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.     

Slow flow past a heterogeneous porous sphere

Stokes’ flow past a heterogeneous porous sphere has been studied, adopting the boundary conditions modified by Jones (1973) for curved surfaces at the interface of the free fluid region and porous material. The porous sphere is made up ofn + 1 concentric spheres of different permeability. The results for drag experienced by the sphere has been discussed and the following cases of interest are deduced:

1. WhenK ₁=K ₂=...=K _n+1=K.
2. WhenK _i is very small for eachi.

     

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.     

Application of p-Adic Wavelets to Model Reaction–Diffusion Dynamics in Random Porous Media

Fourier and more generally wavelet analysis over the fields of p-adic numbers are widely used in physics, biology and cognitive science, and recently in geophysics. In this note we present a model of the reaction–diffusion dynamics in random porous media, e.g., flow of fluid (oil, water or emulsion) in a a complex network of pores with known topology. Anomalous diffusion in the model is represented by the system of two equations of reaction–diffusion type, for the part of fluid not bound to solid’s interface (e.g., free oil) and for the part bound to solid’s interface (e.g., solids–bound oil). Our model is based on the p-adic (treelike) representation of pore-networks. We present the system of two p-adic reaction–diffusion equations describing propagation of fluid in networks of pores in random media and find its stationary solutions by using theory of p-adic wavelets. The use of p-adic wavelets (generalizing classical wavelet theory) gives a possibility to find the stationary solution in the analytic form which is typically impossible for anomalous diffusion in the standard representation based on the real numbers.     

A second‐order backward difference time‐stepping scheme for penalized Navier–Stokes equations modeling filtration through porous media

We investigate the stability and convergence of a fully implicit, linearly extrapolated second‐order backward difference time‐stepping scheme for the penalized Navier–Stokes equations modeling filtration through porous media. In the penalization approach, an extended Navier–Stokes equation is used in the entire computational domain with suitable resistance terms to mimic the presence of porous medium. It is widely used as an alternative to the heterogeneous approach in which different types of partial differential equations (PDEs) are used in fluid and porous subregions along with suitable continuity conditions at the interface. However, the introduction of extra resistance terms makes the penalized Navier–Stokes equations more nonlinear. We prove that the linearly extrapolated scheme is unconditionally stable and derive optimal order error estimates without any stability condition. To show feasibility and applicability of the approach, it is used to numerically solve a passive control problem in which flow around a solid body is controlled by adding porous layers on the surface. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 681–705, 2016     

Uniform stabilization to equilibrium of a nonlinear fluid–structure interaction model

We consider uniform stability to a nontrivial equilibrium of a nonlinear fluid–structure interaction (FSI) defined on a two or three dimensional bounded domain. Stabilization is achieved via boundary and/or interior feedback controls implemented on both the fluid and the structure. The interior damping on the fluid combining with the viscosity effect stabilizes the dynamics of fluid. However, this dissipation propagated from the fluid alone is not sufficient to drive uniformly to equilibrium the entire coupled system. Therefore, additional interior damping on the wave component or boundary porous like damping on the interface is considered. A geometric condition on the interface is needed if only boundary damping on the wave is active. The main technical difficulty is the mismatch of regularity of hyperbolic and parabolic component of the coupled system. This is overcome by considering special multipliers constructed from Stokes solvers. The uniform stabilization result obtained in this article is global for the fully coupled FSI model.     

On a random scaled porous media equation

It is shown that a random scaled porous media equation arising from a stochastic porous media equation with linear multiplicative noise through a random transformation is well-posed in L^∞. In the fast diffusion case we show existence in Lp.     

A cellular automaton technique for modelling of a binary dendritic growth with convection

A two-dimensional model for the simulation of a binary dendritic growth with convection has been developed in order to investigate the effects of convection on dendritic morphologies. The model is based on a cellular automaton (CA) technique for the calculation of the evolution of solid/liquid (s/l) interface. The dynamics of the interface controlled by temperature, solute diffusion and Gibbs–Thomson effects, is coupled with the continuum model for energy, solute and momentum transfer with liquid convection. The solid fraction is calculated by a governing equation, instead of some approximate methods such as lever rule method [A. Jacot, M. Rappaz, Acta Mater. 50 (2002) 1909–1926.] or interface velocity method [L. Nastac, Acta Mater. 47 (1999) 4253; L. Beltran-Sanchez, D.M. Stefanescu, Mat. and Mat. Trans. A 26 (2003) 367.]. For the dendritic growth without convection, mesh independency of simulation results is achieved. The simulated steady-state tip velocity are compared with the predicted values of LGK theory [Lipton, M.E. Glicksmanm, W. Kurz, Metall. Trans. 18(A) (1987) 341.] as a function of melt undercooling, which shows good agreement. The growth of dendrite arms in a forced convection has been investigated. It was found that the dendritic growth in the upstream direction was amplified, due to larger solute gradient in the liquid ahead of the s/l interface caused by melt convection. In the isothermal environment, the calculated results under very fine mesh are in good agreement with the Oseen–Ivanstov solution for the concentration-driven growth in a forced flow.     

Some Self-Similar Two-Phase Flows of Non-Newtonian Fluids Through a Porous Medium

Pascal This paper addresses the question of the rheological effects of non-Newtonian fluids in a flow system, in which a two-phase flow zone is coupled to a single-phase flow zone by a moving fluid interface. This flow system is involved in a technique for oil displacement in a porous medium, where a non-Newtonian displacing fluid (a polymer solution) is used to displace a non-Newtonian heavy oil. The self-similar solutions of the equations governing the dynamics of the moving interface, separating the displacing and displaced fluids, are obtained for the one-dimensional and plane radial flows. The effects associated with the presence of a two-phase flow zone, behind the moving interface, on the interface movement are analyzed. The existence of a pressure front ahead of the moving interface, moving with a finite velocity, is also shown. The relevance of this result to the propagation of pressure disturbances in a non-Newtonian fluid flowing through a porous medium is discussed with regard to interpretation of the transient pressure response in an injection well for polymer-solution floods.     

Motion of a distant solid particle in a shear flow along a porous slab

The motion of a solid and no-slipping particle immersed in a shear flow along a sufficiently porous slab is investigated. The fluid flow outside and inside of the slab is governed by the Stokes and Darcy equations, respectively, and the so-called Beavers and Joseph slip boundary conditions are enforced on the slab surface. The problem is solved for a distant particle with length scale a in terms of the small parameter a/d where d designates the large particle–slab separation. This is achieved by asymptotically inverting a relevant boundary-integral equation on the particle surface, which has been recently proposed for any particle location (distant or close particle) in Khabthani et al. (J Fluid Mech 713:271–306, 2012). It is found that at order O(a/d) the slab behaves for any particle shape as a solid plane no-slip wall while the slab properties (thickness, permeability, associated slip length) solely enter at O((a/d)²). Moreover, for a spherical particle, the numerical results published in Khabthani et al. (J Fluid Mech 713:271–306, 2012) perfectly agree with the present asymptotic analysis.     

Isogeometric Analysis of a Phase Field Model for Darcy Flows with Discontinuous Data

The authors consider a phase field model for Darcy flows with discontinuous data in porous media; specifically,they adopt the Hele-Shaw-Cahn-Hillard equations of[Lee,Lowengrub,Goodman,Physics of Fluids,2002] to model flows in the Hele-Shaw cell through a phase field formulation which incorporates discontinuities of physical data,namely density and viscosity,across interfaces. For the spatial approximation of the problem,the authors use NURBS—based isogeometric analysis in the framework of the Galerkin method,a computational framework which is particularly advantageous for the solution of high order partial differential equations and phase field problems which exhibit sharp but smooth interfaces. In this paper,the authors verify through numerical tests the sharp interface limit of the phase field model which in fact leads to an internal discontinuity interface problem; finally,they show the efficiency of isogeometric analysis for the numerical approximation of the model by solving a benchmark problem,the so-called"rising bubble" problem.     

Waiting Time in a Two Station Series Queueing System: The Effect of Dependent Interarrival Times

An analysis taking into account the dependencies in the departure process from the first station of the M/E _k /1→./M/1 system is conducted. Arrivals to the second station are approximated as a general independent distribution, and the waiting times in this station are compared to those found through computer simulation.     

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.     

A Proof of the Convexity of the Free Boundary for Porous Flow Through a Rectangular Dam Using the Maximum Principle

We consider the steady two-dimensional flow under gravity ofwater from one reservoir (on the left) to a lower reservoir(on the right) through a porous rectangular isotropic homogeneousdam with impervious bottom. Because of gravity the water doesnot flow through the entire dam and the dam is dry near itsupper right corner. The interface separating the dry and wetregions of the dam is a free boundary. Recently, Friedman &Jensen (1977) have proved that the free boundary is convex.We give a different proof which uses only the maximum principleand its generalizations.     

An added-mass free semi-implicit coupling scheme for fluid–structure interaction

In this Note we propose a semi-implicit coupling scheme for the numerical simulation of fluid–structure interaction systems involving a viscous incompressible fluid. The scheme is stable irrespectively of the so-called added-mass effect and allows for conservative time-stepping within the structure. The efficiency of the scheme is based on the explicit splitting of the viscous effects and geometrical/convective non-linearities, through the use of the Chorin–Temam projection scheme within the fluid. Stability relies on the implicit treatment of the pressure stresses and on the Nitsche's treatment of the viscous coupling. To cite this article: M. Astorino et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Periodic solutions for the transport of a pollutant concentration in a porous medium

We study the existence and uniqueness of the periodic solution for a model which describes the transport of a pollutant in a porous medium, with periodic boundary conditions. The prove of existence, is based upon a topological tool as the Schauder fixed point theorem, applied to a PDE. The uniqueness is instead of obtained, assuming thats→d _i,j(x,? ^–1(s)) is Hölder continuous of order γ∈ ∈ (1/2, 1).     

Fractal dimension and complexity in the long-term dynamics of a monomolecular layer

The formation of three-dimensional domains in monomolecular layers (Nucleation Dynamics, ND) of four fatty acids, stearic, arachidic, behenic and lignoceric acids, containing the same carboxylic (–COOH) head and an alkyl chain with 18, 20, 22 and 24 carbon atoms, respectively, on water surface, has been studied through Specific Molecular Area (A) versus time (t) studies from Surface Pressure (π)-A isotherms and Brewster Angle Microscopy (BAM). To investigate the fractal nature, the gray-scale Brewster angle micrographs are converted to binary images containing only two pixel values – 0 for 2D phase and 255 for 3D phase before box-counting method is employed to compute the fractal dimension. The 3D phase in the background of 2D phase is found to be fractal in nature. In fact, 3D phase is an interpenetration of two fractal structures with two different fractal dimensions – one corresponding to smaller (intra-domain) structures and other corresponding to larger (inter-domain) structures. These fractal dimensions are seen to evolve with coverage as 3D phase grows. The two fractal dimensions and their evolution in ND dynamics are identical for the longest-tailed lignoceric acid which has a porous 3D phase.     

Existence of solutions for a finite nonlinearly hyperelastic rod

A theoretical investigation of a mathematical model for the capillary-tissue fluid exchange, including the characteristics and influence of the boundaries and media through which the fluid flows, has been studied. Filtration from the cylindrical capillary into the concentrically surrounding tissue-space and flow from a capillary into the tissue across the thin membrane are analyzed in detail. It has been observed that the filtration efficiency of the functional unit decreases as the peripherallayer viscosity increases, and that contrary to the results of Apelblat, Katziv-Kutchalsky and Silborberg (Biorheology2 (1974), 1–49), the slip velocity plays dominant role on filtration efficiency. It is also noted that he filtration efficiency decreases as the slip velocity at the porous boundary increases.     

Customized sequential designs for random simulation experiments: Kriging metamodeling and bootstrapping

This paper proposes a novel method to select an experimental design for interpolation in random simulation, especially discrete event simulation. (Though the paper focuses on Kriging, this design approach may also apply to other types of metamodels such as non-linear regression models and splines.) Assuming that simulation requires much computer time, it is important to select a design with a small number of observations (or simulation runs). The proposed method is therefore sequential. Its novelty is that it accounts for the specific input/output behavior (or response function) of the particular simulation at hand; i.e., the method is customized or application-driven. A tool for this customization is bootstrapping, which enables the estimation of the variances of predictions for inputs not yet simulated. The method is tested through two classic simulation models, namely the expected steady-state waiting time of the M/M/1 queuing model, and the mean costs of a terminating (s, S) inventory simulation. For these two simulation models the novel design indeed gives better results than a popular alternative design, namely Latin Hypercube Sampling (LHS) with a prefixed sample.     

Domain decomposition for a finite volume method on non-matching grids

We are interested in a robust and accurate domain decomposition method with Robin interface conditions on non-matching grids using a finite volume discretization. We introduce transmission operators on the non-matching grids and define new interface conditions of Robin type. Under a compatibility assumption, we show the equivalence between Robin interface conditions and Dirichlet–Neumann interface conditions and the well-posedness of the global and local problems. Two error estimates are given in terms of the discrete H¹-norm: one in O(h^1/2) with operators based on piecewise constant functions and the other in O(h) (as in the conforming case) with operators using a linear rebuilding. Numerical results are given. To cite this article: L. Saas et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).