. Under a time step restriction of the form △t ≤ C (physical parameters) we prove stability and convergence of the method. Numerical tests are given to show the convergence result and demonstrate the computational efficiency of the partitioned method. They also show that in the expected case of greater fluid velocities in the free‐flow region than in the porous media region, allowing smaller time steps in the subregion with the faster velocities increases both accuracy and efficiency. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Homogenization of a two‐phase incompressible fluid in crossflow filtration through a porous medium

We study the homogenization of a slow viscous two‐phase incompressible flow in a domain consisting of a free fluid domain, a porous medium, and the interface between them. We take into account the capillary forces on the fluid‐fluid interfaces. We construct boundary layers describing the flow at the interface between the free fluid and the porous medium. We derive a macroscopic model with a viscous two‐phase fluid in the free domain, a coupled Darcy law connecting two‐phase velocities in the porous medium, and boundary conditions at the permeable interface between the free fluid domain and the porous medium.     

Uncoupling evolutionary groundwater‐surface water flows using the Crank–Nicolson Leapfrog method

Consider an incompressible fluid in a region Ω_f flowing both ways across an interface into a porous media domain Ω_p saturated with the same fluid. The physical processes in each domain have been well studied and are described by the Stokes equations in the fluid region and the Darcy equations in the porous media region. Taking the interfacial conditions into account produces a system with an exactly skew symmetric coupling. Spatial discretization by finite element method and time discretization by Crank–Nicolson LeapFrog give a second‐order partitioned method requiring only one Stokes and one Darcy subphysics and subdomain solver per time step for the fully evolutionary Stokes‐Darcy problem. Analysis of this method leads to a time step condition sufficient for stability and convergence. Numerical tests verify predicted rates of convergence; however, stability tests reveal the problem of growth of numerical noise in unstable modes in some cases. In such instances, the addition of time filters adds stability. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

First‐Order system least squares for generalized‐Newtonian coupled Stokes‐Darcy flow

The coupled problem for a generalized Newtonian Stokes flow in one domain and a generalized Newtonian Darcy flow in a porous medium is studied in this work. Both flows are treated as a first‐order system in a stress‐velocity formulation for the Stokes problem and a volumetric flux‐hydraulic potential formulation for the Darcy problem. The coupling along an interface is done using the well‐known Beavers–Joseph–Saffman interface condition. A least squares finite element method is used for the numerical approximation of the solution. It is shown that under some assumptions on the viscosity the error is bounded from above and below by the least squares functional. An adaptive refinement strategy is examined in several numerical examples where boundary singularities are present. Due to the nonlinearity of the problem a Gauss–Newton method is used to iteratively solve the problem. It is shown that the linear variational problems arising in the Gauss–Newton method are well posed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1150–1173, 2015     

Decoupled energy‐law preserving numerical schemes for the Cahn–Hilliard–Darcy system

We study two novel decoupled energy‐law preserving and mass‐conservative numerical schemes for solving the Cahn‐Hilliard‐Darcy system which models two‐phase flow in porous medium or in a Hele–Shaw cell. In the first scheme, the velocity in the Cahn–Hilliard equation is treated explicitly so that the Darcy equation is completely decoupled from the Cahn–Hilliard equation. In the second scheme, an intermediate velocity is used in the Cahn–Hilliard equation which allows for the decoupling. We show that the first scheme preserves a discrete energy law with a time‐step constraint, while the second scheme satisfies an energy law without any constraint and is unconditionally stable. Ample numerical experiments are performed to gauge the efficiency and robustness of our scheme. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 936–954, 2016     

Dirichlet problem for a nonlinear generalized Darcy–Forchheimer–Brinkman system in Lipschitz domains

The purpose of this paper is to show existence of a solution of the Dirichlet problem for a nonlinear generalized Darcy–Forchheimer–Brinkman system in a bounded Lipschitz domain in , with small boundary datum in L²‐based Sobolev spaces. A useful intermediary result is the well‐posedness of the Poisson problem for a generalized Brinkman system in a bounded Lipschitz domain in , with Dirichlet boundary condition and data in L²‐based Sobolev spaces. We obtain this well‐posedness result by showing that the matrix type operator associated with the Poisson problem is an isomorphism. Then, we combine the well‐posedness result from the linear case with a fixed point theorem in order to show the existence of a solution of the Dirichlet problem for the nonlinear generalized Darcy–Forchheimer–Brinkman system. Some applications are also included. Copyright © 2014 John Wiley & Sons, Ltd.     

Attractor dimension estimate for plane shear flow of micropolar fluid with free boundary

This research is motivated by a problem from lubrication theory. We consider a free boundary problem of a two‐dimensional boundary‐driven micropolar fluid flow. The existence of a unique global‐in‐time solution of the problem and the global attractor for the associated semigroup are known. In this paper we estimate the dimension of the global attractor in terms of the given data and the geometry of the domain of the flow by establishing a new version of the Lieb–Thirring inequality with constants depending explicitly on the geometry of the domain. We also obtain some new estimates for the Navier–Stokes shear flows. Copyright © 2005 John Wiley & Sons, Ltd.     

Analysis of nonlocal model of compressible fluid in 1‐D

We study a nonlocal modification of the compressible Navier–Stokes equations in mono‐dimensional case with a boundary condition characteristic for the free boundaries problem. From the formal point of view, our system is an intermediate between the Euler and Navier–Stokes equations. Under certain assumptions, imposed on initial data and viscosity coefficient, we obtain the local and global existence of solutions. Particularly, we show the uniform in time bound on the density of fluid. Copyright © 2010 John Wiley & Sons, Ltd.     

Oseen coupling method for the exterior flow. Part II: Well‐posedness analysis

In this paper, we recall the Oseen coupling method for solving the exterior unsteady Navier–Stokes equations with the non‐homogeneous boundary conditions. Moreover, we derive the coupling variational formulation of the Oseen coupling problem by using of the integral representations of the solution of the Oseen equations at an infinity domain. Finally, we provide some properties of the integral operators over the artificial boundary and the well‐posedness of the coupling variational formulation. Copyright © 2004 John Wiley & Sons, Ltd.     

Defect correction method for viscoelastic fluid flows at high Weissenberg number

We study a defect correction method for the approximation of viscoelastic fluid flow. In the defect step, the constitutive equation is computed with an artificially reduced Weissenberg parameter for stability, and the resulting residual is corrected in the correction step. We prove the convergence of the defect correction method and derive an error estimate for the Oseen‐viscoelastic model problem. The derived theoretical results are supported by numerical tests for both the Oseen‐viscoelastic problem and the Johnson‐Segalman model problem. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

An existence result for nonisothermal immiscible incompressible 2‐phase flow in heterogeneous porous media

In the present article, we study the temperature effects on two‐phase immiscible incompressible flow through a porous medium. The mathematical model is given by a coupled system of 2‐phase flow equations and an energy balance equation. The model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy‐Muskat and the capillary pressure laws. The problem is written in terms of the phase formulation; ie, the saturation of one phase, the pressure of the second phase, and the temperature are primary unknowns. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we show the existence of weak solutions with the help of an appropriate regularization and a time discretization. We use suitable test functions to obtain a priori estimates. We prove a new compactness result to pass to the limit in nonlinear terms.     

Numerical simulation of Darcy–Forchheimer flow of third grade liquid with Cattaneo–Christov heat flux model

The main purpose of this research is to evaluate the impact of Darcy–Forchheimer flow in an incompressible third‐grade liquid through Cattaneo–Christov heat flux approach. The Cattaneo–Christov heat flux theory is adopted to govern the mathematical expression of energy, which involves the heat flux relaxation time chracteristics. Time‐dependent thermal conductivity is accounted. The steady problem is reduced to ordinary differential equations via suitable transformation. Numerical solutions for the resulting flow expressions have been computed with the help of Euler's explicit technique. Impact of influential variables on the velocity, temperature and skin‐friction coefficient have been demonstrated and discussed through graphs. Copyright © 2017 John Wiley & Sons, Ltd.     

Strong stability for a fluid–structure interaction model

In this paper we consider the question of stabilization of a fluid–structure model that describes the interaction between a 3‐D incompressible fluid and a 2‐D plate, the interface, which coincides with a flat flexible part of the surface of the vessel containing the fluid. The mathematical model comprises the Stokes equations and the equations for the longitudinal deflections of the plate with inclusion of the shear stress, which the fluid exerts on the plate. We show that the energy associated with the model decays strongly when the interface is equipped with a locally supported dissipative mechanism. Our main tool is an abstract resolvent criterion due to Tomilov. Copyright © 2008 John Wiley & Sons, Ltd.     

Diffusion in poro‐plastic media

A model is developed for the flow of a slightly compressible fluid through a saturated inelastic porous medium. The initial‐boundary‐value problem is a system that consists of the diffusion equation for the fluid coupled to the momentum equation for the porous solid together with a constitutive law which includes a possibly hysteretic relation of elasto‐visco‐plastic type. The variational form of this problem in Hilbert space is a non‐linear evolution equation for which the existence and uniqueness of a global strong solution is proved by means of monotonicity methods. Various degenerate situations are permitted, such as incompressible fluid, negligible porosity, or a quasi‐static momentum equation. The essential sufficient conditions for the well‐posedness of the system consist of an ellipticity condition on the term for diffusion of fluid and either a viscous or a hardening assumption in the constitutive relation for the porous solid. Copyright © 2004 John Wiley & Sons, Ltd.     

Bingham flow in porous media with obstacles of different size

By using the unfolding operators for periodic homogenization, we give a general compactness result for a class of functions defined on bounded domains presenting perforations of two different size. Then we apply this result to the homogenization of the flow of a Bingham fluid in a porous medium with solid obstacles of different size. Next, we give the interpretation of the limit problem in terms of a nonlinear Darcy law. Copyright © 2017 John Wiley & Sons, Ltd.     

An exact solution of start-up flow for the fractional generalized Burgers’ fluid in a porous half-space

Modified Darcy’s law for fractional generalized Burgers’ fluid in a porous medium is introduced. The flow near a wall suddenly set in motion for a fractional generalized Burgers’ fluid in a porous half-space is investigated. The velocity of the flow is described by fractional partial differential equations. By using the Fourier sine transform and the fractional Laplace transform, an exact solution of the velocity distribution is obtained. Some previous and classical results can be recovered from our results, such as the velocity solutions of the Stokes’ first problem for viscous Newtonian, second grade, Maxwell, Oldroyd-B or Burgers’ fluids.     

Convergence of a family of Galerkin discretizations for the Stokes–Darcy coupled problem

In this article we analyze the well‐posedness (unique solvability, stability, and Céa's estimate) of a family of Galerkin schemes for the coupling of fluid flow with porous media flow. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers—Joseph—Saffman law. We consider the usual primal formulation in the Stokes domain and the dual‐mixed one in the Darcy region, which yields a compact perturbation of an invertible mapping as the resulting operator equation. We then apply a classical result on projection methods for Fredholm operators of index zero to show that use of any pair of stable Stokes and Darcy elements implies the well‐posedness of the corresponding Stokes—Darcy Galerkin scheme. This extends previous results showing well‐posedness only for Bernardi—Raugel and Raviart—Thomas elements. In addition, we show that under somewhat more demanding hypotheses, an alternative approach that makes no use of compactness arguments can also be applied. Finally, we provide several numerical results illustrating the good performance of the Galerkin method for different geometries of the problem using the MINI element and the Raviart—Thomas subspace of lowest order. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 721–748, 2011     

A systematic approximation for the equations governing convection–diffusion in a porous medium

In order to take into account thermal effects in flows through porous media, one makes ad hoc modifications to Darcy’s equation by appending a term that is similar to the one that is obtained in the Oberbeck–Boussinesq approximation for a fluid. In this short paper we outline a systematic procedure for obtaining an Oberbeck–Boussinesq type of approximation for the flow of a fluid through a porous medium. In addition to establishing the appropriate equation for a flow governed by Darcy’s equation, we proceed to obtain the approximations for flows governed by equations due to Forchheimer and Brinkman.     

SUPG and grad‐div stabilized finite element methods for steady weakly compressible viscous flow

The extensive application of mathematical and computational methods has become an efficient and powerful approach to the investigation and solution of many problems and processes in fluid dynamics from qualitative as well as quantitative point of view. In this work a new class of advanced numerical approximation schemes to isothermal compressible viscous flow is presented. The schemes are based on an iteration between an Oseen like problem for the velocity and a hyperbolic transport equation for the density. Such schemes seem attractive for computations because they offer a reduction to simpler problems for which highly refined numerical methods either are known or can be built from existing approximation schemes to similar equations, and because of the guidance that can be drawn from an existence theory based on them. For the generalized Oseen subproblem a Taylor–Hood finite element method is proposed that is stabilized by a reduced SUPG and grad‐div technique (cf. [1, 4]) in the convection‐dominated case. Results of theoretical investigations and numerical studies are presented. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Derivation of a contact law between a free fluid and thin porous layers via asymptotic analysis methods

We describe the asymptotic behaviour of an incompressible viscous free fluid in contact with a porous layer flow through the porous layer surface. This porous layer has a small thickness and consists of thin channels periodically distributed. Two scales are present in this porous medium, one associated to the periodicity of the distribution of the channels and the other to the size of these channels. Proving estimates on the solution of this Stokes problem, we establish a critical link between these two scales. We prove that the limit problem is a Stokes flow in the free domain with further boundary conditions on the basis of the domain which involve an extra velocity, an extra pressure and two second-order tensors. This limit problem is obtained using Γ-convergence methods. We finally consider the case of a Navier–Stokes flow within this context.