Estimate of the transient conduction of heat in materials with linear thermal properties based on the solution for constant properties

A method of analysis is described which yields quasianalytical solutions for one and multidimensional unsteady heat conduction problems with linearly dependent thermal properties, such as thermal conductivity and volumetric specific heat. The method accomodates rather general thermal boundary conditions including arbitrary variations in surface temperature or in surface heat flux or a convective exchange with a fluid having even varying temperature. Once the solution for the identical problem but with constant properties has been developed, its practical realization is rather direct, being facilitated by a reduced number of iterations. The four applied examples given in this work show that a wide variety of nonlinear heat conduction problems can be tackled by this procedure without much difficulty. These simple solutions compare favorably with more laborious results reported in the archival heat transfer literature.

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Berechnung nichtstationärer Wärmeleitvorgänge mit linear temperaturabhängigen Stoffwerten aus der Lösung für konstante Stoffwerte

Zusammenfassung Es werden quasi-analytische Lösungen für ein- und mehrdimensionale nichtstationäre Wärmeleitprobleme mit linear temperaturabhängigen Stoffwerten, wie Wärmeleitfähigkeit und volumetrische Wärmekapazität, mitgeteilt. Die Methode gilt für recht allgemeine Randbedingungen wie beliebige Veränderungen der Oberflächentemperatur, der Wärmestromdichte oder auch konvektiven Wärmeaustausch mit veränderlicher Fluidtemperatur. Ist die Lösung für das identische Problem mit konstanten Stoffwerten bekannt, kann die Methode direkt mit einer begrenzten Zahl von Iterationen angewandt werden. Die vier hier mitgeteilten Beispiele zeigen, daß eine große Zahl nichtlinearer Wärmeleitprobleme auf diese Weise ohne Schwierigkeit angepackt werden können. Die einfachen Lösungen stimmen befriedigend mit komplizierteren Ergebnissen aus der Literatur überein.

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Nomenclature a side of square bar - B _i0 reference Biot number,hR/k₀ - B _i0 ^T transformed Biot number, equation (16) - c geometric parameter, equation (8) - h convective coefficient - k thermal conductivity - k ₀ value ofk atT ₀ - K dimensionless thermal conductivity,k/k ₀ - K _i value ofK at _i - K _i+1 value ofK at _i+1 - m _k slope of theK- line, equation (3) - m _s slope of theS- line, equation (4) - R characteristic length - s volumetric specific heat - s ₀ value of s at T₀ - S dimensionless volumetric specific heat, s/s₀ - S _i value ofS at _i - S _i+1 value of S at _i+1 - t time - T temperature - T ₀ reference temperature - x, y cartesian coordinates - X, Y dimensionless cartesian coordinates,x/a andy/a - thermal diffusivity - _k transformed time, equation (11) - _s transformed time, equation (37) - _k dimensionless time for variable conductivity, equation (8) - _s dimensionless time for variable specific heat, equation (34) - dimensionless temperature,T/T ₀ - dimensionless coordinate,r/R - ₀ value of at T₀ - _i lower value of the interval (_i, _i+1) - _i+1 upper value of the interval (_i, _i+1     

Nonlinear rheological behavior of a concentrated spherical silica suspension

Linear and nonlinear viscoelastic properties were examined for a 50 wt% suspension of spherical silica particles (with radius of 40 nm) in a viscous medium, 2.27/1 (wt/wt) ethylene glycol/glycerol mixture. The effective volume fraction of the particles evaluated from zero-shear viscosities of the suspension and medium was 0.53. At a quiescent state the particles had a liquid-like, isotropic spatial distribution in the medium. Dynamic moduli G^* obtained for small oscillatory strain (in the linear viscoelastic regime) exhibited a relaxation process that reflected the equilibrium Brownian motion of those particles. In the stress relaxation experiments, the linear relaxation modulus G(t) was obtained for small step strain (0.2) while the nonlinear relaxation modulus G(t, ) characterizing strong stress damping behavior was obtained for large (>0.2). G(t, ) obeyed the time-strain separability at long time scales, and the damping function h() (–G(t, )/G(t)) was determined. Steady flow measurements revealed shear-thinning of the steady state viscosity () for small shear rates (< ^–1; = linear viscoelastic relaxation time) and shear-thickening for larger (>^–1). Corresponding changes were observed also for the viscosity growth and decay functions on start up and cessation of flow, + (t, ) and _– (t, ). In the shear-thinning regime, the and dependence of ₊(t,) and _–(t,) as well as the dependence of () were well described by a BKZ-type constitutive equation using the G(t) and h() data. On the other hand, this equation completely failed in describing the behavior in the shear-thickening regime. These applicabilities of the BKZ equation were utilized to discuss the shearthinning and shear-thickening mechanisms in relation to shear effects on the structure (spatial distribution) and motion of the suspended particles.Dedicated to the memory of Prof. Dale S. Parson     

Die Torsionsspannungen in Wellen mit tiefen Längsnuten

Übersicht MitF(x, y) als Spannungsfunktion einer Welle ohne Nut und(, y) als Potentialfunktion des Quelle-Senke-Systems erhält man Spannungsfunktionen(, y) =F(x, y) –(, y) für Wellen mit tiefen Längsnuten. Es wird gezeigt, daß sich damit die Schubspannungen in den Läufern von Schraubenverdichtern ermitteln lassen.

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Shearing stresses in shafts with deep longitudinal grooves

Summary The stress functions(, y) of shafts with deep longitudinal grooves may be represented by(, y) =F(x, y) –(, y) whereF(x, y) is the stress function of a cylindrical shaft without grooves and(, y) denotes the potential function of the source-sink system. It is shown that the shearing stresses in rotors of screw-compressors may be obtained in this way.

     

Similar solutions and the asymptotics of filtration equations

In this paper we consider the asymptotic behavior of solutions of the quasilinear equation of filtration as t. We prove that similar solutions of the equation u _t = (u )_xx asymptotically represent solutions of the Cauchy problem for the full equation u _t = [(u)]_xx if (u) is close to u for small u.     

Oscillatory measurements of silicone oils —Loss and storage modulus master curves

The work describes a way to obtain loss modulus and storage modulus master curves from oscillatory measurements of silicone oils.The loss modulus master curve represents the dependence of the viscous flow behavior on · ₀ ^* and the storage modulus master curve — the dependence of the elastic flow behavior on · ₀ ^* .The relation between the values of the loss modulus and storage modulus master curves (at a certain frequency) is a measurement of the viscoelastic behavior of a system. The G/G-ratio depends on · ₀ ^* which leads to a viscoelastic master curve. The viscoelastic master curve represents the relation between the elastic and viscous oscillatory flow behavior.     

Measurement of the relaxation time in Darcy flow

The inertia of a liquid flowing through a porous medium is normally ignored, but if the acceleration is great, it may be important. The relaxation time, defined so that it alone accounts for the inertia, has been determined experimentally with a simple oscillator. A U-Tube is provided with a porous plug and filled with a liquid. During pendulation of the liquid, the frequency and the damping define the relaxation time. The measured value of the relaxation time is about 10 times the theoretical estimate derived from Navier-Stokes equation.Symbols E modulus of elasticity - E _D dissipated energy - E _k kinetic energy - g acceleration of gravity - G pressure gradient - h height - K ₀ permeability - L length of porous plug - n porosity - P dissipated power - pressure - R half the tube length - R _c radius of the tube bend - r radial coordinate - r _o radius of the tube - s coordinate along a streamline in the tube - t time - v flux per unit area - it relaxation time - , auxiliary variables - , v dynamic and kinematic viscosity - , velocity potential for inviscid flow and gravity potential - dissipation function - displacement of the liquid - , _o frequency of damped and undamped oscillations     

On the effects of uniform high suction on the rotationally symmetric flow of a conducting liquid near a stationary disc in the presence of a transverse magnetic field

In the present paper an attempt has been made to find out effects of uniform high suction in the presence of a transverse magnetic field, on the motion near a stationary plate when the fluid at a large distance above it rotates with a constant angular velocity. Series solutions for velocity components, displacement thickness and momentum thickness are obtained in the descending powers of the suction parameter a. The solutions obtained are valid for small values of the non-dimensional magnetic parameter m (= 4 _e ² H ₀ ² /) and large values of a (a2).Nomenclature a suction parameter - E electric field - E _r , E , E _z radial, azimuthal and axial components of electric field - F, G, H reduced radial, azimuthal and axial velocity components - H magnetic field - H _r , H , H _z radial, azimuthal and axial components of magnetic field - H ₀ uniform magnetic field - H* displacement thickness and momentum thickness ratio, */ - h induced magnetic field - h _r , h , h _z radial, azimuthal and axial components of induced magnetic field - J current density - m nondimensional magnetic parameter - p pressure - P reduced pressure - R Reynolds number - U ₀ representative velocity - V velocity - V _r , V , V _z radial, azimuthal and axial velocity components - w ₀ uniform suction through the disc. - density - electrical conductivity - kinematic viscosity - _e magnetic permeability - a parameter, (/)^1/2 z - a parameter, a - * displacement thickness - momentum thickness - angular velocity     

Low power laser Rayleigh probe for flow mixing studies

A laser Rayleigh correlation probe was constructed, which allows the application of low cost, low power (milliwatt) laser sources. It was tested for basic mixing studies in isothermal binary gas flows. Here, it can be used for the time and space resolved measurement of the concentration mean value and of all important statistical quantities, which give information on the distribution around the concentration mean value (rms, skewness, kurtosis) and on the relation of adjecent fluctuations in time or space (autocorrelation function, power spectral density).List of symbols c concentration (mole fraction) of investigated gas species - c time averagered mean concentration - c instantaneous fluctuating concentration - rms concentration - D Rayleigh intensity difference of two gas species (I _R1–I _R2) - d width of the rectangular channels (x-direction), see Fig. 3 - f frequency - G() Rayleigh autocorrelation function (ACF) - I ₀ intensity of irradiated laser light - I _Ri intensity of Rayleigh signal of gas species i - K, k calibration constant of Rayleigh probe - l lenght of observed scattering volume - n(t) temporally fluctuating number density of gas molecules - R() normalized ACF - S Rayleigh intensity of gas components 2 in a binary mixture (I _R2) - T gas temperature - t time - u exit velocity - skewness of the concentration distribution around the mean value - kurtosis of the concentration distribution around the mean value - (d/d)_eff effective scattering cross section of the binary gas mixture - solid angle of collection optics - delay time - sample time     

The moving, plane heat source in an elastic medium

Summary This note presents an exact solution for the stress and displacement field in an unbounded and transversely constrained elastic medium resulting from the motion of a plane heat source travelling through the medium at constant speed in the direction normal to the source plane.Nomenclature mass density - diffusivity - thermal conductivity - Q heat emitted by plane heat source per unit time per unit area - speed of propagation of plane heat source - shear modulus - Poisson's ratio - T temperature - _x, _y, _z normal stress components - u _x, u_y, u_z displacement components - c speed of irrotational waves - t time - x, y, z Cartesian coordinates - =x–vt moving coordinate     

Die Analogie zwischen den Verschiebungen und den Spannungsfunktionen in der Biegetheorie der Kreiszylinderschale

Zusammenfassung Für die Kreiszylinderschale wurde eine Biegetheorie aufgestellt, in der die Gleichgewichtsbedingungen (unter Voraussetzung der Symmetrie des Momententensors M _ik ) durch drei Spannungsfunktionen ₁, ₂, ₃ exakt erfüllt sind. Bei der Definition der Deformationsgrößen und der Einführung der Elastizitätsgesetze war die Reißner-Meißnersche Theorie der symmetrisch belasteten Rotationsschale das Vorbild. Die drei Differentialgleichungen für die Verschiebungen _{1 2}, ₃ unterscheiden sich von den drei Differentialgleichungen für die Spannungsfunktionen ₁, ₂, ₃ formal nur im Vorzeichen der Poissonschen Querkontraktionsziffer v. Die beiden Differentialgleichungen achter Ordnung, die man nach Eliminationsprozessen sowohl für ₃ als auch für ₃ erhält, unterscheiden sich nicht mehr voneinander. So trifft man bei der Zylinderschale die Timpe-Wieghardtsche Analogie zwischen Durchbiegung ₃ der Platte und Airyscher Spannungsfunktion ₃ der Scheibe wieder.Es konnte ferner gezeigt werden, daß unsere neue Biegetheorie der bekannten Flüggeschen Theorie an Genauigkeit nicht nachsteht.Es ist wohl nicht zu bezweifeln, daß auch bei Schalen beliebiger Gestalt unsere Analogie vorhanden ist. Sie scheint uns wertvoll als Ordnungsprinzip inmitten der Fülle von Gleichungen, die nun einmal zu einer Schalentheorie gehören.Die Formulierung des Schalenproblems mit Hilfe der drei Spannungsfunktionen ₁, ₂, ₃ wird sich immer dann empfehlen, wenn die Randbelastung vorgegeben ist. Denn dann lassen sich die Randbedingungen in den Spannungsfunktionen übersichtlicher formulieren als in den Verschiebungen. Auch die Gewißheit, daß selbst durch radikales Streichen lästiger Glieder in den Differentialgleichungen der Spannungsfunktionen die Gleichgewichtsbedingungen nicht verletzt werden, mag manchem Rechner angenehm sein.     

Translatory accelerating motion of a circular disk in a viscous fluid

The exact solution of the equation of motion of a circular disk accelerated along its axis of symmetry due to an arbitrarily applied force in an otherwise still, incompressible, viscous fluid of infinite extent is obtained. The fluid resistance considered in this paper is the Stokes-flow drag which consists of the added mass effect, steady state drag, and the effect of the history of the motion. The solutions for the velocity and displacement of the circular disk are presented in explicit forms for the cases of constant and impulsive forcing functions. The importance of the effect of the history of the motion is discussed.Nomenclature a radius of the circular disk - b one half of the thickness of the circular disk - C dimensionless form of C ₁ - C ₁ magnitude of the constant force - D fluid drag force - f(t) externally applied force - F() dimensionaless form of applied force - F ₀ initial value of F - g gravitational acceleration - H() Heaviside step function - k magnitude of impulsive force - K dimensionless form of k - M a dimensionless parameter equals to (1+37#x03C0;_s/4_f) - S displacement of disk - t time - t ₁ time of application of impulsive force - u velocity of the disk - V dimensionless velocity - V ₀ initial velocity of V - V _t terminal velocity - parameter in (13) - parameter in (13) - (t) Dirac delta function - ratio of b/a - () function given in (5) - dynamical viscosity of the fluid - kinematic viscosity of the fluid - _f fluid density - _s mass density of the circular disk - dimensionless time - _i dimensionless form of t _i - dummy variable - dummy variable     

Some remarks on differential equations of quadratic type

In this paper we study differential equations of the formx(t) + x(t)=f(x(t)), x(0)=x ₀ C HereC is a closed, bounded convex subset of a Banach spaceX,f(C) C, and it is often assumed thatf(x) is a quadratic map. We study the differential equation by using the general theory of nonexpansive maps and nonexpansive, non-linear semigroups, and we obtain sharp results in a number of cases of interest. We give a formula for the Lipschitz constant off: C C, and we derive a precise explicit formula for the Lipschitz constant whenf is quadratic,C is the unit simplex inR ⁿ, and thel ₁ norm is used. We give a new proof of a theorem about nonexpansive semigroups; and we show that if the Lipschitz constant off: CC is less than or equal to one, then lim_tf(x(t))–x(t)=0 and, if {x(t):t 0} is precompact, then lim_tx(t) exists. Iff¦C=L¦C, whereL is a bounded linear operator, we apply the nonlinear theory to prove that (under mild further conditions on C) lim_t f(x(t))–x(t)=0 and that lim_t x(t) exists if {x(t):t 0} is precompact. However, forn 3 we give examples of quadratic mapsf of the unit simplex ofR ⁿ into itself such that lim_t x(t) fails to exist for mostx ₀ C andx(t) may be periodic. Our theorems answer several questions recently raised by J. Herod in connection with so-called model Boltzmann equations.     

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)     

The theory of a vibrating-rod viscometer

The paper presents a complete theory for a viscometer based upon the principle of a circular-section rod, immersed in a fluid, performing transverse oscillations perpendicular to its axis. The theory is established as a result of a detailed analysis of the fluid flow around the rod and is subject to a number of criteria which subsequently constrain the design of an instrument. Using water as an example it is shown that a practical instrument can be designed so as to enable viscosity measurement with an accuracy of ±0.1%, although it is noted that many earlier instruments failed to satisfy one or more of the newly-established constraints.Nomenclature A, D constants in equation (46) - A _m , B _m , C _m , D _m constants in equations (50) and (51) - A _j , B _j constants in equation (14) - a _j ⁺ , a _j ^– wavenumbers given by equation (15) - C _f drag coefficient defined in equation (53) - c speed of sound - D _b drag force of fluid b - D ₀ coefficient of internal damping - E extensional modulus - f(z) initial deformation of rod - f(), F _m () functions of defined in equation (41) - F force in the rod - force per unit length near t=0 - F dimensionless force per unit length near t=0 - g _m amplitude of transient force - G modulus of rigidity - h, h* functions defined by equations (71) and (72) - H functions defined by equation (69) and (70) - I second moment of area - I _0,1, J _0,1, K _0,1 modified Bessel functions - k, k functions defined in equations (2) - L half-length of oscillator - Ma Mach number - m _b added mass per unit length of fluid b - m _s mass per unit length of solid - n _j eigenvalue defined in equations (15) and (16) - R radius of rod - R _c radius of container - r radial coordinate - T tension - T _visc temperature rise due to heat generation by viscous dissipation - t time - v _r , v radial and angular velocity components - y lateral displacement - y ₀ initial lateral displacement - y ₁, y ₂ successive maximum lateral displacement - z axial coordinate - dimensionless tension - dimensionless mass of fluid - dimensionless drag of fluid - amplification factor - logarithmic decrement in a fluid - _a , _b logarithmic decrement in fluids a and b - ₀ logarithmic decrement in vacuo - _j logarithmic decrement in mode j in a fluid - spatial resolution of amplitude - _v voltage resolution - r, , , _s, , increments in R, , , _s , , - dimensionless amplitude of oscillation - dimensionless axial coordinate - angular coordinate - _f thermal conductivity of fluid - viscosity of fluid - viscosity of fluid calculated on assumption that * - _a , _b viscosity of fluids a and b - _m constants in equation (10) - dimensionless displacement - _j j the component of - density of fluid - _a , _b density of fluids a and b - _s density of tube or rod material - dimensionless radial coordinate - * dimensionless radius of container - dimensionless times - spatial component of defined in equation (11) - _j , _tm jth, mth component of - dimensionless streamfunction - ⁰, ¹ components of in series expansion in powers of - streamfunction - dimensionless frequency (based on ) - angular frequency - ₀ angular frequency in absence of fluid and internal damping - _j angular frequency in mode j in a fluid - _a , _b frequencies in fluids a and b     

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.     

The theory of a vibrating-rod densimeter

The paper presents a theory of a device for the accurate determination of the density of fluids over a wide range of thermodynamic states. The instrument is based upon the measurement of the characteristics of the resonance of a circular section tube, or rod, performing steady, transverse oscillations in the fluid. The theory developed accounts for the fluid motion external to the rod as well as the mechanical motion of the rod and is valid over a defined range of conditions. A complete set of working equations and corrections is obtained for the instrument which, together with the limits of the validity of the theory, prescribe the parameters of a practical design capable of high accuracy.Nomenclature A, B, C, D constants in equation (60) - A _j , B _j constants in equation (18) - a _j ⁺ , a _j ^– wavenumbers given by equation (19) - C _f drag coefficient defined in equation (64) - C _f ^/0 , C _f ^/1 components of C _f in series expansion in powers of - c speed of sound - D _b drag force of fluid b - D ₀ coefficient of internal damping - E extensional modulus - force per unit length - F _j ⁺ , F _j ^– constants in equation (24) - f, g functions of defined in equations (56) - G modulus of rigidity - I second moment of area - K constant in equation (90) - k, k constants defined in equations (9) - L half-length of oscillator - Ma Mach number - m _a mass per unit length of fluid a - m _b added mass per unit length of fluid b - m _s mass per unit length of solid - n _j eigenvalue defined in equation (17) - P power (energy per cycle) - P _a , P _b power in fluids a and b - p pressure - R radius of rod or outer radius of tube - R _c radius of container - R _i inner radius of tube - r radial coordinate - T tension - T _visc temperature rise due to heat generation by viscous dissipation - t time - v _r , v radial and angular velocity components - y lateral displacement - z axial coordinate - dimensionless tension - _a dimensionless mass of fluid a - _b dimensionless added mass of fluid b - _b dimensionless drag of fluid b - dimensionless parameter associated with - ₀ dimensionless coefficient of internal damping - dimensionless half-width of resonance curve - dimensionless frequency difference defined in equation (87) - spatial resolution of amplitude - R, , , _s , increments in R, , , _s , - dimensionless amplitude of oscillation - dimensionless axial coordinate - ratio of to - _a , _b ratios of to for fluids a and b - angular coordinate - parameter arising from distortion of initially plane cross-sections - _f thermal conductivity of fluid - dimensionless parameter associated with - viscosity of fluid - _a , _b viscosity of fluids a and b - dimensionless displacement - _j jth component of - density of fluid - _a , _b density of fluids a and b - _s density of tube or rod material - density of fluid calculated on assumption that * - dimensionless radial coordinate - * dimensionless radius of container - dimensionless times - _{rr rr}, _r radial normal and shear stress components - spatial component of defined in equation (13) - _j jth component of - dimensionless streamfunction - ⁰, ¹ components of in series expansion in powers of - phase angle - _r phase difference - _ra , _rb phase difference for fluids a and b - streamfunction - _j jth component defined in equation (22) - dimensionless frequency (based on ) - _a , _b dimensionless frequency in fluids a and b - _s dimensionless frequency (based on _s ) - angular frequency - ₀ resonant frequency in absence of fluid and internal damping - _r resonant frequency in absence of internal fluid - _ra , _rb resonant frequencies in fluids a and b - dimensionless frequency - dimensionless frequency when _a vanishes - dimensionless frequencies when _a vanishes in fluids a and b - dimensionless resonant frequency when _a , _b, _b and ₀ vanish - dimensionless resonant frequency when _a , _b and _b vanish - dimensionless resonant frequency when _b and _b vanish - dimensionless frequencies at which amplitude is half that at resonance     

A phase plane discussion of convergence to travelling fronts for nonlinear diffusion

The paper is concerned with the asymptotic behavior as t of solutions u(x,t) of the equation in the case f(0)=f(1)=0, with f(u) non-positive for u(>0) sufficiently close to zero and f(u) non-negative for u(<1) sufficiently close to 1. This guarantees the uniqueness (but not the existence) of a travelling front solution u;U(x–ct), U(–);0, U();, and it is shown in essence that solutions with monotonic initial data converge to a translate of this travelling front, if it exists, and to a stacked combination of travelling fronts if it does not. The approach is to use the monotonicity to take u and t as independent variables and p = u _x as the dependent variable, and to apply ideas of sub- and super-solutions to the diffusion equation for p.This research was sponsored by the United States Army under Contract No. DAAG29-75-C-0024.     

Reservoir transport equations by compositional approach

The objective of this paper is to present an overview of the fundamental equations governing transport phenomena in compressible reservoirs. A general mathematical model is presented for important thermo-mechanical processes operative in a reservoir. Such a formulation includes equations governing multiphase fluid (gas-water-hydrocarbon) flow, energy transport, and reservoir skeleton deformation. The model allows phase changes due to gas solubility. Furthermore, Terzaghi's concept of effective stress and stress-strain relations are incorporated into the general model. The functional relations among various model parameters which cause the nonlinearity of the system of equations are explained within the context of reservoir engineering principles. Simplified equations and appropriate boundary conditions have also been presented for various cases. It has been demonstrated that various well-known equations such as Jacob, Terzaghi, Buckley-Leverett, Richards, solute transport, black-oil, and Biot equations are simplifications of the compositional model.Notation List B reservoir thickness - B formation volume factor of phase - Cⁱ mass of component i dissolved per total volume of solution - C ⁱ mass fraction of component i in phase - C heat capacity of phase at constant volume - C_p heat capacity of phase at constant pressure - D ⁱ hydrodynamic dispersion coefficient of component i in phase - D_MTf thermal liquid diffusivity for fluid f - F = F(x, y, z, t) defines the boundary surface - f_p fractional flow of phase - g gravitational acceleration - H_p enthalpy per unit mass of phase - J_p volumetric flux of phase - k_rf relative permeability to fluid f - k₀ absolute permeability of the medium - M_p ⁱ mass of component i in phase - n porosity - N rate of accretion - P_f pressure in fluid f - p_ca capillary pressure between phases and =p-p - Rⁱ rate of mass transfer of component i from phase to phase - Rⁱ _source source rate of component i within phase - S saturation of phase - s gas solubility - T temperature - t time - U displacement vector - u velocity in the x-direction - v velocity in the y-direction - V volume of phase - V_s velocity of soil solids - W_i body force in coordinate direction i - x horizontal coordinate - z vertical coordinate Greek Letters _p volumetric coefficient of compressibility - _T volumetric coefficient of thermal expansion - _ij Kronecker delta - volumetric strain - _m thermal conductivity of the whole matrix - internal energy per unit mass of phase - gf suction head - density of phase - _ij tensor of total stresses - _ij tensor of effective stresses - volumetric content of phase - f viscosity of fluid f     

Reflection, transmission and excitation of SH-surface waves by a discontinuity in mass-loading on a semi-infinite elastic medium

The scattering of an SH-wave by a discontinuity in mass-loading on a semi-infinite elastic medium is investigated theoretically. The incident wave is either a plane body wave or a plane SH-surface wave. The problem is reduced to a Wiener-Hopf problem for the scattered wave. In this problem the amplitude spectral density of the particle displacement occurs as unknown function. Special attention is given to the numerical values of the surface wave contributions to the scattered field.Nomenclature x ₁, x ₂, x ₃ Cartesian coordinates - , polar coordinates in x ₁, x ₃-plane - volume mass density - surface mass density of mass-loading - , Lamé constants - U scalar wave function, defined by (2.1) - c _S speed of propagation of uniform shear waves in bulk medium (c _S=(/)^1/2) - angular frequency - t time - k _S wave number of uniform shear waves (k _S=/c _S) - reduced specific acoustic impedance of mass-loading (=k _S /) - k _m wave number of SH-surface wave (k _m=k _S(1+ ²)^1/2) - _1,2,3 partial differentiation with respect to x _1,2,3 - ⁱ angle between x ₃-axis and direction of propagation of incident body wave - ⁱ wave number in horizontal direction of incident body wave ( ⁱ=k _S sin( ⁱ)) - ⁱ wave number in vertical direction of incident body wave ( ⁱ=k _S cos( ⁱ)) - C _1,2 complex amplitude of surface wave excited by a body wave - R reflection factor of surface wave, when surface wave is incident - T transmission factor of surface wave, when surface wave is incident - S particle displacement vector The research presented in this paper has been carried out with partial financial support from the Delfts Hogeschoolfonds.     

Dynamic-mechanical properties of a bitumen-silica composite

Summary The dynamic-mechanical behaviour of bitumensilica composites is described by a linear biparabolic model. Its mathematical expression allows the calculation of the mean relaxation times () either at different temperatures and given filler contents or for diverse filler contents () at imposed temperatures. At fixed filler concentration and within restricted temperature domains, obeys Arrhenius' law. The activation energies are respectively close to 10 kcal/mole (creep) and 30 kcal/mole (glass-transition). varies exponentially with. The mathematical treatment of the expressions ofE , as a function of temperature and of, leads to a general equation relating the complex modulus to temperature, frequency and filler content. A unique master curve, accounting for the viscoelastic behaviour of the composites, in limited ranges, can thus be constructed.

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Zusammenfassung Das dynamisch-mechanische Verhalten von Bitumen-Siliziumdioxyd-Zusammensetzungen kann durch ein lineares biparabolisches Modell beschrieben werden. Sein mathematischer Ausdruck erlaubt die Ausrechnung der mittleren Relaxationszeiten () entweder für verschiedene Temperaturen bei gegebenem Füllstoffgehalt oder für unterschiedliche Siliziumdioxydmengen () bei bekannter Temperatur. Für einen bestimmten Füllstoffgehalt folgt in einem beschränkten Temperaturbereich dem Arrheniusschen Gesetz. Die Aktivierungsenergien betragen näherungsweise 10 kcal/Mol (Fließprozeß) bzw. 30 kcal/Mol (Glasübergang). ändert sich exponentiell mit. Die mathematische Umformung der Ausdrücke fürE und als Funktion der Temperatur und des Parameters ergibt eine allgemeine Gleichung, die den komplexen Modul mit der Temperatur, der Frequenz und dem Füllstoffgehalt verknüpft. Man kann eine einzige Masterkurve bilden, die das viskoelastische Verhalten der Zusammensetzungen zumindest in begrenzten Bereichen beschreibt.

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Résumé Le comportement mécanique dynamique des composites à base de bitume et de silice peut être décrit par un modèle biparabolique linéaire. L'expression mathématique permet le calcul des temps moyens () de relaxation d'une part aux différentes températures, à taux de charge donné, et d'autre part pour diverses valeurs des taux de charge (paramètre) à température imposée. A taux de charge donné, et pour des domaines de température restreints, suit la loi d'Arrhénius. Les énergies apparentes d'activation sont respectivement voisines de 10 kcal/mole (processus de fluage) et de 30 kcal/mole (passage à l'état vitreux). Avec, varie exponentiellement. L'évaluation mathématique deE , de en fonction deT et de conduit à une expression générale du module complexe en fonction de la température, de la fréquence et du taux de charge. On peut donc construire une courbe maitresse unique qui décrit entièrement, mais dans des domaines restreints, le comportement viscoélastique des composites.

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With 6 figures