Beitrag zur Auswertung von Anlauf-Vorgängen in viskoelastischen Flüssigkeiten

Zusammenfassung In diesem Bericht wird die Kraftübertragung zwischen zwei unbegrenzt ausgedehnten und durch eineMaxwell-Flüssigkeit gekoppelten Platten bei beliebigem Zeitgesetz der Bewegung der oberen auf die elastisch aufgehängte untere Platte unter Einbeziehung der Trägheit von unterer Platte undMaxwell-Flüssigkeit berechnet.Diese Fragestellung führt mathematisch auf ein Randwertproblem, dessen eine Randbedingung die Form ³ u/x ² y + ₁ u + ₁ u/y – ₂ ² u/xy = 0 fürx = 0 annimmt, also gemischte Ableitungen nach 2 unabhängigen Veränderlichen enthält.Es wird gezeigt, daß sich die Lösung als unendliche Reihe darstellen läßt, deren 1. Glied aus abklingenden Exponentialfunktionen, alle weiteren hingegen aus gedämpften Schwingungen bestehen.Als spezielle Zeitgesetze für die obere Platte werden die impulsförmige Erregung, die zeitproportionale und die periodische Bewegung untersucht.Es wird abgeleitet, wie sich die Kenngrößen undG (Viskosität und Schubmodul) aus Eigenfrequenz und Dämpfung bzw. aus den beiden Exponenten dere-Funktionen in den beiden ersten Fällen berechnen lassen. Bei periodischer Beanspruchung werden undG aus der Resonanzfrequenz und der Halbwertbreite bzw. dem Phasenwinkel ermittelt.Abschließend wird gezeigt, daß und wie ein Rotationsviskosimeter auf den behandelten Fall ebener Platten zurückgeführt werden kann.

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Summary In this report the force transmission between two infinitely extended plates coupled by aMaxwell-Fluid is calculated. In this calculation the inertia of the fluid and the influence of the elastically suspended lower plate is included, while the upper plate can perform any time-dependent movement.Mathematically this question leads to a boundery value problem; one boundery condition has the form ³ u/ ² y + ₁ u + ₁ u/y – ₂ ² u/xy = 0 (x = 0) ( ₁, ₂, = constant), that means it contains mixed differential-quotients of two independent variables.The solution is demonstrated as an infinite series, the form of the first term is of two negative exponential functions, all the other terms are damped oscillations.The excitation from a single pulse, the uniform timeproportional and the periodic movement as timedependent laws for the upper plate are analysed.The values of andG (viscosity and shear modulus) are derived from eigenfrequency and damping-constant respectively from the values of the two exponentials. In the periodic case andG are calculated from the resonance frequency and halfwidth or phase-angle respectively.Finally the conditions of the rotation-viscometer are reduced to the case of parallel-plate-viscometer.

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Vorgetragen auf der Jahrestagung der Deutschen Rheologen vom 7.–9. Mai 1973 in Berlin.

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Mit 6 Abbildungen und 1 Tabelle     

On the measurement of lateral velocity derivatives in turbulent flows

Direct numerical simulation data for the lateral velocity derivative u/y at the centreline of a fully developed turbulent channel flow provide reasonable support for Wyngaard's analysis of the error involved in measuring this quantity using parallel hot wires. Numerical data in the wall region of the channel flow also provide a useful indication of how to select the separation between the wires. Justification for this choice is obtained by comparing several measured statistics of u/y with the corresponding numerical data.     

Limitation and improvement of PIV

In this second part of the paper, the Particle Image Distortion (PID) technique is described. It is proposed to overcome the limitations of conventional PIV due to the local deformations u/x, u/y, v/x and v/y in two-dimensional flows. Both simulation and experiment demonstrate that high accuracy and high spatial resolution are possible with this technique. The large time required to compute the cross-correlations, however, limits its wide applications at present.     

An application of nonsymmetric Lax-Milgram lemma to nonassociated plasticity

Based on a general assumption for plastic potential and yield surface, some properties of the nonassociated plasticity are studied, and the existence and uniqueness of the distribution of incremental stress and displacement for work-hardening materials are proved by using nonsymmetric Lax-Milgram lemma, when the work-hardening parameter A>F/Q/–F/, Q/.     

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.     

Approximate method for calculating turbulent boundary layer with positive pressure gradient

A single-parameter integral method is proposed for calculating the turbulent boundary layer with positive pressure gradient which makes it possible to calculate the friction, thermal flux, and layer thickness both ahead of the separation point and in some region behind the separation point.Notation u velocity - density - ^* displacement thickness - ^** momentum thickness - energy thickness - M Mach number - r radius - dynamic viscosity - c_p specific heat at constant pressure - Reynolds number based on initial boundary layer thickness - P Prandtl number - p₁ static pressure at point of initial interaction - p₂ static pressure at pressureplateau - p₀ stagnation pressure - T₀ stagnation temperature - I enthalpy - T_e recovery temperature - T_w ⁰ temperature factor - H form parameter - r₁ recovery coefficient Indices 0 denotes initial section of boundary layer - 1 parameters taken at edge of boundary layer - w parameters taken at the wall temperature - * parameters referred to flow on a flat plate with =0     

Analysis of melt spinning transients in Lagrangian coordinates

Transients in melt spinning of isothermal power law and Newtonian fluids were found to be governed by an extremely simple partial differential equation ² ( ^1/n )/() = 0 in Lagrangian coordinates where is the cross-sectional area,n the power law exponent, the time and the the time at which a fluid molecule constituting the spinline left the spinneret. The general integral ^1/n =f() +g () of the above governing equation containing two arbitrary functions represents physically attainable spinline transients. Hitherto unknown analytical transient solutions of the above governing equation were obtained for the response of isothermal constant tension spinlines to a stepwise change in tension, spinneret hole area, extrusion speed or extrusion viscosity and for the starting transient in gravitational spinning. Linearized perturbation solutions and the stability limit of the spinline derived from the above new found nonlinear solutions were in agreement with previous findings and the above nonlinear response of the spinline to a step increase in the spinneret hole area was found to be equivalent to Orowan's tandem cylinder model of dent growth in filament stretching.     

Filament stretching equation in Lagrangian coordinates

The general integral of the very simple equation ^21/n/() was found to describe the cross sectional area of filaments of isothermal power law fluids while in transient stretching where is time and is the initial location of fluid molecules at time = 0 given as the distance from a reference point fixed in space. Any such stretching transient given as a solution of the above equation is physically realizable subject to the restrictions > 0 and/ < 0.     

LDA measurements in low Reynolds number turbulent boundary layer

LDA measurements of the mean velocity in a low Reynolds number turbulent boundary layer allow a direct estimate of the friction velocity U from the value of /y at the wall. The trend of the Reynolds number dependence of / is similar to the direct numerical simulations of Spalart (1988).     

Modeling pressure-strain rate correlations in turbulence theory

On the basis of a spectral representation of the rapid part _ij,2 of the correlation tensor p(u _i /x _j ) using Cramer's theorem the inequality _ij,2(U _j /x _i )0 is obtained. As distinct from the realizability conditions, it can serve as a direct and very rigorous test of the adequacy of model expressions for _ij,2. In particular, it is shown that the best known of such expressions do not satisfy this test.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 42–46, March–April, 1992.     

Bäcklund transformations for the equation ∂2u/∂x1∂x1 + ∂2u/∂x2∂x2−f (u)

Bäcklund transformations for the equation ² u/x¹x ¹+ ²u/x ² x ²=f (u)here f is an arbitrary function) is studied in this paper, using the procedure of Wahlquist and Estabrook (WEP). We conclude that the condition d ² f/du ²=f is sufficient for the existence of Bäcklund transformations for the equation of our interest. A special case of our results leads to the conclusion of Leibbrandt ^[1,2].     

Three-dimensional numerical simulation of free convection in a plane-parallel flow

The results are given of numerical simulation of convection in Couette and Poiseuille flow for stationary (T*/t = 0) and uniformly increasing (T*/t = = const) mean temperature of the convective layer. Numerical experiments were made for air (Prandtl number Pr = 0.7) in the range of Rayleigh numbers 2 000 Ra 44 000. The results confirm the conclusion drawn in earlier studies that at slightly supercritical Ra the dominant forms in the convection are cylindrical rolls with helical circulation. The rolls are oriented along the direction of the flow and are stationary formations. When a certain value of Ra, which depends on the vertical distribution of the temperature and velocity, is reached, the roll structure is deformed by transverse perturbations. All the considered flow forms have a stabilizing influence on the transverse modes, occurring at larger Rayleigh numbers than is the case for convection in a fluid at rest. The perturbations are displaced at a phase velocity close to the mean velocity of the undisturbed flow. In the considered range of Rayleigh numbers, a shear flow does not have an appreciable influence on the heat transfer, although there is a certain tendency for the Nusselt number to be larger in a shear flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 127–135, May–June, 1979.We thank N. M. Sazanovich and L. I. Derevich for assistance in the calculations.     

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.     

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.     

The rheology of powders

This paper studies the slow flow of powders. It is argued that since powders can flow like liquids, there must be equations similar to those of liquids. The phenomenon of a variable density, dilatancy, is described by an analogue of temperature called the compactivity X. Whereas, in thermal physicsT = E/S, powders are controlled byX = V/S. The equations for, v, T of a liquid are replaced by, v, X. An analogy for free energy is described, and the solution to some simple problems of packing and mixing are offered. As an example of rheology, it is shown that the simplest flow equations produce a transition to plug flow in appropriate circumstances.Delivered as a Gold Medal Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

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     

Optical co-ordinates in general relativity 2 — the problem of distance

Summary Let denote the congruence of null geodesics associated with a given optical observer inV ₄. We prove that determines a unique collection of vector fieldsM₍₎ ( =1, 2, 3) and ₍₀₎ overV ₄, satisfying a weak version of Killing's conditions.This allows a natural interpretation of these fields as the infinitesimal generators of spatial rotations and temporal translation relative to the given observer. We prove also that the definition of the fieldsM₍₎ and ₍₀₎ is mathematically equivalent to the choice of a distinguished affine parameter f along the curves of, playing the role of a retarded distance from the observer.The relation between f and other possible definitions of distance is discussed.

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Sommario Sia la congruenza di geodetiche nulle associata ad un osservatore ottico assegnato nello spazio-tempoV ₄. Dimostriamo che determina un'unica collezione di campi vettorialiM₍₎ ( =1, 2, 3) e ₍₀₎ inV ₄ che soddisfano una versione in forma debole delle equazioni di Killing. Ciò suggerisce una naturale interpretazione di questi campi come generatori infinitesimi di rotazioni spaziali e traslazioni temporali relative all'osservatore assegnato. Dimostriamo anche che la definizione dei campiM₍₎, ₍₀₎ è matematicamente equivalente alla scelta di un parametro affine privilegiato f lungo le curve di, che gioca il ruolo di distanza ritardata dall'osservatore. Successivamente si esaminano i legami tra f ed altre possibili definizioni di distanza in grande.

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Work performed in the sphere of activity of: Gruppo Nazionale per la Fisica Matematica del CNR.     

Stability criterion for steady-state burning of powders

A model of a powder with a variable surface temperature is considered. A stability criterion for the steady-state burning of powders at constant pressure is found on the assumption that all the processes in the gas phase and the reaction layer of the condensed phase are inertialess. It is shown that the stability region is determined by two parameters: K=(T₁ – t₀) ( ln m/T₀)p and r=(T₁/T₀)p (here T₁ is the surface temperature, T₁ is the initial temperature of the powder, m is the mass burning rate, p is pressure). Burning is always stable if < 1. If > 1, the burning process is stable only when r > ( – 1)²/ / + 1.The author expresses his appreciation to A. S. Kompaneits, O. I. Leipunskii, A. G. Istratov, V. B. Librovich, and S. S. Novikov for discussing his work.     

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.     

Neck propagation as a shock wave

Neck propagation in the stretching of elastic solid filaments having a yield point was analyzed using the space one-dimensional thin filament governing equations developed previously by the authors and other researchers. Constitutive model for the filament was assumed to be expressible as engineering tensile stress(X) (tensile force) given as a function of elongational strain with the(X) curve having a yield point maxima followed by a minima and a breaking point greater than the yield point maxima. Also incorporated into the model is the hysteresis of irreversible plastic deformation. When inertia is taken into consideration, the thin filament equations were found to reduce to the nonlinear wave equation ² (X)/ ² =C ₁ ² X/ ² where is Lagrangean space coordinate, is time, andC ₁ is inertia coefficient. The above nonlinear wave equation yields a solutionX(, ) having a stepwise discontinuity inX which propagates along the axis. The zero speed limit of the step wave solution was found to describe the above neck propagation occurring in solid filaments. Furthermore, it was recognized that the nonlinear wave equation was known for many years to also govern the plastic shock wave which propagates axially within a metal rod subjected to a very strong impact on its end. The one-dimensional atmospheric shock wave also was known to be governed by the nonlinear wave equation upon making certain simplifying assumptions. The above and other evidences lead to the conclusion that neck propagation occurring in the extension of solid filament obeying the above(X) function can be formally described as a shock wave.