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.     

On nonlinear problems of mixed type: A qualitative theory using infinite-dimensional center manifolds

We consider the equation a(y)u_xx+div_y(b(y)_yu)+c(y)u=g(y, u) in the cylinder (–l,l)×, being elliptic where b(y)>0 and hyperbolic where b(y)<0. We construct self-adjoint realizations in L₂() of the operatorAu= (1/a) div_y(b_yu)+(c/a) in the case ofb changing sign. This leads to the abstract problem u_xx+Au=g(u), whereA has a spectrum extending to + as well as to –. For l= it is shown that all sufficiently small solutions lie on an infinite-dimensional center manifold and behave like those of a hyperbolic problem. Anx-independent cross-sectional integral E=E(u, u_x) is derived showing that all solutions on the center manifold remain bounded forx ±. For finitel, all small solutionsu are close to a solution on the center manifold such that u(x)-(x) Ce ^-(1-|x|) for allx, whereC and are independent ofu. Hence, the solutions are dominated by hyperbolic properties, except close to the terminal ends {±1}×, where boundary layers of elliptic type appear.     

Chaos and integrability in a nonlinear wave equation

We consider the parametrized family of equations _tt ,u- _xx u-au+u ₂ ² u=O,x(0,L), with Dirichlet boundary conditions. This equation has finite-dimensional invariant manifolds of solutions. Studying the reduced equation to a four-dimensional manifold, we prove the existence of transversal homoclinic orbits to periodic solutions and of invariant sets with chaotic dynamics, provided that =2, 3, 4,.... For =1 we prove the existence of infinitely many first integrals pairwise in involution.     

Eine analytische Näherungslösung für die eingefrorene laminare Grenzschichtströmung eines binären Gemisches

Zusammenfassung Für die eingefrorene laminare Grenzschichtströmung eines teilweise dissoziierten binären Gemisches entlang einer stark gekühlten ebenen Platte wird eine analytische Näherungslösung angegeben. Danach läßt sich die Wandkonzentration als universelle Funktion der Damköhler-Zahl der Oberflächenreaktion angeben. Für das analytisch darstellbare Konzentrationsprofil stellt die Damköhler-Zahl den Formparameter dar. Die Wärmestromdichte an der Wand bestehend aus einem Wärmeleitungs- und einem Diffusionsanteil wird angegeben und diskutiert. Das Verhältnis beider Anteile läßt sich bei gegebenen Randbedingungen als Funktion der Damköhler-Zahl ausdrücken.

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An analytical approximation for the frozen laminar boundary layer flow of a binary mixture

An analytical approximation is derived for the frozen laminar boundary layer flow of a partially dissociated binary mixture along a strongly cooled flat plate. The concentration at the wall is shown to be a universal function of the Damkohler-number for the wall reaction. The Damkohlernumber also serves as a parameter of shape for the concentration profile which is presented in analytical form. The heat transfer at the wall depending on a conduction and a diffusion flux is derived and discussed. The ratio of these fluxes is expressed as a function of the Damkohler-number if the boundary conditions are known.

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Formelzeichen A Atom - A₂ Molekül - C Konstante in Gl. (20) - c₁=1/(2C) Konstante in Gl. (35) - c_p spezifische Wärme bei konstantem Druck - D binärer Diffusionskoeffizient - Ec=u ² /(2h_f) Eckert-Zahl - h spezifische Enthalpie - h_t=h+u²/2 totale spezifische Enthalpie - h _A ⁰ spezifische Dissoziationsenthalpie - K_w Reaktionsgeschwindigkeitskonstante der heterogenen Wandreaktion - 1= /( ) Champman-Rubesin-Parameter - Le=Pr/Sc Lewis-Zahl - M Molmasse - p statischer Druck - Pr= c_pf/ Prandtl-Zahl - q_w Wärmestromdichte an der Wand - q_cw, q_dw Wärmeleitungsbzw. Diffusionsanteil der Wärmestromdichte an der Wand - universelle Gaskonstante - R=/(2M_a) individuelle Gaskonstante der molekularen Komponente - Re_x= u x/ Reynolds-Zahl - Sc=/( D) Schmidt-Zahl - T absolute Temperatur - T_d=h _A ⁰ /R charakteristische Dissoziationstemperatur - u, v x- und y-Komponenten der Geschwindigkeit - U=u/u normierte x-Komponente der Geschwindigkeit - x, y Koordinaten parallel und senkrecht zur Platte Griechische Symbole - =A/ Dissoziationsgrad - Grenzschichtdicke - ₂ Impulsverlustdicke - Damköhler-Zahl der Oberflächenreaktion - =T/T normierte Temperatur - =y/ normierter Wandabstand - Wärmeleitfähigkeit - dynamische Viskosität - , ^* Ähnlichkeitskoordinaten - Dichte - Schubspannung Indizes A auf ein Atom bezogen - M auf ein Molekül bezogen - f auf den eingefrorenen Zustand bezogen - w auf die Wand bezogen - auf den Außenrand der Grenzschicht bezogen     

Analysis of suddenly started laminar flow in the entrance region of a circular tube

Suddenly started laminar flow in the entrance region of a circular tube, with constant inlet velocity, is investigated analytically by using integral momentum approach. A closed form solution to the integral momentum equation is obtained by the method of characteristics to determine boundary layer thickness, entrance length, velocity profile, and pressure gradient.Nomenclature M(, , ) a function - N(, , ) a function - p pressure - p* p/1/2U ², dimensionless pressure - Q(, , ) a function - R radius of the tube - r radial distance - Re 2RU/, Reynolds number - t time - U inlet velocity, constant for all time, uniform over the cross section - u velocity in the boundary layer - u* u/U, dimensionless velocity - u ₁ velocity in the inviscid core - x axial distance - y distance perpendicular to the axis of the tube - y* y/R, dimensionless distance perpendicular to the axis - boundary layer thickness - * displacement thickness - /R, dimensionless boundary layer thickness - momentum thickness - absolute viscosity of the fluid - /, kinematic viscosity of the fluid - x/(R Re), dimensionless axial distance - density of the fluid - tU/(R Re), dimensionless time - _w wall shear stress     

Investigations of mean and turbulent flow characteristics of a two dimensional plane diffuser

This paper reports the investigation of mean and turbulent flow characteristics of a two-dimensional plane diffuser. Both experimental and theoretical details are considered. The experimental investigation consists of the measurement of mean velocity profiles, wall static pressure and turbulence stresses. Theoretical study involves the prediction of downstream velocity profiles and the distribution of turbulence kinetic energy using a well tested finite difference procedure. Two models, viz., Prandtl's mixing length hypothesis and k- model of turbulence, have been used and compared. The nondimensional static pressure distribution, the longitudinal pressure gradient, the pressure recovery coefficient, percentage recovery of static pressure, the variation of U _max/U _bar along the length of the diffuser and the blockage factor have been valuated from the predicted results and compared with the experimental data. Further, the predicted and the measured value of kinetic energy of turbulence have also been compared. It is seen that for the prediction of mean flow characteristics and to evaluate the performance of the diffuser, a simple turbulence model like Prandtl's mixing length hypothesis is quite adequate.List of symbols C ₁ , C ₂ ,C turbulence model constants - F _x body force - k kinetic energy of turbulence - l _m mixing length - L length of the diffuser - u, v, w rms value of the fluctuating velocity - u, v, w turbulent component of the velocity - mean velocity in the x direction - _A average velocity at inlet - U _bar average velocity in any cross section - U _max maximum velocity in any cross section - V mean velocity in the y direction - W local width of the diffuser at any cross section - x, y coordinates - dissipation rate of turbulence - _m eddy diffusivity - Von Karman constant - mixing length constant - _l laminar viscosity - _eff effective viscosity - v kinematic viscosity - density - _k effective Schmidt number for k - effective Schmidt number for - stream function - non dimensional stream function     

Unsteady convective diffusion in a rotating parallel plate channel

The paper presents an exact analysis of the dispersion of a passive contaminant in a viscous fluid flowing in a parallel plate channel driven by a uniform pressure gradient. The channel rotates about an axis perpendicular to its walls with a uniform angular velocity resulting in a secondary flow. Using a generalized dispersion model which is valid for all time, we evaluate the longitudinal dispersion coefficientsK _i (i=1, 2, ...) as functions of time. It is shown thatK ₁=0 andK ₃,K ₄, ... decay rapidly in comparison withK ₂. ButK ₂ decreases with increasing (the dimensionless rotation parameter) for values of upto approximately =2.2. ThereafterK ₂ increases with further increase in and its value gets saturated for large values of (say, 500) and does not change any further with increase in . A physical explanation of this anomalous behaviour ofK ₂ is given.

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Instationäre konvektive Diffusion in einem rotierenden Parallelplattenkanal

Zusammenfassung In dieser Untersuchung wird eine exakte Analyse der Ausbreitung eines passiven Kontaminierungsstoffes in einer zähen Flüssigkeit gegeben, die, befördert durch einen gleichförmigen Druckgradienten, in einem Parallelplattenkanal strömt. Der Kanal rotiert mit gleichförmiger Winkelgeschwindigkeit um eine zu seinen Wänden senkrechte Achse, wodurch sich eine Sekundärströmung ausbildet. Unter Verwendung eines generalisierten, für alle Zeiten gültigen Dispersionsmodells werden die longitudinalen DispersionskoeffizientenK _i (i=1, 2, ...) als Funktionen der Zeit ermittelt. Es wird gezeigt, daßK ₁=0 gilt und dieK ₃,K ₄, ... gegenüberK ₂ schnell abnehmen.K ₂ nimmt ab, wenn , der dimensionslose Rotationsparameter, bis etwa zum Wert 2,2 ansteigt. Danach wächstK ₂ mit bis auf einem Endwert an, der etwa ab =500 erreicht wird. Dieses anomale Verhalten vonK ₂ findet eine physikalische Erklärung.

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List of symbols C solute concentration - D molecular diffusivity - K _i longitudinal dispersion coefficients - 2L depth of the channel - P ₀ dimensionless pressure gradient along main flow - Pe Péclet number - q velocity vector - Q _x,Q _y mass flux along the main flow and the secondary flow directions - dimensionless average velocity along the main flow direction - (x, y, z) Cartesian co-ordinates Greek symbols dimensionless rotation parameter - the inclination of side walls withx-axis - kinematic viscosity - fluid density - dimensionless time - angular velocity of the channel - dimensionless distance along the main flow direction - dimensionless distance along the vertical direction - dimensionless solute concentration - integral of the dispersion coefficientK ₂() over a time interval     

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     

Stochastic response of structures with small geometric imperfections

Summary A probabilistic model of the geometric imperfections of a real structure is proposed, in order to provide a general theory of the stochastic response of structures in presence of small random deviations from the perfect scheme. The main statistical measures of the stochastic response are derived and an application to the study of a particular conservative elastic system is developed.

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Sommario Si propone una teoria generale della risposta probabilistica di strutture, in presenza di piccole deviazioni aleatorie dei dati iniziali rispetto allo schema geometrico perfetto. Si deducono le principali proprietà statistiche della risposta della struttura a sollecitazioni esterne deterministiche, e si sviluppa una applicazione riguardante il comportamento aleatorio di un particolare sistema elastico conservativo.

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List of symbols element of the sample space of events - _kn random variables modelling the structural imperfections - P(o) probability density of random variables - random imperfection of the unloaded structure - u additional displacement of the loaded structure - u_o deterministic fundamental solution for the perfect structure - difference between the additional displacement of the loaded structure and the deterministic fundamental solution for the perfect structure - V₁=u₁ buckling mode of the perfect structure - _i intrinsic coordinates of the structure - suitable measure of the magnitude of the random imperfections - scalar geometric variable representing the internal product - random imperfection divided by - single scalar variable denoting the magnitude of the prescribed loads - potential energy of the structure - potential energy of the perfect structure - difference between and - _c lowest critical load - _s real local maximum for the magnitude of the prescribed loads - _c divided by _S - E{} expected value of a random variable - ² variance of a random variable - , random variables defined by Eq. (21)     

Parameter determination for thermodynamical models of the pseudoelastic behaviour of shape memory alloy by resistivity measurements and infrared thermography

Two thermodynamical models of pseudoelastic behaviour of shape memory alloys have been formulated. The first corresponds to the ideal reversible case. The second takes into account the hysteresis loop characteristic of this shape memory alloys.Two totally independent techniques are used during a loading-unloading tensile test to determine the whole set of model parameters, namely resistivity and infrared thermography measurements. In the ideal case, there is no difficulty in identifying parameters.Infrared thermography measurements are well adapted for observing the phase transformation thermal effects.Notations 1 austenite 2 martensite - ₍₎ Macroscopic infinitesimal strain tensor of phase - ₍₂₎ ^f Traceless strain tensor associated with the formation of martensite phase - Macroscopic infiniesimal strain tensor - Macroscopic infinitesimal strain tensor deviator - _f Trace - Equivalent strain - ^pe Macroscopic pseudoelastic strain tensor - x Distortion due to parent (austenite =1)product (martensite =2) phase transformation (traceless symmetric second order tensor) - M Total mass of a system - M() Total mass of phase - V Total volume of a system - V() Total volume of phase - z=M(2)/M Weight fraction of martensite - 1-z=M(1)/M Weight fraction of austenite - u ₀ ^* () Specific internal energy of phase (=1,2) - s ₀ ^* () Specific internal entropy of phase - Specific configurational energy - Specific configurational entropy - ₀ ^f (T) Driving force for temperature-induced martensitic transformation at stress free state ( ₀ ^f T) = T ^*–Ts ^*) - Kirchhoff stress tensor - Kirchhoff stress tensor deviator - Equivalent stress - Cauchy stress tensor - Mass density - K Bulk moduli (K ₀=K) - L Elastic moduli tensor (order 4) - E Young modulus - Energetic shear (₀ = ) - Poisson coefficient - M _s ^o (M _F ^o ) Martensite start (finish) temperature at stress free state - A _s ^o (A _F ^o ) Austenite start (finish) temperature at stress free state - C _v Specific heat at constant volume - k Conductivity - Pseudoelastic strain obtained in tensile test after complete phase transformation (AM) (unidimensional test) - ₀ Thermal expansion tensor - r Resistivity - 1MPa 10⁶ N/m ² - ⁽⁾ Specific free energy of phase - _n Specific free energy at non equilibrium (R model) - _n ^eq Specific free energy at equilibrium (R model) - _n ^v Volumic part of ^eq - Specific free energy at non equilibrium (R _L model) - _conf Specific coherency energy (R _L model) - _c Specific free energy at constrained equilibria (R _L model) - _it (T) Coherency term (R _L model)     

Theoretical derivation of tangential velocity profiles in a flat vortex chamber-influence of turbulence and wall friction

Summary As part of a study on the hydrodynamics of a cyclone separator, a theoretical investigation of the flow pattern in a flat box cyclone (vortex chamber) has been carried out. Expressions have been derived for the tangential velocity profile as influenced by internal friction (eddy viscosity) and wall friction. The most important parameter controlling the tangential velocity profile is = –u ₀ R/(v+ ), where u ₀ is the radial velocity at the outer radius R of the cyclone, the kinematic liquid viscosity and is the kinematic eddy viscosity. For values of greater than about 10 the tangential velocity profile is nearly hyperbolic, for smaller than 1 the tangential velocity even decreases towards the centre. It is shown how and also the wall friction coefficient may be obtained from experimental velocity profiles with the aid of suitable graphs. Because of the close relation between eddy viscosity and eddy diffusion, measurements of velocity profiles in flat box cyclones will also provide information on the eddy motion of particles in a cyclone, a motion reducing its separation efficiency.List of symbols A cross-sectional area of cyclone inlet - h height of cyclone - p static pressure in cyclone - p static pressure difference in cyclone between two points on different radius - r radius in cyclone - r ₁ radius of cyclone outlet - R radius of cyclone circumference - u radial velocity in cyclone - u ₀ radial velocity at circumference of flat box cyclone - v tangential velocity - v ₀ tangential velocity at circumference of flat box cyclone - w axial velocity - z axial co-ordinate in cyclone - friction coefficient in flat box cyclone (for definition see § 5) - ₁ value of friction coefficient for ₁<< ₂ - ₂ value of friction coefficient for ₂<<1 - = - ₁ value of for ₁<< ₂ - ₂ value of for ₂<<1 - thickness of laminar boundary layer - =/h - turbulent kinematic viscosity - ratio of z to h - k ratio of height of cyclone to radius R of cyclone - parameter describing velocity profile in cyclone =–u ₀ R/(+) - kinematic viscosity of fluid - density of fluid - ratio of r to R - ₁ value of at outlet of cyclone - ₂ value of at inner radius of cyclone inlet - _w shear stress at cyclone wall - angular momentum in cyclone/angular momentum in cyclone inlet - ₁ value of at = ₁ - ₂ value of at = ₂     

Determination of permeability tensors for two-phase flow in homogeneous porous media: Theory

In this paper we continue previous studies of the closure problem for two-phase flow in homogeneous porous media, and we show how the closure problem can be transformed to a pair of Stokes-like boundary-value problems in terms of pressures that have units of length and velocities that have units of length squared. These are essentially geometrical boundary value problems that are used to calculate the four permeability tensors that appear in the volume averaged Stokes' equations. To determine the geometry associated with the closure problem, one needs to solve the physical problem; however, the closure problem can be solved using the same algorithm used to solve the physical problem, thus the entire procedure can be accomplished with a single numerical code.Nomenclature a a vector that maps V onto , m^-1. - A a tensor that maps V onto . - A area of the - interface contained within the macroscopic region, m². - A area of the -phase entrances and exits contained within the macroscopic region, m². - A area of the - interface contained within the averaging volume, m². - A area of the -phase entrances and exits contained within the averaging volume, m². - Bo Bond number (= (=(–)g²/). - Ca capillary number (= v/). - g gravitational acceleration, m/s². - H mean curvature, m^-1. - I unit tensor. - permeability tensor for the -phase, m². - viscous drag tensor that maps V onto V. - ^* dominant permeability tensor that maps onto v , m². - ^* coupling permeability tensor that maps onto v , m². - characteristic length scale for the -phase, m. - l characteristic length scale representing both and , m. - L characteristic length scale for volume averaged quantities, m. - n unit normal vector directed from the -phase toward the -phase. - n unit normal vector representing both n and n . - n unit normal vector representing both n and n . - P pressure in the -phase, N/m². - p superficial average pressure in the -phase, N/m². - p intrinsic average pressure in the -phase, N/m². - p –p , spatial deviation pressure for the -phase, N/m². - r ₀ radius of the averaging volume, m. - r position vector, m. - t time, s. - v fluid velocity in the -phase, m/s. - v superficial average velocity in the -phase, m/s. - v intrinsic average velocity in the -phase, m/s. - v –v , spatial deviation velocity in the -phase, m/s. - V volume of the -phase contained within the averaging volmue, m³. - averaging volume, m³. Greek Symbols V /, volume fraction of the -phase. - viscosity of the -phase, Ns/m². - density of the -phase, kg/m³. - surface tension, N/m. - (v +v ^T ), viscous stress tensor for the -phase, N/m².     

Loss of real characteristics for models of three-phase flow in a porous medium

In this paper we examine the generalized Buckley-Leverett equations governing threephase immiscible, incompressible flow in a porous medium, in the absence of gravitational and diffusive/dispersive effects. We consider the effect of the relative permeability models on the characteristic speeds in the flow. Using a simple idea from projective geometry, we show that under reasonable assumptions on the relative permeabilities there must be at least one point in the saturation triangle at which the characteristic speeds are equal. In general, there is a small region in the saturation triangle where the characteristic speeds are complex. This is demonstrated with the numerical results at the end of the paper.Symbols and Notation a, b, c, d entries of Jacobian matrix - A, B, C, D coefficients in Taylor expansion of _t, _v, _a - det J determinant of matrix J - dev J deviator of matrix J - J Jacobian matrix - L linear term in Taylor expansion for J near (s _v, s_a) = (0, 1) - m slope of r ₊ - p pressure - r± eigenvectors of Jacobian matrix - R real line - S intersection of saturation triangle with circle of radius centered at (1, 0) - S intersection of saturation triangle with circle of radius centered at (0, 1) - s _l, s_v, s_a saturations of phases (liquid, vapor, aqua) - tr J trace of matrix J - v _l , v _v , v _a phase flow rates (Darcy velocities) - v _T total flow rate - X, Y, Z entries of dev J - smooth closed curve inside saturation triangle - saturation triangle - _l, _v, _a phase density times gravitational acceleration times resevoir dip angle - K total permeability - _l, _v, _a three-phase relative permeabilities - _lv>, _la liquid phase relative permeabilities from two-phase data - _l, _v, _a mobilities of phases - _T total mobility - _l Corey mobility - _l, _v, _a phase viscosities - _± eigenvalues of Jacobian matrix - porosity Supported in part by National Science Foundation grant No. DMS-8701348, by Air Force Office of Scientific Research grant No. AFOSR-87-0283, and by Army Research Office grant No. DAAL03-88-K-0080.This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.     

Two-phase flow in heterogeneous porous media II: Numerical experiments for flow perpendicular to a stratified system

In this work, we make use of numerical experiments to explore our original theoretical analysis of two-phase flow in heterogeneous porous media (Quintard and Whitaker, 1988). The calculations were carried out with a two-region model of a stratified system, and the parameters were chosen be consistent with practical problems associated with groundwater flows and petroleum reservoir recovery processes. The comparison between theory (the large-scaled averaged equations) and experiment (numerical solution of the local volume averaged equations) has allowed us to identify conditions for which the quasi-static theory is acceptable and conditions for which a dynamic theory must be used. Byquasi-static we mean the following: (1) The local capillary pressure,everywhere in the averaging volume, can be set equal to the large-scale capillary pressure evaluated at the centroid of the averaging volume and (2) the large-scale capillary pressure is given by the difference between the large-scale pressures in the two immiscible phases, and is therefore independent of gravitational effects, flow effects and transient effects. Bydynamic, we simply mean a significant departure from the quasi-static condition, thus dynamic effects can be associated with gravitational effects, flow effects and transient effects. To be more precise about the quasi-static condition we need to refer to the relation between the local capillary pressure and the large-scale capillary pressure derived in Part I (Quintard and Whitaker, 1990). Herep _c ¦_y represents the local capillary pressure evaluated at a positiony relative to the centroid of the large-scale averaging volume, and {p _c }¦_x represents the large-scale capillary pressure evaluated at the centroid.In addition to{p _c } ^c being evaluated at the centroid, all averaged terms on the right-hand side of Equation (1) are evaluated at the centroid. We can now write the equations describing the quasi-static condition as , , This means that the fluids within an averaging volume are distributed according to the capillary pressure-saturation relationwith the capillary pressure held constant. It also means that the large-scale capillary pressure is devoid of any dynamic effects. Both of these conditions represent approximations (see Section 6 in Part I) and one of our main objectives in this paper is to learn something about the efficacy of these approximations. As a secondary objective we want to explore the influence of dynamic effects in terms of our original theory. In that development only the first four terms on the right hand side of Equation (1) appeared in the representation for the local capillary pressure. However, those terms will provide an indication of the influence of dynamic effects on the large-scale capillary pressure and the large-scale permeability tensor, and that information provides valuable guidance for future studies based on the theory presented in Part I.Roman Letters A scalar that maps {}*/t onto - A scalar that maps {}*/t onto - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - a vector that maps ({}*/t) onto , m - a vector that maps ({}*/t) onto , m - b vector that maps ({p}– g) onto , m - b vector that maps ({p}– g) onto , m - B second order tensor that maps ({p}– g) onto , m² - B second order tensor that maps ({p}– g) onto , m² - c vector that maps ({}*/t) onto , m - c vector that maps ({}*/t) onto , m - C second order tensor that maps ({}*/t) onto , m² - C second order tensor that maps ({}*/t) onto . m² - D third order tensor that maps ( ) onto , m - D third order tensor that maps ( ) onto , m - D second order tensor that maps ( ) onto , m² - D second order tensor that maps ( ) onto , m² - E third order tensor that maps () onto , m - E third order tensor that maps () onto , m - E second order tensor that maps () onto - E second order tensor that maps () onto - p _c =(), capillary pressure relationship in the-region - p _c =(), capillary pressure relationship in the-region - g gravitational vector, m/s² - largest of either or - - - i unit base vector in thex-direction - I unit tensor - K local volume-averaged-phase permeability, m² - K local volume-averaged-phase permeability in the-region, m² - K local volume-averaged-phase permeability in the-region, m² - {K } large-scale intrinsic phase average permeability for the-phase, m² - K –{K }, large-scale spatial deviation for the-phase permeability, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K ^* large-scale permeability for the-phase, m² - L characteristic length associated with local volume-averaged quantities, m - characteristic length associated with large-scale averaged quantities, m - I _i i = 1, 2, 3, lattice vectors for a unit cell, m - l characteristic length associated with the-region, m - ; characteristic length associated with the-region, m - l _H characteristic length associated with a local heterogeneity, m - - n unit normal vector pointing from the-region toward the-region (n =–n ) - n unit normal vector pointing from the-region toward the-region (n =–n ) - p pressure in the-phase, N/m² - p local volume-averaged intrinsic phase average pressure in the-phase, N/m² - {p } large-scale intrinsic phase average pressure in the capillary region of the-phase, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - P _c p –{p }, capillary pressure, N/m² - {p_c}^c large-scale capillary pressure, N/m² - r ₀ radius of the local averaging volume, m - R ₀ radius of the large-scale averaging volume, m - r position vector, m - , m - S /, local volume-averaged saturation for the-phase - S ^* {}*{}*, large-scale average saturation for the-phaset time, s - t time, s - u , m - U , m² - v -phase velocity vector, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - {v } large-scale intrinsic phase average velocity for the-phase in the capillary region of the-phase, m/s - {v } large-scale phase average velocity for the-phase in the capillary region of the-phase, m/s - v –{v }, large-scale spatial deviation for the-phase velocity, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - V local averaging volume, m³ - V volume of the-phase in, m³ - V large-scale averaging volume, m³ - V capillary region for the-phase within, m³ - V capillary region for the-phase within, m³ - V _c intersection of m³ - V volume of the-region within, m³ - V volume of the-region within, m³ - V () capillary region for the-phase within the-region, m³ - V () capillary region for the-phase within the-region, m³ - V () , region in which the-phase is trapped at the irreducible saturation, m³ - y position vector relative to the centroid of the large-scale averaging volume, m Greek Letters local volume-averaged porosity - local volume-averaged volume fraction for the-phase - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region (This is directly related to the irreducible saturation.) - {} large-scale intrinsic phase average volume fraction for the-phase - {} large-scale phase average volume fraction for the-phase - {}* large-scale spatial average volume fraction for the-phase - –{}, large-scale spatial deviation for the-phase volume fraction - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - a generic local volume-averaged quantity associated with the-phase - mass density of the-phase, kg/m³ - mass density of the-phase, kg/m³ - viscosity of the-phase, N s/m² - viscosity of the-phase, N s/m² - interfacial tension of the - phase system, N/m - , N/m - , volume fraction of the-phase capillary (active) region - , volume fraction of the-phase capillary (active) region - , volume fraction of the-region ( + =1) - , volume fraction of the-region ( + =1) - {p }– g, N/m³ - {p }– g, N/m³     

Mass transport in flow of sources

Steady and unsteady local concentration has been determined analytically for two- und three-dimensional sources and is presented for various boundary-concentrations, volumetric flows and diffusion coefficients. The steady cases have been evaluated numerically. In addition an unsteady two-dimensional mass transport has been evaluated.

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Stofftransport in Quellströmungen

Zusammenfassung Es wurden die stationäre und instationäre örtliche Konzentration von einer zwei- und drei-dimensionalen Quellströmung als Funktion verschiedener Randkonzentrationen, verschiedener Stromvolumen und Diffusionskoeffizienten analytisch bestimmt. Die stationären Fälle wurden numerisch ausgewertet. Außerdem wurde ein zwei-dimensionaler instationärer Stofftransport behandelt.

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Nomenclature a inner radius of circle (2-dimensional case), inner radius of sphere (three-dimensional case) - b } >a outer radius of circle (2-dimensional case), outer radius of sphere (three-dimensional case) - c concentration - c ₁,c ₂ given concentration at the boundariesr=a andb resp - c _i initial concentration at the timet=0 - D diffusion coefficient - I _n +1/2 modified spherical Bessel function - J _v ,Y _v Bessel function ofv-th order and first and second kind resp - k =b/a} > 1 diameter ratio - P _n ^o () Legendre polynomials - ¯ r, polar coordinates - r, , spherical coordinates - t time - u velocity in radial direction - V ₀ volumetric flow - ₀ V/4D flow parameter for two-dimensional flow - ₀ V ₀/8 D flow parameter for three-dimensional flow - _mn eigenvalues - _mn te] ² =n ² + ₀ ² ,=cos =r/a roots of determinant (28)     

Thin-Walled Beams: the Case of the Rectangular Cross-Section

In this paper we present an asymptotic analysis of the three-dimensional problem for a thin linearly elastic cantilever =×(0,l) with rectangular cross-section of sides and ², as goes to zero. Under suitable assumptions on the given loads, we show that the three-dimensional problem converges in a variational sense to the classical one-dimensional model for extension, flexure and torsion of thin-walled beams. Mathematics Subject Classifications (2000) 474K20, 74B10, 49J45.     

On Perturbative Expansions to the Stochastic Flow Problem

When analyzing stochastic steady flow, the hydraulic conductivity naturally appears logarithmically. Often the log conductivity is represented as the sum of an average plus a stochastic fluctuation. To make the problem tractable, the log conductivity fluctuation, f, about the mean log conductivity, lnK _G, is assumed to have finite variance, _f ². Historically, perturbation schemes have involved the assumption that _f ²<1. Here it is shown that _f may not be the most judicious choice of perturbation parameters for steady flow. Instead, we posit that the variance of the gradient of the conductivity fluctuation, _f ², is more appropriate hoice. By solving the problem withthis parameter and studying the solution, this conjecture can be refined and an even more appropriate perturbation parameter, , defined. Since the processes f and f can often be considered independent, further assumptions on f are necessary. In particular, when the two point correlation function for the conductivity is assumed to be exponential or Gaussian, it is possible to estimate the magnitude of f in terms of _f and various length scales. The ratio of the integral scale in the main direction of flow ( _x ) to the total domain length (L^*), _x ²=_x/L^*, plays an important role in the convergence of the perturbation scheme. For _x smaller than a critical value _c, _x < _c, the scheme's perturbation parameter is =_f/_x for one- dimensional flow, and =_f/_x ² for two-dimensional flow with mean flow in the x direction. For _x > _c, the parameter =_f/_x ³ may be thought as the perturbation parameter for two-dimensional flow. The shape of the log conductivity fluctuation two point correlation function, and boundary conditions influence the convergence of the perturbation scheme.     

An exact solution for non-darcy free convection from a horizontal line source of heat

In this paper we consider the free convection from a horizontal line source of heat which is embedded in an unbounded porous medium saturated with a fluid at rest under gravity. The convective fluid and the porous medium are in local thermal equilibrium.

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Eine exakte Lösung der nicht-darcy'schen freien Konvektion von einer horizontalen, linienförmigen Wärmequelle

Zusammenfassung In dem Aufsatz wird die freie Konvektion von einer horizontalen, linienförmigen Wärmequelle untersucht, die in ein unbegrenztes poröses Medium eingebettet ist. Die Poren des porösen Mediums sind mit einem Fluid gefüllt, das unter Schwerkrafteinfluß ruht. Das strömende Fluid und das poröse Medium sind örtlich im thermischen Gleichgewicht.

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Nomenclature c _p specific heat of convective fluid - F _o parameter,=/(vl>g - g acceleration due to gravity - k thermal conductivity of the saturated porous medium - l typical length scale of body - Q heat flux per unit length of a line source - Ra Rayleigh number, =gQl/²c_p - Ra _x local Rayleigh number, =xg Qx/ a²c_p - T temperature - T temperature of ambient fluid - u, x andy components of velocity - x, y coordinates vertically upwards and normal to axis of plume - X, Y non-dimensional coordinates vertically upwards and normal to axis of plume Greek symbols equivalent themal diffusivity - coefficient of thermal expansion - similarity variable - non-dimensional temperature - x permeability of porous medium - viscosity of convective fluid - v kinematic viscosity of convective fluid - density of convective fluid - stream function - non-dimensional stream function - the Forchheimer's coefficient     

Effects of MHD free convection and mass transfer on the flow past an oscillating infinite coaxial vertical circular cylinder

The effects of MHD free convection and mass transfer are taken into account on the flow past oscillating infinite coaxial vertical circular cylinder. The analytical expressions for velocity, temperature and concentration of the fluid are obtained by using perturbation technique.

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Einwirkungen von freier MHD-Konvektion und Stoffübertragung auf eine Strömung nach einem schwingenden unendlichen koaxialen vertikalen Zylinder

Zusammenfassung Die Einwirkungen der freien MHD-Konvektion und Stoffübertragung auf eine Strömung nach einem schwingenden, unendlichen, koaxialen, vertikalen Zylinder wurden untersucht. Die analytischen Ausdrücke der Geschwindigkeit, Temperatur und Fluidkonzentration sind durch die Perturbationstechnik erhalten worden.

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Nomenclature C _p Specific heat at constant temperature - C the species concentration near the circular cylinder - C _w the species concentration of the circular cylinder - C the species concentration of the fluid at infinite - * dimensionless species concentration - D chemical molecular diffusivity - g acceleration due to gravity - Gr Grashof number - Gm modified Grashof number - K thermal conductivity - Pr Prandtl number - r _a ,r _b radius of inner and outer cylinder - a, b dimensionless inner and outer radius - r,r coordinate and dimensionless coordinate normal to the circular cylinder - Sc Schmidt number - t time - t dimensionless time - T temperature of the fluid near the circular cylinder - T _w temperature of the circular cylinder - T temperature of the fluid at infinite - u velocity of the fluid - u dimensionless velocity of the fluid - U ₀ reference velocity - z,z coordinate and dimensionless coordinate along the circular cylinder - coefficient of volume expansion - * coefficient of thermal expansion with concentration - dimensionless temperature - H ₀ magnetic field intensity - coefficient of viscosity - _e permeability (magnetic) - kinematic viscosity - electric conductivity - density - M Hartmann number - dimensionless skin-friction - frequency - dimensionless frequency