The temperature field in rotational rheometers and flow curve correction for viscous dissipation

A solution is presented for incompressible non-Newtonian liquids of the one-dimensional stationary temperature field which arises due to heat dissipation between two concentric cylinders, the outer fixed and thermostated, the inner rotating at a constant angular velocity. The object of the study is to outline a simple procedure for determining the temperature rise of the liquid and, primarily, to ascertain the corrections of the consistent variables and D which enable the experimenter to rectify the rheogram on the basis of measurement of the shear stress and the angular velocity . The results obtained are summarized in graphical form as diagrams of the temperature and velocity fields and, to facilitate practical application of the correction procedure, in a table relating the dimensionless temperature function (, n, ) to the geometry , the flow behaviour index n, and the coefficient of temperature rise and showing the function (1) as well.List of symbols a radius of the inner cylinder - b radius of the outer cylinder - constant angular velocity of the inner cylinder - r* dimensionless radial coordinate r/b - * dimensionless angular velocity of the liquid - K fluid consistency index - n flow behaviour index - dimensionless temperature rise (T–T ₀)/T ₀ - T temperature of measured liquid (K) - T ₀ temperature of the thermostated bath - Br Brinkman criterion - _f thermal conductivity of liquid - C constant of integration - coefficient of sensitivity in consistency-temperature law - coefficient of sensitivity divided by flow behaviour index: /n - (r*) dimensionless temperature function - coefficient of temperature rise; =Br· - ratio of the radii of inner and outer cylinder - T(1) temperature on the inner wall of the outer cylinder, i.e. for r*=1 - outer cylinder wall thickness - coefficient of heat transfer - q heat flux - k overall heat transfer coefficient - h height of measured liquid - _s thermal conductivity of the outer cylinder - (1) derivative of the dimensionless temperature function at point r*=1 - dimensionless heat transfer constant - _i (r*) dimensional temperature function calculated for isothermal wall; T(1)=T ₀ - dynamic viscosity - _i () maximum value of the dimensionless temperature function - dimensionless symbol — ratio of C/C ₀ - D rate of shear - shear stress - rate of shear (not considering dissipation) - shear stress (not considering dissipation) - D ⁺ rate of shear corrected for the inner cylinder temperature - ⁺ shear stress on the inner cylinder obtained by measurement on the rheometer used - _j rate of shear on the inner cylinder for j-th measurement referred to a single constant temperature     

Non-Linear response of a long,discontinuous fiber/melt system in elongational flows

The authors investigated the transient elongational behavior of a highly-aligned 600% volume fraction long, discontinuous fiber filled poly-ether-ketone-ketone melt with a computer-controlled extensional rheometer at 370°C. Prior experiments at controlled strain rate and stress produced _E ⁺ (t, ) and ^– (t, _E) similar to a shear dominated flow of a non-linear viscoelastic fluid. Stress relaxation following steady extension showed nonlinear effects in the change in stress decay rate with increasing strain rate. Continuous relaxation spectra showed a shift in the spectral peak to smaller values of with increasing strain rate. The Giesekus nonlinear constitutive relation modeled the elongation and stress relaxation with shearing rate at the fiber surface set by a strain rate magnification factor. Suitable for elongation, the model produced insufficient shift in the stress relaxation spectrum to account for the large change in stress decay rate exhibited in the experiments.English alphabet a _r aspect ratio of the fibers or l/d - A ₀ initial uniform cross-section area of the specimen - d fiber diameter - f fiber volume fraction - H() relaxation spectrum found by the method of Ferry and William l length of the fiber - L(t) time function specimen length - L ₀ initial specimen length - r radial coordinate across the shear cell - R _i fiber radius and inner cell dimension - R _o outer cell radius - t time in s - t _max duration of the extension - T _g glass transition temperature of the polymer - v velocity of the moving end of the test specimen - x axial position where is calculated Greek alphabet nonlinearity parameter in the Giesekus relation - axial mass distribution along the specimen major axis - shear strain rate - strain tensor - ₍₁₎ first convected derivative of the strain tensor - ₍₂₎ second convected derivative of the strain tensor - average strain at the end of extension as determined from - extension strain rate - average extension strain rate determined from - ^– transient strain rate under controlled stress, creep, test - _E elongational viscosity - _Eapp apparent elongational viscosity determined from - _E ⁺ transient elongational viscosity - ₀ zero shear rate viscosity - relaxation parameter - ₁ relaxation parameter in either Jeffrey's or Giesekus fluid - ₂ retardation parameter in either Jeffrey's or Giesekus fluid - _max relaxation value at which 99.9% of the H spectrum had occurred - _p relaxation value at which H reaches a maximum - volumetric composite density - _E elongational stress - _E ⁺ transient elongational stress - _E ^– controlled elongational stress, creep stress - _E ^y peak elongational stress in controlled experiment - shear stress at surface of the fiber in a shear cell - _yx simple shear component of the strain rate tensor - stress tensor - ₁ first convected derivative of the stress tensor     

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.     

Elastico-viscous properties of deflocculated china clay suspensions — concentration effects

Summary The physical properties of deflocculated china clay suspensions are studied in a combined steady and low-amplitude oscillatory shear flow. Concentration effects are examined and it is shown that, with increasing concentration, an initial shear thinning region is followed by a shear thickening one. Qualitative agreement is obtained between theory and experiment for a range of concentrations of suspensions, all of which exhibit marked elastic properties. The experimental results were obtained using a Weissenberg Rheogoniometer.

---

Zusammenfassung Es werden die physikalischen Eigenschaften deflockulierter Suspensionen von Porzellanerde in einer kombinierten stationären und oszillatorischen Scherströmung mit niedriger Amplitude studiert. Der Einfluß der Konzentration wird untersucht, und es wird gezeigt, daß mit wachsender Konzentration sich an den anfänglich allein vorhandenen Bereich mit Scherentzähung ein Bereich mit Scherverzähung anschließt. Zwischen Theorie und Experiment wird eine qualitative Übereinstimmung in einem Konzentrationsbereich gefunden, in dem ausgeprägte viskoelastische Eigenschaften vorhanden sind. Die experimentellen Ergebnisse werden mit Hilfe eines Weissenberg-Rheogoniometers erhalten.

---

c phase lag in oscillatory testing - D(t – t) deformation history - F, G non-dimensional complex functions of - complex conjugate ofF - G dynamic rigidity - i - I % increase in mean couple under superposed shear rates - I ₁ moment of inertia of the top platen (i.e. cone) - J amplitude ratio, ₁/ ₁ - K ₁ restoring constant of the torsion bar - q steady shear rate - r, , spherical polar coordinates - t current time - v _i velocity vector - w/w concentration by weight - W a function of andt - ₁ angular amplitude of the motion of the plate - shear rate - /q - apparent viscosity - dynamic viscosity - ^* complex dynamic viscosity - ₀ limiting viscosity at small rates of shear - ₀ gap angle in cone and plate system - ₁, ₂, ₃, ₄,µ ₀ relaxation time constants - shear stress - ₀ unperturbed shear stress - ₁, ₂ kernel functions - angular frequency of oscillation - steady angular velocity of the plate With 16 figures     

Rheological properties of reactive polymer melts.

A new method for describing the rheological properties of reactive polymer melts, which was presented in an earlier paper, is developed in more detail. In particular, a detailed derivation of the equation of a first-order rheometrical flow surface is given and a procedure for determining parameters and functions occurring in this equation is proposed. The experimental verification of the presented approach was carried out using our data for polyamide-6.Notation E Dimensionless reduced viscosity, eq. (34) - E ₀ Newtonian asymptote of the function (36) - E power-law asymptote of the function (36) - E _{= 1} the value ofE at = 1 - k degradation reaction rate constant, s^–1 - k ₁ rate constant of function (t), eq. (26), s^–1 - k ₂ rate constant of function (t), eq. (29), s^–1 - K(t) residence-time-dependent consistency factor, eq. (22) - M _w weight-average molecular weight - M _x x-th moment of the molecular weight distribution - R gas constant - S _x M _x /M _w - t residence time in molten state, s - t _j thej-th value oft, s - T temperature, K - % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xd9vqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieGaceWFZo% Gbaiaaaaa!3B4E!\[\dot \gamma \] shear rate, s^–1 - _i thei-th value of , s^–1 - _r =1 the value of at = 1, s^–1 - ^* reduced shear rate, eq. (44), s^–1 - dimensionless reduced shear rate, eq. (35) - viscosity, Pa · s - shear-rate and residence-time dependent viscosity, Pa · s - zero-shear-rate degradation curve - degradation curve at - _{t0 (t)} zero-residence-time flow curve - Newtonian asymptote of the RFS - instantaneous flow curve - power-law asymptote of the RFS - _0,0 zero-shear-rate and zero-residence-time viscosity, Pa · s - _E=1 value of viscosity atE=1, Pa · s - ^* reduced viscosity, eq. (43), Pa · s - zero-residence-time rheological time constant, s - density, kg/m³ - (t),(t) residence time functions     

Model analysis of nonlinear viscoelastic behaviour by use of a single integral constitutive equation: Stresses and birefringence of a polystyrene melt in intermittent shear flows

Summary A single integral constitutive equation with strain dependent and factorized memory function is applied to describe the time dependence of the shear stress, the primary normal-stress difference, and, by using the stress-optical law, also the extinction angle and flow birefringence of a polystyrene melt in intermittent shear flows. The theoretical predictions are compared with measurements. The nonlinearity of the viscoelastic behaviour which is represented by the so called damping function, is approximated by a single exponential function with one parametern. The damping constantn as well as a discrete relaxation time spectrum of the melt can be determined from the frequency dependence of the loss and storage moduli.

---

Zusammenfassung Eine Zustandsgleichung vom Integraltyp mit einer deformationsabhängigen und faktorisierten Gedächtnisfunktion wird zur Beschreibung der Zeitabhängigkeit der Schubspannung, der ersten Normalspannungsdifferenz und, unter Verwendung des spannungsoptischen Gesetzes, auch des Auslöschungswinkels und der Strömungsdoppelbrechung einer Polystyrol-Schmelze bei Scherströmungen herangezogen. Die theoretischen Voraussagen werden mit Messungen verglichen. Die Nichtlinearität des viskoelastischen Verhaltens, repräsentiert durch die sogenannte Dämpfungsfunktion, wird durch eine einfache Exponentialfunktion mit nur einem Parametern angenähert. Die Dämpfungskonstanten kann, wie auch ein diskretes Relaxationszeitspektrum der Schmelze, aus der Frequenzabhängigkeit der Speicher- und Verlustmoduln bestimmt werden.

---

a _i weight factor of thei-th relaxation time - a _T shift factor - C stress-optical coefficient - n flow birefringence in the shear flow plane - shear relaxation modulus - G() shear storage modulus - () shear loss modulus - H() relaxation time spectrum - h( _t,t ² ) damping function - M _w weight-average molecular weight - M _n number-average molecular weight - n damping constant - p ₁₂ shear stress - p ₁₁ –p ₂₂ primary normal stress difference - t current time - t past time - extinction angle - ( — _i) delta function - time and shear rate dependent viscosity - | ^*| absolute value of the complex viscosity - shear rate - _t,t relative shear strain between the statest andt - memory function - angular frequency - relaxation time - _i i-th relaxation time of the line spectrum - time and shear rate dependent primary normal stress coefficient - s steady-state value - t time dependence - ° linear viscoelastic behaviour With 6 figures and 1 table     

Yield stress: A time-dependent property and how to measure it

This paper reviews the different aspects of the yield stress phenomenon and attempts a synthesis of knowledge. Yield stress can be probed using constant shear stress or shear rate. The magnitude of the result depends on the time allowed to determine whether the sample has developed continuous flow or has ceased flowing. It is closely associated with creep, stress growth and thixotropic breakdown and recovery, and the characteristic times of these transient responses play a part in yield stress measurement. In thixotropic fluids, yield stress is a function of structure and hence of time. In simple thixotropy, the yield stress derived from the equilibrium flow curve is the same as that for the fully built-up structure. But in many materials, the static yield stress obtained after prolonged rest is much higher than the dynamic yield stress from the equilibrium flow curve. This is associated with the phenomenon in which the equilibrium flow curve bends upwards as the shear rate is reduced to very low values. The paper also reviews the many methods that can be used to measure yield stress. It is pointed out that the choice of observation time or shear rate to use should be related to the characteristic time of the flow process to which the result is to be applied. Examples discussed are solids suspension capability of fluids, levelling and sagging, pipeline flow and start-up power requirement of mixers. CS constant structure - D diameter of Gun Rheometer tube - EFC equilibrium flow curve - G measured torque - L length of Gun Rheometer tube - P _min minimum pressure to cause flow - t time - form factor for shear stress - - _y - shear rate - a particular value of shear rate - reference shear rate - test shear rate - shear stress - _y yield stress - _yd dynamic yield stress - _ys static yield stress - 0 initial value after speed change - e equilibrium Paper presented at the British Society of Rheology Conference on New Techniques in Experimental Rheology, University of Reading, 9–11 September 1985.     

Heat conduction in a cylinder crossed by an electric current with skin effect

The steady periodic temperature distribution in an infinitely long solid cylinder crossed by an alternating current is evaluated. First, the time dependent and non-uniform power generated per unit volume by Joule effect within the cylinder is determined. Then, the dimensionless temperature distribution is obtained by analytical methods in steady periodic regime. Dimensionless tables which yield the amplitude and the phase of temperature oscillations both on the axis and on the surface of copper or nichrome cylindrical electric resistors are presented.

---

Wärmeleitung in einem stromdurchflossenen Zylinder unter Berücksichtigung des Skin-Effektes

Zusammenfassung Es wird die periodische Temperaturverteilung für den eingeschwungenen Zustand in einem unendlich langen, von Wechselstrom durchflossenen Vollzylinder ermittelt. Zuerst erfolgt die Bestimmung der zeitabhängigen, nichgleichförmigen Energiefreisetzung pro Volumeneinheit des Zylinders infolge Joulescher Wärmeentwicklung und anschließend die Ermittlung der quasistationären Temperaturverteilung auf analytischem Wege. Amplitude und Phasenverzögerung der Temperaturschwingungen werden für die Achse und die Oberfläche eines Kupfer- oder Nickelchromzylinders tabellarisch in dimensionsloser Form mitgeteilt.

---

Nomenclature A integration constant introduced in Eq. (2) - ber, bei Thomson functions of order zero - Bi Biot numberhr ₀/ - c speed of light in empty space - c ₁,c ₂ integration constants introduced in Eq. (46) - c _p specific heat at constant pressure - E electric field - E _z component ofE alongz - E time independent part ofE, defined in Eq. (1) - f function ofs and defined in Eq. (11) - g function ofs and defined in Eq. (37) - h convection heat transfer coefficient - H magnetic field - i imaginary uniti=(–1)^1/2 - I electric current - I _eff effective electric currentI _eff=I/2^1/2 - Im imaginary part of a complex number - J _n Bessel function of first kind and ordern - J electric current density - q _g power generated per unit volume - time average of the power generated per unit volume - time averaged power per unit length - r radial coordinate - R electric resistance per unit length - r ₀ radius of the cylinder - Re real part of a complex number - s dimensionless radial coordinates=r/r ₀ - s, s integration variables - t time - T temperature - time averaged temperature - T _f fluid temperature outside the boundary layer - time average of the surface temperature of the cylinder - u, functions ofs, and defined in Eqs. (47) and (48) - W Wronskian - x position vector - x real variable - Y _n Bessel function of second kind and ordern - z unit vector parallel to the axis of the cylinder - z axial coordinate - · modulus of a complex number - equal by definition Greek symbols amplitude of the dimensionless temperature oscillations - electric permittivity - dimensionless temperature defined in Eq. (16) - ₀, ₁, ₂ functions ofs defined in Eq. (22) - thermal conductivity - dimensionless parameter=(2)^1/2 - magnetic permeability - ₀ magnetic permeability of free space - function of defined in Eq. (59) - dimensionless parameter=c _p/() - mass density - electric conductivity - dimensionless time=t - phase of the dimensionless temperature oscillations - function ofs:= ₁+i ₂ - angular frequency - dimensionless parameter=()^1/2 r ₀     

Influence of the formation of associations of macromolecules on the rheo-optical properties of their dilute solutions. I. Theory

Assuming the formation of doublets in the flow according to a mass action law, the shear rate and the concentration dependence of the extinction angle, of the birefringence, and of the average coil expansion are calculated for dilute solutions of flexible macromolecules. It is shown that this reversible association process has a strong influence on the measurable parameters in a flow birefringence experiment. c concentration (g/cm³) - h ² mean square end-to-end distance at shear rate - h ₀ ² mean-square end-to-end distance at zero-shear rate - n refractive index of the solution (not very different from the solvent for a very dilute solution) - E mean coil expansion - K ₀,K constant of the mass action law - M molecular weight - R _G gas constant - T absolute temperature - ₁ – ₂ optical anisotropy of the segment - ₀ Deborah number: - Deborah number: - shear rate - ₀, reduced concentration - _s viscosity of the solvent - [] ₀ intrinsic viscosity at zero-shear rate - [] intrinsic viscosity at shear rate - extinction angle - N _a Avodagro's number - _n magnitude of the birefringence     

A three-parameter model describing the behaviour of a viscoelastic liquid in a tangential annular flow

A three-parameter model describing the shear rate-shear stress relation of viscoelastic liquids and in which each parameter has a physical significance, is applied to a tangential annular flow in order to calculate the velocity profile and the shear rate distribution. Experiments were carried out with a 5000 wppm aqueous solution of polyacrylamide and different types of rheometers. In a shear-rate range of seven decades (5 10^–3 s^–1 < < 1.2 10⁵ s^–1) a good agreement is obtained between apparent viscosities calculated with our model and those measured with three different types of rheometers, i.e. Couette rheometers, a cone-and-plate rheogoniometer and a capillary tube rheometer. a physical quantity defined by:a = {1 – ( / ₀)}/ ₀ (Pa^–1) - C constant of integration (1) - r distancer from the center (m) - r ₁,r ₂ radius of the inner and outer cylinder (m) - v _r local tangential velocity at a distancer from the center (v _r = _r r) (m s^–1) - v ₂ local tangential velocity at a distancer ₂ from the center (m s^–1) - shear rate (s^–1) - local shear rate (s^–1) - ₁ wall shear rate at the inner cylinder (s^–1) - dynamic viscosity (Pa s) - _a apparent viscosity (_a = / ) (Pa s) - _a1 apparent viscosity at the inner cylinder (Pa s) - ₀ zero-shear viscosity (Pa s) - infinite-shear viscosity (Pa s) - shear stress (Pa) - _r local shear stress at a distancer from the center (Pa) - ₀ yield stress (Pa) - ₁, ₂ wall shear-stress at the inner and outer cylinder (Pa) - _r local angular velocity (s^–1) - ₂ angular velocity of the outer cylinder (s^–1)     

Mixed convection flow in narrow vertical ducts

The mixed convection flow in a vertical duct is analysed under the assumption that , the ratio of the duct width to the length over which the wall is heated, is small. It is assumed that a fully developed Poiseuille flow has already been set up in the duct before heat from the wall causes this to be changed by the action of the buoyancy forces, as measured by a buoyancy parameter . An analytical solution is derived for the case when the Reynolds numberRe, based on the duct width, is of 0 (1). This is extended to the case whenRe is 0 (^–1) by numerical integrations of the governing equations for a range of values of representing both aiding and opposing flows. The limiting cases, || 1 andR=Re of 0 (1), andR and both large, with of 0 (R ^1/3) are considered further. Finally, the free convection limit, large with R of 0 (1), is discussed.

---

Mischkonvektion in engen senkrechten Rohren

Zusammenfassung Mischkonvektion in einem senkrechten Rohr wird unter der Voraussetzung untersucht, daß das Verhältnis der Rohrbreite zur Länge, über welche die Wand beheizt wird, klein ist. Es wird angenommen, daß sich bereits eine voll entwickelte Poiseuille-Strömung in dem Rohr eingestellt hat, bevor Antriebskräfte, gemessen mit dem Auftriebsparameter , aufgrund der Wandbeheizung die Strömung verändern. Es wird eine analytische Lösung für den Fall erhalten, daß die mit der Rohrbreite als charakteristische Länge gebildete Reynolds-ZahlRe konstant ist. Dies wird mittels einer numerischen Integration der wichtigsten Gleichungen auf den FallRe =f (^–1) sowohl für Gleich- als auch für Gegenstrom ausgedehnt. Weiterhin werden die beiden Grenzfälle betrachtet, wenn || 1 undR=Re konstant ist, sowieR und beide groß mit proportionalR ^1/3. Schließlich wird der Grenzfall der freien Konvektion, großes mit konstantem R, diskutiert.

---

Nomenclature g acceleration due to gravity - Gr Grashof number - G modified Grashof number - h duct width - l length of the heated section of the duct wall - p pressure - Pr Prandtl number - Q flow rate through the duct - Q ₀ heat transfer on the wally=0 - Q ₁ heat transfer on the wally=1 - Re Reynolds number - R modified Reynolds number - T temperature of the fluid - T ₀ ambient temperature - T applied temperature difference - u, velocity component in thex-direction - v, velocity component in they-direction - x, co-ordinate measuring distance along the duct - y, co-ordinate measuring distance across the duct - buoyancy parameter - ₀ modified buoyancy parameter, ₀=R ^–1/3 - coefficient of thermal expansion - ratio of duct width to heated length, =h/l - (non-dimensional) temperature - _w applied temperature on the wally=0 - kinematic viscosity - density of the fluid - ₀ shear stress on the wally=0 - ₁ shear stress on the wally=1 - stream function     

Modelling of heat transfer by conduction in a transition region between a porous medium and an external fluid

We study the modelling of purely conductive heat transfer between a porous medium and an external fluid within the framework of the volume averaging method. When the temperature field for such a system is classically determined by coupling the macroscopic heat conduction equation in the porous medium domain to the heat conduction equation in the external fluid domain, it is shown that the phase average temperature cannot be predicted without a generally negligible error due to the fact that the boundary conditions at the interface between the two media are specified at the macroscopic level.Afterwards, it is presented an alternative modelling by means of a single equation involving an effective thermal conductivity which is a function of point inside the interfacial region.The theoretical results are illustrated by means of some numerical simulations for a model porous medium. In particular, temperature fields at the microscopic level are presented.Roman Letters _sf interfacial area of thes-f interface contained within the macroscopic system m² - A _sf interfacial area of thes-f interface contained within the averaging volume m² - C _p mass fraction weighted heat capacity, kcal/kg/K - g vector that maps to _s , m - h vector that maps to _f , m - K _eff effective thermal conductivity tensor, kcal/m s K - l _s,l _f microscopic characteristic length m - L macroscopic characteristic length, m - n _fs outwardly directed unit normal vector for thef-phase at thef-s interface - n outwardly directed unit normal vector at the dividing surface. - R ₀ REV characteristic length, m - T _i macroscopic temperature at the interface, K - error on the external fluid temperature due to the macroscopic boundary condition, K - T ^* macroscopic temperature field obtained by solving the macroscopic Equation (3), K - V averaging volume, m³ - V _s,V _f volume of the considered phase within the averaging volume, m³. - _mp volume of the porous medium domain, m³ - _ex volume of the external fluid domain, m³ - _s , _f volume of the considered phase within the volume of the macroscopic system, m³ - dividing surface, m² - x, z spatial coordinates Greek Letters _s, _f volume fraction - ratio of the effective thermal conductivity to the external fluid thermal conductivity - ^* macroscopic thermal conductivity (single equation model) kcal/m s K - _s, _f microscopic thermal conductivities, kcal/m s K - spatial average density, kg/m³ - microscopic temperature, K - ^* microscopic temperature corresponding toT ^*, K - spatial deviation temperature K - error in the temperature due to the macroscopic boundary conditions, K - ^* _i macroscopic temperature at the interface given by the single equation model, K - spatial average - ^s , ^f intrinsic phase average.     

Viscosity of suspensions modeled with a shear-dependent maximum packing fraction

A large amount of data from the literature on viscosity of concentrated suspensions of rigid spherical particles are analyzed to support the new concept that the maximum packing fraction ( _M ) is shear-dependent. Incorporation of this behavior in a rheological model for viscosity () as a function of particle volume fraction () succeeds in describing virtually all non-Newtonian effects over the entire concentration range and also accounts for a yield stress. The most successful model is one proposed by Krieger and Dougherty for Newtonian viscosities, (, _M ), but with _M varying from a low-shear limit _M0 to a high-shear limit _M. Microstructural interpretations of this behavior are advanced, with arguments suggesting that similar rheological models should apply to suspensions of nonspherical and irregular particles.Symbols a particle size scale (for spheres, the diameter) - A lumped kinetic parameter in eqs. (23) and (24) - BS butadiene-styrene - C coefficient in Arrhenius model, eq. (2) - D coefficient in Mooney model, eq. (3) - e _i parameter representing one of the three electroviscous effects (i = 1, 2, or 3) - f fraction of total particulates that exist in the dispersed phase, eq. (22) - h solution factor, in Arrhenius model, eq. (2) - k crowding factor, in Mooney model, eq. (3) - k _D ,k _F kinetic rate coefficient for producing particles of dispersed or flocculated type, respectively - K Einstein coefficient for particles of any shape, eq. (1); equal to [] - KD Krieger-Dougherty model, eq. (6) - m exponent to characterize shear-dependence in viscosity models of Cross, eq. (10), and eq. (23), and also in yield stress prediction eq. (24) - N number of monodisperse components in a blend of spheres with different diameters - PD polydispersity (in size) parameter - S generalized shape parameter - T temperature - V _c volume of chamber in figure 6, representing the entire volume of the sample - V _P total volume of particles in the sample - V _D ,V _F sample volumes in which dispersed particles or flocculated particles, respectively, prevail; volumes of the dispersed phase or flocculated phase, containing both particles and carrier fluid - V _PD ,V _PF particle volume within the phase volumeV _D orV _F , respectively Greek coefficient in definition of _c in eq. (8); of order unity - coefficient regulating -sensitivity in eq. (10) - shear rate,dv ₁/dx ₂ in simple shear - shear viscosity of the suspension - ₀, low-shear and high-shear limiting values of - _s viscosity of the suspending fluid - [] intrinsic viscosity, - _r reduced viscosity,/ _s - Boltzmann's constant; in _c - shear stress - _c parameter characterizing sensitivity of viscosity to stress, in eq. (8) - _B dynamic yield stress in the floc model - _y yield stress - volume fraction occupied by solids in a suspension - _M maximum value of attainable by a given collection of particles under given conditions of flow - _M0, _M limiting values of _M at the low- and high- conditions, respectively     

Rheology of concentrated coagulating suspensions in non-aqueous media

The rheology of concentrated coagulating suspensions is analysed on the basis of the following model: (i) at low shear rates, the shear is not distributed homogeneously but limited to certain shear planes; (ii) the energy dissipation during steady flow is due primarily to the overcoming of viscous drag by the suspended particles during motion caused by encounters of particles in the shear planes. This model is called the giant floc model.With increasing shear rate the distance between successive shear planes diminishes, approaching the suspended particles' diameter at average shear stresses of 88–117 Pa in suspensions of 78 µm particles (glass ballotini coated by a hydrophobic layer) in glycerol — water mixtures, at solid volume fractions between 0.35 and 0.40. Smaller particles form a more persistent coagulation structure. The average force necessary to separate two touching 78 µm particles is too large to be accounted for by London-van der Waels forces; thus coagulation is attributed to bridging connections between polymer chains protruding from the hydrophobic coatings.The frictional ratio of the glass particles in these suspensions is of the order of 10. Coagulation leads to build-up of larger structural units at lower shear rates; on doubling the shear rate the average distance between the shear planes decreases by a factor of 0.81 to 0.88. A inter-shear plane distance - A Hamaker constant - b radius of primary particles - f frictional ratio - F _A attractive force between two particles - g acceleration due to gravity - H distance between the surfaces of two particles - K proportionality constant in power law - l fraction of distance by which a moving particle entrains its neighbours - l effective length of inner cylinder in the rheometer - M torque experienced by inner cylinder during measurements - n exponent in power law - n ₀ ,n ₁ ,n ₂ constants in extended power law - NC _hex number of contacts, per mm², between particles in adjacent layers with an average degree of occupation, assuming a hexagonal arrangement of the particles within the layers - NC _cub asNC _hex, but with a cubical arrangement - p () d increase of slippage probability when the shear stress increases from to + d - q average coordination number of a particle in a coagulate - R _i radius of inner cylinder of rheometer - R _u radius of outer cylinder of rheometer - t _i time during which particlei moves - t ₀ time during which a particle bordering a shear plane moves from its rectilinear course, on meeting another particle - u angle between the direction of motion, and the line connecting the centers of two successive particles bordering a shear plane - V _A attractive energy between two particles - x, y, z Cartesian coordinates:x — the direction of motion;y — the direction of the velocity gradient - y ₀ ,z ₀ y, z value of a particle meeting another particle, when both are far removed from each other - y ₀ spread iny ₀ values - —2/n - ₀ capture efficiency - shear rate - average shear rate calculated for a Newtonian liquid - _i distance by which particlei moves - ₀ distance by which a particle bordering a shear plane moves from its rectilinear course, when it encounters another particle - square root of area occupied by a particle bordering a shear plane, in this plane - _c energy dissipated during one encounter of two particles bordering a shear plane - _p energy dissipated by one particle - energy dissipated per unit of volume and time during steady flow - viscosity - _app calculated as if the liquid is Newtonian - ₀ viscosity of suspension medium - _PL lim - [] intrinsic viscosity - _diff - _{diff, rel diff}/ ₀ - standard deviation of distribution ofy ₀ values - shear stress - _n average shear stress at the highest values applied - mass average particle diameter - _n number average particle diameter - solid volume fraction - _eff effective solid volume fraction in Dougherty-Krieger relation - _max maximum solid volume fraction permitting flow - _i angular velocity of inner cylinder in rheometer during measurements     

An approach to the estimation of polymer melt elasticity

An effective method has been proposed to estimate the primary normalstress difference versus shear rate curves at temperatures relevant to the processing conditions only from the knowledge of the melt flow index, the molecular-weight distribution and the glass transition temperature of the polymer. The method involves the use of a unified curve obtained by coalescing the elastic response curves of various grades in terms of the modified normal-stress coefficient ₁ (MFI)² and a modified shear rate . Unified curves have been reported for low density polyethylene, high density polyethylene, polypropylene and nylon.Nomenclature C ₁ constant in eq. (4) - J _e steady state compliance (cm²/dyne) - proportionality constant in eq. (6) - L load (kg) - L ₁ load (kg) at ASTM test conditions - L ₂ load (kg) at required conditions - MFI melt flow index (gm/10 min) - number average molecular weight - weight average molecular weight - z-average molecular weight - (z+1)-average molecular weight - n slope of the shear stress vs. shear curve on a log-log scale - N ₁ primary normal-stress difference (dynes/cm²) - Q molecular weight distribution expressed as - T ₁is> temperature (K) at condition 1 - T ₂is> temperature (K) at condition 2 - T _g glass transition temperature (K) - T _s standard reference temperature equal toT _g + 50 K - shear rate (s^–1) - ₀ zero-shear viscosity (poise) - apparent viscosity (poise) - density (g/cm³) - ₁₂ shear stress (dynes/cm²) - ₁₁– ₂₂ primary normal-stress difference (dynes/cm²) - _1,0 zero shear rate primary normal-stress coefficient (dynes/cm² · sec²) - ₁ primary normal-stress coefficient (dynes/cm² · sec²) - ₂ secondary normal-stress coefficient (dynes/cm² · sec²) NCL-Communication No. 3106     

Impingement heat transfer from wedge surface

An experimental investigation is made to study the heat transfer characteristics of slot jet impingement on a wedge whose included angle is 90°. Local and average heat transfer rates from the wedge surfaces have been measured. The experiments have been conducted with isothermal wedge surface at Reynolds numbers ranging from 5 680 to 16 600. The effects of varying the flow rate, width of the nozzle, distance of the wedge vertex from the nozzle exit, eccentricity of the wedge vertex to the jet axis on the flow properties of the fluid have been investigated. A correlation has been proposed considering the relevant dimensionless parameters and then compared with experimental data.

---

Wärmeübertragung einer Strömung, die auf einen Keil aufprallt

Zusammenfassung Ein experimenteller Versuch wurde durchgeführt, um die Wärmeübertragungseigenschaften bei der Spaltung eines Strahls, die beim Aufprall auf einen rechtwinkligen Keil entsteht, zu untersuchen. Gemessen wurde die lokale und mittlere Wärmeübertragungsrate der Keiloberfläche. Dieses Experiment wurde mit einer isothermen Keiloberfläche bei einer Reynolds-Zahl zwischen 5 680–16 600 durchgeführt. Untersucht wurden die Einwirkungen bei der Änderung der Strömungsrate, Düsenbreite, Abstand zwischen Keilscheitel und Düsenausgang, Exzentrizität von Keilscheitel und Strahlachse, sowie die Strömungseigenschaften des Fluids. Unter Betracht der maßgebenden dimensionslosen Parameter wurden Berechnungen durchgeführt und diese mit den experimentellen Daten verglichen.

---

Nomenclature A surface area of the wedge - B breadth of the wedge or nozzle - C face length of the wedge - c _p specific heat at constant pressure - Gr _C Grashof number - H height of the wedge=C sin - h local heat transfer coefficient - average heat transfer coefficient - h _f free convective heat transfer coefficient - K thermal conductivity - L distance from the nozzle exit to the wedge vertex - Nu _f free convection Nusselt number - Nu _W local Nusselt number=hW/K - average Nusselt number= W/K - Pr Prandtl number=c _p /K - q heat loss due to forced convection - q _i heat input - q _f heat loss due to free convection - q _L heat loss due to conduction and radiation - Re _W Reynolds number=u _j W/ _a - average temperature in °C - T _j jet temperature in °C - T _a ambient air temperature in °C - T _m mean temperature=( +T _a )/2 - u mean velocity in m/s - u _j average jet exit velocity in m/s - W nozzle width - x distance measured from wedge vertex - angle between upper surface of the wedge and jet center line - angle between lower surface of the wedge and jet center line - eccentricity of the wedge vertex with the jet center line - absolute viscosity - kinematic viscosity     

Laminar source flow between parallel porous disks

An analysis is presented for laminar source flow between infinite parallel porous disks. The solution is in the form of a perturbation from the creeping flow solution. Expressions for the velocity, pressure, and shear stress are obtained and compared with the results based on the assumption of creeping flow.Nomenclature a half distance between disks - radial coordinate - r dimensionless radial coordinate, /a - axial coordinate - z dimensionless axial coordinate, /a - radial coordinate of a point in the flow - R dimensionless radial coordinate of a point in the flow, /a - velocity component in radial direction - u =a/, dimensionless velocity component in radial direction - velocity component in axial direction - v = a/}, dimensionless velocity component in axial direction - static pressure - p = (a ²/ ²), dimensionless static pressure - =p(r, z)–p(R, z), dimensionless pressure drop - V magnitude of suction or injection velocity - Q volumetric flow rate of the source - Re source Reynolds number, Q/4a - reduced Reynolds number, Re/r ² - critical Reynolds number - R _w wall Reynolds number, Va/ - viscosity - density - =/, kinematic viscosity - shear stress at upper disk - ₀ = (a ²/ ²), dimensionless shear stress at upper disk - shear stress ratio, ₀/( ₀)_inertialess - u = , dimensionless average radial velocity - u/u, ratio of radial velocity to average radial velocity - dimensionless stream function     

The virtual origin of riblets in an open channel flow

In this paper, a method using the mean velocity profiles for the buffer layer was developed for the estimation of the virtual origin over a riblets surface in an open channel flow. First, the standardized profiles of the mixing length were estimated from the velocity measurement in the inner layer, and the location of the edge of the viscous layer was obtained. Then, the virtual origins were estimated by the best match between the measured velocity profile and the equations of the velocity profile derived from the mixing length profiles. It was made clear that the virtual origin and the thickness of the viscous layer are the function of the roughness Reynolds number. The drag variation coincided well with other results.Nomenclature f _r skin friction coefficient - f _ro skin friction coefficient in smooth channel at the same flow quantity and the same energy slope - g gravity acceleration - H water depth from virtual origin to water surface - H ⁺ u*H/ - H false water depth from top of riblets to water surface - H ⁺ u*H/ - I _e streamwise energy slope - I _b bed slope - k riblet height - k ⁺ u*k/ - l mixing length - l _s standardized mixing length - Q flow quantity - Re Reynolds number volume flow/unit width/v - s riblet spacing - u mean velocity - u* friction velocity = - u* false friction velocity = - y distance from virtual origin - y distance from top of riblet - y ₀ distance from top of riblet to virtual origin - y _v distance from top of riblet to edge of viscous layer - y ⁺ u*y/ - y ⁺ u*y/ - y ₀ ⁺ u*y ₀/ - u ⁺ u*y/ - shifting coefficient for standardization - thickness of viscous layer=y ₀+y - ⁺ u*/ - ⁺ u*/ - eddy viscosity - ridge angle - v kinematic viscosity - density - shear stress     

Zur energetischen Optimierung von Schleppströmungspumpen mit durchgehend konstantem Arbeitsspalt für Medien mit nicht-newtonschem Fließverhalten

Zusammenfassung Die zum kontinuierlichen Austragen und Ausformen von strukturviskosen und anderen nicht-newtonschen Medien dienenden Schleppströmungspumpen lassen sich bei vorgegebenem Volumendurchsatz und Betriebsdruckp durch Anpassung des Arbeitsspaltesh und der Arbeitsdrehzahln energetisch optimal auslegen und betreiben. Die entsprechende Kennzahl ist der als Quotient aus Austriebs-Leistung p und AntriebsleistungP definierte Pumpwirkungsgrad . — Die optimalen (h, n, )-Werte werden unter der Voraussetzung berechnet, daß sich das Fließverhalten des geförderten Mediums durch einen Polynomansatz nachRabinowitsch beschreiben läßt. Dabei ergibt sich für die optimalen-Werte ein Bereich zwischen etwa 20% und 33%. Rheologische Ansätze mit einer auf eine mittlere Schergeschwindigkeit bezogenen konstanten scheinbaren Viskosität, welche in jedem Fall auf den für newtonsche Medien charakteristischen Idealwert=33% führen, sind hiernach für strukturviskose und andere nicht-newtonsche Medien unzulässig.

---

Summary Drag-flow pumps, as used for the continuous extrusion of non-Newtonian fluids, can be operated with minimum drive powerP at a given volume throughput and pressurep, if the radial dimensionh of the drag channel and working speedn are optimized. The key number of this optimization is the efficiency . — Appropriate (h, n, )-values are calculated on the basis of the rheological equation proposed byRabinowitsch. The optimum range of-values is found to be between 20% and 33%, whilst former calculations with an average apparent viscosity resulted in _opt = 33% generally. Obviously, here is one of the causes of discrepancy between theoretical and actual efficiencies of such pumps.

---

Symbole a Stoffkennzahl, Gl. [3] - b Breite des Schleppspalts, Abb. 2 - c Stoffkennzahl, Gl. [3] - C ₁ Integrationskonstante, Gl. [5] - C ₂ Integrationskonstante, Gl. [8] - d Durchmesser des rotierenden Elements, Abb. 1 - e spezifische Antriebsleistung, Gl. [18] - h Höhe (Radialmaß) des Schleppspalts, Abb. 1 - k Anzahl der Schleppspalte - m Fließexponent im Potenzansatz - Massedurchsatz - M _d Drehmoment - n Umdrehungsgeschwindigkeit, Arbeitsdrehzahl des rotierenden Elements, Abb. 1 - p Betriebsdruck - p Druckgradient, Gl. [6] - P aufgenommene Antriebsleistung - r radiale Koordinate - r _i=d/2 – h Innenradius des rotierenden Elements - r _a=d/2 Außenradius des rotierenden Elements - s zirkulare Länge des Schleppspalts - t (mittlere) Verweilzeit des Mediums im Schleppspalt - T Temperatur - v lokale zirkulare Geschwindigkeit - v ₀ Umfangsgeschwindigkeit des rotierenden Elements, Abb. 1 - V Volumen des Schleppspalts - Volumendurchsatz der Schleppströmungspumpe - Volumendurchsatz der Druck(gradienten)strömung - Volumendurchsatz der Schleppströmung - dimensionslose Kennzahl, Gl. [22] - Schergeschwindigkeit, Gl. [2] - dimensionsloser Pumpwirkungsgrad, Gl. [1] - µ Scherviskosität - Dichte - Schubspannung, Gl. [2] - zirkulare Koordinate - Fluidität im Potenzansatz - Winkelgeschwindigkeit Erweiterte Fassung eines Vortrages anläßlich des 5. Stuttgarter Kunststoff-Kolloquiums vom 2. März 1977.Mit 14 Abbildungen     

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)