A sketch of continuum thermodynamics

Different formulations of non-equilibrium continuum thermodynamics are discussed: Thermodynamics of Irreversible Processes (TIP), Rational Thermodynamics (RAT), Extended Thermodynamics (ET), Mesoscopic Continuum Thermodynamics (MT), and the GENERIC version of thermodynamics. Concepts as constitutive quantity, state space, material frame indifference, exploiting dissipation inequality, mesoscopic variables, and GENERIC balance equations are taken into consideration.     

Non-gaussian diffusion modeling in composite porous media by homogenization: Tail effect

In the paper anomalous diffusion appearing in a porous medium composed of two porous components of considerably different diffusion characteristics is examined. The differences in diffusivities are supposed to result either from two medium types being present or from variations in pore size (double porosity media). The long-tail effect is predicted using the homogenization approach based on the application of multiple scale asymptotic developments. It is shown that, if the ratio of effective diffusion coefficients of two porous media is of the order of magnitude smaller or equal O( ²), where is a homogenization parameter, then the macroscopic behaviour of the composite may be affected by the presence of tail-effect. The results of the theoretical analysis were applied to a problem of diffusion in a bilaminate composite. Analytical calculations were performed to show the presence of the long-tail effect in two particular cases.Notations c _i the concentration of chemical species in water within the medium i - D _i the effective diffusion coefficient for the medium i - D _ij ^eff the macroscopic (or effective) diffusion tensor in the composite - ERV the elementary representative volume - h the thickness of the period - l a chracteristic length of the ERV or the periodic cell - L a characteristic macroscopic length - n the volumetric fraction of the material 2 - 1–n the volumetric fraction of the material 1 - N the unit vector normal to - t the time variable - x the macroscopic (or slow) space variable - y the microscopic (or fast) space variable - c _1c ,C _2c ,D _1c ,D _2c the characteristic quantities - T,T _1L ,T _2L ,T _1l ,T _2l the characteristic times - c ₁ ^* ,c ₂ ^* ,D ₁ ^* ,D ₂ ^* ,t ^* the non-dimensional variables - the homogenization parameter - ₁ the domain occupied by the material 1 - ₂ the domain occupied by the material 2 - the interface between the domains 1 and 2 - the total volume of the periodic cell - /x_i the gradient operator - the gradient operator     

Inhomogeneous ‘longitudinal’ plane waves in a deformed elastic material

By definition, a homogeneous isotropic compressible Hadamard material has the property that an infinitesimal longitudinal homogeneous plane wave may propagate in every direction when the material is maintained in a state of arbitrary finite static homogeneous deformation. Here, as regards the wave, homogeneous means that the direction of propagation of the wave is parallel to the direction of eventual attenuation; and longitudinal means that the wave is linearly polarized in a direction parallel to the direction of propagation. In other words, the displacement is of the form u = ncos k(n · x – ct), where n is a real vector. It is seen that the Hadamard material is the most general one for which a longitudinal inhomogeneous plane wave may also propagate in any direction of a predeformed body. Here, inhomogeneous means that the wave is attenuated, in a direction distinct from the direction of propagation; and longitudinal means that the wave is elliptically polarized in the plane containing these two directions, and that the ellipse of polarization is similar and similarly situated to the ellipse for which the real and imaginary parts of the complex wave vector are conjugate semi-diameters. In other words, the displacement is of the form u = {S exp i(S · x – ct)}, where S is a complex vector (or bivector). Then a Generalized Hadamard material is introduced. It is the most general homogeneous isotropic compressible material which allows the propagation of infinitesimal longitudinal inhomogeneous plane circularly polarized waves for all choices of the isotropic directional bivector. Finally, the most general forms of response functions are found for homogeneously deformed isotropic elastic materials in which longitudinal inhomogeneous plane waves may propagate with a circular polarization in each of the two planes of central circular section of the ⁿ -ellipsoid, where is the left Cauchy-Green strain tensor corresponding to the primary pure homogeneous deformation.     

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     

Onset of viscous fingering for miscible liquid-liquid displacements in porous media

The displacement of one fluid by another miscible fluid in porous media is an important phenomenon that occurs in petroleum engineering, in groundwater movement, and in the chemical industry. This paper presents a recently developed stability criterion which applies to the most general miscible displacement. Under special conditions, different expressions for the onset of fingering given in the literature can be obtained from the universally applicable criterion. In particular, it is shown that the commonly used equation to predict the stable velocity ignores the effects of dispersion on viscous fingering.Nomenclature C Solvent concentration - Unperturbed solvent concentration - D _L Longitudinal dispersion coefficient [m²/s] - D _T Transverse dispersion coefficient [m²/s] - g Gravitational acceleration [m/s²] - I _sr Instability number - k Permeability [m²] - K Ratio of transverse to longitudinal dispersion coefficient - L Length of the porous medium [m] - L _x Width of the porous medium [m] - L _y Height of the porous medium [m] - M Mobility ratio - V Superficial velocity [m/s] - V _c Critical velocity [m/s] - V _s Velocity at the onset of instability [m/s] - µ Viscosity [Pa/s] - Unperturbed viscosity [Pa/s] - µ ₀,µ _s Viscosities of oil and solvent, respectively [Pa/s] - Density [kg/m³] - ₀, _s Densities of oil and solvent, respectively [kg/m³] - Porosity - Dimensionless length     

New planforms in systems of partial differential equations with Euclidean symmetry

Dionne & Golubitsky [10] consider the classification of planforms bifurcating (simultaneously) in scalar partial differential equations that are equivariant with respect to the Euclidean group in the plane. In particular, those planforms corresponding to isotropy subgroups with one-dimensional fixed-point space are classified.Many important Euclidean-equivariant systems of partial differential equations essentially reduce to a scalar partial differential equation, but this is not always true for general systems. We extend the classification of [10] obtaining precisely three planforms that can arise for general systems and do not exist for scalar partial differential equations. In particular, there is a class of one-dimensional pseudoscalar partial differential equations for which the new planforms bifurcate in place of three of the standard planforms from scalar partial differential equations. For example, the usual rolls solutions are replaced by a nonstandard planform called anti-rolls. Scalar and pseudoscalar partial differential equations are distinguished by the representation of the Euclidean group.     

Sensitivity analysis of the non-linear dynamic response of beams using space-time perturbation modes

The focus of the present work is directed towards the development of an effective reduced basis technique for calculating the sensitivity of the non-linear dynamic structural response of mechanical systems with respect to variations in the design variables.The proposed methodology is formulated within the context of a mixed space-time finite element method, which naturally allows the treatment of initial and boundary value problems. The time dependency of the solutions is implied in the assumed space-time modal shapes, and hence the partial differential equations of motion are directly reduced to a set of non-linear simultaneous equations of a purely algebraic nature.The independent field variables are approximated in terms of perturbations modes or path derivatives with respect to a load control parameter. These modes, extracting information about the kinematic and dynamic behavior of the structural system through the higher order derivatives of the strain and kinetic energies, are appropriate bases for non-linear dynamic problems. The sensitivity derivatives of the field variables are then approximated using a combination of perturbation modes and of their sensitivity derivatives.The resulting computational procedure offers high potential for the effective and numerically efficient sensitivity analysis of dynamic systems exhibiting periodic-in-time response. The proposed methodology is illustrated addressing non-linear beam problems subjected to harmonic loading and the results obtained are compared with those of a full finite-element model.Nomenclature (O, I _i), (is1, 2, 3) Inertial frame of origin O - (P, s _i), (is1, 2, 3) Local frame in the undeformed configuration - (Q, s _i ^*), (is1, 2, 3) Local frame in the deformed configuration - t Time - l Abscissa along the beam reference line - L Beam length - (·)s(·)/t Partial derivative with respect to time - (·)s(·)/l Partial derivative with respect to space - u Position vector of the beam reference line - r Rotation parameters - ds(u, r) Generalized displacement vector - R(r) Rotation tensor associated with r - (r) Tensor defined in equations (4) and (5) - s· Finite rotation vector - as(a _s, a _v) Conformal rotation vector - Angular velocity - ws(u, ) Generalized velocity vector - k Curvature - e Generalized strains - ps(h, l) Generalized momenta - fs(s, m) Generalized sectional stress resultants - f _es(S _e, m _e) Applied external loads - M Inertia tensor     

Hyperbolicity in Extended Thermodynamics of Fermi and Bose gases

We consider the balance system of Extended Thermodynamics with 13 Moments in the case of Fermi and Bose gases, for processes not far from equilibrium. In this case, the hyperbolicity of the differential system holds only in a neighborhood of the equilibrium state. The main aim of the paper is to evaluate the hyperbolicity region of the differential system. The knowledge of this region in the state variables is mandatory to check the admissibility of the solutions and the corresponding boundary and Cauchy data in the limit of the approximation considered. The results are obtained through numerical evaluations of the Fermi and Bose integral functions that appear in the characteristic polynomial. Particular attention is devoted to the completely degenerate case when Fermi gas reaches the 0 K and when the Bose gas is in proximity of the transition temperature T _c . In these limiting cases, the hyperbolicity requirement is lost according to previous results. In the last section we make use of the Maxwellian iteration in order to evaluate the heat conductivity and the viscosity for the degenerate Fermi and Bose gas.Received: 2 March 2004, Accepted: 26 March 2004, Published online: 25 June 2004 Correspondence to: T. Ruggeri     

Structural optimization with parameter-transfer finite element

A mathematical method to solve structural problems, using parameter-transfer finite elements (P-TFE) was recently proposed by the authors [1] [2] [3]. The proposed transfer finite element approach is able to create a mathematical model of a structure, taking into account directly the whole behaviour of the structure under dynamic, aerodynamic, and thermal actions, and not by assembling, in a separate fashion, the stiffness and the mass matrix on one side and the external load vector as performed by the classical finite element procedure.The purpose of this paper is to apply the above methodology to optimization problems, in particular to obtain the minimum structural weight for a beam, under primary constraints on buckling load or natural frequencies.The use of P-TFE in the field of structural optimization overcomes most difficulties of the usual techniques of solution and the element is particularly useful in the evaluation of the sensitivity matrix.The formulation of the optimization problem based on P-TFE is presented and some applications are studied. The numerical results obtained are compared with other existing methodologies and briefly discussed.

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Sommario Gli autori hanno già proposto un metodo per studiare problemi strutturali [1] [2] [3], introducendo una nuova metodologia di discretizzazione, basata sull'impiego di elementi finiti di trasferimento, funzioni esplicite di un parametro, indicati come P-TFE. Tali elementi sono in grado di rappresentare, in similitudine alla funzione di trasferimento, il comportamento completo dell'elemento strutturale in esame, soggetto ad azioni dinamiche, aerodinamiche e termiche; sono parimenti in grado di produrre, in similitudine al metodo degli elementi finiti, un modello matematico discreto di un continuo.Scopo del presente lavoro è di applicare detta metodologia a problemi di ottimizzazione, in particolare alla ricerca del minimo peso per una trave che mantenga inalterate le sue caratteristiche di carico critico o le frequenze naturali di vibrazione.Vengono quindi presentati alcuni risultati numerici dei casi esaminati e confrontati con quelli ottenuti da altri autori con l'impiego di altre metodologie.

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List of Symbols {B} _m vector of the generalized state variables - {C} _m vector of integration constants - [I] unit matrix - EI bending stiffness - A cross-sectional area - u adimensional thickness - l beam length - M,M bending moment - [N] _m shape function ofm-th order - [N*] shape function atx ₀ - P axial load - [R] _i transfer matrix of thei-th element - T,T shear force - w transverse displacement - x adimensional independent variable - x ₀ value ofx at the left of the element - {Y} vector of state variables - {Y*} imposed condition atx ₀ - _0m Kronecker delta with the first pedix always set equal to zero - normalized eigenfrequency - normalized buckling load - mass density     

Fourier Transform in Bounded Domains

The functional analysis, the concept of distributionsu in the sense of Schwartz [7] andtheir extension given by Gelfand and Shilov [5]to ultradistributions u ,enables us to find by the means of the Fourier transform a secondlanguage to characterize physical behaviour. Almost any expressionwith physical meaning can be transformed, even if it isformulated in domains with complicated boundaries and evenif it is not integrable.Numerical procedures in the transformed space can bedeveloped in analogy to those well known in engineeringmechanics like the methods of Finite or BoundaryElements (FEM or BEM). Basis of the approaches presentedhere is the analytical representation of characteristicdistribution of a domain and the theorem of Parseval whichstates the invariance of energy in respect to thetransformation. In addition, the concept of thecharacteristic distribution leads to a very simplederivation of the Green-Gauss formulas fundamental for theBoundary or Finite Elements (e.g. [6]).     

Experimental study of similarity subgrid-scale models of turbulence in the far-field of a jet

Several versions of similarity subgrid-scale turbulence models are testeda-priori using high Reynolds number experimental data. Measurements are performed by two-dimensional Particle Image Velocimetry (PIV) in the far field of a turbulent round jet. It is first verified that the usual Smagorinsky model is poorly correlated with the real stress _ij . On the other hand, a similarity subgrid-scale model based on the resolved stress tensorL _ij , which is obtained by filtering products of resolved velocities at a scale equal to twice the grid scale, displays a much higher level of correlation. Several variants of this model are examined: the mixed model, and the global and local dynamic procedure. Model coefficients are measured, based on the condition that the subgrid models dissipate energy at the correct rate. The experimental data are employed to show that the dynamic procedure [4] yields appropriate model coefficients based only on the resolved portion of the velocity field. Some features of the dynamic procedure in its local formulation are also explored.     

Wirkungsgrad und Stabilitätsverhalten von Schleppströmungs-Pumpen für nicht-newtonsche Medien,insbesondere von Kunststoff-Extrudern

Zusammenfassung Basierend auf dem Fließgesetz nachL. Prandtl werden Einschneckenpressen und Einwalzen-Extruder als Austragsmaschinen für schmelzflüssige Kunststoffe hinsichtlich ihres Pumpwirkungsgrades und dessen Stabilität als Funktionen von zwei dimensionslosen Kennzahlen untersucht. Dabei ergeben sich grundsätzliche Vorteile für (einfache und multiple) Einwalzen-Extruder: Reduzierte Baulänge, größere Stabilität des energetischen Verhaltens, niedrigere Antriebsdrehmomente und entsprechend kleinere Getriebe, günstigere Kühlungsmöglichkeiten sowie größere Flexibilität der modellmäßigen Ansätze (relative Schleppspaltbreiteb/D als freie Variable)

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Summary The paper gives an analysis of hot-melt extrusion with drag-flow pumps, especially referring to the efficiency ( volume throughput,p extrusion pressure,P drive power requirement) calculated on the basis of the rheological equation proposed byPrandtl. It is shown that is a unique function of two dimensionless variables, and. The(, )-relief is used to introduce another dimensionsless figure,=1/|grad|, describing the operational stability of the pump at the selected design and working conditions. — Generally it is found that, with respect to operational stability at a given volume throughput, so-called single-roller extruders with a comparatively small length may offer advantages over conventional single-screw pumps.

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Symbole a dimensionsloser Drosselquotient - A stoffspezifische Schubspannung, Gl. [1] - b Breite des Schleppspalts, Gl. [8] - C stoffspezifische Schergeschwindigkeit, Gl. [1] - D Durchmesser des rotierenden Elements der Schleppströmungspumpe - e spezifische Antriebsleistung, Gl. [3] - F Mantelfläche des Schleppspalts, Gl. [9] - g Gangzahl der Schnecke bzw. Anzahl der Schleppspalte, Gl. [11b] - h Höhe (Radialmaß) des Schleppspalts, Gl. [6] - K Abkürzung, Gl. [7] - l Länge des Schleppspalts, allgemein, Gl. [5] - L wirksame axiale Schneckenlänge, Gl. [11a] - M _d Drehmoment, Gl. [4] - n Umdrehungsgeschwindigkeit, Arbeitsdrehzahl des rotierenden Elements, Gl. [4] - p Betriebsdruck, Gl. [2] - P aufgenommene Antriebsleistung, Gl. [2] - q dimensionsloser Quotient, Gl. [1b] - s zirkulare Länge des Schleppspalts, Gl. [11b] - S Abkürzung, Gl. [15] - v ₀ Umfangsgeschwindigkeit des rotierenden Elements, Gl. [6] - Volumendurchsatz, Gl. [2] - Volumendurchsatz der Druck(gradienten)strömung - Volumendurchsatz der Schleppströmung - y Ortskoordinate - Schergeschwindigkeit, Gl. [1] - generalisierter Drosselquotient, Gl. [14] - dimensionslose Systemkennzahl, Gl. [5] - dimensionsloser Pumpwirkungsgrad, Gl. [2] - µ ₀ Nullviskosität (A/C) - Stabilitätskennzahl, Gl. [20] - Schubspannung - _w Wandschubspannung - Gangsteigungswinkel - Funktionszeichen, Gl. [17] - dimensionslose Kennzahl, Gl. [21] - dimensionslose Systemkennzahl, Gl. [6] P.S.: Nach Eingang des Manuskripts dieses Beitrages bei der Redaktion (8.12.1977) wurde der Verfasser durch ein Schreiben von Prof. Dr.E. Becker/Darmstadt (datiert 21.12.1977) auf dessen Vorveröffentlichung der Abbildung 3 a hingewiesen:Mech. Res. Comm.4 (4), 235–240 (1977), dort. Figure 3.Mit 9 Abbildungen     

The effects of span position of winglet vortex generator on local heat/mass transfer over a three-row flat tube bank fin

The naphthalene sublimation method was used to study the effects of span position of vortex generators (VGs) on local heat transfer on three-row flat tube bank fin. A dimensionless factor of the larger the better characteristics, JF, is used to screen the optimum span position of VGs. In order to get JF, the local heat transfer coefficient obtained in experiments and numerical method are used to obtain the heat transferred from the fin. A new parameter, named as staggered ratio, is introduced to consider the interactions of vortices generated by partial or full periodically staggered arrangement of VGs. The present results reveal that: VGs should be mounted as near as possible to the tube wall; the vortices generated by the upstream VGs converge at wake region of flat tube; the interactions of vortices with counter rotating direction do not effect Nusselt number (Nu) greatly on fin surface mounted with VGs, but reduce Nu greatly on the other fin surface; the real staggered ratio should include the effect of flow convergence; with increasing real staggered ratio, these interactions are intensified, and heat transfer performance decreases; for average Nu and friction factor (f), the effects of interactions of vortices are not significant, f has slightly smaller value when real staggered ratio is about 0.6 than that when VGs are in no staggered arrangement. A cross section area of flow passage [m²] - A _mim minimum cross section area of flow passage [m²] - a width of flat tube [m] - b length of flat tube [m] - B _pT lateral pitch of flat tube: B _pT = S ₁/T _p - d _h hydraulic diameter of flow channel [m] - D _naph diffusion of naphthalene [m²/s] - f friction factor: f = pd _h/(Lu ² _max/2) - h mass transfer coefficient [m/s] - H height of winglet type vortex generators [m] - j Colburn factor [–] - JF a dimensionless ratio, defined in Eq. (23) [–] - L streamwise length of fin [m] - L _PVG longitudinal pitch of vortex generators divided by fin spacing: L _pVG = l _VG/T _p - l _VG pitch of in-line vortex generators [m] - m mass [kg] - m mass sublimation rate of naphthalene [kg/m²·s] - Nu Nusselt number: Nu = d _h/ - P pressure of naphthalene vapor [Pa] - p non-dimensional pitch of in-line vortex generators: p = l _VG/S ₂ - Pr Prandtl number [–] - Q heat transfer rate [W] - R universal gas constant [m²/s²·K] - Re Reynolds number: Re = ·u _max·d _h/ - S ₁ transversal pitch between flat tubes [m] - S ₂ longitudinal pitch between flat tubes [m] - Sc Schmidt number [–] - Sh Sherwood number [–]: Sh = hd _h/D _naph - Sr staggered ratio [–]: Sr = (2Hsin – C)/(2Hsin) - T _p fin spacing [m] - T temperature [K] - u _max maximum velocity [m/s] - u average velocity of air [m/s] - V volume flow rate of air [m³/s] - x,y,z coordinates [m] - z sublimation depth[m] - heat transfer coefficient [W/m²·K] - heat conductivity [W/m·K] - viscosity [kg/m²·s] - density [kg/m³] - attack angle of vortex generator [°] - time interval for naphthalene sublimation [s] - fin thickness, distance between two VGs around the tube [m] - small interval - C distance between the stream direction centerlines of VGs - p pressure drop [Pa] - 0 without VG enhancement - 1, 2, I, II fin surface I, fin surface II, respectively - atm atmosphere - f fluid - fin fin - local local value - m average - naph naphthalene - n,b naphthalene at bulk flow - n,w naphthalene at wall - VG with VG enhancement - w wall or fin surface     

Existence and Uniqueness of Time-Periodic Physically Reasonable Navier-Stokes Flow Past a Body

Let be a three-dimensional exterior domain of class C^2,, 0<<1. Assume that a Navier-Stokes liquid is moving in under the action of a body force F that is time-periodic of period T, and that the velocity of the liquid is zero at spatial infinity. In this paper we show that, if F satisfies suitable conditions, and its norm, in appropriate function spaces, is sufficiently small, there is at least one time-periodic strong solution. Furthermore, the velocity field v of such a solution decays to zero for large |x| as |x|^–1 and its spatial gradient decays as |x|^–2, both uniformly in time. In addition, the pressure p decays like |x|^–2 and its gradient like |x|^–3, for almost all t[0,T]. In the special case where F is time-independent, these solutions are also time-independent and coincide with Finns physically reasonable solutions [4]. Moreover, we show that our time-periodic solutions are unique in a very large class, namely, the class of time-periodic weak solutions satisfying the energy inequality and with corresponding pressure fields verifying mild summability conditions in ×[0,T].     

Isotherme und nicht-isotherme Spannungsrelaxation nach stationärer Scherströmung

Summary A review of the different possibilities for formulating viscoelastic models or theories is given. In steady shear flow such theories allow one to interrelate the various viscometric parameters for a given polymer.The relaxation time model proposed byBogue was chosen because of its relative simplicity. With this choice no independent parameter is introduced into the theory.In the original model an effective relaxation time, based on an integration of the strain rate history, was used. In the present work, a generalized averaging mode for the relaxation time is proposed to allow nonsteady deformation histories and non-isothermal temperature histories to be analysed. The advantage of the new mode becomes clear when either isothermal or non-isothermal stress relaxation following isothermal steady state flow is considered. The effect of the steady shear persists into the relaxation period even though no shear is being imposed then.The relaxation times and moduli for a high density polyethylene were determined and used to calculate the isothermal shear stress relaxation following cessation of steady state shear flow. The calculated results are in good agreement with the experimental data ofMenges and coworkers (50, 51).

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Zusammenfassung Es wird ein Überblick über die Struktur der viskoelastischen Modelltheorien vom Integral-Typ gegeben. Anhand der stationären einfachen Scherströmung wird gezeigt, daß sich alle viskosimetrischen Größen eines Kunststoffes im Rahmen einer solchen Theorie miteinander verknüpfen lassen.Zur Ausarbeitung eines Modells wählen wir wegen ihrer Einfachheit die Boguesche Relaxationszeit. Im Rahmen unserer Untersuchungen stellen wir fest, daß mit der Wahl dieser Relaxationszeit kein unabhängiger Parameter in die Modelltheorie eingeführt wird.Um auch instationäre Strömungsvorgänge analysieren zu können, wird eine neue mittlere Relaxationszeit definiert. Für isotherme Strömungen führt diese Mittelwertbildung zum selben Resultat wie mit einer gemittelten zweiten Invarianten des Deformationsgeschwindigkeitstensors. Der Vorteil dieser Mittelwertbildung zeigt sich deutlich bei der nicht-isothermen Spannungsrelaxation nach stationärer isothermer Scherströmung. In den dafür abgeleiteten Gleichungen ist auch weiterhin ein Einfluß der Deformationsgeschwindigkeit bzw. des Schergradienten enthalten.Schließlich werden noch Relaxationszeiten und -moduln eines Polyäthylens hoher Dichte bestimmt und daraus anschließend die isotherme Relaxation der Schubspannung nach einer stationären Scherströmung berechnet. Die erzielte Übereinstimmung mit den experimentellen Ergebnissen vonMenges und Mitarbeitern (50, 51) ist gut.

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Nomenclature a ₁,a ₂ Faktoren - a _T WLF-Faktor - b Parameter in der Bogueschen Relaxationszeit - c Cauchyscher Deformationstensor - c ^–1 Fingerscher Deformationstensor - d [s^–1] Deformationsgeschwindigkeitstensor - g _ij metrischer Tensor - i, l Summationsparameter - m ₁,m ₂ [Pa s^–1] Relaxationsfunktionen - t [s] Zeit - t _w [s] gewichtete Zeit - t,t,s [s] Integrationsparameter - C ₁ Konstante des WLF-Faktors - C ₂ [°C] Konstante des WLF-Faktors - E dimensionslose Schubspannung - G _i [Pa] Relaxationsmodul - H() [Pa] Relaxationsspektrum - N ₁,N ₂ [Pa] erste bzw. zweite Normalspannungsdifferenz - P [Pa] hydrostatischer Druck - T [°C] Temperatur - T ₀ [°C] Bezugstemperatur des WLF-Faktors - U [Pa s^–1] skalare Funktion - W [Pa] Verformungsenergie - Schergeschwindigkeit - [Pa s] Scherviskosität - ₁₂ [Pa] Schubspannung - ₁₁, ₂₂, ₃₃ [Pa] Normalspannungen in Richtung der drei RaumkoordinatenX ₁,X ₂,X ₃ - _i [s] Relaxationszeit - _i ⁰ [s] Relaxationszeit bei Bezugstemperatur - _ieff [s] effektive Relaxationszeit - mittlere Relaxationszeit - I,II,III Invarianten des Fingerschen Deformationstensors - I _d [s^–1] Invarianten des Deformationsgeschwindigkeitstensors - II _d [s^–2] Invarianten des Deformationsgeschwindigkeitstensors - III _d [s^–3] Invarianten des Deformationsgeschwindigkeitstensors Mit 6 Abbildungen und 1 Tabelle     

Stability of Large-Amplitude Viscous Shock Profiles of Hyperbolic-Parabolic Systems

We establish nonlinear L¹H³L^p orbital stability, 2p, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a class of symmetric hyperbolic-parabolic systems including compressible gas dynamics and magnetohydrodynamics (MHD) under the necessary conditions of strong spectral stability, i.e., a stable point spectrum of the linearized operator about the wave, transversality of the profile, and hyperbolic stability of the associated ideal shock. This yields in particular, together with the spectral stability results of [50], the nonlinear stability of arbitrarily large-amplitude shock profiles of isentropic Navier–Stokes equations for a gamma-law gas as 1: the first complete large-amplitude stability result for a shock profile of a system with real (i.e., partial) viscosity. A corresponding small-amplitude result was established in [53, 54] for general systems of Kawashima class by a combination of Kawashima-type energy estimates and pointwise Green function bounds, where the small-amplitude assumption was used only to close the energy estimates. Here, under the mild additional assumption that hyperbolic characteristic speeds (relative to the shock) are not only nonzero but of a common sign, we close the estimates instead by use of a Goodman-type weighted norm [25, 26] designed to control estimates in the crucial hyperbolic modes.     

Injection moulding of plastics

Summary In continuation of a previous investigation a simple analytical expression is derived in closed form for the thickness distribution of the freeze-off layer which is vitrified at the (flat) wall of an oblong rectangular cavity. As has been pointed out previously, this layer is marked for amorphous polymers by the molecular orientation (birefringence pattern) in the moulded sample. One can show that a more detailed study with the aid of the coupled equations of energy and of motion will not furnish essential improvements. Problems of polymer physics like glass transition or crystallization kinetics at extreme rates of cooling and shearing must be solved first.

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Zusammenfassung In Fortsetzung einer früheren Untersuchung wurde ein einfacher analytischer Ausdruck in geschlossener Form für die Dickenverteilung der eingefrorenen Schicht abgeleitet, die an der (flachen) Wand eines langgestreckten rechteckigen Formnestes während des Einspritzvorgangs glasig erstarrt. Wie früher auseinandergesetzt wurde, wird diese Schicht bei amorphen Polymeren durch die Molekülorientierung (Doppelbrechungsmuster) im gespritzten Formteil markiert. Man kann zeigen, daß eine eingehendere Studie mit Hilfe der gekoppelten Energie- und Impulsgleichungen keine essentiellen Verbesserungen bringt. Probleme der Polymerphysik, wie Glasübergang oder Kristallisationskinetik bei extremen Abkühlungs- und Schergeschwindigkeiten, müssen erst gelöst werden.

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List of Symbols a heat diffusivity of polymer melt (averaged overT) [m²s^–1] - B breadth of mould cavity [m] - Br Brinkman number ( ) - c heat capacity of polymer melt (averaged overT) [J kg^–1 K^–1] - F ₀ Fourier number (at _i/4H ²) - h heat transfer coefficient by melt flow [J K^–1 s^–1 m^–2] - h heat transfer coefficient by layer growth [J K^–1 s^–1 m^–2] - H half height of mould cavity [m] - L length of mould cavity [m] - n exponent in eq. [18] (= 0.417) - Nu Nußelt number (2Hh/) - P pressure gradientdP/dz in mould [N m^–3] - t time [s] - t _i injection time [s] - T _g glass transition temperature of polymer [K] - T _i injection temperature of polymer melt [K] - T _l stagnation temperature [K] - T _m mould wall temperature [K] - speed of flow front during mould filling [m s^–1] - x coordinate perpendicular to mould wall [m] - z coordinate in the injection direction [m] - thickness of stagnant layer (atT _l) [m] - ₀ optically detectable freeze-off thickness [m] - ⁺ apparent layer thickness (atT _i) [m] - dimensionless freeze-off thickness (= ₀/2H) - dimensionless distance from entrance (=z/L) - _m dimensionless coordinate of layer maximum - _g dimensionless temperature (= (T _i –T _l)/(T _g –T _m)) - _i dimensionless temperature (= (T _i –T _l)/(T _i –T _m)) - _l dimensionless temperature (= (T _i –T _l)/(T _l –T _m)) - _i viscosity of polymer atT _i [N s m^–3] - _l viscosity of polymer atT _l [N s m^–3] - heat conductivity of polymer melt (averaged) [J K^–1 s^–1 m^–1] - density of polymer melt (averaged) [kg m^–3] - dimensionless time (eq. [11]) - ⁺ dimensionless parameter (eqs. [19a] and [19b]) - dimensionless layer thickness (eq. [12]) - ⁺ dimensionless parameter (eq. [20a]) - dimensionless parameter (eqs. [11a] and [11b]) Formerly at Delft University of Technology, Delft (The Netherlands).Paper presented at the Conference on Chemical Engineering Rheology, Annual Meeting of the Deutsche Rheologische Gesellschaft in Aachen, March 5–7, 1979.With 3 figures and 1 table     

Sedimentation of solid discs in viscous medium

Free sedimentation velocities of thin discs have been measured in castor oil and liquid paraffin at constant temperature. Wall and end corrections have been applied to the measured velocities and the corrected values have been compared with those calculated fromOverbeck andGans' equations. Sedimentation velocities have been measured in both edgewise and broadsideon positions. Good agreements have been obtained with the theory in both types of sedimentation. The broadsideon position has been found to be the preferred orientation and all particles deviating even slightly from the edgewise position take the preferred orientation quickly. Results of other investigators have been discussed in the light of the present findings.Based on a dissertation submitted by (1) in partial fulfilment of the requirements for M. Sc. degree of the Dacca University in 1961.     

Analysis and computation for nonlinear elastic surface waves of permanent form

Linear elastic surface waves are nondispersive. All wavelengths travel at the Rayleigh wave speed c _R. This absence of frequency dispersion means that nonlinear waves of permanent form cannot be determined as a small perturbation from a sinusoidal wavetrain. By representing the general Rayleigh wave of the linear theory in terms of a pair of conjugate harmonic functions, waves which propagate without distortion are characterized as those having surface elevation profiles which satisfy a certain nonlinear functional equation. In the small-strain limit, this reduces to a quadratic functional equation. Methods for the analysis of this equation are presented for both periodic and nonperiodic waveforms. For periodic waveforms, the infinite system of quadratic equations for the Fourier coefficients of the profile is solved numerically in the case of a certain harmonic elastic material. Two distinct families of profiles having phase speed differing from the linearized Rayleigh wave speed are found. Additionally, two families of exceptional waveforms are found, describing profiles which travel at the Rayleigh wave speed.     

Helical flow of molten polymers in a cylindrical annulus

Summary The paper is concerned with an analytical investigation of helical flow of a non-Newtonian fluid through an annulus with a rotating inner cylinder. The shear dependence of viscosity is described by a power law and the temperature dependence by an exponential function.Velocity and temperature profiles, energy input and shear along the stream lines, pressure drop, and torque are presented for the range of input parameters encountered in polymer extrusion.The results of the study can be applied to a mixing element in a screw extruder and for a device to control extrudate temperature and output.Nomenclature a thermal diffusivity [m²/s] - b temperature coefficient [K^–1], see eq. [4] - c heat capacity [J/kg K] - h slot width [m] - I ₁,I ₂,I ₃ invariants of the rate of deformation tensor, see eq. [5] - k thermal conductivity [J/m s K] - l, L = 1/h length of the slot - l _T ,l _K thermal and kinematic entrance length - m power law exponent, see eq. [3] - M torque [m N] - p pressure [N/m²] - P dimensionless pressure gradient, see eq. [24] - P _R,P _RZ dimensionless components of the shear stress tensor, see eq. [25] and eq. [26] - r, R = r/r _wa radial coordinate - r _wa, r_wi outer and inner radius of annulus [m] - t time [s]; dwell time in the annulus - T temperature [K] - v , v_r, V_z velocity components [m/s] - v ₀ angular velocity at inner wall [m/s] - average velocity inz-direction [m/s] - V , V_R, V_Z dimensionless velocity components,v /v₀, v_r/v₀, v_z/v₀ - V _z velocity ratio, helical parameter - Y coordinate inr-direction, see eq. [20] - z, Z = z/h Pe axial coordinate - deformation - rate of deformation tensor [s^–1] - apparent viscosity [N s/m²], see eq. [3] - dimensionless temperature,b (T – T ₀) - azimuth coordinate - ratio of radii,r _wi/r_wa - density [kg/m³] - , kl shear stress tensor [N/m²] - fluidity [m²w/Nw s], see eq. [4] - Gf Griffith number, see eq. [12] - Pe Péclet number, see eq. [13] - Re Reynolds number, - 0 initial state, reference state - equilibrium state - e entrance - wi, wa at surface of inner or outer wall - r, R, z, Z, coordinates - i, j radial and axial position of nodal point in the grid - k, l tensor components Presented at Euromech 37, Napoli 6. 20–23. 1972.With 15 figuresDedicated to Prof. Dr.-Ing. G. Schenkel on his 60th birthday