Transient Free Convection from a Horizontal Surface in a Porous Medium Subjected to a Sudden Change in Surface Heat Flux

This paper presents an analytical and numerical study of transient free convection from a horizontal surface that is embedded in a fluid-saturated porous medium. It is assumed that for time steady state velocity and temperature fields are obtained in the boundary-layer which occurs due to a uniform flux dissipation rate q ₁ on the surface. Then, at the heat flux on the surface is suddenly changed to q ₂ and maintained at this value for . Firstly, solutions which are valid for small and large are obtained. The full boundary-layer equations are then integrated step-by-step for the transient regime from the initial unsteady state ( ) until such times at which this forward marching approach is no longer well posed. Beyond this time no valid solutions could be obtained which matched the final solution from the forward integration to the steady state profiles at large times .     

Flow in Random Porous Media

Flow in a porous medium with a random hydraulic conductivity tensor K(x) is analyzed when the mean conductivity tensor (x) is a non-constant function of position x. The results are a non-local expression for the mean flux vector (x) in terms of the gradient of the mean hydraulic head (x), an integrodifferential equation for (x), and expressions for the two point covariance functions of q(x) and (x). When K(x) is a Gaussian random function, the joint probability distribution of the functions q(x) and (x) is determined.     

Dimensional analysis of pore scale and field scale immiscible displacement

A basic re-examination of the traditional dimensional analysis of microscopic and macroscopic multiphase flow equations in porous media is presented. We introduce a macroscopic capillary number which differs from the usual microscopic capillary number Ca in that it depends on length scale, type of porous medium and saturation history. The macroscopic capillary number is defined as the ratio between the macroscopic viscous pressure drop and the macroscopic capillary pressure. can be related to the microscopic capillary number Ca and the LeverettJ-function. Previous dimensional analyses contain a tacit assumption which amounts to setting = 1. This fact has impeded quantitative upscaling in the past. Our definition for , however, allows for the first time a consistent comparison between macroscopic flow experiments on different length scales. Illustrative sample calculations are presented which show that the breakpoint in capillary desaturation curves for different porous media appears to occur at 1. The length scale related difference between the macroscopic capillary number for core floods and reservoir floods provides a possible explanation for the systematic difference between residual oil saturations measured in field floods as compared to laboratory experiment.     

Investigation of the Geometrical Dispersion Regime in a Single Fracture Using Positron Emission Projection Imaging

The transport properties of a natural fracture crossing a limestone block of 36 cm × 26 cm × 60 cm is studied using positron emission projection imaging. This non-invasive technique allows to measure the spatial distribution of the activity of a radioactive solution (here irradiated-copper-EDTA solution) within the fracture. The fracture aperture is measured from the spatial distribution of the activity as the fracture is completely filled with the tracer. The experiment consists in injecting the tracer at a constant flow rate in the plane of the fracture filled with an identical non-radioactive solution. Every 10 min, a two-dimensional grey scale image of the concentration field is recorded. The heterogeneity of the tracer distribution increases with time in relation with the spatial heterogeneity of the aperture field, and favours only slightly the region of larger aperture. The correlation length of the aperture distribution is larger than the correlation length of the concentration distribution of the tracer within the sample. Consequently, the concentration distribution cannot be modelled using a classical advection–dispersion equation; the mixing process has not reached a stationary Fickian dispersion regime in the finite size domain of the experiment. Nevertheless, the transversally averaged concentration profiles evaluated along the flow direction x rescale adequately with an advective variable , where is the mean velocity and t the time. This result is explained in the context of the geometrical dispersion regime where the mixing dispersion zone grows proportionally with time. Different approaches are proposed to characterise this anomalous dispersion regime.     

Effect of a Short Roughness Strip on a Turbulent Boundary Layer

The response of a turbulent boundary layer to a short roughness strip is investigated using laser Doppler velocimetry (LDV) and laser induced fluorescence (LIF). Skin friction coefficients are inferred from accurate near-wall measurements. There is an undershoot in , where is the undisturbed smooth wall skin friction coefficient, immediately after the strip. Downstream of the strip, overshoots before relaxing back to unity in an oscillatory manner. The roughness strip has a major effect on the turbulent stresses ; these quantities increase, relative to the undisturbed smooth wall, in the region between the two internal layers originating at the upstream and downstream edges of the strip. The increase in the ratio suggests a decrease in near-wall anisotropy. From the flow visualizations, it is inferred that streamwise vortical structures are weakened immediately downstream of the strip. Consistently, streamwise length scales are also reduced; direct support for this is provided by measured two-point velocity correlations.     

Global Attractors: Topology and Finite-Dimensional Dynamics

Many dissipative evolution equations possess a global attractor with finite Hausdorff dimension d. In this paper it is shown that there is an embedding X of into , with N=[2d+2], such that X is the global attractor of some finite-dimensional system on with trivial dynamics on X. This allows the construction of a discrete dynamical system on which reproduces the dynamics of the time T map on and has an attractor within an arbitrarily small neighborhood of X. If the Hausdorff dimension is replaced by the fractal dimension, a similar construction can be shown to hold good even if one restricts to orthogonal projections rather than arbitrary embeddings.     

On the dynamic behaviour of an elastically supported beam of infinite length, loaded by a concentrated force

Summary An elastically supported beam of infinite length, initially at rest, carries a variable concentrated force at a prescribed point A. General expressions are given for the deflection and the bending moment at A (6.3 and 6.4). Three special cases are considered; the first one is defined by =0 for and =K=const. for ; the second one by =0 for 0 > > , given function of for 0 ; the third one applies to problems in which, during the period of impact, itself is an unknown. The results given here may be of use in those railway-engineering problems in which a rail can be considered as a beam of infinite length, and in which the supporting ground has the required properties.     

Optimal cupolas of uniform strength: Spherical M-shells and axisymmetric T-shells

Summary The first part of this paper is concerned with the optimal design of spherical cupolas obeying the von Mises yield condition. Five different load combinations, which all include selfweight, are investigated. The second part of the paper deals with the optimal quadratic meridional shape of cupolas obeying the Tresca yield condition, considering selfweight plus the weight of a non-carrying uniform cover. It is established that at long spans some non-spherical Tresca cupolas are much more economical than spherical ones.

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Optimale Kuppeln gleicher Festigkeit: Kugelschalen und axialsymmetrische Schalen

Übersicht Im ersten Teil dieser Arbeit wird der optimale Entwurf sphärischer Kuppeln behandelt, wobei die von Misessche Fließbewegung zugrunde gelegt wird. Fünf verschiedene Lastkombinationen werden untersucht. Der zweite Teil befaßt sich mit der optimalen quadratischen Form des Meridians von Kuppeln, die der Fließbedingung von Tresca folgen.

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List of Symbols a_k, b_k, c_k, A_k, B_k, C_k coefficients used in series solutions - A, B constants in the nondimensional equation of the meridional curve - normal component of the load per unit area of the middle surface - meridional and circumferential forces per unit width - radial pressure per unit area of the middle surface, - skin weight per unit area of the middle surface, - vertical external load per unit horizontal area, - base radius, - R radius of convergence - s - cupola thickness, - u, w subsidiary functions for quadratic cupolas - vertical component of the load per unit area of middle surface - resultant vertical force on a cupola segment - structural weight of cupola, - combined weight of cupola and skin, - distance from the axis of rotation, - vertical distance from the shell apex, - z auxiliary variable in series solutions - specific weight of structural material of cupola - radius of the middle surface, - uniaxial yield stress - meridional stress, - circumferential stress, - _a, _b, _c, _d, _e subsidiary variables used in evaluating the meridional stress - auxiliary function used in series solutions This paper constitutes the third part of a study of shell optimization which was initiated and planned by the late Prof. W. Prager     

On the continuum modelling of porous media containing fluid: a molecular viewpoint with particular attention to scale

Mass conservation and linear momentum balance relations for a porous body and any fluid therein, valid at any given length scale in excess of nearest-neighbour molecular separations, are established in terms of local weighted averages of molecular quantities. The mass density field for the porous body at a given scale is used to identify its boundary at this scale, and a porosity field is defined for any pair of distinct length scales. Specific care is paid to the interpretation of the stress tensor associated with each of the body and fluid at macroscopic scales, and of the force per unit volume each exerts on the other. Consequences for the usual microscopic and macroscopic viewpoints are explored.Nomenclature material system; Section 2.1. - porous body (example of a material system); Sections 2.1, 3.1, 4.1 - fluid body (example of a material system); Sections 2.1, 3.1, 4.1 - weighting function; Sections 2.1, 2.3 - ,h weighting function corresponding to spherical averaging regions of radius and boundary mollifying layer of thicknessh; Section 3.2 - Euclidean space; Section 2.1 - V space of all displacements between pairs of points in; Section 2.1 - mass density field corresponding to; (2.3)₁ - ^P , ^f mass density fields for , ; (4.1) - P momentum density field corresponding to; (2.3)₂ - v velocity field corresponding to; (2.4) - S _r (X) interior of sphere of radiusr with centre at pointx; (3.3) - boundary ofany region - region in which ^p > 0 with = ,h; (3.1) - subset of whose points lie at least+h from boundary of ; (3.4) - abbreviated versions of ; Section 3.2, Remark 4 - strict interior of ; (3.7) - analogues of for fluid system ; Section 3.2 - general version of corresponding to any choice of weighting function; (4.6) - interfacial region at scale; (3.8) - ₀ scale of nearest-neighbour separations in ; Section 3.2. Remark 1 - porosity field at scales ( ₁; ₂); (3.9) - pore space at scales ( ₁; ₂); (3.12)     

An Approximate Solution for Fluid Infiltration near a Circular Opening in Unsaturated Media

A regular perturbation technique is employed to approximate the solution for fluid infiltration from a circular opening into an unsaturated medium. Introducing two empirical constitutive relations and relating the permeability k and water content with pore fluid pressure p, a nonlinear diffusion equation in terms of pore pressure is established. After rearranging the nonlinear diffusion equation, a parameter perturbation on is performed and an approximate solution with an error of is obtained, which correlate to a condition in which = . This approximate solution is verified by a finite difference solution and compared also with a linear solution in which the diffusivity is constant. It is shown that the perturbation solution with terms up to and including first-order can give a reasonably accurate solution for the parameter range for p ₀ selected in this paper. The solution procedure provided in this paper also avoids the numerical problem normally encountered for a small time solution. The solution may also be used to overcome difficulties arising in solution procedure by the similarity transformation (Boltzmann), commonly conducted on diffusion equation, which cannot be applied for a finite wellbore problem.     

Strong Solutions for Generalized Newtonian Fluids

We consider the motion of a generalized Newtonian fluid, where the extra stress tensor is induced by a potential with p-structure (p = 2 corresponds to the Newtonian case). We focus on the three dimensional case with periodic boundary conditions and extend the existence result for strong solutions for small times from \tfrac{5}{3}$$ " align="middle" border="0"> (see [16]) to \tfrac{7}{5}.$$ " align="middle" border="0"> Moreover, for we improve the regularity of the velocity field and show that for all 0.$$ " align="middle" border="0"> Within this class of regularity, we prove uniqueness for all \tfrac{7}{5}.$$ " align="middle" border="0"> We generalize these results to the case when p is space and time dependent and to the system governing the flow of electrorheological fluids as long as      

On Approximations of First Integrals for a Strongly Nonlinear Forced Oscillator

In this paper a strongly nonlinear forced oscillator will be studied. It will be shown that the recently developed perturbation method based on integrating factors can be used to approximate first integrals. Not onlyapproximations of first integrals will be given, butit will also be shown how, in a rather efficient way, the existence and stability oftime-periodic solutions can be obtained from these approximations. In additionphase portraits, Poincaré-return maps, and bifurcation diagrams for a set of values of the parameters will be presented. In particularthe strongly nonlinear forced oscillator equation will be studied in this paper. It will be shown that the presentedperturbation method not onlycan be applied to a weakly nonlinear oscillator problem (that is, when the parameter ) but also to a strongly nonlinear problem (that is, when ). The model equation as considered in this paper is related to the phenomenon of galloping ofoverhead power transmission lines on which ice has accreted.     

Dispersion of inert solutes in spatially periodic,two-dimensional model porous media

Taylor dispersion of a passive solute within a fluid flowing through a porous medium is characterized by an effective or Darcy scale, transversely isotropic dispersitivity , which depends upon the geometrical microstructure, mean fluid velocity, and physicochemical properties of the system. The longitudinal, and lateral, dispersivity components for two-dimensional, spatially periodic arrays of circular cylinders are here calculated by finite element techniques. The effects of bed voidage, packing arrangement, and microscale Péclet and Reynolds numbers upon these dispersivities are systematically investigated.The longitudinal dispersivity component is found to increase with the microscale Péclet number at a rate less than Pe². This accords with previous calculations by Eidsath et al. (1983), although the latter calculations were found to yield significantly lower longitudinal dispersivities than those obtained with the present numerical scheme. With increasing Péclet number, a Pe² dependence is, however, approached asymptotically, particularly for square cylindrical arrays - owing to the creation of a linear streamline zone between cylinders.     

High Reynolds Number Flow past a Plate Mounted at a Small Angle of Attack

A plane incompressible fluid flow past a plate mounted in a homogeneous stream at a small angle of attack ^* is investigated on the basis of an asymptotic analysis of the Navier-Stokes equations at high Reynolds numbers (Re). In the neighborhood of the leading edge the flow structure is studied in detail. It is found that separation is initiated in a small vicinity of the leading edge at and the length of the slow reverse stream zone is of the order O(1) at . The nonuniqueness of the solution is detected at and the hysteresis phenomenon is explained. It is shown that under certain conditions the solutions obtained also hold for flows past bodies of small thickness.     

An Elliptic Lindstedt--Poincaré Method for Certain Strongly Non-Linear Oscillators

An elliptic Lindstedt--Poincaré (L--P) method is presented for the steady-state analysis of strongly non-linear oscillators of the form , in which the Jacobian elliptic functions are employed instead of the usual circular functions in the classical L--P perturbation procedure. This method can be viewed as a generalization of the L--P method. As an application of this method, three types of the generalized Van der Pol equation with are studied in detail.     

A technique for evaluation of three-dimensional behavior in turbulent boundary layers using computer augmented hydrogen bubble-wire flow visualization

A system is described which allows the recreation of the three-dimensional motion and deformation of a single hydrogen bubble time-line in time and space. By digitally interfacing dualview video sequences of a bubble time-line with a computer-aided display system, the Lagrangian motion of the bubble-line can be displayed in any viewing perspective desired. The u and v velocity history of the bubble-line can be rapidly established and displayed for any spanwise location on the recreated pattern. The application of the system to the study of turbulent boundary layer structure in the near-wall region is demonstrated.List of Symbols Reynolds number based on momentum thickness u /v - t⁺ nondimensional time - u shear velocity - u local streamwise velocity, x-direction - u ⁺ nondimensional streamwise velocity - v local normal velocity, -direction - x ⁺ nondimensional coordinate in streamwise direction - ⁺ nondimensional coordinate normal to wall - ₊ ^wire wire nondimensional location of hydrogen bubble-wire normal to wall - z ⁺ nondimensional spanwise coordinate - momentum thickness - v kinematic viscosity - _W wall shear stress     

On the Capillary Coupling between Two Phases in a Droplet Train Model

A droplet train model proposed by Foulser {\it et al.} ({\it Transport in Porous Media} (1991), 223) is modified with addition of capillary resistance. It is shown that linear transport equations for this model can be represented in the Onsager form, where the generalized thermodynamic forces are pressure gradients of corresponding phases. In particular, the onset of capillary interactions give rise to the nonzero and equal cross term coefficients.     

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.     

Modeling of contaminant transport resulting from dissolution of nonaqueous phase liquid pools in saturated porous media

A mathematical model for transient contaminant transport resulting from the dissolution of a single component nonaqueous phase liquid (NAPL) pool in two-dimensional, saturated, homogeneous porous media was developed. An analytical solution was derived for a semi-infinite medium under local equilibrium conditions accounting for solvent decay. The solution was obtained by taking Laplace transforms to the equations with respect to time and Fourier transforms with respect to the longitudinal spatial coordinate. The analytical solution is given in terms of a single integral which is easily determined by numerical integration techniques. The model is applicable to both denser and lighter than water NAPL pools. The model successfully simulated responses of a 1,1,2-trichloroethane (TCA) pool at the bottom of a two-dimensional porous medium under controlled laboratory conditions.Notation a,a ₁ defined in (45a) and (45b), respectively - b defined in (45c) - b vector of true model parameters (n×1) - vector of estimated model parameters (n×1) - c liquid phase solute concentration (solute mass/liquid volume), M/L³ - c _s aqueous saturation concentration (solubility), M/L³ - C dimensionless liquid phase solute concentration, equal toc/c _s - molecular diffusion coefficient, L²/t - ^e effective molecular diffusion coefficient, equal to / ^*, L²/t - D _x longitudinal hydrodynamic dispersion coefficient, L²/t - D _z hydrodynamic dispersion coefficient in the vertical direction, L²/t - e random vector with zero mean (m×1) - erf[x] error function, equal to (2/ ^1/2) - f vector of fitting errors or residuals (m×1) - Fourier operator - _-1 Fourier inverse operator - g vector of model simulated data (m×1) - k mass transfer coefficient, L/t - average mass transfer coefficient, L/t - K _d partition or distribution coefficient (liquid volume/solids mass), L³/M - pool length, L - _o distance between the pool and the origin of the specified Cartesian coordinate system, L - Laplace operator - _-1 Laplace inverse operator - m number of observations - M Laplace/Fourier function defined in (38) - n number of model parameters - N Laplace/Fourier function defined in (39) - p defined in (46) - Pe _x Péclet number, equal toU _x /D _x - Pe _z Péclet number, equal toU _x /D _z - q defined in (47) - R retardation factor - s Laplace transform variable - S objective function - Sh local Sherwood number, equal tok/ _e - Sh _o overall Sherwood number, equal to l/ _e - t time,t - T dimensionless time, equal toU _x t/ - u dummy integration variable - u vector of independent variables - U _x average interstitial velocity, L/t - x spatial coordinate in the longitudinal direction, L - X dimensionless longitudinal length, equal to (x–)/ - y vector of observed data (m×1) - z spatial coordinate in the vertical direction, L - Z dimensionless vertical length, equal toz/ - Fourier transform variable - defined in (37) - defined in (50) - porosity (liquid volume/aquifer volume), L³/L³ - defined in (52a) and (52b), respectively - decay coefficient, t^–1 - dimensionless decay coefficient, equal to /U _x - bulk density of the solid matrix (solids mass/aquifer volume), M/L³ - dummy integration variable - ^* tortuosity     

On the Denseness of the Set of Unsolvable Cauchy Problems in the Set of All Cauchy Problems in the Case of an Infinite-Dimensional Banach Space

We prove the following statement: Theorem 1. Let E and be an arbitrary infinite-dimensional Banach space and a continuous mapping, respectively. Then, for every and > 0, there exists a continuous mapping such that