Deformationsbeständige,elastische und Relaxations-Eigenschaften der flüssigplastischen kolloiden Systeme

Zusammenfassung Es wurde gezeigt, daß es zweckmäßig ist, die strukturierten kolloiden Systeme in fest-plastische und flüssig-plastische Systeme einzuteilen, weil beide Systeme einen übereinstimmenden Eigenschaftskomplex aufweisen, sich jedoch quantitativ durch die RelaxationszeitenP >P _k im Schubspannungsbereich unterhalb der unteren FließgrenzeP < P _k unterscheiden. Es wurde darauf hingewiesen, daß für beide Systeme die Spannungsdeformationskurven sehr charakteristisch sind.Die Kurven werden unter der Bedingung der konstanten Deformationsgeschwindigkeit erhalten, wobei ihre Form von der gegenseitigen Beziehung von und abhängt und mit den Strukturelementetypen, die durch und _i charakterisiert sind, im Zusammenhang stehen.Die Methoden,die zur Messung der elastischen Deformation im breiten Bereich längs der KurveP () bei sowohl kleiner als auch größer _r entsprechend der kritischen SchubspannungP _r angewandt werden können, wurden entwickelt. Dabei wurde gezeigt, daß die Kurveine() durch das Maximum bei _m hindurchgeht.Der Einfluß von auf die kritische Deformation _r der Strukturzerstörung und auf die maximale Rückfederung _e max, die ihrerseits wiederum von der Gelkonzentration abhängen, wurde eingehend untersucht.Es wurden die Zahlenwerte der Grenzviskosität der Nachwirkung bestimmt und die Abhängigkeit der Geschwindigkeit (der Zeit) der Relaxation der elastischen Deformation von der gesamten und der elastischen Deformation ermittelt.Weiter wurde gezeigt, daß die größte elastische Deformation _e max des Systems größer als die kritische Deformation _r der Strukturzerstörung, die dem Maximum der kritischen Schubspannung der Struktur entspricht, sein kann.     

Transcendentally small transversality in the rapidly forced pendulum

The rapidly forced pendulum equation with forcing sin((t/), where =<₀^p,p = 5, for ₀, sufficiently small, is considered. We prove that stable and unstable manifolds split and that the splitting distanced(t) in the ( ,t) plane satisfiesd(t) = sin(t/) sech(/2) +O( ₀ exp(–/2)) (2.3a) and the angle of transversal intersection,, in thet = 0 section satisfies 2 tan/2 = 2S _s = (/2) sech(/2) +O(( ₀ /) exp(–/2)) (2.3b) It follows that the Melnikov term correctly predicts the exponentially small splitting and angle of transversality. Our method improves a previous result of Holmes, Marsden, and Scheuerle. Our proof is elementary and self-contained, includes a stable manifold theorem, and emphasizes the phase space geometry.     

Spin-up in a saturated porous medium

It is shown that, on the Brinkman model, spin-up is confined to boundary layers whose thickness is of order k ^1/2, and the spin-up is established in a time of order k/, where k, , and denote permeability, density, porosity and dynamic viscosity, respectively.     

Homogenization of the Robin problem in a thick multilevel junction

We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction that is the union of a domain ₀ and a large number 2N of thin rods with variable thickness of order = (N ^–1). The thin rods are divided into two levels, depending on their length. In addition, the thin rods from each level are -periodically alternated. We investigate the asymptotic behavior of the solution as 0 under the Robin conditions on the boundaries of the thin rods. By using some special extension operators, a convergence theorem is proved.Published in Neliniini Kolyvannya, Vol. 7, No. 3, pp. 336–355, July–September, 2004.     

Scheme of Investigation of the Spectrum of a Family of Perturbed Operators and Its Application to Spectral Problems in Thick Junctions

We develop a scheme for the investigation of the asymptotic behavior of eigenvalues and eigenvectors of a family of self-adjoint compact operators {A: > 0} that act in different spaces and lose their compactness in the limit case 0. We prove the Hausdorff convergence of the spectrum of the operator A to the spectrum of the limit operator A₀, obtain asymptotic estimates for this convergence both to points of the discrete spectrum and to points of the essential spectrum of the operator A₀, and prove asymptotic estimates for eigenvectors of A. This scheme is applied to the investigation of the asymptotic behavior of eigenvalues and eigenfunctions of the Neumann problem in a thick singularly degenerate junction that consists of two domains connected by an -periodic system of thin rods of fixed length.     

On the indentation of an inhomogeneous plastic solid

Summary The problem considered here is that of the indentation of a semi infinite, inhomogeneous rigid-plastic solid by a smooth, flat ended punch under conditions of plane strain. It is assumed that the yield stress of the solid k(x, y) has the form k ⁰+k(x, y) where k ⁰ is a constant and is small. A perturbation method of solution developed by Spencer [1] is used, and general results are obtained for arbitrary values of k(x, y). Some particular cases are then considered.     

Some measurements in a binary gas jet

Simultaneous measurements of species volume concentration and velocities in a helium/air binary gas jet with a jet Reynolds number of 4,300 and a jet-to-ambient fluid density ratio of 0.64 were carried out using a laser/hot-wire technique. From the measurements, the turbulent axial and radial mass fluxes were evaluated together with the means, variances and spatial gradients of the mixture density and velocity. In the jet near field (up to ten diameters downstream of the jet exit), detailed measurements of u/ ₀ U ₀, v/ ₀ U₀, u v/ ₀ U ₀ ² , u ² / ₀ U ₀ ² and v ² / ₀ U ₀ ² reveal that the first three terms are of the same order of magnitude, while the last two are at least one order of magnitude smaller than the first three. Therefore, the binary gas jet in the near field cannot be approximated by a set of Reynolds-averaged boundary-layer equations. Both the mean and turbulent velocity and density fields achieve self-preservation around 24 diameters. Jet growth and centerline decay measurements are consistent with existing data on binary gas jets and the growth rate of the velocity field is slightly slower than that of the scalar field. Finally, the turbulent axial mass flux is found to follow gradient diffusion relation near the center of the jet, but the relation is not valid in other regions where the flow is intermittent.     

On laminar flow through a uniformly porous pipe

Numerous investigations ([1] and [4–9]) have been made of laminar flow in a uniformly porous circular pipe with constant suction or injection applied at the wall. The object of this paper is to give a complete analysis of the numerical and theoretical solutions of this problem. It is shown that two solutions exist for all values of injection as well as the dual solutions for suction which had been noted by previous investigators. Analytical solutions are derived for large suction and injection; for large suction a viscous layer occurs at the wall while for large injection one solution has a viscous layer at the centre of the channel and the other has no viscous layer anywhere. Approximate analytic solutions are also given for small values of suction and injection.

Nomenclature

General r distance measured radially - z distance measured along axis of pipe - u velocity component in direction of z increasing - v velocity component in direction of r increasing - p pressure - density - coefficient of kinematic viscosity - a radius of pipe - V velocity of suction at the wall - r ²/a ² - R wall or suction Reynolds number, Va/ - f() similarity function defined in (6) - u ₀() eigensolution - U(0) a velocity at z=0 - K an arbitrary constant - B _K Bernoulli numbers Particular Section 5 perturbation parameter, –2/R - ² a constant, –K - x / - g(x) f()/ Section 6 perturbation parameter, –R/2 - ² a constant, –K - g() f() - g _c ()=g() near centre of pipe - * point where g()=0 Section 7 2/R - ² K - t (1–)/ - w(t, ) [1–f(t)]/ - ₀, ₁ constants - g() f()– ₀ - ₀/ - ₀ a constant - * point where f()=0     

Asymptotic behavior of one-dimensional discrete-velocity models in a slab

We prove results on the asymptotic behavior of solutions to discrete-velocity models of the Boltzmann equation in the one-dimensional slab 0x<1 with=" general=" stochastic=" boundary=" conditions=" at=" x="0" and=" x="1." assuming=" that=" there=" is=" a=" constant=">wall Maxwellian M=(M _i) compatible with the boundary conditions, and under a technical assumption meaning strong thermalization at the boundaries, we prove three types of results:

Calculation of the asymptotic behaviour of the TDR step response related to the asymptotic behaviour of dielectrics in the frequency domain

The asymptotic behaviour of the TDR step response is compared with the asymptotic behaviour of dielectrics in the frequency domain. For non conducting materials the asymptotic behaviour of the TDR step response appears to be related to the angles of intersection in the Cole-Cole plot. In the case of conducting materials the asymptotic behaviour for t depends on the low frequency conductivity, which suggests a new method of determining this conductivity from TDR experiments. Consequences are discussed for the accuracy of the determination of and from the TDR response obtained experimentally.     

A model of stress dissipation in second-moment closures

The paper presents a modified expression for the dissipation rate tensor _ij in the second-moment closure models, which employs the dissipation flatness parameterE and the turbulenceRe number. The expression reproduced the distribution among the three diagonal components of _ij in agreement with the direct numerical simulation of a plane channel flow ofMansour, Kim and Moin, 1988. Implemented in a low-Re-number differentialRe-stress model the relationship yielded predictions of dissipative components better than other models, albeit spoiled by still unsatisfactory modelling of the equation for the energy dissipation rate . on leave from Mainski Fakultet, University of Sarajevo, Bosnia Hercegovina.     

Application of the Chapman-Enskog method to the case of a binary two-temperature gas mixture

An interesting property of the flows of a binary mixture of neutral gases for which the molecular mass ratio =m/M1 is that within the limits of the applicability of continuum mechanics the components of the mixture may have different temperatures. The process of establishing the Maxwellian equilibrium state in such a mixture divides into several stages, which are characterized by relaxation times _i which differ in order of magnitude. First the state of the light component reaches equilibrium, then the heavy component, after which equilibrium between the components is established [1]. In the simplest case the relaxation times differ from one another by a factor of ^*.Here the mixture component temperature difference relaxation time _T /, where is the relaxation time for the light component. If 1, 1, so that _T ~1, then for the characteristic hydrodynamic time scale t~1 the relative temperature difference will be of order unity. In the absence of strong external force fields the component velocity difference is negligibly small, since its relaxation time _vt1.In the case of a fully ionized plasma the Chapman-Enskog method is quite easily extended to the case of the two-temperature mixture [3], since the Landau collision integral is used, which decomposes directly with respect to . In the Boltzmann cross collision integral, the quantity appears in the formulas relating the velocities before and after collision, which hinders the decomposition of this integral with respect to , which is necessary for calculating the relaxation terms in the equations for temperatures differing from zero in the Euler approximation [4] (the transport coefficients are calculated considerably more simply, since for their determination it is sufficient to account for only the first (Lorentzian [5]) terms of the decomposition of the cross collision integrals with respect to ). This led to the use in [4] for obtaining the equations of the considered continuum mixture of a specially constructed model kinetic equation (of the Bhatnagar-Krook type) which has an undetermined degree of accuracy.In the following we use the Boltzmann equations to obtain the equations of motion of a two-temperature binary gas mixture in an approximation analogous to that of Navier-Stokes (for convenience we shall term this approximation the Navier-Stokes approximation) to determine the transport coefficients and the relaxation terms of the equations for the temperatures. The equations in the Burnett approximation, and so on, may be obtained similarly, although this derivation is not useful in practice.     

The influence of blending polystyrenes of narrow molecular weight distribution on melt creep flow and creep recovery in elongation

Creep and creep recovery experiments in elongation were performed with melts of anionically polymerized polystyrenes (PS) and with their blends at a temperature of 150 °C. For stresses ₀ < 10 000 N/m² the samples with narrow molecular weight distribution show linear viscoelastic behavior up to the maximum Hencky strain = 3.5, achievable in a newly developed elongational rheometer for polymer melts. The compliances,D (t), of the blends are linear-viscoelastic only up to a strain limit _L . For strains beyond _L the compliance of each blend depends on the stress ₀. For a series of binary blends, prepared from the same components of narrow MWD, the linear-viscoelastic limit _L seems to be independent of the mixing ratio and stress. _L seems to be a function only of the molecular weights of the original components, the blends investigated were made from.Paper presented at the Annual Conference of the German Society of Rheology at Ulm, March 7–10, 1983.     

Second Order Multiple Scales for Oscillators with Large Delay

We use the method of multiple scales (MMS) to study small perturbations, governed by a parameter , of a harmonic oscillator by a small term with a large delay. These systems differ significantly from others where small terms have delays; or an term has delay in a system near a Hopf bifurcation. Here, the slow flow in time t depends strongly on even at lowest order, and itself has an delay. The MMS has already been applied elsewhere for such systems, but only to first order and with attention restricted to periodic and quasiperiodic solutions. Here, we address transients as well as proceed to second order. The second order analysis holds unless a special resonance occurs (we assume it does not). Several numerical examples are presented. In each case, the slow flows are infinite-dimensional, show strong -dependence, require significantly less computation time than the full solutions, yet agree well with the same.     

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

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

Propagation of a spherical wave in nonlinearly compressible and elastoplastic materials

The problem of spherical wave propagation in soil under the action of an intense uniformly decreasing load ₀(t) applied to the boundary of a cavity with radius r₀ is considered. Soil with a high stress level is modeled either by ideally nonlinearly compressible or elastoplastic material, taking account of linear irreversible unloading for the material. In contrast to [1–7], in order to describe material movement use is made of strain theory [8] with determining functions = (), _i=_i(_i), where , _i, , _i are the first and second invariants of strain and stress tensors. During material loading these functions are presented in the form of polynomials ()=(_i+₂¦¦), _ii)=(_i-_2i)_i, in which constant coefficients _i, _i=1, 2) are determined by experiment, taking account of the triaxial stressed state of soil. Solution of the problem is constructed by an analytically reversible method, with prescribed shape for the shock-wave (SW) surface in the form of a second-degree polynomial relating to time t and a numerical method of characteristics for a prescribed arbitrarily decreasing load _i(t). On the basis of the analytical equations obtained, calculations are carried out for material parameters (including loading profile) in a computer and stresses and mass velocity of plastic and elastoplastic materials are compared.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 95–100, July–August, 1986.The authors express their sincere thanks to Kh. A. Rakhmatulin for discussing the results of this work.     

Numerical computation of high Rayleigh number natural convection and prediction of hot radiator induced room air motion

The two-dimensional stationary turbulent buoyant flow and heat transfer in a cavity at high Rayleigh numbers was computed numerically. The k– turbulence model was used. The time-averaged equations for momentum, energy and continuity, which are coupled to the turbulence equations, were solved using a finite difference formulation. In order to validate the computer code, a comparison exercise was carried out. The test results are in good agreement with the internationally accepted benchmark solution. Grid-refinement shows the necessity of a very fine grid at high Rayleigh numbers with especially small grid-distances in the near-wall region. The computed boundary layer velocity profiles are in excellent agreement with available experimental data. The local heat transfer in the turbulent part of the boundary layers is predicted 20% too high. Computations were carried out for the natural convective flow in a room induced by a hot radiator and a cold window. Various radiator configurations and types of thermal boundary conditions were applied including thermal radiation interaction between surfaces.Nomenclature a thermal diffusivity (m²/s) - C constant in _t expression - D cavity dimensions (m) - g acceleration of gravity (m/s²) - G _k production/destruction of k by buoyancy (kg/ms³) - h enthalpy (J/kg) - IX index of grid point - k turbulent kinetic energy (m²/s²) - m dimensionless stratification parameter - Nu overall Nusselt number - Nu _y local Nusselt number - NX total number of grid points - p pressure (N/m²) - P _k production of k by shear stress (kg/ms³) - Q heat flux through wall (W/m) - Ra overall Rayleigh number - Ra _y local Rayleigh number - Re _t turbulent Reynolds number - S source term in -equation (kg/ms⁴) - S source term for - T _c, T _h temperatures of cold and hot walls (K) - T _s (y) stratification temperature on vertical mid-line (K) - T ₀ mean cavity temperature (K) - u, v horizontal and vertical velocity components (m/s) - u ₀ Brunt-Vaisälä velocity scale (m/s) - x, y horizontal and vertical coordinates (m) - non-linearity parameter for grid - coefficient of thermal expansion (l/K) - jet angle (°) - diffusivity for - S dissipation rate for turbulent kinetic energy (m²/s³) - variable to be solved - thermal conductivity (W/mK) - , _t kinematic and eddy viscosities (m²/s) - stream function (kg/ms) - density (kg/m³) - _k, , _t constants in k– model     

Fluctuating flow in a rotating fluid

Summary Fluctuating flow of a viscous fluid rotating over a disk whose angular velocity oscillates about a nonzero mean is investigated. Initially the disk and the fluid rotate in the same sense with different angular velocities ₁ and ₂ ( ₂> ₁) and at a particular instant of time, the angular velocity of the disk becomes ₁[1+ sin( )]. The problem is solved as an initial boundary value problem and it is found that for small values of the results of analytical and numerical methods are in excellent agreement. The effect of frequency parameter on surface skin frictions has been analysed for various values of angular velocity ratio s and amplitude parameter .

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Fluktuierende Strömung in einer rotierenden Flüssigkeit

Übersicht Untersucht wird die fluktuierende Strömung einer viskosen Flüssigkeit, die über einer Scheibe, deren Winkelgeschwindigkeit um einen von Null verschiedenen Mittelwert schwankt, rotiert. Anfangs drehen sich die Scheibe und die Flüssigkeit gleichsinnig, aber mit verschiedenen Winkelgeschwindigkeiten ₁ und ₂ ( ₂> ₁). Zu einem Anfangszeitpunkt geht die Winkelgeschwindigkeit der Scheibe über in ₁[1+ sin ( )]. Die Aufgabe wird als Anfangs-/Randwertproblem gelöst. Für kleine Werte stimmen die analytischen und numerischen Ergebnisse hervorragend überein. Für verschiedene Werte des Winkelgeschwindigkeitsverhältnisses und des Amplitudenparameters wurde der Einfluß des Frequenzparameters auf die Reibspannungen an der Scheibe untersucht.