On the thermodynamics of elastic-plastic materials with temperature-dependent moduli and yield stresses

Summary The subject of this article is the thermodynamics of perfect elastic-plastic materials undergoing unidimensional, but not necessarily isothermal, deformations. The first and second laws of thermodynamics are employed in a form in which only the following quantities appear: the temperature , the elastic strain _e, the plastic strain _p, the elastic modulus (gq), the yield strain (gq), the heat capacity (_e, _p,), the latent elastic heat _e(_e, _p, ), and the latent plastic heat _p(_e, _p, ). Relations among the response functions , , , _e, and _p are derived, and it is shown that a set of these relations gives a necessary and sufficient condition for compliance with the laws of thermodynamics. Some observations are made about the existence and uniqueness of energy and entropy as functions of state.Dedicated to Clifford Truesdell on the occasion of his 60th birthdayThis research was supported by the U.S. National Science Foundation.     

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.     

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).     

A direct formulation of correlations for wall-proximity corrections of hot-wire measurements

Correlations for corrections to hot-wire data for the effects of wall proximity within the viscous sublayer are usually presented in the form u/u = F (y u /). The application of such correlations requires a prior knowledge of the wall shear stress; alternatively, the correlation must be used in an iterative fashion. It is shown in the present note that any such correlation may be recast with no loss of generality in the explicit form u/u _m = f (y u _m/), which is more convenient for use.List of symbols u difference between measured and actual velocities, u _m – u - u _m measured velocity - u shear velocity, - u ⁺ on-dimensional velocity, u/u - y distance from wall - y ⁺ non-dimensional distance from wall, y u / - fluid density - fluid kinematic viscosity - _s wall shear stress     

Approximate solutions for drag coefficient of bubbles moving in shear-thinning elastic fluids

The drag coefficient for bubbles with mobile or immobile interface rising in shear-thinning elastic fluids described by an Ellis or a Carreau model is discussed. Approximate solutions based on linearization of the equations of motion are presented for the highly elastic region of flow. These solutions are in reasonably good agreement with the theoretical predictions based on variational principles and with published experimental data. C _D Drag coefficient - E ^* Differential operator [E ^{* 2} = ²/² + (sin/ ²)/(1/sin /)] - El Ellis number - F _D Drag force - K Consistency index in the power-law model for non-Newtonian fluid - n Flow behaviour index in the Carreau and power-law models - P Dimensionless pressure [=(p – p ₀)/₀ (U /R)] - p Pressure - R Bubble radius - Re ₀ Reynolds number [= 2R U /₀] - Re Reynolds number defined for the power-law fluid [= (2R) ⁿ U ^2–n /K] - r Spherical coordinate - t Time - U Terminal velocity of a bubble - u Velocity - Wi Weissenberg number - Ellis model parameter - Rate of deformation - Apparent viscosity - ₀ Zero shear rate viscosity - Infinite shear rate viscosity - Spherical coordinate - Parameter in the Carreau model - ^* Dimensionless time [=/(U /R)] - Dimensionless length [=r/R] - Second invariant of rate of deformation tensors - ^* Dimensionless second invariant of rate of deformation tensors [=/(U /R)²] - Second invariant of stress tensors - ^* Dimensionless second invariant of second invariant of stress tensor [= / ₀ ² (U /R)²] - Fluid density - Shear stress - ^* Dimensionless shear stress [=/ ₀ (U /R)] - _1/2 Ellis model parameter - ₁ ^2/* Dimensionless Ellis model parameter [= _1/2/ ₀(U /R)] - Stream function - ^* Dimensionless stream function [=/U R ²]     

Flow through a helically coiled annulus

In this paper the flow is studied of an incompressible viscous fluid through a helically coiled annulus, the torsion of its centre line taken into account. It has been shown that the torsion affects the secondary flow and contributes to the azimuthal component of velocity around the centre line. The symmetry of the secondary flow streamlines in the absence of torsion, is destroyed in its presence. Some stream lines penetrate from the upper half to the lower half, and if is further increased, a complete circulation around the centre line is obtained at low values of for all Reynolds numbers for which the analysis of this paper is valid, being the ratio of the torsion of the centre line to its curvature.Nomenclature A =constant - a outer radius of the annulus - b unit binormal vector to C - C helical centre line of the pipe - D rL - g 1000 - K Dean number=Re² - L 1+r sin - M (L ²+ ² r ²)^1/2 - n unit normal vector to C - P, P pressure and nondimensional pressure - p ₀, p pressures of O(1) and O() - Re Reynolds number=aW ₀/ - (r, , s), (r, , s) coordinates and nondimensional coordinates - nonorthogonal unit vectors along the coordinate directions - r ₀ radius of the projection of C - t unit tangent vector to C - V _r, V , V _s velocity components along the nonorthogonal directions - V_r, V, V _s nondimensional velocity components along - W ₀ average velocity in a straight annulus Greek symbols , curvature and nondimensional curvature of C - U, V, W lowest order terms for small in the velocity components along the orthogonal directions t - _r, , _s first approximations to V _r , V, V _s for small - =/=/ - kinematic viscosity - density of the fluid - , torsion and nondimensional torsion of C - , stream function and nondimensional stream function - nondimensional streamfunction for U, V - a inner radius of the annulus After this paper was accepted for publication, a paper entitled On the low-Reynolds number flow in a helical pipe, by C.Y. Wang, has appeared in J. Fluid. Mech., Vol 108, 1981, pp. 185–194. The results in Wangs paper are particular cases of this paper for =0, and are also contained in [9].     

On the effects of a periodic, spanwise variation of suction upon asymptotic suction flow

Summary The effects of superposing streamwise vorticity, periodic in the lateral direction, upon two-dimensional asymptotic suction flow are analyzed. Such vorticity, generated by prescribing a spanwise variation in the suction velocity, is known to play an important role in unstable and turbulent boundary layers. The flow induced by the variation has been obtained for a freestream velocity which (i) is steady, (ii) oscillates periodically in time, (iii) changes impulsively from rest. For the oscillatory case it is shown that a frequency can exist which maximizes the induced, unsteady wall shear stress for a given spanwise period. For steady flow the heat transfer to, or from a wall at constant temperature has also been computed.Nomenclature (x, y, z) spatial coordinates - (u, v, w) corresponding components of velocity - (, , ) corresponding components of vorticity - t time - stream function for v and w - v _w mean wall suction velocity - nondimensional amplitude of variation in wall suction velocity - characteristic wavenumber for variation in direction of z - T temperature - P pressure - density - coefficient of kinematic viscosity - coefficient of thermal diffusivity - (/v _w)² - frequency of oscillation of freestream velocity - nondimensional amplitude of freestream oscillation - /v _w ² - z z - y –v _w y/ - v _w ² t/4 - /v _w - U ₀ characteristic freestream velocity - u/U ₀ - coefficient of viscosity - _w wall shear stress - Prandtl number (/) - q heat transfer to wall - T _w wall temperature - T (T _w–T)/(T _w–)     

The onset of Taylor-like vortices in the flow induced by an impulsively started rotating cylinder

The onset of instability in the flow by an impulsively started rotating cylinder is analyzed under linear theory. It is well-known that at the critical Taylor number T_c=1695 the secondary flow in form of Taylor vortices sets in under the narrow-gap approximation. Here the dimensionless critical time _c to mark the onset of instability for TT_c is presented as a function of the Taylor number T. Available experimental data of water indicate that deviation of the velocity profiles from the primary flow occurs starting from a certain time 4_c. It seems evident that during _c4_c the secondary flow is very weak and the primary state of time-dependent annular Couette flow is maintained.     

Correlations between porous medium flow data and turbulent tube flow results for aqueous hydroxypropylguar solutions (Part 2)

Turbulent tube flow and the flow through a porous medium of aqueous hydroxypropylguar (HPG) solutions in concentrations from 100 wppm to 5000 wppm is investigated. Taking the rheological flow curves into account reveals that the effectiveness in turbulent tube flow and the efficiency for the flow through a porous medium both start at the same onset wall shear stress of 1.3 Pa. The similarity of the curves = ( _w ) and = ( _w ), respectively, leads to a simple linear relation / =k, where the constantk or proportionality depends uponc. This offers the possibility to deduce (for turbulent tube flow) from (for flow through a porous medium). In conjunction with rheological data, will reveal whether, and if yes to what extent, drag reduction will take place (even at high concentrations).The relation of our treatment to the model-based Deborah number concept is shown and a scale-up formula for the onset in turbulent tube flow is deduced as well.     

Stability of stationary traveling waves on the surface of a vertical film of viscous fluid

The stability of stationary traveling waves of the first and second families with respect to infinitesimal perturbations of arbitrary wavelength is subjected to a detailed numerical investigation. The existence of a unique region of stability of the first family is established for wave numbers (₁, ₁) corresponding to the optimal wave regime. There are several regions of stability of the second family ( _k , _k),k=2,3,..., lying close to the local flow rate maxima. In the regions of instability the growth rates of perturbations of the first family are several times greater than for the second family. This difference increases with increase in the Reynolds number. The calculations make it possible to explain a number of experimental observations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 33–41, May–June, 1989.The authors are grateful to V. Ya. Shkadov for his constant interest, and to A. G. Kulikovskii, A. A. Barmin and their seminar participants for useful discussions and suggestions.     

Testing viscometric predictions of the internal viscosity model,using dilute viscous theta solutions

A stress-symmetrized internal viscosity (I.V.) model for flexible polymer chains, proposed by Bazua and Williams, is scrutinized for its theoretical predictions of complex viscosity ^* () = – i and non-Newtonian viscosity (), where is frequency and is shear stress. Parameters varied are the number of submolecules,N (i.e., molecular weightM = NM _s ); the hydrodynamic interaction,h ^*; and/f, where andf are the I.V. and friction coefficients of the submolecule. Detailed examination is made of the eigenvalues _p (N, h ^*) and how they can be estimated by various approximations, and property predictions are made for these approximations.Comparisons are made with data from our preceding companion paper, representing intrinsic properties [], [], [] in very viscous theta solutions, so that theoretical foundations of the model are fulfilled. It is found that [ ()] data can be predicted well, but that [ ()] data cannot be matched at high. The latter deficiency is attributed in part to unrealistic predictions of coil deformation in shear.     

Analysis of suddenly started laminar flow in the entrance region of a circular tube

Suddenly started laminar flow in the entrance region of a circular tube, with constant inlet velocity, is investigated analytically by using integral momentum approach. A closed form solution to the integral momentum equation is obtained by the method of characteristics to determine boundary layer thickness, entrance length, velocity profile, and pressure gradient.Nomenclature M(, , ) a function - N(, , ) a function - p pressure - p* p/1/2U ², dimensionless pressure - Q(, , ) a function - R radius of the tube - r radial distance - Re 2RU/, Reynolds number - t time - U inlet velocity, constant for all time, uniform over the cross section - u velocity in the boundary layer - u* u/U, dimensionless velocity - u ₁ velocity in the inviscid core - x axial distance - y distance perpendicular to the axis of the tube - y* y/R, dimensionless distance perpendicular to the axis - boundary layer thickness - * displacement thickness - /R, dimensionless boundary layer thickness - momentum thickness - absolute viscosity of the fluid - /, kinematic viscosity of the fluid - x/(R Re), dimensionless axial distance - density of the fluid - tU/(R Re), dimensionless time - _w wall shear stress     

Dynamic-mechanical properties of a bitumen-silica composite

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

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

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

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With 6 figures     

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.     

Viscometric properties of viscous theta solutions of monodisperse polystyrene

Theta solvents for polystyrene are prepared from high-viscosity blends of styrene and low-molecular-weight polystyrene, and then used to make dilute solutions with monodisperse polystyrene solutes of high-M = 2.3, 6.0, 9.0, 18.0 · 10⁵. A Weissenberg rheogoniometer is used to measure the non-Newtonian viscosity as a function of shear stress, for low values, and also the complex viscosity components and as functions of frequency. A capillary viscometer is used for high- measurements of(). Viscometric properties, at room temperature, are analyzed as functions of high-molecular-weight solute concentrationc with parameters of constant or to obtain [()], [ ()], and [ ()]. Such a collection of data has apparently not previously been available for polymers in theta solvents (in which Gaussian chain statistics prevail). Also unique is the achievement of high stress ( = 2 10⁴ Pa) at low shear rate, by virtue of high solvent viscosity which is not characteristic of other known theta solvents.     

Effect of measuring volume length on two-component laser velocimeter measurements in a turbulent boundary layer

The effects of finite measuring volume length on laser velocimetry measurements of turbulent boundary layers were studied. Four different effective measuring volume lengths, ranging in spanwise extent from 7 to 44 viscous units, were used in a low Reynolds number (Re=1440) turbulent boundary layer with high data density. Reynolds shear stress profiles in the near-wall region show that u v strongly depends on the measuring volume length; at a given y-position, u v decreases with increasing measuring volume length. This dependence was attributed to simultaneous validations on the U and V channels of Doppler bursts coming from different particles within the measuring volume. Moments of the streamwise velocity showed a slight dependence on measuring volume length, indicating that spatial averaging effects well known for hot-films and hot-wires can occur in laser velocimetry measurements when the data density is high.List of symbols time-averaged quantity - u wall friction velocity, ( _w /)^1/2 - v kinematic viscosity - d _p pinhole diameter - l _eff spanwise extent of LDV measuring volume viewed by photomultiplier - l ⁺ non-dimensional length of measuring volume, l _eff u /v - y ⁺ non-dimensional coordinate in spanwise direction, y u /v - z ⁺ non-dimensional coordinate in spanwise direction, z u /v - U ⁺ non-dimensional mean velocity, /u - u instantaneous streamwise velocity fluctuation, U &#x2329;U - v instantaneous normal velocity fluctuation, V–V - u RMS streamwise velocity fluctuation, u ^21/2 - v RMS normal velocity fluctuation, v ^21/2 - Re Reynolds number based on momentum thickness, U 0/v - R _uv cross-correlation coefficient, u v/u v - R₁₂(0, 0, z) two point correlation between u and v with z-separation, <u(0, 0, 0) v (0, 0, z)>/<u(0, 0, 0) v (0, 0, 0)> - N rate at which bursts are validated by counter processor - T Taylor time microscale, u (dv/dt²)^–1/2     

Cooperative diffusion in a temporary network

Summary Local rearrangements following connectivity changes in a temporary network are taken into explicit account by introducing a friction coefficient for non-affine adaptive motions of the general junction. In this way a simple model of internal diffusion is obtained, in which connectivity changes act as driving forces. A transport equation is then derived from the microscopic equation of motion. This transport equation contains a regeneration term, a convective term and an additional diffusive contribution, depending on past deformation history. Retaining the basic assumptions of the simple network theory on regeneration kinetics, a calculation of the diffusive term is performed for tetrafunctional networks; preliminary results obtained for semi-permanent networks undergoing regeneration pulses are extended to temporary systems on the basis of mean field arguments. Two characteristic time constants appear in the constitutive equation: the chain lifetime ₀ and the diffusive constant ₁. A preliminary application to elongational flow shows that the elastic modulusG and the lifetime ₀ practically determine the limit response of the material at low flow rates, whereas the magnitude of the diffusive time constant ₁ relative to ₀ is responsible for the ultimate properties of the fluid. If ₀/ ₁ > _c 0.933 the elongational viscosity curve shows a vertical asymptote, whereas a turbulent transition occurs at finite velocity gradients if ₀/ ₁ < _c . The latter range may correspond to the physical situation actually found in very concentrated systems, such as gels and polymer melts.

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Zusammenfassung Lokale Umordnungen als Folge von Platzwechseln in temporären Netzwerken werden durch Einführung eines Reibungskoeffizienten für die nicht-affinen Anpassungsbewegungen der Bindungen explizit in die Betrachtung einbezogen. Auf diese Weise erhält man ein einfaches Modell für die innere Diffusion, bei der die Platzwechsel als Antriebskräfte wirken. Aus der mikroskopischen Bewegungsgleichung wird eine Transportgleichung abgeleitet. Diese enthält einen Regenerierungsterm, einen konvektiven Term und einen zusätzlichen Diffusionsbeitrag, der von der Vorgeschichte abhängt. Unter Voraussetzung der Erhaltung der Grundannahmen der Theorie einfacher Netzwerke für die Regenerierungskinetik wird der Diffusionsterm für tetrafunktionale Netzwerke berechnet. Vorläufige Ergebnisse für semi-permanente Netzwerke bei Einwirkung von Regenerierungs-Impulsen werden mit Hilfe von Gleichgewichts-Argumenten (mean field arguments) auf temporäre Systeme übertragen. In der konstitutiven Gleichung treten zwei charakteristische Zeitkonstanten auf: die Lebensdauer der Ketten ₀ und die Diffusionskonstante ₁. Eine vorläufige Anwendung auf Dehnströmungen zeigt, daß der ElastizitätsmodulG und die Lebensdauer ₀ das Grenzverhalten des Stoffes bei kleinen Dehngeschwindigkeiten praktisch allein bestimmen, wohingegen das Verhältnis ₀/ ₁ für das Grenzverhalten bei großen Dehngeschwindigkeiten verantwortlich ist. Wenn ₀/ ₁ > _c 0,933 wird, zeigt die Dehnviskositätskurve eine vertikale Asymptote, wohingegen bei ₀/ ₁ < _c für endliche Geschwindigkeiten ein Übergang zur turbulenten Strömung vorhanden ist. Dieser turbulente Bereich entspricht möglicherweise physikalischen Realisierungen, wie sie in sehr konzentrierten Systemen, z. B. Gelen oder Polymerschmelzen, vorgefunden werden.

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The response of a turbulent boundary layer to different shaped transverse grooves

The response of a turbulent boundary layer to three different shaped transverse grooves was investigated at two values of momentum thickness Reynolds numbers ( R =1000 and 3000). A 20-mm wide square, semicircular and triangular groove with depth to width ( d / w) ratio of unity was used. In general, the effects of the grooves are more significant at the higher R , with the most pronounced effects caused by the square groove. An increase in wall shear stress _w was observed just downstream of the groove for all three shapes. The increase in _w is followed by a small decrease in _w below the smooth-wall value before it relaxes back to the corresponding smooth-wall value at x / ₀3. At the higher R , the maximum increase in _w for the square groove is about 50% higher than for the semicircular groove and almost twice that for the triangular groove. The effect of the square groove on U / U ₀, u / U ₀ and v / U ₀ is much more significant than the effect of the semicircular and triangular grooves. There is an increase in the bursting frequency ( f _B⁺) on the grooved-wall compared to the smooth-wall case. The distribution of f _B⁺ downstream of the different shaped grooves is similar to the _w distribution.Symbols C _f skin friction coefficient, C _f2 _w/( ( U ₀)²) - C _f,0 skin friction coefficient on the smooth wall - d groove depth - D _h diameter of the idealized primary eddy inside the groove - D _h,s diameter of the idealized secondary eddies inside the groove - d _i internal layer thickness - E turbulent energy spectrum - f _B bursting frequency - f _B⁺ normalized bursting frequency, f _B⁺ f _B/( u )² - k wave number, k =2f/ U - q _i ⁺ contributing quadrant to the total Reynolds stress – uv , q _i ⁺ – uv _i /( u )², i =1, 2, 3, 4 - R Reynolds number based on , R U ₀ / - R Reynolds number based on , R U ₀ / - U mean velocity in the streamwise direction - U ₀ free stream velocity - U ⁺ normalized U by inner variable, U ⁺ U / u - u root-mean-square of velocity fluctuation in the streamwise direction - u ⁺ normalized u by inner variable, u ⁺ u / u - u friction velocity, u ( _w/ )^0.5 - – uv Reynolds stress - v root-mean-square of velocity fluctuation in the wall-normal direction - w groove width - x streamwise coordinate measured from the groove trailing edge - y wall-normal coordinate - y ⁺ normalized y by inner variables, y ⁺ yu / Greek symbols boundary layer thickness - ₀ boundary layer thickness just upstream of the groove, unless otherwise stated - fluid kinematic viscosity - momentum thickness - fluid density - _w wall shear stress     

Approximate First Integrals of a Galaxy Model

First-order approximate first integrals (conserved quantities)of a Hamiltonian dynamical system with two degrees of freedomwhich arises in the modeling of central part of a deformed galaxy [1] havebeen obtained based on the approximate Noether symmetries for resonances₁=₂, ₁=2₂ and 2₁=3₂. Furthermore,KAM curves have been obtained analytically and they have been compared with thenumerical ones on the Poincaré surface of section.     

Flow in porous media II: The governing equations for immiscible,two-phase flow

The Stokes flow of two immiscible fluids through a rigid porous medium is analyzed using the method of volume averaging. The volume-averaged momentum equations, in terms of averaged quantities and spatial deviations, are identical in form to that obtained for single phase flow; however, the solution of the closure problem gives rise to additional terms not found in the traditional treatment of two-phase flow. Qualitative arguments suggest that the nontraditional terms may be important when / is of order one, and order of magnitude analysis indicates that they may be significant in terms of the motion of a fluid at very low volume fractions. The theory contains features that could give rise to hysteresis effects, but in the present form it is restricted to static contact line phenomena.Roman Letters (, = , , and ) A interfacial area of the- interface contained within the macroscopic system, m² - A _e area of entrances and exits for the -phase contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - g gravity vector, m²/s - H mean curvature of the- interface, m^–1 - H area average of the mean curvature, m^–1 - H – H , deviation of the mean curvature, m^–1 - I unit tensor - K Darcy's law permeability tensor, m² - K permeability tensor for the-phase, m² - K viscous drag tensor for the-phase equation of motion - K viscous drag tensor for the-phase equation of motion - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - n unit normal vector pointing from the-phase toward the-phase (n = –n ) - p _c p – P , capillary pressure, N/m² - p pressure in the-phase, N/m² - p intrinsic phase average pressure for the-phase, N/m² - p – p , spatial deviation of the pressure in the-phase, N/m² - r ₀ radius of the averaging volume, m - t time, s - v velocity vector for the-phase, m/s - v phase average velocity vector for the-phase, m/s - v intrinsic phase average velocity vector for the-phase, m/s - v – v , spatial deviation of the velocity vector for the-phase, m/s - V averaging volume, m³ - V volume of the-phase contained within the averaging volume, m³ Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - viscosity of the-phase, Nt/m² - surface tension of the- interface, N/m - viscous stress tensor for the-phase, N/m² - / kinematic viscosity, m²/s