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21.
In this study, we use the method of homogenization to develop a filtration law in porous media that includes the effects of
inertia at finite Reynolds numbers. The result is much different than the empirically observed quadratic Forchheimer equation.
First, the correction to Darcy’s law is initially cubic (not quadratic) for isotropic media. This is consistent with several
other authors (Mei and Auriault, J Fluid Mech 222:647–663, 1991; Wodié and Levy, CR Acad Sci Paris t.312:157–161, 1991; Couland
et al. J Fluid Mech 190:393–407, 1988; Rojas and Koplik, Phys Rev 58:4776–4782, 1988) who have solved the Navier–Stokes equations
analytically and numerically. Second, the resulting filtration model is an infinite series polynomial in velocity, instead
of a single corrective term to Darcy’s law. Although the model is only valid up to the local Reynolds number, at the most,
of order 1, the findings are important from a fundamental perspective because it shows that the often-used quadratic Forchheimer
equation is not a universal law for laminar flow, but rather an empirical one that is useful in a limited range of velocities.
Moreover, as stated by Mei and Auriault (J Fluid Mech 222:647–663, 1991) and Barree and Conway (SPE Annual technical conference
and exhibition, 2004), even if the quadratic model were valid at moderate Reynolds numbers in the laminar flow regime, then
the permeability extrapolated on a Forchheimer plot would not be the intrinsic Darcy permeability. A major contribution of
this study is that the coefficients of the polynomial law can be derived a priori, by solving sequential Stokes problems.
In each case, the solution to the Stokes problem is used to calculate a coefficient in the polynomial, and the velocity field
is an input of the forcing function, F, to subsequent problems. While numerical solutions must be utilized to compute each coefficient in the polynomial, these
problems are much simpler and robust than solving the full Navier–Stokes equations. 相似文献
22.
Thierry Clopeau Angiolo Farina Antonio Fasano Andro Mikelić 《Nonlinear Analysis: Real World Applications》2010,11(6):4533-4545
In this article, we study mathematical modeling of thermal drawing of glass fibers. We give a derivation of the effective model from the generalized Oberbeck–Boussinesq equations with free boundary, using singular perturbation expansion. We generalize earlier approaches by taking the isochoric compressible model, with density depending on the temperature, and we handle correctly the viscosity, which changes over several orders of magnitude. For the obtained effective system of nonlinear differential equations, we prove the existence of a stationary solution for the boundary value problem. We impose only physically realistic assumptions on the data (viscosity taking large values with cooling). Finally we present numerical simulations with realistic data. 相似文献
23.
Andro Mikeli? 《Journal of Differential Equations》2010,248(6):1561-1577
In this paper we investigate the pseudoparabolic equation
24.
The duality theory of Mosco, Capuzzo-Dolcetta, and Matzeu for variational and quasi-variational inequalities is extended. Then it is applied to two problems of contact with friction of an elastic body with a rigid foundation. The more realistic normal compliance condition is used in place of Signorini's condition on the contact surface.We (A.M. and M.S.) are grateful for the financial support and hospitality of the Department of Mechanical Engineering in the Linköping Institute for Technology during our two weeks stay in August 1988. In addition M.S. was partially supported by the Oakland University Faculty Research Award. 相似文献
25.
We consider the evolution of a reactive soluble substance introduced into the Poiseuille flow in a slit channel. The reactive transport happens in presence of dominant Péclet and Damköhler numbers. We suppose Péclet numbers corresponding to Taylor’s dispersion regime. The two main results of the paper are the following. First, using the anisotropic perturbation technique, we derive rigorously an effective model for the enhanced diffusion. It contains memory effects and contributions to the effective diffusion and effective advection velocity, due to the flow and chemistry reaction regime. Error estimates for the approximation of the physical solution by the upscaled one are presented in the energy norms. Presence of an initial time boundary layer allows only a global error estimate in L 2 with respect to space and time. We use the Laplace’s transform in time to get optimal estimates. Second, we explicit the retardation and memory effects of the adsorption/desorption reactions on the dispersive characteristics and show their importance. The chemistry influences directly the characteristic diffusion width. 相似文献
26.
In this chapter of the special issue of the journal “Transport in Porous Media,” on the topic “Flow and transport above permeable
domains,” we present modeling of flow and transport above permeable domains using the homogenization method. Our goal is to
develop a heuristic approach which can be used by the engineering community for treating this type of problems and which has
a solid mathematical background. The rigorous mathematical justification of the presented results is given in the corresponding
articles of the authors. The plan is as follows: We start with the section “Introduction” where we give an overview and comparison
with interface conditions obtained using other approaches. In Sect. 2, we give a very short derivation of the Darcy law by
homogenization, using the two-scale expansion in the typical pore size parameter ε. It gives us the definition of various auxiliary functions and typical effective properties as permeability. In Sect. 3,
we introduce our approach to the effective interface laws on a simple 1D example. The approximation is obtained heuristically
using the two steps strategy. For the 1D problem we calculate the approximation and the effective interface law explicitly
and show that it is valid at order O(ε
2). Next, in Sect. 4 we give a derivation of the Beavers–Joseph–Saffman interface condition and of the pressure jump condition,
using homogenization. We construct the corresponding boundary layer and present a heuristic calculation, leading to the interface
law and being based on the rigorous mathematical result. In addition, we show the invariance of the law with respect to the
small variations in the choice of the interface position. Finally, there is a short concluding section.
The research of A.M. was partially supported by the GDR MOMAS (Modélisation Mathématique et Simulations numériques liées aux
problèmes de gestion des déchets nucléaires) (PACEN/CNRS, ANDRA, BRGM, CEA, EDF, IRSN). 相似文献
27.
Oleg Boiarkine Dmitri Kuzmin Sunčica Čanić Giovanna Guidoboni Andro Mikelić 《Journal of computational physics》2011,230(8):2896-2914
A new high-resolution scheme is developed for convection–diffusion problems in domains with moving boundaries. A finite element approximation of the governing equation is designed within the framework of a conservative Arbitrary Lagrangian Eulerian (ALE) formulation. An implicit flux-corrected transport (FCT) algorithm is implemented to suppress spurious undershoots and overshoots appearing in convection-dominated problems. A detailed numerical study is performed for P1 finite element discretizations on fixed and moving meshes. Simulation results for a Taylor dispersion problem (moderate Peclet numbers) and for a convection-dominated problem (large Peclet numbers) are presented to give a flavor of practical applications. 相似文献
28.
29.
Andro Mikelić 《Annali dell'Universita di Ferrara》2007,53(1):95-106
Abstract We consider the isentropic compressible flow through a tiny pore. Our approach is to adapt the recent results by N. Masmoudi
on the homogenization of compressible flows through porous media to our situation. The major difference is in the a priori
estimates for the pressure field. We derive the appropriate ones and then Masmoudi’s results allow to conclude the convergence.
In this way the compressible Reynolds equation in the lubrication theory is rigorously justified.
Keywords: Compressible Navier-Stokes equations, Lubrication, Pressure estimates
Mathematics Subject Classification (2000): 35B27, 76M50, 35D05 相似文献
30.
We investigate in this paper existence of a weak solution for a stationary incompressible Navier-Stokes system with non-linear viscosity and with non-homogeneous boundary conditions for velocity on the boundary. Our concern is with the viscosity obeying the power-law dependence ν(ξ) = ∣Tr(ξξ*)∣r/2?1, r < 2, on shear stress ξ. It is corresponding to most quasi-Newtonian flows with injection on the boundary. Since for r ? 2 the inertial term precludes any a priori estimate in general, we suppose the Reynolds number is not too large. Using the specific algebraic structure of the Navier-Stokes system we prove existence of at least one approximate solution. The constructed approximate solution turns out to be uniformly bounded in W1,r (Omega;)n and using monotonicity and compactness we successfully pass to the limit for r ≥ 3n/(n + 2). For 3n/(n + 2) > r > 2n/(n + 2) our construction gives existence of at least one very weak solution. Furthermore, for r ≥ 3n/(n + 2) we prove that all weak solutions lying in the ball in W of radius smaller than critical are equal. Finally, we obtain an existence result for the flow through a thin slab. 相似文献