Inertia Effects in High-Rate Flow Through Heterogeneous Porous Media

The paper deals with the effects of large scale permeability–heterogeneity on flows at high velocities through porous media. The media is made of a large number of homogeneous blocks where the flow is assumed to be governed by the Forchheimer equation with a constant inertial coefficient. By assuming the validity of the Forchheimer equation at the large scale, an effective inertial coefficient is deduced from numerical simulations. Different media are investigated: serial-layers, parallel-layers and correlated media. The numerical results show that: (i) for the serial-layers, the effective inertial coefficient is independent of the Reynolds number and decreases when the variance and the mean permeability ratio increases; (ii) for the parallel-layers and the correlated media, the effective inertial coefficient is function of the Reynolds number and increases when the variance and the mean permeability ratio increases. Theoretical relationships are proposed for the inertial coefficient as function of the Reynolds number and the characteristics of the media.     

The Devil’s Staircase Dimensions and Measure-Theoretical Entropy of Maps with Horizontal Gap

This work elucidates the measure-theoretical entropy and dimensions of a unimodal map with a horizontal gap. The measure-theoretical entropy and dimensions of the F_t (which is defined later) are shown to form a devils staircase structure with respect to the gap size t. Pesins formula for gap maps is also considered.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.     

Iterative Process for Certain Nonlinear Mappings with Lipschitz Condition

1　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＰｒｅｌｉｍｉｎａｒｉｅｓＴｈｒｏｕｇｈｏｕｔｔｈｉｓｐａｐｅｒ,ｗｅａｓｓｕｍｅｔｈａｔＸｉｓａｒｅａｌＢａｎａｃｈｓｐａｃｅａｎｄＸ ｉｓｔｈｅｄｕａｌｓｐａｃｅｏｆＸ ,〈· ,·〉ｄｅｎｏｔｅｓｔｈｅｐａｉｒｉｎｇｏｆＸａｎｄＸ .ＴｈｅｍａｐｐｉｎｇＪ:Ｘ → 2 Ｘ ｄｅｆｉｎｅｄｂｙＪ(ｘ) =ｊ∈Ｘ :〈ｘ ,ｊ〉=‖ｘ‖·‖ｊ‖ ,‖ｊ‖ =‖ｘ‖ ,　　ｘ∈Ｘｉｓｃａｌｌｅｄｔｈｅｎｏｒｍａｌｉｚｅｄｄｕａｌｉｔｙｍａｐｐｉｎｇ .Ｄｅｆｉｎｉｔｉｏｎ 1 1　ＬｅｔＸｂｅａ…     

A correlation for yield stress fluid flow through packed beds

Relatively few correlations are available for non-Newtonian fluid flows through packed beds, even though such fluids are frequently used in industry. In this paper, a correlation is presented for yield stress fluid flow through packed beds. The correlation is developed by introducing the yield stress model in place of the Newtonian model used in deriving Erguns equation. The resulting model has three parameters that are functions of the geometry and roughness of the particle surfaces. Two of the parameters can be deduced in the limit as the yield stress becomes negligible and the model reduces to Erguns equation for Newtonian fluids. The third model parameter is determined from experimental data. The correlation relates a defined friction factor to the dimensionless Reynolds and Hedstrom numbers and can be used to predict pressure drop for flow of a yield stress fluid through a packed bed of spherical particles. Conditions for flow or no-flow are also determined in the correlation. Comparison of model calculations, between a Newtonian and a yield stress fluid for flow penetration into a packed bed of spheres, shows the yield stress fluid initially performs similar to the Newtonian fluid, at large Reynolds numbers. At lower Reynolds numbers the yield stress effect becomes important and the flow rate significantly decreases when compared to the Newtonian fluid.     

Numerical Experiments in Scalar Diffusion under Stochastic Convection

We study scalar diffusion, both from Eulerian and Lagrangian perspectives, advected by two dimensional flows. Emphasis is devoted to the problem of scalar diffusion under a synthetic turbulent flow. We present numerical and analytical results for the turbulent diffusion coefficient either from the influence of the turbulent synthetic field and a periodic array of eddies. Preliminary results concerning Lagrangian dispersion are also reported.     

Ši’lnikov Chaos in the Generalized Lorenz Canonical Form of Dynamical Systems

This paper studies the generalized Lorenz canonical form of dynamical systems introduced by elikovský and Chen [International Journal of Bifurcation and Chaos 12(8), 2002, 1789]. It proves the existence of a heteroclinic orbit of the canonical form and the convergence of the corresponding series expansion. The ilnikov criterion along with some technical conditions guarantee that the canonical form has Smale horseshoes and horseshoe chaos. As a consequence, it also proves that both the classical Lorenz system and the Chen system have ilnikov chaos. When the system is changed into another ordinary differential equation through a nonsingular one-parameter linear transformation, the exact range of existence of ilnikov chaos with respect to the parameter can be specified. Numerical simulation verifies the theoretical results and analysis.     

On Ericksen’s Theorem for Unconstrained Hyperelastic Materials

After reviewing the proof of Ericksens theorem for the set of unconstrained, homogeneous, isotropic, hyperelastic materials, Ericksens result is sharpened to cover the subset of materials that possess a natural configuration and satisfy the empirical inequalities. Mathematics Subject Classifications (2000) 74B20.     

Symbolic Sequences Generated by a Special Class of Discrete Systems With Discontinuity and Input

In this paper, we study the symbolic sequences generated by aclass of discrete systems. This class contains the double-modulators, a typical example of discretetime electronic systems with discontinuity and input. First we develop ageneral theory and then we apply it to some examples in order to obtainsets of inadmissible sequences.     

The modified Casson＇s equation and its application to pipe flows of shear-thickening fluid

A few additional data from our previous experiments were plotted to emphasize the shear-thickening behavior of deoxy sickle erythrocyte (SS) suspension. A constitutive equation (named as FX equation) was developed and applied to a cylindrical pipe flow of a shear-thickening fluid. A blunt velocity profile and its volume flow rate were calculated. The flow was non-viscous (potential) in the central part of the pipe (i.e. the central core or the central plug-flow), and became more and more viscous towards the wall of the pipe after a specific radial distance, which was determined by a critical shear rate of (named as Fungs shear rate). Furthermore, combining the FX equation with the original Cassons equation, the author obtained a modified Cassons equation by introducing .The English text was polished by Yunming Chen.     

“Thermal layer” effect in MHD: Interaction of a parallel shock with a layer of decreased density

Interaction of a parallel fast MHD shock with a layer of decreased density is discussed using ideal MHD approach. This is an extrapolation of gas dynamic thermal layer effect on ideal MHD. Computer simulations show that a magnetic field of a moderate intensity ( 1) may change the character of the flow for intermediate Mach numbers (M 5) and a new raking regime may occur which is not observed in the absence of a magnetic field. Self similar precursor analogous to that in gas dynamics may develop in the case of highM and low density in the layer but magnetic forces essentially decrease its growth rate. This problem appears in connection with cosmical shock propagation where planetary magnetic tails play the role of the thermal layer, and it may also be observed in the laboratory when the shock is strong enough to heat the walls ahead of it.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

The Experimental Approach to Effective Stress Law of Coal Mass by Effect of Methane

Effective stress law of all kinds of coal samples, including steam coal, fat coal, corking coal, thin coal and anthracite, under pore pressure of gas, is experimentally studied using a newly developed test machine. These samples are taken from Coal Mines in Wuda, Hebi, Yanzhou, Yangquan, Qingshui, and Gujiao in China. The experiment results show that, under pore pressure of gas, the tested coal samples comply with Biots effective stress law, where the Biots coefficient is not a constant, and is bilinear function of volumetric stress () and pore pressure (p), that is, We define four areas according to the numerical feature of , that is, functionless area of pore pressure, normal function area, fracturing function area, and quasi-soil function area. The effective stress law of coal mass introduced by this paper is a constitutive equation in the study of coupled solid and fluid. This has significance in the drainage and outburst of methane in coal seam.     

Nonaqueous Phase Liquid Dissolution in Porous Media: Current State of Knowledge and Research Needs

Our understanding of nonaqueous phase liquid (NAPL) dissolution in the subsurface environment has been increasing rapidly over the past decade. This knowledge has provided the basis for recent developments in the area of NAPL recovery, including cosolvent and surfactant flushing. Despite these advances toward feasible remediation technologies, there remain a number of unresolved issues to motivate environmental researchers in this area. For example, the lack of an effective NAPLlocation methodology precludes effective deployment of NAPL recovery technologies. The objectives of this paper are to critically review the state of knowledge in the area of stationary NAPL dissolution in porous media and to identify specific research needs. The review first compares NAPL dissolutionbased mass transfer correlations reported for environmental systems with more fundamental results from the literature involving model systems. This comparison suggests that our current understanding of NAPL dissolution in smallscale (on the order of cm) systems is reasonably consistent with fundamental mass transfer theory. The discussion then expands to encompass several issues currently under investigation in NAPL dissolution research, including: characterizing NAPL morphology (i.e. effective size and surface area); multicomponent mixtures; scale-related issues (dispersion, flow by-passing); locating NAPL in the subsurface and enhanced NAPL recovery. Research needs and potential approaches are discussed throughout the paper. This review supports the following conclusions: (1) Our knowledge related to local dissolution and remediation issues is maturing, but should be brought to closure with respect to the link between NAPL emplacement theory (as it impacts NAPL morphology) and NAPL dissolution; (2) The role of nonideal NAPL mixtures, and intra-NAPL mass transfer processes must be clarified; (3) Valid models for quantifying and designing NAPL recovery schemes with chemical additives need to be refined with respect to chemical equilibria, mass transfer and chemical delivery issues; (4) Computational and large-scale experimental studies should begin to address parameter up-scaling issues in support of model application at the field scale; and (5) Inverse modeling efforts aimed at exploiting the previous developments should be expanded to support field-scale characterization of NAPL location and strength as a dissolving source.     

Stochastic Response of Offshore Structures via Statistical Cubicization

This paper deals with the stochastic response of single-degree-of-freedom structures with polynomial restoring force excited by random Morisons forces with mean current. The problem is recast by expressing the excitation by means of a cubic polynomial of the wave elevation, which in turn is assumed as a stationary zero-mean Gaussian process, whose spectral density is given by the output of a cascade of two second order linear filters having a Gaussian white noise as primary excitation. Thus, Itôs stochastic differential calculus becomes applicable, and the solution is pursued with a moment equation approach by using a suitable closure scheme. The results of the applications compare well with digital simulation.     

Upscaling in Subsurface Transport Using Cluster Statistics of Percolation

Transport/flow problems in soils have been treated in random resistor network representations (RRNs). Two lines of argument can be used to justify such a representation. Solute transport at the pore-space level may probably be treated using a system of linear, first-order differential equations describing inter-pore probability fluxes. This equation is equivalent to a random impedance network representation. Alternatively, Darcys law with spatially variable hydraulic conductivity is equivalent to an RRN. Darcys law for the hydraulic conductivity is applicable at sufficiently low pressure head in saturated soils, but only for steady-state flow in unsaturated soils. The result given here will have two contributions, one of which is universal to any linear conductance problem, i.e., requires only the applicability of Darcys (or Ohms) law. The second contribution depends on the actual distribution of linear conductances appropriate. Although nonlinear effects in RRNs (including changes in resistance values resulting from current, analogous to changes in matric potential resulting from flow) have been treated within the framework of percolation theory, the theoretical development lags the corresponding development of the linear theory, which is, in principle, on a solid foundation. In practice, calculations of the nonlinear conductivity in relatively (compared with soils) well characterized solid-state systems such as amorphous or impure semiconductors, do not agree with each other or with experiment. In semiconductors, however, experiments do at least appear consistent with each other.In the limit of infinite system size the transport properties of a sufficiently inhomogeneous medium are best calculated through application of critical rate analysis with the system resistivity related to the critical (percolating) resistance value, R_c. Here well-known cluster statistics of percolation theory are used to derive the variability, W (R,x) in the smallest maximal resistance, R of a path spanning a volume x³ as well as to find the dependence of the mean value of the conductivity, (x). The functional form of the cluster statistics is a product of a power of cluster size, and a scaling function, either exponential or Gaussian, but which, in either case, cuts off cluster sizes at a finite value for any maximal resistance other than R_c. Either form leads to a maximum in W (R,x) at R=R_c. When the exponential form of the cluster statistics is used, and when individual resistors are exponential functions of random variables (as in stochastic treatments of the unsaturated zone by the McLaughlin group [see Graham and MacLaughlin (1991), or the series of papers by Yeh et al. (1985, 1995), etc.], or as is known for hopping conduction in condensed matter physics), then W (R,x) has a power law decay in R/R_c (or R_c/R, the power being an increasing function of x. If the statistics of the individual resistors are given by power law functions of random variables (as in Poiseiulles Law), then an exponential decay in R for W (R,x) is obtained with decay constant an increasing function of x. Use, instead, of the Gaussian cluster statistics alters the case of power law decay in R to an approximate power, with the value of the power a function of both R and x.     

Analytical solutions of one-dimensional discrete dynamical systems with chaotic behaviour

This paper studies exact analytical solutions in closed form of the difference equation

The stress state of an elastic orthotropic medium with a penny-shaped crack

The static-equilibrium problem for an elastic orthotropic space with a circular (penny-shaped) crack is solved. The stress state of an elastic medium is represented as a superposition of the principal and perturbed states. To solve the problem, Willis approach is used, which is based on the triple Fourier transform in spatial variables, the Fourier-transformed Greens function for an anisotropic material, and Cauchys residue theorem. The contour integrals obtained are evaluated using Gauss quadrature formulas. The results for particular cases are compared with those obtained by other authors. The influence of anisotropy on the stress intensity factors is studied.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 76–83, December 2004.     

Lagrange equation of a class of nonholonomic systems

Making use of conclusions from[1]:(1)d-δoperations are commutative;(2)theAppell-Chetaev condition restricting virtual displacements is superfluous,the present paperderives the Lagrange equation without multipliers for a class of first-order nonlinearnonholonomic dynamical systems by means of variational principle.This kind of equationsis new.     

Non-gaussian diffusion modeling in composite porous media by homogenization: Tail effect

In the paper anomalous diffusion appearing in a porous medium composed of two porous components of considerably different diffusion characteristics is examined. The differences in diffusivities are supposed to result either from two medium types being present or from variations in pore size (double porosity media). The long-tail effect is predicted using the homogenization approach based on the application of multiple scale asymptotic developments. It is shown that, if the ratio of effective diffusion coefficients of two porous media is of the order of magnitude smaller or equal O( ²), where is a homogenization parameter, then the macroscopic behaviour of the composite may be affected by the presence of tail-effect. The results of the theoretical analysis were applied to a problem of diffusion in a bilaminate composite. Analytical calculations were performed to show the presence of the long-tail effect in two particular cases.Notations c _i the concentration of chemical species in water within the medium i - D _i the effective diffusion coefficient for the medium i - D _ij ^eff the macroscopic (or effective) diffusion tensor in the composite - ERV the elementary representative volume - h the thickness of the period - l a chracteristic length of the ERV or the periodic cell - L a characteristic macroscopic length - n the volumetric fraction of the material 2 - 1–n the volumetric fraction of the material 1 - N the unit vector normal to - t the time variable - x the macroscopic (or slow) space variable - y the microscopic (or fast) space variable - c _1c ,C _2c ,D _1c ,D _2c the characteristic quantities - T,T _1L ,T _2L ,T _1l ,T _2l the characteristic times - c ₁ ^* ,c ₂ ^* ,D ₁ ^* ,D ₂ ^* ,t ^* the non-dimensional variables - the homogenization parameter - ₁ the domain occupied by the material 1 - ₂ the domain occupied by the material 2 - the interface between the domains 1 and 2 - the total volume of the periodic cell - /x_i the gradient operator - the gradient operator     

The ‘Missing’ Self-Similar Free Convection Boundary-Layer Flow Over a Vertical Permeable Surface in a Porous Medium

The free convection boundary-layer flow of a Darcy–Boussinesq fluid from a vertical permeable plate with an inverse-linear temperature distribution is considered. The outstanding characteristics of this self-similar flow which, according to the usual reduction procedure of pseudo-similarity to full similarity, should not exist at all, are analyzed in detail. Thus it is shown that this flow only exists if a lateral suction with a sufficiently large suction parameter _min = 1.079131 is applied. For the threshold value _min the solution is unique but above it multiple solutions are encountered for every given value of .