Two-Phase Inertial Flow in Homogeneous Porous Media: A Theoretical Derivation of a Macroscopic Model

The purpose of this article is to derive a macroscopic model for a certain class of inertial two-phase, incompressible, Newtonian fluid flow through homogenous porous media. Starting from the continuity and Navier–Stokes equations in each phase β and γ, the method of volume averaging is employed subjected to constraints that are explicitly provided to obtain the macroscopic mass and momentum balance equations. These constraints are on the length- and time-scales, as well as, on some quantities involving capillary, Weber and Reynolds numbers that define the class of two-phase flow under consideration. The resulting macroscopic momentum equation relates the phase-averaged pressure gradient to the filtration or Darcy velocity in a coupled nonlinear form explicitly given by

Homogenization of the nonlinear Kelvin–Voigt model of viscoelasticity and of the Prager model of plasticity

Denoting by the stress tensor, by the linearized strain tensor, by A the elasticity tensor, and assuming that is a convex potential, the inclusion accounts for nonlinear viscoelasticity, and encompasses both the linear Kelvin–Voigt model of solid-type viscoelasticity and the Prager model of rigid plasticity with linear kinematic strain-hardening. This relation is assumed to represent the constitutive behavior of a space-distributed system, and is here coupled with the dynamical equation. An initial- and boundary-value problem is formulated, and the existence and uniqueness of the solution are proved via classical techniques based on compactness and monotonicity. A composite material is then considered, in which the function and the tensor A rapidly oscillate in space. A two-scale model is derived via Nguetseng’s notion of two-scale convergence. This provides a detailed account of the mesoscopic state of the system. Any dependence on the fine-scale variable is then eliminated, and the existence of a solution of a new single-scale macroscopic model is proved. The final outcome is at variance with the nonlinear extension of the generalized Kelvin–Voigt model, which is based on an apparently unjustified mean-field-type hypothesis.     

A Jordan Curve Spanned by a Complete Minimal Surface

In this paper we construct complete (conformal) minimal immersions which admit continuous extensions to the closed disk, . Moreover, is a homeomorphism and is a (non-rectifiable) Jordan curve with Hausdorff dimension 1. It turns out that the set of Jordan curves constructed by the above procedure is dense in the space of Jordan curves of with the Hausdorff metric.     

A Helmholtz/Weyl Decomposition in Energy Space

For a bounded region in a Helmholtz/Weyl decomposition of the Sobolev space is given,with orthogonality with respect to the strain-energy inner product of elasticity (anisotropic or isotropic).     

Explicit Effective Constants for an Inhomogeneous Porothermoelastic Medium

An arbitrary anisotropic micro-inhomogeneous (composite) poroelastic medium is considered, containing a random set of ellipsoidal inhomogeneities with different poroelastic characteristics. The properties of these constituents are described by the linear porothermoelastic theory of Biot. One of the self-consistent schemes named effective field method is used to develop explicit expressions for the effective porothermoelastic constants (tensor of the frame elastic compliances , tensor of the generalized Skempton’s coefficients , tensor of thermal expansion coefficients , Biot’s constants , and the heat capacity at constant stress for the static porothermoelastic theory. It is shown that for two components composite porous material these expressions are interconnected and can be expressed only via the components of tensor . Some special cases are considered for the isotropic main material (matrix).     

A Numerical Study of the Applicability of the Shear Compression Specimen to Parabolic Hardening Materials

This paper addresses quasi-static loading of the shear compression specimen (SCS), that has been especially developed to investigate the shear response of materials at various strain rates. Previous work [4, 5] addressed bi-linear hardening materials, whereas the present work concerns parabolic hardening materials. The investigation is done numerically using three-dimensional elastoplastic finite element simulations. The analyses show that the averaged von Mises stress ( ) and strain ( ) on a mid-section of the gauge reflect accurately the prescribed parabolic hardening model. A method for finding the parabolic hardening coefficients and reducing the measured load, P, and displacement, d, into equivalent stress and strain is introduced and tested. A very good agreement is observed, thus confirming the potential of the technique for large strain testing of parabolic hardening materials.     

The Semigeostrophic Equations Discretized in Reference and Dual Variables

We study the evolution of a system of n particles in . That system is a conservative system with a Hamiltonian of the form , where W ₂ is the Wasserstein distance and μ is a discrete measure concentrated on the set . Typically, μ(0) is a discrete measure approximating an initial L ^∞ density and can be chosen randomly. When d  =  1, our results prove convergence of the discrete system to a variant of the semigeostrophic equations. We obtain that the limiting densities are absolutely continuous with respect to the Lebesgue measure. When converges to a measure concentrated on a special d–dimensional set, we obtain the Vlasov–Monge–Ampère (VMA) system. When, d = 1 the VMA system coincides with the standard Vlasov–Poisson system.     

Onset of Buoyancy-driven Instability in Porous Medium Solidified from Above

When a porous melt layer saturated by liquid is solidified from above, convection often sets in due to buoyancy forces. In this study, the onset of buoyancy-driven convection during time-dependent solidification is investigated by using the similarly transformed disturbance equations. The thermal disturbance distribution of the solid phase is approximated by the WKB method and effects of various parameters on the stability condition of the melt phase are analyzed theoretically. For the limiting case of λ → 0 and finite k _r, the critical conditions approach asymptotically and . This study presenting a constant-temperature cooling model predicts greater instability and gives more unstable results than those obtained from the constant solidification rate model.     

Shock wave kinetics of Fe + NO based on Fe, O, and N atom measurements

The reaction of Fe atoms with NO was studied behind incident shock waves in the temperature range of 780–1,020 K at pressures between 0.3 and 1.2 bar. Atomic-resonance-absorption spectroscopy (ARAS) was applied for the time-resolved measurement of Fe , N, and O atoms in gas mixtures containing Fe(CO)₅ and NO, highly diluted in argon. The experiments showed a Fe-atom consumption without an associated O- or N-atom formation which can be explained by a recombination of Fe and NO:

The Toda System and Clustering Interfaces in the Allen–Cahn equation

We consider the Allen–Cahn equation in a bounded, smooth domain Ω in , under zero Neumann boundary conditions, where is a small parameter. Let Γ₀ be a segment contained in Ω, connecting orthogonally the boundary. Under certain nondegeneracy and nonminimality assumptions for Γ₀, satisfied for instance by the short axis in an ellipse, we construct, for any given N ≥ 1, a solution exhibiting N transition layers whose mutual distances are and which collapse onto Γ₀ as . Asymptotic location of these interfaces is governed by a Toda-type system and yields in the limit broken lines with an angle at a common height and at main order cutting orthogonally the boundary.     

The MSF model: relation of nonlinear parameters to molecular structure of long-chain branched polymer melts

The elongational viscosity data of model PS combs (Hepperle J, Einfluss der Molekularen Struktur auf Rheologische Eigenschaften von Polystyrol- und Polycarbonatschmelzen. Doctoral Thesis, University Erlangen-Nürnberg, 2003) are reconsidered by including the interchain pressure term of Marrucci and Ianniruberto [Macromolecules 37:3934–3942, 2004] in the Molecular Stress Function model [Wagner et al., J Rheol 47(3):779–793, 2003, Wagner et al., J Rheol 49:1317–1327, 2005d]. Two nonlinear model parameters are needed to describe elongational flow, β and . The parameterβ determines the slope of the elongational viscosity after the inception of strain hardening. It is directly related to the molecular structure of the polymer and represents the ratio of the molar mass of the (branched) polymer to the molar mass of the backbone alone. β follows from the hypothesis of Wagner et al. [J Rheol 47(3):779–793, 2003] that side chains are compressed onto the backbone. We consider also the case that side chains are oriented by deformation, but not stretched, and found little difference in the model predictions. The parameter represents the maximum strain energy stored in the polymeric system and determines the steady-state value of the viscosity in extensional flows. The relation of this energy parameter to the molecular structure is discussed. Good correlations between the energy parameter and different coil contraction ratios, as determined either experimentally or calculated theoretically by considering the topology of the macromolecule, are found. The smaller the relative size of the polymer coil, the larger is the energy parameter and the more strain energy can be stored in the polymeric system. Presented at the 3rd Annual European Rheology Conference, AERC2006, Crete, Greece.     

Counterflow flames of air and methane,propane and ethylene,with and without periodic forcing

The extinction of forced and unforced turbulent premixed counterflow flames has been quantified with lean mixtures of air and each of methane, propane and ethylene. Symmetric flames were produced with two streams of equal equivalence ratios between 0.6 and 1.0, and nozzle separations from 0.2 to 2.5 D, while acoustic drivers were used to force the flow at discrete frequencies. Photographs confirmed visual observation of unforced twin flames and their merging with increasing strain rate into one reaction zone at the stagnation plane before extinction. Propane flames merged at velocities closer to the extinction limit. At separations less than 0.4 D local quenching and extinction and relight occurred at equivalence ratios less than 0.7, independent of fuel type. Unforced extinction times were determined by igniting mixtures with equivalence ratios of 0.6 to 0.9 and bulk velocities above the extinction limit, and observing the extinction process with high-speed video: they were found to increase quasi-exponentially with reduction in strain rate, and were strongly dependent on equivalence ratio and fuel type. Forced extinction times also increased with decrease in strain rate and with reduction in forcing amplitude and instantaneous strain rates greater than the unforced limit were observed. Ethylene flames were more sensitive to the cyclic weakening with more rapid temperature decay rates and shorter extinction times.Abbreviations f Forcing frequency (Hz) - H Nozzle separation (m) - D Nozzle diameter (m) - Bulk strain rate, 2U _b/H, (s^-1) - Bulk strain rate at extinction (s^-1) - Maximum instantaneous forced strain rate (s^-1) - Maximum instantaneous unforced strain rate (s^-1) - Forcing time to extinction (s) - Time of one period of forcing oscillation (s) - Bulk velocity, flow rate/nozzle exit area (ms^-1) - Bulk velocity at extinction (ms^-1) - u Fluctuating component of turbulent velocity (ms^-1) - Fluctuating component of forced velocity (ms^-1) - Equivalence ratio (dimensionless)     

Fluid flow and heat transfer in microchannels with rectangular cross section

Forced convective heat transfer coefficients and friction factors for flow of water in microchannels with a rectangular cross section were measured. An integrated microsystem consisting of five microchannels on one side and a localized heater and seven polysilicon temperature sensors along the selected channels on the other side was fabricated using a double-polished-prime silicon wafer. For the microchannels tested, the friction factor constant obtained are values between 53.7 and 60.4, which are close to the theoretical value from a correlation for macroscopic dimension, 56.9 for D _h  = 100 μm. The heat transfer coefficients obtained by measuring the wall temperature along the micro channels were linearly dependent on the wall temperature, in turn, the heat transfer mechanism is strongly dependent on the fluid properties such as viscosity. The measured Nusselt number in the laminar flow regime tested could be correlated by which is quite different from the constant value obtained in macrochannels.     

Role of the Pressure for Validity of the Energy Equality for Solutions of the Navier–Stokes Equation

We prove that a weak solution of the Navier–Stokes system satisfies the energy equality if the associated pressure is locally square integrable. A similar statement is shown to hold for the Euler system.     

Lagrangian Approach to the Study of Level Sets: Application to a Free Boundary Problem in Climatology

We study the dynamics and regularity of level sets in solutions of the semilinear parabolic equation

Non-autonomous 2D Navier–Stokes System with Singularly Oscillating External Force and its Global Attractor

We study the global attractor of the non-autonomous 2D Navier–Stokes (N.–S.) system with singularly oscillating external force of the form . If the functions g ₀(x, t) and g ₁ (z, t) are translation bounded in the corresponding spaces, then it is known that the global attractor is bounded in the space H, however, its norm may be unbounded as since the magnitude of the external force is growing. Assuming that the function g ₁ (z, t) has a divergence representation of the form where the functions (see Section 3), we prove that the global attractors of the N.–S. equations are uniformly bounded with respect to for all . We also consider the “limiting” 2D N.–S. system with external force g ₀(x, t). We have found an estimate for the deviation of a solution of the original N.–S. system from a solution u ⁰(x, t) of the “limiting” N.–S. system with the same initial data. If the function g ₁ (z, t) admits the divergence representation, the functions g ₀(x, t) and g ₁ (z, t) are translation compact in the corresponding spaces, and , then we prove that the global attractors converges to the global attractor of the “limiting” system as in the norm of H. In the last section, we present an estimate for the Hausdorff deviation of from of the form: in the case, when the global attractor is exponential (the Grashof number of the “limiting” 2D N.–S. system is small).      

Numerical Study of Natural Convection Heat Transfer in an Inclined Porous Cavity with Time-Periodic Boundary Conditions

Kalabin et al. (Numer. Heat Transfer A 47, 621-631, 2005) studied the unsteady natural convection for the sinusoidal oscillating wall temperature on one side wall and constant average temperature on the opposing side wall. The present article is on the unsteady natural convective heat transfer in an inclined porous cavity with similar temperature boundary conditions as those of Kalabin et al. The inclined angle of the cavity is varied from 0° to 80°. The flow field is modeled with the Brinkman-extended Darcy model. The combined effects of inclination angle of the enclosure and oscillation frequency of wall temperature are studied for Ra* = 10³, Da = 10⁻³, , and Pr=1. Some results are also obtained with the Darcy–Brinkman–Forchheimer model and Darcy’s law and are compared with the present Brinkman-extended Darcy model. The maximal heat transfer rate is attained at the oscillating frequency f = 46.7π and the inclined angle .     

Rarefied Pure Gas Transport in Non-isothermal Porous Media: Validation and Tests of the Model

Viscous flow, effusion, and thermal transpiration are the main gas transport modalities for a rarefied gas in a macro-porous medium. They have been well quantified only in the case of simple geometries. This paper presents a numerical method based on the homogenization of kinetic equations producing effective transport properties (permeability, Knudsen diffusivity, thermal transpiration ratio) in any porous medium sample, as described, e.g. by a digitized 3D image. The homogenization procedure—neglecting the effect of gas density gradients on heat transfer through the solid—leads to closure problems in for the obtention of effective properties; they are then simplified using a Galerkin method based on a 21-element basis set. The kinetic equations are then discretized in space with a finite- volume scheme. The method is validated against experimental data in the case of a closed test tube. It shows to be coherent with past approaches of thermal transpiration. Then, it is applied to several 3D images of increasing complexity. Another validation is brought by comparison with other distinct numerical approaches for the evaluation of the Darcian permeability tensor and of the Knudsen diffusion tensor. Results show that thermal transpiration has to be described by an effective transport tensor which is distinct from the other tensors.