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.     

Singular solutions of some quasilinear elliptic equations

We study isolated singularities of the quasilinear equation in an open set of ^N , where 1 < p N, p -1 q < N(p — 1)/ (N -p). We prove that, for any positive solution, if a singularity at the origin is not removable then either or u(x)/(x) any positive constant as x 0 where is the fundamental solution of the p-harmonic equation: . Global positive solutions are also classified.     

Pulsatile blood flow through arterioles

An analytical study was made to examine the effect of vascular deformability on the pulsatile blood flow in arterioles through the use of a suitable mathematical model. The blood in arterioles is assumed to consist of two layers — both Newtonian but with differing coefficients of viscosity. The flow characteristics of blood as well as the resistance to flow have been determined using the numerical computations of the resulting expressions. The applicability of the model is illustrated using numerical results based on the existing experimental data. r, z coordinate system - u, axial/longitudinal velocity component of blood - p pressure exerted by blood - _b density of blood - µ viscosity of blood - t time - , displacement components of the vessel wall - T _t0,T ₀ known initial stresses - density of the wall material - h thickness of the vessel wall - T _t,T stress components of the vessel - K _l,K _r components of the spring coefficient - C _l,C _r components of the friction coefficient - M _a additional mass of the mechanical model - r ₁ outer radius of the vessel - thickness of the plasma layer - r ₁ inner radius of the vessel - circular frequency of the forced oscillation - k wave number - E ₀,E _t, , _t material parameters for the arterial segment - µ _p viscosity of the plasma layer - Q total flux - Q _p flux across the plasma zone - Q _h flux across the core region - Q mean flow rate - resistance to flow - P pressure difference - l length of the segment of the vessel     

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     

Response of an elastic Bingham fluid to oscillatory shear

The response of an elastic Bingham fluid to oscillatory strain has been modeled and compared with experiments on an oil-in-water emulsion. The newly developed model includes elastic solid deformation below the yield stress (or strain), and Newtonian flow above the yield stress. In sinusoidal oscillatory deformations at low strain amplitudes the stress response is sinusoidal and in phase with the strain. At large strain amplitudes, above the yield stress, the stress response is non-linear and is out of phase with strain because of the storage and release of elastic recoverable strain. In oscillatory deformation between parallel disks the non-uniform strain in the radial direction causes the location of the yield surface to move in-and-out during each oscillation. The radial location of the yield surface is calculated and the resulting torque on the stationary disk is determined. Torque waveforms are calculated for various strains and frequencies and compared to experiments on a model oil-in-water emulsion. Model parameters are evaluated independently: the elastic modulus of the emulsion is determined from data at low strains, the yield strain is determined from the phase shift between torque and strain, and the Bingham viscosity is determined from the frequency dependence of the torque at high strains. Using these parameters the torque waveforms are predicted quantitatively for all strains and frequencies. In accord with the model predictions the phase shift is found to depend on strain but to be independent of frequency.Notation A plate strain amplitude (parallel plates) - A _R plate strain amplitude at disk edge (parallel disks) - G elastic modulus - m torque (parallel disks) - M normalized torque (parallel disks) = 2m/R ³₀ - N ratio of viscous to elastic stresses (parallel plates) =µ A/ ₀ ratio of viscous to elastic stresses (parallel disks) =µ A _R/₀ - r normalized radial position (parallel disks) =r/R - r radial position (parallel disks) - R disk radius (parallel disks) - t normalized time = t — /2 - t time - _E elastic strain - _P plate strain (displacement of top plate or disk divided by distance between plates or disks) - _PR plate strain at disk edge (parallel disks) - ₀ yield strain - _E normalized elastic strain = _E/₀ - _P normalized plate strain = _P/₀ - _PR normalized plate strain at disk edge (parallel disks) = _PR/₀ - ₀ normalized plate strain amplitude (parallel plates) =A/ ₀ — normalized plate strain amplitude at disk edge (parallel disks) =A _R/₀ - phase shift between _P andT (parallel plates) — phase shift between _PR andM (parallel disks) - µ Bingham viscosity - stress - ₀ yield stress - T normalized stress =/ ₀ - frequency     

Über eine Anwendung der Hillschen Minimalbedingung in der Plastizitätstheorie

Zusammenfassung Auf dem gezeigten Weg wurden die Spannungen _r , , _z berechnet, wobei an Stelle der Veränderlichen r und die dimensionlosen Größen x _i = r _i /, x=r/ und x _a = r/ in die Rechnung eingeführt wurden. Die Funktion (r, ) wurde dann für den Bereich 0,45x_i1,0, 1x_a 2 tabuliert. Hierbei zeigte sich, daß der Rechenaufwand bei der Durchrechnung eines Einzelbeispiels nach der Charakteristikenmethode wesentlich geringer ist. Bei der Anlage von Zahlentafeln zur Berechnung von Spannungen für beliebige Durchmesserverhältnisse ergab sich, daß der aufgezeigte Wege zu geringerem Rechenaufwand führt. Für das Beispiel r _i /r _a=1/2 wurden die Rohraufweitungen bestimmt und diese Werte noch durch praktische Versuche nachgeprüft. Hierbei ergab sich, daß die theoretisch bestimmten Rohraufweitungen in dem Streubereich der gemessenen Rohraufweitungen lagen, wobei Messungen an drei Rohren aus demselben Material und demselben Rohrverhältnis durchgeführt wurden. Insbesondere stimmten die theoretischen Rohraufweitungen auch mit den gemessenen Rohraufweitungen überein, wenn das Rohr entlastet wurde und die Restdeformationen bestimmt wurden.Daraus kann geschlossen werden, daß durch die berücksichtigte lineare Verfestigung die tatsächlichen Verhältnisse außerordentlich gut erfaßt werden.Der sogenannte Platzdruek eines Rohres kann auf rein rechnerischem Weg nicht erfaßt werden, da für =r_a die geometrische Gestalt des Rohres instabil wird. Bei den Versuchen zeigt sich, wenn der Innendruck über p _i ( =r **** _a ) gesteigert wird, daß das Rohr schon bei geringen Überschreitungen aufzubauchen beginnt.Meinem Lehrer Herrn Prof. Dr. Dr. R. Grammel zum 65. Geburtstag gewidmet.     

Theory of single well tests in a leaky aquifer. Analytical solution for non steady flow

The evaluation of a pump test or a slug test in a single well that completely penetrates a leaky aquifer does not yield a unique relation between the hydraulic properties of the aquifer, independent of the testing conditions. If the flow is transient, the drawdown is characterized by a single similarity parameter that does not distinguish between the storativity and the leakage factor. If the flow is quasi stationary, the drawdown is characterized by a single similarity parameter that does not distinguish between the transmissivity and the leakage factor. The general non steady solution, which is derived in closed form, is characterized bythree similarity parameters.Nomenclature a e 0.8905 = auxiliary parameter - b thickness of the aquifer - b _c thickness of the semipervious stratum - B() auxiliary function - f(s),g(s) auxiliary functions in the complex plane - F(t),G(t) auxiliary functions of time - h undisturbed level of the phreatic surface - K conductivity of the aquifer - K _c conductivity of the semipervious stratum - m ₀ leakage factor - m dimensionless leakage factor - N(s) auxiliary function in the complex plane - Q _w (t) discharge flux - Q steady discharge flux - Q ₀ constant discharge flux during limited time - Q(t) dimensionless discharge flux - r ₀ radius of the well - r radial coordinate - r dimensionless radial coordinate - s complex variable - s ₀ pole - S storativity of the aquifer - S _n n'th part of an integration contour - t time - t dimensionless time - T transmissivity of the aquifer - ,,,,, dimensionless parameters - Euler's number - dummy variable - ₁(), ₂() auxiliary functions - (r, t) drawdown - ₀(t) drawdown in the well - (r, t) dimensionless drawdown - ₀(t) dimensionless drawdown in the well     

A class of hypo-elastic non-elastic materials and their thermodynamics

Thermodynamics is developed for a class of thermo-hypo-elastic materials. It is shown that materials of this class obey the laws of thermodynamics, but are not elastic.

Table of Symbols

Latin Letters A _ijkl tensor-valued function of t _ij appearing in hypo-elastic constitutive relation - B _ijkl another tensor-valued function. See equation (4.2) - B the square of - d _ij rate of deformation tensor - d _ij deviator of rate of deformation - f, k functions of pressure, p - g, h functions of the invariant - p pressure - q _i heat flux vector - s _ij stress deviator - _ij co-rotational derivative of stress deviator - t time - t ₁ t ₂ specific values of time - t _ij stress tensor - t _ij ⁰ a specific value of stress - T Temperature - T ₀ a specific value of temperature - u _i velocity - V(t) a material volume as a function of time, t - V ₀ a material volume at a reference configuration - W work (W = work done in a deformation—section 5) Sript Letters Specific internal energy - Specific Helmholtz free energy - G Specific Gibbs function Greek Letters an invariant of the stress deviator—see eq. (2.4) - _ij kroneker delta - (W = work done in a deformation—section 5) - specific entropy - hypo-elastic potential - hypo-elastic potential - mass density - ₀ mass density in a reference configuration - specific volume = 1/ - a function of p - _ijkl a constant tensor—see eq. (2.5) - –G/ - _ij rate of rotation tensor This work is dedicated to Jerald L. Ericksen, without whose influence it would not have been possible     

The extension of Beltrami's ellipsoid to anisotropic bodies

Summary The spectral decomposition of the compliance, stiffness, and failure tensors for transversely isotropic materials was studied and their characteristic values were calculated using the components of these fourth-rank tensors in a Cartesian frame defining the principal material directions. The spectrally decomposed compliance and stiffness or failure tensors for a transversely isotropic body (fiber-reinforced composite), and the eigenvalues derived from them define in a simple and efficient way the respective elastic eigenstates of the loading of the material. It has been shown that, for the general orthotropic or transversely isotropic body, these eigenstates consist of two double components, ₁ and ₂ which are shears (₂ being a simple shear and ₁, a superposition of simple and pure shears), and that they are associated with distortional components of energy. The remaining two eigenstates, with stress components ₃, and ₄, are the orthogonal supplements to the shear subspace of ₁ and ₂ and consist of an equilateral stress in the plane of isotropy, on which is superimposed a prescribed tension or compression along the symmetry axis of the material. The relationship between these superimposed loading modes is governed by another eigenquantity, the eigenangle .The spectral type of decomposition of the elastic stiffness or compliance tensors in elementary fourth-rank tensors thus serves as a means for the energy-orthogonal decomposition of the energy function. The advantage of this type of decomposition is that the elementary idempotent tensors to which the fourth-rank tensors are decomposed have the interesting property of defining energy-orthogonal stress states. That is, the stress-idempotent tensors are mutually orthogonal and at the same time collinear with their respective strain tensors, and therefore correspond to energy-orthogonal stress states, which are therefore independent of each other. Since the failure tensor is the limiting case for the respective _x, which are eigenstates of the compliance tensor S, this tensor also possesses the same remarkable property.An interesting geometric interpretation arises for the energy-orthogonal stress states if we consider the projections of _x in the principal₃D stress space. Then, the characteristic state ₂ vanishes, whereas stress states ₁, ₃ and ₄ are represented by three mutually orthogonal vectors, oriented as follows: The ₃ and ₄ lie on the principal diagonal plane (₃₁₂) with subtending angles equaling (–/2) and (-), respectively. On the positive principal ₃-axis, is the eigenangle of the orthotropic material, whereas the ₁-vector is normal to the (₃₁₂)-plane and lies on the deviatoric -plane. Vector ₂ is equal to zero.It was additionally conclusively proved that the four eigenvalues of the compliance, stiffness, and failure tensors for a transversely isotropic body, together with value of the eigenangle , constitute the five necessary and simplest parameters with which invariantly to describe either the elastic or the failure behavior of the body. The expressions for the _x-vector thus established represent an ellipsoid centered at the origin of the Cartesian frame, whose principal axes are the directions of the ₁-, ₃- and ₄-vectors. This ellipsoid is a generalization of the Beltrami ellipsoid for isotropic materials.Furthermore, in combination with extensive experimental evidence, this theory indicates that the eigenangle alone monoparametrically characterizes the degree of anisotropy for each transversely isotropic material. Thus, while the angle for isotropic materials is always equal to _i = 125.26° and constitutes a minimum, the angle || progressively increases within the interval 90–180° as the anisotropy of the material is increased. The anisotropy of the various materials, exemplified by their ratiosE _L/2G_L of the longitudinal elastic modulus to the double of the longitudinal shear modulus, increases rapidly tending asymptotically to very high values as the angle approaches its limits of 90 or 180°.     

Existence of fast traveling waves for some parabolic equations: A dynamical systems approach

We study semilinear elliptic equationsu + cu _x =f(u,u) and ² u + cu _x =f(u,u, ² u) in infinite cylinders (x,y) × ⁿ⁺¹ using methods from dynamical systems theory. We construct invariant manifolds, which contain the set of bounded solutions and then study a singular limitc, where the equations change type from elliptic to parabolic. In particular we show that on the invariant manifolds, the elliptic equation generates a smooth dynamical system, which converges to the dynamical system generated by the parabolic limit equation. Our results imply the existence of fast traveling waves for equations like a viscous reactive 2d-Burgers equation or the Cahn-Hillard equation in infinite strips.     

Supersonic nonequilibrium flow past segmental bodies of a mixture simulating the atmosphere of venus

Calculations of the flow of the mixture 0.94 CO₂+0.05 N₂+0.01 Ar past the forward portion of segmentai bodies are presented. The temperature, pressure, and concentration distributions are given as a function of the pressure ahead of the shock wave and the body velocity. Analysis of the concentration distribution makes it possible to formulate a simplified model for the chemical reaction kinetics in the shock layer that reflects the primary flow characteristics. The density distributions are used to verify the validity of the binary similarity law throughout the shock layer region calculated.The flow of a CO₂+N₂+Ar gas mixture of varying composition past a spherical nose was examined in [1]. The basic flow properties in the shock layer were studied, particularly flow dependence on the free-stream CO₂ and N₂ concentration.New revised data on the properties of the Venusian atmosphere have appeared in the literature [2, 3] One is the dominant CO₂ concentration. This finding permits more rigorous formulation of the problem of blunt body motion in the Venus atmosphere, and attention can be concentrated on revising the CO₂ thermodynamic and kinetic properties that must be used in the calculation.The problem of supersonic nonequilibrium flow past a blunt body is solved within the framework of the problem formulation of [4].Notation V body velocity - shock wave standoff - universal gas constant - ratio of frozen specific heats - hRt/m enthalpy per unit mass undisturbed stream P pressure - density - T temperature - m molecular weight - c_p specific heat at constant pressure - (X) concentration of component X (number of particles in unit mass) - R body radius of curvature at the stagnation point - _j rate of j-th chemical reaction shock layer P V ² pressure - density - TT temperature - mm molecular weight Translated from Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 5, No. 2, pp. 67–72, March–April, 1970.The author thanks V. P. Stulov for guidance in this study.     

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.     

The value of the critical exponent for reaction-diffusion equations in cones

Let D R ^N be a cone with vertex at the origin i.e., D = (0, )x where S ^N–1 and x D if and only if x = (r, ) with r=¦x¦, . We consider the initial boundary value problem: u _t = u+u ^p in D×(0, T), u=0 on Dx(0, T) with u(x, 0)=u ₀(x) 0. Let ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on and let ₊ denote the positive root of (+N–2) = ₁. Let p ^* = 1 + 2/(N + ₊). If 1 < p < p ^*, no positive global solution exists. If p>p ^*, positive global solutions do exist. Extensions are given to the same problem for u _t=+¦x¦ u ^p .This research was supported in part by the Air Force Office of Scientific Research under Grant # AFOSR 88-0031 and in part by NSF Grant DMS-8 822 788. The United States Government is authorized to reproduce and distribute reprints for governmental purposes not withstanding any copyright notation therein.     

Green's functions,bi-linear forms,and completeness of the eigenfunctions for the elastostatic strip and wedge

The bi-harmonic Green's functionG(r,r) for the infinite strip region -1y1, -<x<, with the boundary conditionsG=G/y ony=±1, is obtained in integral form. It is shown thatG has an elegant bi-linear series representation in terms of the (Papkovich-Fadle) eigenfunctions for the strip. This representation is then used to show that any function bi-harmonic in arectangle, and satisfying the same boundary conditions asG, has a unique representation in the rectangle as an infinite sum of these eigenfunctions. For the case of the semi-infinite strip, we investigate conditions on sufficient to ensure that is exponentially small asx. In particular it is proved that this is so, solely under the condition that be bounded asx.A corresponding pattern of results is established for the wedge of general angle. The Green's function is obtained in integral form and expressed as a bilinear series of the (Williams) eigenfunctions. These eigenfunctions are proved to be complete for all functions bi-harmonic in anannular sector (and satisfying the same boundary conditions as the Green's function). As an application it is proved that if an elastostatic field exists in a corner region with free-free boundaries, and with either (i) the total strain energy bounded, or (ii) the displacement field bounded, then this field has a unique representation as a sum of those Williams eigenfunctions whichindividually posess the properties (i), (ii).The methods used here extend to all other linear homogeneous boundary conditions for these geometries.On leave of absence at the University of British Columbia, Vancouver, B.C. Canada, during 1977–79. This work was supported in part by N.R.C. grants Nos. A9259 and A9117.     

A rotating hot-wire technique for spatial sampling of disturbed and manipulated duct flows

The documentation and control of flow disturbances downstream of various open inlet contractions was the primary focus with which to evaluate a spatial sampling technique. An X-wire probe was rotated about the center of a cylindrical test section at a radius equal to one-half that of the test section. This provided quasi-instantaneous multi-point measurements of the streamwise and azimuthal components of the velocity to investigate the temporal and spatial characteristics of the flowfield downstream of various contractions. The extent to which a particular contraction is effective in controlling ingested flow disturbances was investigated by artificially introducing disturbances upstream of the contractions. Spatial as well as temporal mappings of various quantities are presented for the streamwise and azimuthal components of the velocity. It was found that the control of upstream disturbances is highly dependent on the inlet contraction; for example, reduction of blade passing frequency noise in the ground testing of jet engines should be achieved with the proper choice of inlet configurations.List of symbols K _uv correlation coefficient= - P percentage of time that an azimuthal fluctuating velocity derivative dv/d is found - U streamwise velocity component U=U (, t) - V azimuthal or tangential velocity component due to flow and probe rotation V=V (, t) - mean value of streamwise velocity component - U _m resultant velocity from and - mean value of azimuthal velocity component induced by rotation - u fluctuating streamwise component of velocity u=u(, t) - v fluctuating azimuthal component of velocity v = v (, t) - u phase-averaged fluctuating streamwise component of velocity u=u(0) - v phase-averaged fluctuating azimuthal component of velocity v=v() - û average of phase-averaged fluctuating streamwise component of velocity (u()) over cases I-1, II-1 and III-1 û = û() - average of phase-averaged fluctuating azimuthal component of velocity (v()) over cases I-1, II-1 and III-1 - u fluctuating streamwise component of velocity corrected for non-uniformity of probe rotation and/or phase-related vibration u = u(0, t) - v fluctuating azimuthal component of velocity corrected for non-uniformity or probe rotation and/or phase-related vibration v=v (, t) - u ² rms value of corrected fluctuating streamwise component of velocity - rms value of corrected fluctuating azimuthal component of velocity - phase or azimuthal position of X-probe     

Boundary layer effect on thermal NO decomposition behind incident shock waves

The thermal decomposition of nitric oxide (diluted in Argon) has been measured behind incident shock waves by means of IR diode laser absorption spectroscopy. In two independent runs the diode laser was tuned to the=0 =1² _3/2 R(18.5)-rotational vibrational transition and the=1 =2² _3/2 R(20.5)-rotational vibrational transition of nitric oxide, respectively. These two transitions originating from the vibrational ground state (=0) and the first excited vibrational state (=1) were selected in order to probe the homogeneity along the absorption path. The measured NO decomposition could satisfactorily be described by a chemical reaction mechanism after taking into account boundary layer corrections according to the theory of Mirels. The study forms a further proof of Mirels' theory including his prediction of the laminar-turbulent transition. It also shows, that the inhomogeneities from the boundary layer do not affect the IR linear absorption markedly.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.     

Heteroclinic orbits in singular systems: A unifying approach

We consider singularly perturbed systems , such that=f(, _o, 0). _o ^m , has a heteroclinic orbitu(t). We construct a bifurcation functionG(, ) such that the singular system has a heteroclinic orbit if and only ifG(, )=0 has a solution=(). We also apply this result to recover some theorems that have been proved using different approaches.     

A Direct Method for the Identification of the Permeability Field Based on Flux Assimilation by a Discrete Analog of Darcy's Law

Inverse models to determine the permeability are generally based on existing forward models for the pressure. The permeabilities are adapted in such a way that the calculated pressures match the specified pressures in a number of points. To assimilate a priori knowledge about the flux, we introduce the flux assimilation method, which is based on the vector potential–pressure formulation of Darcy's law. Thanks to an unconventional discretization technique – the edge-based face element method – not only the specified pressures, but also specified information about the flux density can easily be assimilated. A relatively simple, but insightful analytical example illustrates the potential of this method.     

Motion of large gas bubbles ascending in a liquid

An equation is derived for the ascent velocity of large gas bubbles in a liquid. This velocity is assumed to be governed by the propagation of a wavelike perturbation caused by the bubble in the liquid.Notation w bubble (or drop) velocity - specific gravity - dynamic viscosity - kinematic viscosity - r bubble (or drop) radius - surface tension - coefficient of friction - g gravitational acceleration - D bubble (or drop) diameter - p pressure - c propagation velocity of the wavelike perturbation - wavelength     

Injection moulding of thermoplastics: Freezing during mould filling

The injection moulding of thermoplastic polymers involves, during mould filling, flows of hot melts into mould networks, the walls of which are so cold that frozen layers form on them. Theoretical analyses of such flows are presented here. Br Brinkman number - c _L polymer melt specific heat capacity - c _S frozen polymer specific heat capacity - e exponential function - erf() error function - Gz Graetz number in thermal entrance region - Gz ^* modified Graetz number in thermal entrance region - Gz overall Graetz number - h channel half-height - h ^* half-height of polymer melt region - H mean heat transfer coefficient - k _L polymer melt thermal conductivity - k _S frozen polymer thermal conductivity - ln( ) natural logarithm function - L length of thermal entrance region in pipe or channel - m viscosity shear rate exponent - M(,,) Kummer function - Nu Nusselt number - p pressure - P pressure drop in thermal entrance region - P _f pressure drop in melt front region - Pe Péclet number - Pr Prandtl number - Q volumetric flow rate - r radial coordinate in pipe - R pipe radius - R ^* radius of polymer melt region - Re Reynolds number - Sf Stefan number - t time - T temperature - T _i inlet polymer melt temperature - T _m melting temperature of polymer - T _w pipe or channel wall temperature - U(,,) Kummer function - u _r radial velocity in pipe - u _x axial velocity in channel - u _y cross-channel velocity - u _z axial velocity in pipe - V melt front velocity - w channel width - x axial coordinate in channel - x _f melt front position in channel - y cross-channel coordinate - z axial coordinate in pipe - z _f melt front position in pipe - () gamma function - dimensionless thickness of frozen polymer layer - _i i-th term (i = 1,2,3) in power series expansion of - dimensionless axial coordinate in pipe - _f dimensionless melt front position in pipe - dimensionless cross-channel coordinate - ^* dimensionless half-height of polymer melt region - dimensionless temperature - _i i-th term (i = 0, 1, 2, 3) in power series expansion of - _i first derivative of _i with respect toø - _i second derivative of _i with respect toø - ^* dimensionless wall temperature - thermal diffusivity ratio - - latent heat of fusion - µ viscosity - µ ^* unit shear rate viscosity - dimensionless axial coordinate in channel - _f dimensionless melt front position in channel - dimensionless pressure drop in thermal entrance region - _f dimensionless pressure drop in melt front region - _L polymer melt density - _s frozen polymer density - dimensionless radial coordinate in pipe - ^* dimensionless radius of polymer melt region - ø dimensionless similarity variable in thermal entrance region - dummy variable - dimensionless contracted axial coordinate in thermal entrance region - dimensionless similarity variable in melt front region - ^* constant