On the well-posed coupling between free fluid and porous viscous flows

We present a well-posed model for the Stokes/Brinkman problem with a family of jump embedded boundary conditions (J.E.B.C.) on an immersed interface with weak regularity assumptions. It arises from a general framework recently proposed for fictitious domain problems. Our model is based on algebraic transmission conditions combining the stress and velocity jumps on the interface $\Sigma$ separating the fluid and porous domains. These conditions are well chosen to get the coercivity of the operator. Then, the general framework allows us to prove new results on the global solvability of some models with physically relevant stress or velocity jump boundary conditions for the momentum transport at a fluid–porous interface. The Stokes/Brinkman problem with Ochoa-Tapia and Whitaker (1995) [9], [10] interface conditions and the Stokes/Darcy problem with Beavers and Joseph (1967) [13] conditions are both proved to be well-posed, by an asymptotic analysis. Up to now, only the Stokes/Darcy problem with Saffman (1971) [15] approximate interface conditions with negligible tangential porous velocity was known to be well-posed.     

Stokes flow of an assemblage of porous particles: stress jump condition

The present article investigates the overall bed permeability of an assemblage of porous particles. For the bed of porous particles, the fluid-particle system is represented as an assemblage of uniform porous spheres fixed in space. Each sphere, with a surrounding envelope of fluid, is uncoupled from the system and considered separately. This model is popularly known as cell model. Stokes equations are employed inside the fluid envelope and Brinkman equations are used inside the porous region. The stress jump boundary condition is used at the porous-liquid interface together with the continuity of normal stress and continuity of velocity components. On the surface of the fluid envelope, three different possible boundary conditions are tested. The obtained expression for the drag force is used to estimate the overall bed permeability of the assemblage of porous particles and the behavior of overall bed permeability is analyzed with various parameters like modified Darcy number (Da*), stress jump coefficient (??), volume fraction (??), and effective viscosity.     

Interface boundary value problems of Robin‐transmission type for the Stokes and Brinkman systems on n‐dimensional Lipschitz domains: applications

In this paper, we describe a layer potential analysis in order to show an existence result for an interface boundary value problem of Robin‐transmission type for the Stokes and Brinkman systems on Lipschitz domains in Euclidean setting, when the given boundary data belong to some L^p or Sobolev spaces associated to such domains. Applications related to an exterior three‐dimensional Stokes flow past two concentric porous spheres with stress jump conditions on the fluid‐porous interface are also considered. Copyright © 2013 John Wiley & Sons, Ltd.     

A nonconforming rectangular finite element pair for the Darcy–Stokes–Brinkman model

In this article, we consider the Darcy–Stokes–Brinkman model that can be sorted into three problems: the Darcy problems, the Stokes–Brinkman interface problems and the coupled Darcy–Stokes problems. We study finite element approximation of the model with Dirichlet boundary conditions and make a unified analysis of the three problems based on nonconforming element. Optimal error estimates for the fluid velocity and pressure are derived. Finally, we present some numerical examples verifying the theoretical predictions. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Expansions at small Reynolds numbers for the flow past a porous circular cylinder

The purpose of this article is to use the method of matched asymptotic expansions (MMAE) in order to study the two-dimensional steady low Reynolds number flow of a viscous incompressible fluid past a porous circular cylinder. We assume that the flow inside the porous body is described by the continuity and Brinkman equations, and the velocity and boundary traction fields are continuous across the interface between the fluid and porous media. Formal expansions for the corresponding stream functions are used. We show that the force exerted by the exterior flow on the porous cylinder admits an asymptotic expansion with respect to low Reynolds numbers, whose terms depend on the characteristics of the porous cylinder. In addition, by considering Darcy's law for the flow inside the porous circular cylinder, an asymptotic formula for the force on the cylinder is obtained. Also, a porous circular cylinder with a rigid core inside is considered with Brinkman equation inside the porous region. Stress jump condition is used at the porous–liquid interface together with the continuity of velocity components and continuity of normal stress. Some particular cases, which refer to the low Reynolds number flow past a solid circular cylinder, have also been investigated.     

A domain embedding method for mixed boundary value problems

We propose a domain embedding (fictitious domain) method for elliptic equations subject to mixed boundary conditions, and prove the sharp convergence rate. The theory provides a unified treatment for Dirichlet, Neumann, and Robin boundary conditions. To cite this article: S. Zhang, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

A unified fictitious domain model for general embedded boundary conditions

This Note addresses the analysis of a new fictitious domain method for elliptic problems in order to handle general embedded boundary conditions (E.B.C.): Fourier, Neumann and Dirichlet conditions on an immersed interface. Our method is based on a recent model of fracture combining flux and solution jumps on the interface Σ separating the original domain $\stackrel{?}{\Omega }$ from the auxiliary exterior domain ${\Omega }_e$. A class of methods is derived within the same unified formulation with either no penalty or exterior control in ${\Omega }_e$, or surface penalty on Σ, volume $H^1$ or $L^2$ penalty in ${\Omega }_e$, or both. The consistency (no penalty) or optimal error estimates with respect to the penalty parameter are proved for such methods. To cite this article: Ph. Angot, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Existence of Weak Solutions Up to Collision for Viscous Fluid‐Solid Systems with Slip

We study in this paper the movement of a rigid solid inside an incompressible Navier‐Stokes flow within a bounded domain. We consider the case where slip is allowed at the fluid/solid interface through a Navier condition. Taking into account slip at the interface is very natural within this model, as classical no‐slip conditions lead to unrealistic collisional behavior between the solid and the domain boundary. We prove for this model existence of weak solutions of Leray type, up to collision, in three dimensions. The key point is that, due to the slip condition, the velocity field is discontinuous across the fluid/solid interface. This prevents obtaining global H¹ bounds on the velocity, which makes many aspects of the theory of weak solutions for Dirichlet conditions inappropriate. © 2014 Wiley Periodicals, Inc.     

On interfacial transition conditions in two phase gravity flow

A layer of ice and sediment is modelled as a mixture of two nonlinear, very viscous, constant density fluids interacting mechanically via Darcy- and Pick-type forces. An inclined layer of this mixture overlain by a layer of ice modelled as a viscous fluid is considered with boundary conditions of no-slip or viscous sliding at the base and no stress at the free surface. The interface is treated as a singular surface across which the jump conditions of mass and momentum for the constituents are assumed to hold. Furthermore, because the components are viscous fluids, a kinematic condition for the continuity of the tangential velocity is formulated. The momentum jump conditions involve surface production terms requiring additional surfacial constitutive relations.We show that the posed physical problem admits a mathematical solution only in the case that the interface momentum production is non-zero.Dedicated to Hans Roethlisberger on the occasion of his seventieth birthday.     

Sensitivity of the slip rate coefficient in fluid flow – poroelastic coupling conditions

To model flow-induced structural vibrations, an interface to couple fluid flow and poroelastic material in a finite element formulation has been developed. One parameter of this interface condition is the slip rate coefficient, resulting from the so-called Beavers-Joseph-Saffman condition. This condition states that the jump in tangential velocity at a fluid flow – porous interface is proportional to the shear stress. Up to now no a priori determination of this parameter exists, and the known parameter range has been deducted from measurements, i. e., in our case from the results of the pore-resolving simulations. When modeling realistic problems assuming incompressible fluids, vectorial flow velocity and scalar pressure interact with the poroelastic material. As the slip rate coefficient only influences the tangential contributions, its overall influence is not clear. In this work, the sensitivity of the slip rate coefficient regarding the interface's coupling conditions is evaluated. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two-layer muti-parameterized Schwarz Alternating Method

The convergence rate of a numerical procedure based on Schwarz Alternating Method (SAM) for solving elliptic boundary value problems (BVP’s) depends on the selection of the interface conditions applied on the interior boundaries of the overlapping subdomains. It has been observed that theRobin condition(mixed interface condition), controlled by a parameter, can optimize SAM’s convergence rate. Since the convergence rate is very sensitive to the parameter, Tang[17] suggested another interface condition calledover- determined interface condition. Based on the over-determined interface condition, we formulate thetwo-layer multi-parameterized SAM. For the SAM and the one-dimensional elliptic model BVP’s, we determine analytically the optimal values of the parameters. For the two-dimensional elliptic BVP’s, we also formulate the two-layer multiparameterized SAM and suggest a choice of multi-parameter to produce good convergence rate.     

An interface-fitted adaptive mesh method for elliptic problems and its application in free interface problems with surface tension

This work presents a novel two-dimensional interface-fitted adaptive mesh method to solve elliptic problems of jump conditions across the interface, and its application in free interface problems with surface tension. The interface-fitted mesh is achieved by two operations: (i) the projection of mesh nodes onto the interface and (ii) the insertion of mesh nodes right on the interface. The interface-fitting technique is combined with an existing adaptive mesh approach which uses addition/subtraction and displacement of mesh nodes. We develop a simple piecewise linear finite element method built on this interface-fitted mesh and prove its almost optimal convergence for elliptic problems with jump conditions across the interface. Applications to two free interface problems, a sheared drop in Stokes flow and the growth of a solid tumor, are presented. In these applications, the interface surface tension serves as the jump condition or the Dirichlet boundary condition of the pressure, and the pressure is solved with the interface-fitted finite element method developed in this work. In this study, a level-set function is used to capture the evolution of the interface and provide the interface location for the interface fitting.     

Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition

In this paper the boundary layer flow over a flat plat with slip flow and constant heat flux surface condition is studied. Because the plate surface temperature varies along the x direction, the momentum and energy equations are coupled due to the presence of the temperature gradient along the plate surface. This coupling, which is due to the presence of the thermal jump term in Maxwell slip condition, renders the momentum and energy equations non-similar. As a preliminary study, this paper ignores this coupling due to thermal jump condition so that the self-similar nature of the equations is preserved. Even this fundamental problem for the case of a constant heat flux boundary condition has remained unexplored in the literature. It was therefore chosen for study in this paper. For the hydrodynamic boundary layer, velocity and shear stress distributions are presented for a range of values of the parameter characterizing the slip flow. This slip parameter is a function of the local Reynolds number, the local Knudsen number, and the tangential momentum accommodation coefficient representing the fraction of the molecules reflected diffusively at the surface. As the slip parameter increases, the slip velocity increases and the wall shear stress decreases. These results confirm the conclusions reached in other recent studies. The energy equation is solved to determine the temperature distribution in the thermal boundary layer for a range of values for both the slip parameter as well as the fluid Prandtl number. The increase in Prandtl number and/or the slip parameter reduces the dimensionless surface temperature. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow properties, and the applied heat flux.     

Convergence estimates for an optimized Schwarz method for PDEs with discontinuous coefficients

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. When the subdomains are overlapping or nonoverlapping, these methods employ the optimal value of parameter(s) in the boundary condition along the artificial interface to accelerate its convergence. In the literature, the analysis of optimized Schwarz methods rely mostly on Fourier analysis and so the domains are restricted to be regular (rectangular). As in earlier papers, the interface operator can be expressed in terms of Poincaré–Steklov operators. This enables the derivation of an upper bound for the spectral radius of the interface operator on essentially arbitrary geometry. The problem of interest here is a PDE with a discontinuous coefficient across the artificial interface. We derive convergence estimates when the mesh size h along the interface is small and the jump in the coefficient may be large. We consider two different types of Robin transmission conditions in the Schwarz iteration: the first one leads to the best estimate when h is small, whereas for the second type, we derive a convergence estimate inversely proportional to the jump in the coefficient. This latter result improves upon the rate of popular domain decomposition methods such as the Neumann–Neumann method or FETI-DP methods, which was shown to be independent of the jump. In memory of Gene Golub.     

Classical solutions of two-dimensional stokes problems on non-smooth domains. Part 1: The radon integral operators

The applicability of the Neumann indirect method of potentials to the Dirichlet and Neumann problems for the two-dimensional Stokes operator on a non-smooth boundary Γ is subject to two kinds of sufficient and/or necessary conditions on Γ. The first one, occurring in electrostatic, is equivalent to the boundedness on C(Γ) of the velocity double-layer potential W as well as to the existence of jump relations of potentials. The second condition, which forces Γ to be a simple rectifiable curve and which, compared to the Laplacian, is a stronger restriction on the corners of Γ, states that the Fredholm radius of W is greater than 2. Under these conditions, the Radon boundary integral equations defined by the above-mentioned jump relations are solvable by the Fredholm theory; the double- (for Dirichlet) and the single- (for Neumann) layer potentials corresponding to their solutions are classical solutions of the Stokes problems.     

Homogenization of the Navier-Stokes equations with a slip boundary condition

This paper deals with the homogenization of the Stokes or Navier-Stokes equations in a domain containing periodically distributed obstacles, with a slip boundary condition (i.e., the normal component of the velocity is equal to zero, while the tangential velocity is proportional to the tangential component of the normal stress). We generalize our previous results (see [1]) established in the case of a Dirichlet boundary condition; in particular, for a so-called critical size of the obstacles (equal to ε³ in the three-dimensional case, ε being the inter-hole distance), we prove the convergence of the homogenization process to a Brinkman-type law.     

Shape sensitivity of curvilinear cracks on interface to non-linear perturbations

The 2D-model of an anisotropic, non-homogeneous, bonded elastic solid with a crack on the interface is considered. We state the linear problem with the stress-free boundary condition at the crack faces in addition to the transmission condition at the connected part of the interface. The sensitivity of the model to non-linear perturbations of the curvilinear crack along the interface is investigated. We obtain the asymptotic expansion and the corresponding derivatives of the potential energy functional with respect to the crack length via the material derivatives of the solution. This allows us to describe the growth or stationarity, and the local optimality conditions by the Griffith rupture criterion. The integral expression of the energy release rate for the considered problems is obtained, and the Cherepanov-Rice integral is discussed.     

On non-Newtonian p-fluids. The pseudo-plastic case

In the following we study a class of stationary Navier-Stokes equations with shear dependent viscosity, under the non-slip (Dirichlet) boundary condition. We consider pseudo-plastic fluids. A fluid is said pseudo-plastic, or shear thinning, if in Eq. (1.1) below one has p<2. We are interested in global (i.e., up to the boundary) regularity results, in dimension n=3, for the second order derivatives of the velocity and the first order derivatives of the pressure. We consider a cubic domain Ω and impose the non-slip boundary condition only on two opposite faces. On the other faces we assume periodicity, as a device to avoid effective boundary conditions. This choice is made so that we work in a bounded domain Ω and simultaneously with a flat boundary.     

A new augmented immersed finite element method without using SVD interpolations

Augmented immersed interface methods have been developed recently for interface problems and problems on irregular domains including CFD applications with free boundaries and moving interfaces. In an augmented method, one or several augmented variables are introduced along the interface or boundary so that one can get efficient discretizations. The augmented variables should be chosen such that the interface or boundary conditions are satisfied. The key to the success of the augmented methods often relies on the interpolation scheme to couple the augmented variables with the governing differential equations through the interface or boundary conditions. This has been done using a least squares interpolation (under-determined) for which the singular value decomposition (SVD) is used to solve for the interpolation coefficients. In this paper, based on properties of the finite element method, a new augmented immersed finite element method (IFEM) that does not need the interpolations is proposed for elliptic interface problems that have a piecewise constant coefficient. Thus the new augmented method is more efficient and simple than the old one that uses interpolations. The method then is extended to Poisson equations on irregular domains with a Dirichlet boundary condition. Numerical experiments with arbitrary interfaces/irregular domains and large jump ratios are provided to demonstrate the accuracy and the efficiency of the new augmented methods. Numerical results also show that the number of GMRES iterations is independent of the mesh size and nearly independent of the jump in the coefficient.     

Evaluation of mixed mode stress intensity factors for interface cracks using EFGM

This paper presents the implementation of element free Galerkin method for the stress analysis of structures having cracks at the interface of two dissimilar materials. The material discontinuity at the interface has been modeled using a jump function with a jump parameter that governs its strength. The jump function enriches the approximation by the addition of special shape function that contains discontinuities in the derivative. The trial and test functions of the weak form are constructed using moving least-square interpolants in each material domain. An intrinsic enrichment criterion with enriched basis has been used to model the crack tip stress fields. The mixed mode (complex) stress intensity factors for bi-material interface cracks are numerically evaluated using the modified domain form of interaction integral. The numerical results are obtained for edge and center cracks lying at the bi-material interface, and are found to be in good agreement with the reference solutions for the interfacial crack problems.