Brinkman System Research Articles

The mixed problem for the Darcy-Forchheimer-Brinkman system

This paper studies the mixed problems for the Brinkman system in Hs(Ω;Rm)×Hs−1(Ω) with 1≤s≤2. Here Ω⊂Rm is a bounded domain with Lipschitz boundary (not necessarily connected). Then, using a fixed point theorem we prove the existence of solutions of the mixed problems for the Darcy-Forchheimer-Brinkman system in these spaces.

On some Robin–transmission problems for the Brinkman system and a Navier–Stokes type system

The aim of this paper is twofold. The first goal is to give well‐posedness results for a Robin–transmission problem and a Robin–Dirichlet problem for the Brinkman system in bounded Lipschitz domains in . The Robin–Dirichlet problem appears as a special limit case of the Robin–transmission problem. This aim is achieved by employing the layer potential theoretical methods. The second goal is to provide a well‐posedness result for a Robin–transmission problem and an existence result for the Robin–Dirichlet problem for a Navier–Stokes type system in bounded Lipschitz domains in . Again, the Robin–Dirichlet problem appears as a special limit case of the Robin–transmission problem. This aim is achieved by using the well‐posedness results in the linear case combined with a fixed point argument. In order to illustrate our theoretical results, we also consider a special case of boundary value problems of Robin–Dirichlet type, which models a fluid flow in a porous square cavity. We solve numerically this problem.

Pressure-correction projection method for modelling the incompressible fluid flow in porous media

Abstract This work is dedicated to the pressure-correction projection method for the volume-averaged Navier–Stokes system for porous media. A set of parameters controlling the presence of inertia and viscosity is introduced into the system. Switching parameters allows us to reduce the system to either the Brinkman system or the Darcy equation. Considering the jump in the parameters between mesh cells allows capturing the contact of media of different types, such as free-flow and porous media flow. We apply Chorin’s projection method to decouple the system. The splitting of the system yields a momentum conservation equation and an anisotropic pressure correction equation. We propose a combination of collocated finite-volume methods to solve the problem.

Effective governing equations for dual porosity Darcy–Brinkman systems subjected to inhomogeneous body forces and their application to the lymph node

We derive the homogenized governing equations for a double porosity system where the fluid flow within the individual compartments is governed by the coupling between the Darcy and the Darcy–Brinkman equations at the microscale , and are subjected to inhomogeneous body forces. The homogenized macroscale results are obtained by means of the asymptotic homogenization technique and read as a double Darcy differential model with mass exchange between phases. The role of the microstructure is encoded in the effective hydraulic conductivities which are obtained by solving periodic cell problems whose properties are illustrated and compared. We conclude by solving the new model by means of a semi-analytical approach under the assumption of azimuthal axisymmetry to model the movement of fluid within a lymph node.

Cahn–Hilliard–Brinkman model for tumor growth with possibly singular potentials

We analyze a phase field model for tumor growth consisting of a Cahn–Hilliard–Brinkman system, ruling the evolution of the tumor mass, coupled with an advection-reaction-diffusion equation for a chemical species acting as a nutrient. The main novelty of the paper concerns the discussion of the existence of weak solutions to the system covering all the meaningful cases for the nonlinear potentials; in particular, the typical choices given by the regular, the logarithmic, and the double obstacle potentials are admitted in our treatise. Compared to previous results related to similar models, we suggest, instead of the classical no-flux condition, a Dirichlet boundary condition for the chemical potential appearing in the Cahn–Hilliard-type equation. Besides, abstract growth conditions for the source terms that may depend on the solution variables are postulated.

Numerical Simulation for Brinkman System with Varied Permeability Tensor

The aim of this paper is to study a stationary Brinkman problem in an anisotropic porous medium by using a mini-element method with a general boundary condition. One of the important aspects of the P1−Bubble/P1 method is satisfying the inf-sup condition, which allows us the existence and the uniqueness of the weak solution to our problem. To go further in this theoretical study, an a priori error estimate is established. To see the importance of this method in reality, we applied this method to a real problem. The numerical simulation studies support our results and demonstrate the effectiveness of this method.

Numerical methods with particular solutions for nonhomogeneous Stokes and Brinkman systems

Numerical methods with particular solutions for nonhomogeneous Stokes and Brinkman systems

New efficient and unconditionally energy stable schemes for the Cahn–Hilliard–Brinkman system

In this paper, we construct several efficient numerical schemes based on two types of scalar auxiliary variable (SAV) approaches for the Cahn–Hilliard–Brinkman system. The temporal discretizations are built upon the first-order, second-order, and higher-order methods, respectively. The unconditional energy stability analyses for the constructed schemes are rigorously derived. Finally various numerical simulations are showed to demonstrate the accuracy and performance for the constructed schemes.

The method of fundamental solutions for Brinkman flows. Part II. Interior domains

In part I, we considered the application of the method of fundamental solutions (MFS) for solving numerically the Brinkman fluid flow in the unbounded porous medium outside obstacles of known or unknown shapes. In this companion paper we consider the corresponding interior problem for the Brinkman flow in a bounded porous medium which contains an unknown rigid inclusion $$D \subset \Omega $$ . The inclusion D is to be identified by a pair of Cauchy data represented by the fluid velocity and traction on the boundary $$\partial \Omega $$ . The fluid velocity and pressure of the incompressible viscous flow in the porous medium $$\Omega \backslash \overline{D}$$ are approximated by linear combinations of fundamentals solutions for the Brinkman system with sources (singularities) placed outside the closure of the solution domain, i.e. in $$D \cup \big ({\mathbb {R}}^2\backslash \overline{\Omega } \big )$$ , assuming, for simplicity, that we analyse planar domains. By further assuming that the unknown obstacle D is star-shaped (with respect to the origin), the inverse problem recasts as the minimization of the nonlinear Tikhonov’s regularization functional with respect to the MFS expansion coefficients and the discretized polar radii defining D. This minimization subject to simple bounds on the variables is solved numerically using the MATLAB optimization toolbox routine lsqnonlin.

Cahn–Hilliard–Brinkman systems for tumour growth

<p style='text-indent:20px;'>A phase field model for tumour growth is introduced that is based on a Brinkman law for convective velocity fields. The model couples a convective Cahn–Hilliard equation for the evolution of the tumour to a reaction-diffusion-advection equation for a nutrient and to a Brinkman–Stokes type law for the fluid velocity. The model is derived from basic thermodynamical principles, sharp interface limits are derived by matched asymptotics and an existence theory is presented for the case of a mobility which degenerates in one phase leading to a degenerate parabolic equation of fourth order. Finally numerical results describe qualitative features of the solutions and illustrate instabilities in certain situations.</p>

Inertia and Roughness-Induced Effects on the Porous Medium Flow Through a Corrugated Channel

In this paper, we study the flow through a corrugated channel filled with a fluid-saturated sparsely packed porous medium. The porous medium flow is described by the Darcy–Lapwood–Brinkman system taking into account the Brinkman extension of the Darcy law and the flow inertia. We assume the periodicity of the roughness in the longitudinal direction and that the flow is governed by the prescribed pressure drop between the edges of the channel. Introducing average boundary roughness as the small parameter, we propose a higher-order correction of the Darcy–Brinkman filtration velocity via asymptotic analysis.

Effects of Cilia Movement on Fluid Velocity: II Numerical Solutions Over a Fixed Domain

Cilia are hair-like structures that move in unison with the purpose of propelling fluid. They are found, for example, in the human bronchiole respiratory system and molluscs. Here, we validate a novel model of fluid flow due to the movement of cilia in a fixed computational domain. We consider two domains, a porous medium and a free-fluid domain and numerically solve the Stokes–Brinkman system of equations where the cilia geometry and velocity are input and the velocity of fluid due to the movement of cilia is determined. The cilia velocities and geometry are approximated using human lung cilia experimental data available in the literature. We use a mixed finite element method of Taylor-Hood type to calculate the fluid velocities in a three-dimensional domain. The results are validated in a simple case by comparison with an exact solution with good agreement. This problem can be used as a benchmark for the movement of fluid phases due to the self-propelled movement of a solid phase in a porous medium.

Weak solutions for the stationary anisotropic and nonlocal compressible Navier-Stokes system

In this paper, we prove the existence of weak solutions for the stationary compressible Navier-Stokes equations with an anisotropic and nonlocal viscous stress tensor in a periodic domain T3. This gives an answer to an open problem which is important for applications for instance in geophysics or in microfluidics. When dealing with weak solutions for such non-linear PDE system, the most delicate aspect is the stability analysis: Given a sequence of weak solutions for some well-chosen approximated systems, show that this sequence converges to a solution for the initial system. One of the key ingredients in the proof of stability is the adaptation of a new identity discovered by the authors [2] which was developed to study the quasi-stationary anisotropic compressible Brinkman system. This identity is used in order to recover strong convergence properties for the sequence of velocities and recover a posteriori strong convergence for the sequence of densities.

Weak and stationary solutions to a Cahn–Hilliard–Brinkman model with singular potentials and source terms

Abstract We study a phase field model proposed recently in the context of tumour growth. The model couples a Cahn–Hilliard–Brinkman (CHB) system with an elliptic reaction-diffusion equation for a nutrient. The fluid velocity, governed by the Brinkman law, is not solenoidal, as its divergence is a function of the nutrient and the phase field variable, i.e., solution-dependent, and frictionless boundary conditions are prescribed for the velocity to avoid imposing unrealistic constraints on the divergence relation. In this paper we give a first result on the existence of weak and stationary solutions to the CHB model for tumour growth with singular potentials, specifically the double obstacle potential and the logarithmic potential, which ensures that the phase field variable stays in the physically relevant interval. New difficulties arise from the interplay between the singular potentials and the solution-dependent source terms, but can be overcome with several key estimates for the approximations of the singular potentials, which maybe of independent interest. As a consequence, included in our analysis is an existence result for a Darcy variant, and our work serves to generalise recent results on weak and stationary solutions to the Cahn–Hilliard inpainting model with singular potentials.

Boundary integral approach for the mixed Dirichlet–Robin boundary value problem for the nonlinear Darcy–Forchheimer–Brinkman system

The purpose of this paper is to provide the solvability and uniqueness result to the mixed Dirichlet–Robin boundary value problem for the nonlinear Darcy–Forchheimer–Brinkman system in a bounded, two-dimensional Lipschitz domain. First we obtain a well-posedness result for the linear Brinkman system with Dirichlet–Neumann boundary conditions, by reducing the problem to the system of boundary integral equations based on the fundamental solution of the Brinkman system and by analyzing this system employing a variational approach. The result is extended afterwards to the Poisson problem for the Brinkman system and to Dirichlet–Robin boundary conditions, using the Newtonian potential and the linearity of the solution operator. Further, we study the nonlinear Darcy–Forchheimer–Brinkman boundary value problem of Dirichlet and Robin type.

A note on a transmission problem for the Brinkman system and the generalized Darcy-Forchheimer-Brinkman system in Lipschitz domains in R3

The purpose of this paper is to treat a nonlinear transmission-type problem for a generalized version of the Darcy- Forchheimer - Brinkman system and the classical Brinkman system in complementary Lipschitz domains in R^3 . First of all, we define the required spaces in which we seek our solution. Next, we describe the generalized Brinkman and the generalized Darcy- Forchheimer - Brinkman systems. Further, we give important lemmas that allow us to introduce the trace and conormal derivative operators that appear in the formulation of our transmission problem. We invoke a result regarding the well- posedness of the (linear) transmission problem for the generalized and classical Brinkman systems in complementary Lipschitz domains in R^3 . The above mentioned well- posedness result in the linear case combined with Banach's fixed point theorem will allow us to establish the main result of the paper, the well- posedness of the transmission problem for the Brinkman system and the nonlinear generalized Darcy- Forchheimer - Brinkman system in Lipschitz domains in R^3 .

A layer potential analysis for transmission problems for Brinkman‐type systems in Lipschitz domains in

AbstractThe aim of this paper is to establish a well‐posedness result for a boundary value problem of transmission‐type for the standard and generalized Brinkman systems in two Lipschitz domains in , the former being bounded, and the latter, its complement in . As a first step, we establish a well‐posedness result for a transmission problem for the standard Brinkman systems on complementary Lipschitz domains in by making use of the Potential theory developed for such a system. As a second step, we prove our desired result (in L2‐based Sobolev spaces) by using a method based on Fredholm operator theory and the well‐posedness result from the previous step.

A Poisson Problem of Transmission-type for the Stokes and Generalized Brinkman Systems in Complementary Lipschitz Domains in $\mathbb{R}^3$

The purpose of this paper is to give a well-posedness result for a boundary value problem of transmission type for the Stokes and generalized Brinkman systems in two complementary Lipschitz domains in $\mathbb{R}^3$. In the first part of the paper, we have introduced the classical and weighted $L^2$-based Sobolev spaces on Lipschitz domains in $\mathbb{R}^3$. Afterwards, the trace and conormal derivative operators are defined in the case of both Stokes and generalized Brinkman systems. Also, a summary of the main properties of the layer potential operators for the Stokes system, is provided. In the second part of the work, we exploit the well-posedness of another transmission problem concerning the Stokes system on two complementary Lipschitz domains in $\mathbb{R}^3$ which is based on the Potential Theory for the Stokes system. Then, certain properties of Fredholm operators will allow us to show our main well-posedness result in $L^2$-based Sobolev spaces.

Boundary value problems for the Brinkman system with L∞ coefficients in Lipschitz domains on compact Riemannian manifolds. A variational approach

The purpose of this paper is to show well-posedness results in L2-based Sobolev spaces for transmission, Dirichlet, Neumann, and mixed boundary value problems for the Brinkman system with L∞ coefficients in Lipschitz domains on a compact Riemannian manifold of dimension m≥2. The Dirichlet, transmission, and mixed problems for the nonlinear Darcy-Forchheimer-Brinkman system with L∞ coefficients are also analyzed. First, we focus on the well-posedness of linear transmission, Dirichlet and mixed boundary value problems for the Brinkman system with L∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using a variational approach that reduces such a boundary value problem to a mixed variational formulation defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. Further, we show the equivalence between each boundary value problem for the Brinkman system with L∞ coefficients and its mixed variational counterpart, and then the well posedness in L2-based Sobolev spaces by using the Nečas-Babuška-Brezzi technique. The second goal of this paper is the construction of the Newtonian and layer potential operators for the Brinkman system with L∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using the well-posedness results for the analyzed linear transmission problems. Various mapping properties of these operators are also obtained and used to describe the weak solutions of the Poisson problems with Dirichlet and Neumann conditions for the nonsmooth Brinkman system in terms of such potentials. Finally, we combine the well-posedness results of the Poisson problems of Dirichlet, transmission, and mixed type for the nonsmooth Brinkman system with a fixed point theorem in order to show the existence of a weak solution of the Poisson problem of Dirichlet, transmission, or mixed type for the (nonlinear) Darcy-Forchheimer-Brinkman system with L∞ coefficients in L2-based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds of dimension m∈{2,3}.

Existence results for a monophasic compressible Darcy–Brinkman’s flow in porous media

In this paper, we are interested in the displacement of a single compressible phase in Darcy–Brinkman’s flow in porous media. The equations are obtained by the conservation of mass and by considering the Brinkman regularization velocity of the standard Darcy infiltration velocity. This model is treated in its general form with the whole nonlinear terms. First, we prove the existence and uniqueness of a strong solution in one dimensional space for the Darcy–Brinkman system. Second, we treat this system with Bear hypothesis in multidimensional spaces.