Impacts of local dispersion and first-order decay on solute transport in randomly heterogeneous porous media

Stochastic subsurface transport theories either disregard local dispersion or take it to be constant. We offer an alternative Eulerian-Lagrangian formalism to account for both local dispersion and first-order mass removal (due to radioactive decay or biodegradation). It rests on a decomposition of the velocityv into a field-scale componentv , which is defined on the scale of measurement support, and a zero mean sub-field-scale componentv _s , which fluctuates randomly on scales smaller than. Without loss of generality, we work formally with unconditional statistics ofv _s and conditional statistics ofv . We then require that, within this (or other selected) working framework,v _s andv be mutually uncorrelated. This holds whenever the correlation scale ofv is large in comparison to that ofv _s . The formalism leads to an integro-differential equation for the conditional mean total concentration c which includes two dispersion terms, one field-scale and one sub-field-scale. It also leads to explicit expressions for conditional second moments of concentration cc. We solve the former, and evaluate the latter, for mildly fluctuatingv by means of an analytical-numerical method developed earlier by Zhang and Neuman. We present results in two-dimensional flow fields of unconditional (prior) mean uniformv . These show that the relative effect of local dispersion on first and second moments of concentration dies out locally as the corresponding dispersion tensor tends to zero. The effect also diminishes with time and source size. Our results thus do not support claims in the literature that local dispersion must always be accounted for, no matter how small it is. First-order decay reduces dispersion. This effect increases with time. However, these concentration moments c and cc of total concentrationc, which are associated with the scale below, cannot be used to estimate the field-scale concentrationc directly. To do so, a spatial average over the field measurement scale is needed. Nevertheless, our numerical results show that differences between the ensemble moments ofc and those ofc are negligible, especially for nonpoint sources, because the ensemble moments ofc are already smooth enough.     

Flow in porous media II: The governing equations for immiscible,two-phase flow

The Stokes flow of two immiscible fluids through a rigid porous medium is analyzed using the method of volume averaging. The volume-averaged momentum equations, in terms of averaged quantities and spatial deviations, are identical in form to that obtained for single phase flow; however, the solution of the closure problem gives rise to additional terms not found in the traditional treatment of two-phase flow. Qualitative arguments suggest that the nontraditional terms may be important when / is of order one, and order of magnitude analysis indicates that they may be significant in terms of the motion of a fluid at very low volume fractions. The theory contains features that could give rise to hysteresis effects, but in the present form it is restricted to static contact line phenomena.Roman Letters (, = , , and ) A interfacial area of the- interface contained within the macroscopic system, m² - A _e area of entrances and exits for the -phase contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - g gravity vector, m²/s - H mean curvature of the- interface, m^–1 - H area average of the mean curvature, m^–1 - H – H , deviation of the mean curvature, m^–1 - I unit tensor - K Darcy's law permeability tensor, m² - K permeability tensor for the-phase, m² - K viscous drag tensor for the-phase equation of motion - K viscous drag tensor for the-phase equation of motion - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - n unit normal vector pointing from the-phase toward the-phase (n = –n ) - p _c p – P , capillary pressure, N/m² - p pressure in the-phase, N/m² - p intrinsic phase average pressure for the-phase, N/m² - p – p , spatial deviation of the pressure in the-phase, N/m² - r ₀ radius of the averaging volume, m - t time, s - v velocity vector for the-phase, m/s - v phase average velocity vector for the-phase, m/s - v intrinsic phase average velocity vector for the-phase, m/s - v – v , spatial deviation of the velocity vector for the-phase, m/s - V averaging volume, m³ - V volume of the-phase contained within the averaging volume, m³ Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - viscosity of the-phase, Nt/m² - surface tension of the- interface, N/m - viscous stress tensor for the-phase, N/m² - / kinematic viscosity, m²/s     

Nonlinear analysis of the forced response of a beam with three mode interaction

An analysis is presented for the primary resonance of a clamped-hinged beam, which occurs when the frequency of excitation is near one of the natural frequencies,_n . Three mode interaction (₂ 3₁ and _{3 1} + 2₂) is considered and its influence on the response is studied. The case of two mode interaction (₂ 3₁) is also considered to compare it with the case of three mode interaction. The straight beam experiencing mid-plane stretching is governed by a nonlinear partial differential equation. By using Galerkin's method the governing equation is reduced to a system of nonautonomous ordinary differential equations. The method of multiple scales is applied to solve the system. Steady-state responses and their stability are examined. Results of numerical investigations show that there exists no significant difference between both modal interactions' influences on the responses.     

Representation of the rheological properties of polymer melts in terms of complex fluidity

In dynamic rheological experiments melt behavior is usually expressed in terms of complex viscosity ^* () or complex modulusG ^* (). In contrast, we attempted to use the complex fluidity ^* () = 1/µ ^* () to represent this behavior. The main interest is to simplify the complex-plane diagram and to simplify the determination of fundamental parameters such as the Newtonian viscosity or the parameter of relaxation-time distribution when a Cole-Cole type distribution can be applied. ^* () complex shear viscosity - () real part of the complex viscosity - () imaginary part of the complex viscosity - G ^* () complex shear modulus - G() storage modulus in shear - G() loss modulus in shear - J ^* () complex shear compliance - J() storage compliance in shear - J() loss compliance in shear - shear strain - rate of strain - angular frequency (rad/s) - shear stress - loss angle - ^* () complex shear fluidity - () real part of the complex fluidity - () imaginary part of the complex fluidity - ₀ zero-viscosity - ₀ average relaxation time - h parameter of relaxation-time distribution     

Concentration dependence of the critical viscoelastic properties of gelatin at the gel point

Gelatin gel properties have been studied through the evolution of the storage [G()] and the loss [G()] moduli during gelation or melting near the gel point at several concentrations. The linear viscoelastic properties at the percolation threshold follow a power-law G()G() and correspond to the behavior described by a rheological constitutive equation known as the Gel Equation. The critical point is characterized by the relation: tan = G/G = cst = tan ( · /2) and it may be precisely located using the variations of tan versus the gelation or melting parameter (time or temperature) at several frequencies. The effect of concentration and of time-temperature gel history on its variations has been studied. On gelation, critical temperatures at each concentration were extrapolated to infinite gel times. On melting, critical temperatures were determined by heating step by step after a controlled period of aging. Phase diagrams [T = f(C)] were obtained for gelation and melting and the corresponding enthalpies were calculated using the Ferry-Eldridge relation. A detailed study of the variations of A with concentration and with gel history was carried out. The values of which were generally in the 0.60–0.72 range but could be as low as 0.20–0.30 in some experimental conditions, were compared with published and theoretical values.     

Vorticity characteristics of the turbulent intermediate wake

Measurements of the lateral components _j (j=2 and 3) of the vorticity fluctuation vector have been made, using a vorticity probe consisting of two X-wires, in the intermediate wake of a circular cylinder. The effect of the spatial resolution of the probe on the measurement of _j has been studied. As the spatial resolution impairs, the variance and flatness factor of _j decrease whereas the skewness of _j increases. Reasonably accurate values of _j ² can be obtained by applying spectral corrections for the spatial resolution effect.Near the beginning of the intermediate wake, the variance of ₂ is larger than that of ₃ due to the significant contribution from ribs which connect consecutive spanwise roll vortices. This difference decreases with downstream distance. Also, the presence of the rolls is reflected by a local extremum in the skewness of ₃ on each side of the wake centerline. The magnitude of the extremum decreases with downstream distance.The support of the Australian Research Council is gratefully acknowledged.     

The dynamic performance of the Weissenberg rheogoniometer III. Large amplitude oscillatory shearing — harmonic analysis

The harmonic content of the nonlinear dynamic behaviour of 1% polyacrylamide in 50% glycerol/water was studied using a standard Model R 18 Weissenberg Rheogoniometer. The Fourier analysis of the Oscillation Input and Torsion Head motions was performed using a Digital Transfer Function Analyser.In the absence of fluid inertia effects and when the amplitude of the (fundamental) Oscillation Input motion _I is much greater than the amplitudes of the Fourier components of the Torsion Head motion _Tn empirical nonlinear dynamic rheological propertiesG _n (, ⁰),G _n (, ⁰) and/or _n (, ⁰), _n (, ⁰) may be evaluated without a-priori-knowledge of a rheological constitutive equation. A detailed derivation of the basic equations involved is presented.Cone and plate data for the third harmonic storage modulus (dynamic rigidity)G ₃ (, ⁰), loss modulusG ₃ (, ⁰) and loss angle ₃ (, ⁰) are presented for the frequency range 3.14 × 10^–2 1.25 × 10² rad/s at two strain amplitudes, _CP ⁰ = 2.27 and 4.03. Composite cone and plate and parallel plates data for both the third and fifth harmonic dynamic viscosities ₃ (, ⁰), _S (, ⁰) and dynamic rigiditiesG ₃ (, ⁰),G ₅ (, ⁰) are presented for strain amplitudes in the ranges 1.10 _CP ⁰ 4.03 and 1.80 _PP ⁰ 36 for a single frequency, = 3.14 × 10^–1 rad/s. Good agreement was obtained between the results from both geometries and the absence of significant fluid inertia effects was confirmed by the superposition of the data for different gap widths.     

Approximate First Integrals of a Galaxy Model

First-order approximate first integrals (conserved quantities)of a Hamiltonian dynamical system with two degrees of freedomwhich arises in the modeling of central part of a deformed galaxy [1] havebeen obtained based on the approximate Noether symmetries for resonances₁=₂, ₁=2₂ and 2₁=3₂. Furthermore,KAM curves have been obtained analytically and they have been compared with thenumerical ones on the Poincaré surface of section.     

Evaluation of the dynamic-mechanical properties of solids from a cooperative model for stress relaxation

For many solid materials the stress relaxation process obeys the universal relationF = – (d/d lnt)_max = (0.1 ± 0.01) ( ₀ — _i ), regardless of the structure of the material. Here denotes the stress,t the time, ₀ the initial stress of the experiment and _i the internal stress. A cooperative model accounting for the similarity in relaxation behaviour between different materials was developed earlier. Since this model has a spectral character, the concepts of linear viscoelasticity are used here to evaluate the corresponding prediction of the dynamic mechanical properties, i.e. the frequency dependence of the storageE () and lossE () moduli. Useful numerical approximations ofE () andE () are also evaluated. It is noted that the universal relation in stress relaxation had a counterpart in the frequency dependence ofE (). The theoretical prediction of the loss factor for high-density polyethylene is compared with experimental results. The agreement is good.     

Generation of vortices in a partially filled,rapidly rotating cylinder

We describe a system in which vortices are shed from a cylindrical free surface approximately centered in a rotating flow. Shedding is controlled by the parameter =2 g/ ² d, where g, , d denote gravity, rotation rate and the diameter of the free surface. We find vortex shedding for >0.162 and no vortex shedding for < 0.0847. The range depends on the aspect ratio L/d, where L is the column length, in a nonmonotonic fashion. These results are independent of viscosity and surface tension for small values of these parameters.Now at Martin Marietta, Orlando Aerospace, PO Box 5837, Mail Point 150, Orlando, FL 32855, USA     

On the Ellipticity of the Middle Surface of a Shell and its Application to the Asymptotic Analysis of ‘Membrane’ Shells

We consider a surface S = (), where ² is a bounded, connected, open set with a smooth boundary and : ³ is a smooth map; let () denote the components of the two-dimensional linearized strain tensor of S and let 0 with length 0 > 0. We assume the the norm ,|| ()||0, in the space V0() = { H¹() × H¹() × L²(); = 0 on 0 } is equivalent to the usual product norm on this space. We then establish that this assumption implies that the surface S is uniformly elliptic and that we necessarily have 0 = .     

On the stability of periodic motions of an unbalanced rigid rotor on lubricated journal bearings

A theoretical investigation is carried out on the orbital motions of a symmetrical, unbalanced, rigid rotor subjected to a constant vertical load and supported on two lubricated journal bearings. In order to determine the fluid film forces, the short bearing theory is adopted.A method is illustrated that makes it possible to determine the analytical equation of the orbit as an approximated solution of the system of non-linear differential equations of motion of the journal axis. A procedure is also described for evaluating the stability of the solution found. Diagrams of the curves delimiting, in the working plane of the rotor -m, the areas of stability of the various periodic solutions determined are provided.Finally, the results obtained are compared and combined with those provided by a direct integration of the motion equation made using the Runge-Kutta method.Nomenclature C radial clearance - D = 2R bearing diameter - E mass unbalance cecentricity - Fx, Fy fluid film force components - fi = Fi/W dimensionless fluid film force components - L bearing length - M one half rotor mass - % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaad2gacqGH9aqpcaWGnbGaam4qaiabeM8a% 3naaCaaaleqabaGaaGOmaaaakiaac+cacqaHdpWCcaWGxbaaaa!471F!\[m = MC\omega ^2 /\sigma W\] dimensionless one half rotor mass - R bearing radius - T = 2 synchronous orbit period - t time - W load per bearing - X, Y, Z coordinates - x = X/C; y = y/C; z = Z/L dimensionless coordinates - oil dynamic viscosity - = E/C dimensionless mass unbalance eccentricity - = (RL/W)/(R/C) ² (L/D) ² modified Sommerfeld number - = t dimensionless time = periodic orbit frequency - = 2/ frequency ratio - journal angular velocity - (·) dimensionless time derivative     

Primary and secondary bifurcation in baroclinic instability

In modelling atmospheric flows the baroclinic instability of the flow in a differentially heated rotating annulus plays a central role. This paper deals with an experimental study using LDV and flow visualization techniques. Usually the temperature difference, T, was kept fixed while the angular velocity, , was varied. On crossing the stability boundary, the primary bifurcation, the basic flow gives way to a baroclinic wave flow. For a given annulus geometry the wave number, m, of the first wave pattern was found to be uniquely defined by T. The measured critical values of , _crit, agree reasonably well with those obtained by other authors. On increasing above _crit the wave number changed, this process showing hysteresis. The situation might indicate secondary bifurcation phenomena. Flow visualization using aluminium particles shows surface flow details.This paper is dedicated to Prof. Dr. K. Gersten on the occasion of his 60th birthday     

Generalized Rouse model for a dilute solution of clustered polymers

A theory analogue to tha of Rouse is given, to describe the rheological behavior of dilute solutions consisting of clusters of crosslinked polymers. The frequency-dependent behavior of the dynamic moduli of these fluids differs substantially from that of the well-known Rouse-like fluid (GG^1/2). In our case the storage modulus becomes proportional to ^3/2, while the loss modulus is proportional to . The loss modulus dominates the dynamic behavior for frequencies smaller than the largest normal frequency of the clusters.     

A non-affine network model for polymer melts

In this paper, a network model of polymer melts is proposed in which network junction points move non-affinely. In this non-affine motion, junction points follow particle paths as seen by an observer rotating at the fluid element's net rigid-rotation rate. The speed at which junction points move is reduced as the network segments near their maximum extensions. In order to maintain a frame invariant model, it is necessary that the vorticity be decomposed into two portions, such that, = _R + _D . The deformational vorticity, _D , arises from shear deformation and is frame invariant while the rigid vorticity, _R , is frame dependent. A constitutive equation based on this finitely extensible network strand (FENS) motion is developed. The model illustrates how rotations that cause changes in the relative orientation of a fluid element with its surroundings can be incorporated into a constitutive equation using the deformational vorticity. The FENS model predicts a shear-thinning viscosity, and the Trouton viscosity predicted by the model is finite for all elongation rates. Finally, stochastic simulation results are presented to justify a mathematical approximation used in deriving the constitutive equation.     

Normal forms for random diffeomorphisms

Given a dynamical system (,, ,) and a random diffeomorphism (): ^{d d} with fixed point at x=0. The normal form problem is to construct a smooth near-identity nonlinear random coordinate transformation h() to make the random diffeomorphism ()=h()^–1() h() as simple as possible, preferably linear. The linearization D(, 0)=:A() generates a matrix cocycle for which the multiplicative ergodic theorem holds, providing us with stochastic analogues of eigenvalues (Lyapunov exponents) and eigenspaces. Now the development runs pretty much parallel to the deterministic one, the difference being that the appearance of turns all problems into infinite-dimensional ones. In particular, the range of the homological operator is in general not closed, making the conceptof-normal form necessary. The stochastic versions of resonance and averaging are developed. The case of simple Lyapunov spectrum is treated in detail.     

Harmonic solutions of micropolar elastodynamics

The equations of micropolar elastodynamics are considered for an unbounded continuum subjected to a body force and a body couple. These act harmonically with the same real frequency , but with individual arbitrary spatial distributions. Over a harmonic state, the displacement and microrotation are related to two radiation conditioned harmonic vectors, each acquiring three eigenvalue contributions, assuming a noncritical -frequency. Altogether, four distinct eigenvalues are admissible. If ²<2² ₀, ₀ being a frequency parameter of the continuum, two of these are real while two are purely imaginary. But if ²<2² ₀, then all admissible eigenvalues are real. Each eigenvalue contribution resolves into a series of Hankel and Bessel functions coupled to Hankel type transforms of: (i) spherical integrals which, in turn, can be expanded via spherical harmonics for the 3-dimensional problem, (ii) circular integrals for the 2-dimensional problem. Axisymmetric and spherically symmetric results are deduced in 3-dimensions. Asymptotic solutions are also established; they disclose long-range formation of radially attenuated spherical (or circular) waves propagating with, generally, anisotropic amplitudes but, invariably, isotropic eikonals.If, in the absence of a body couple, a body force acts radially in 3-dimensions with a spherically symmetric strength, then the elastic displacement behaves likewise while the microrotation vanishes identically. Another application is made to a 2-dimensional problem for a 1 × 3 source system of body force plus body couple without longitudinal variation but with magnitudes symmetric about a longitudinal axis.As approaches a certain critical frequency , dependent solely on the continuum, at least two eigenvalues approach the same value. The phenomenon is explored for a continuum consistent with ²<2² ₀ and under the hypothesis ²<2² ₀. All admissible eigenvalues are then real throughout an -neighbourhood of . Here, two associated eigenvalue contributions behave singularly. Nevertheless, their essential singularities cancel out within the relevant combination. Examination of a far-field suggests that critical frequency attainment sets off a slow instability in the 2-dimensional configuration. In the 3-dimensional configuration, however, it preserves stability and eliminates radial attenuation; an exact solution is formulated for this case.     

Heat conduction in a cylinder crossed by an electric current with skin effect

The steady periodic temperature distribution in an infinitely long solid cylinder crossed by an alternating current is evaluated. First, the time dependent and non-uniform power generated per unit volume by Joule effect within the cylinder is determined. Then, the dimensionless temperature distribution is obtained by analytical methods in steady periodic regime. Dimensionless tables which yield the amplitude and the phase of temperature oscillations both on the axis and on the surface of copper or nichrome cylindrical electric resistors are presented.

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Wärmeleitung in einem stromdurchflossenen Zylinder unter Berücksichtigung des Skin-Effektes

Zusammenfassung Es wird die periodische Temperaturverteilung für den eingeschwungenen Zustand in einem unendlich langen, von Wechselstrom durchflossenen Vollzylinder ermittelt. Zuerst erfolgt die Bestimmung der zeitabhängigen, nichgleichförmigen Energiefreisetzung pro Volumeneinheit des Zylinders infolge Joulescher Wärmeentwicklung und anschließend die Ermittlung der quasistationären Temperaturverteilung auf analytischem Wege. Amplitude und Phasenverzögerung der Temperaturschwingungen werden für die Achse und die Oberfläche eines Kupfer- oder Nickelchromzylinders tabellarisch in dimensionsloser Form mitgeteilt.

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Nomenclature A integration constant introduced in Eq. (2) - ber, bei Thomson functions of order zero - Bi Biot numberhr ₀/ - c speed of light in empty space - c ₁,c ₂ integration constants introduced in Eq. (46) - c _p specific heat at constant pressure - E electric field - E _z component ofE alongz - E time independent part ofE, defined in Eq. (1) - f function ofs and defined in Eq. (11) - g function ofs and defined in Eq. (37) - h convection heat transfer coefficient - H magnetic field - i imaginary uniti=(–1)^1/2 - I electric current - I _eff effective electric currentI _eff=I/2^1/2 - Im imaginary part of a complex number - J _n Bessel function of first kind and ordern - J electric current density - q _g power generated per unit volume - time average of the power generated per unit volume - time averaged power per unit length - r radial coordinate - R electric resistance per unit length - r ₀ radius of the cylinder - Re real part of a complex number - s dimensionless radial coordinates=r/r ₀ - s, s integration variables - t time - T temperature - time averaged temperature - T _f fluid temperature outside the boundary layer - time average of the surface temperature of the cylinder - u, functions ofs, and defined in Eqs. (47) and (48) - W Wronskian - x position vector - x real variable - Y _n Bessel function of second kind and ordern - z unit vector parallel to the axis of the cylinder - z axial coordinate - · modulus of a complex number - equal by definition Greek symbols amplitude of the dimensionless temperature oscillations - electric permittivity - dimensionless temperature defined in Eq. (16) - ₀, ₁, ₂ functions ofs defined in Eq. (22) - thermal conductivity - dimensionless parameter=(2)^1/2 - magnetic permeability - ₀ magnetic permeability of free space - function of defined in Eq. (59) - dimensionless parameter=c _p/() - mass density - electric conductivity - dimensionless time=t - phase of the dimensionless temperature oscillations - function ofs:= ₁+i ₂ - angular frequency - dimensionless parameter=()^1/2 r ₀     

The Sedov self-similar point blast solutions in nonuniform media

The character of a spherical blast wave propagating radially outward in a pressureless medium with density decreasing as ^– changes dramatically as and the adiabatic index vary. Plots of the Sedov formulas for the density, velocity, and pressure profiles behind the shock front for a selection of different parameters illustrate this and suggest that some of the solutions satisfy the conditions for the Rayleigh-Taylor or convective instablity.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.     

Evolution of Domains of Attraction of a Forced Beam with Two-Mode Interaction

When a dissipative dynamical system has multiple attractors, it is a task to determine and recognize the global domain of attraction of each attractor. In this paper we study the global behavior of a forced hinged-clamped beam with two-mode interaction. The equation of motion of the beam is reduced to four first-order autonomous ordinary differential equations. The system has an internal resonance condition of 2 31, where 1 and 2 denote natural frequencies of the first and second modes, respectively. When the excitation frequency is near 1, the system can have three equilibrium solutions, among which two are asymptotically stable and one is unstable. We examine how the domains of attraction of two stable equilibrium solutions evolve as the forcing frequency is varied across jump points. By using a special plane which contains all equilibrium points, called the principal plane, the global domains of attraction can be discussed more effectively. Results show that knowledge of this evolution helps us better understand the jump phenomenon.