Stochastic response of structures with small geometric imperfections

Summary A probabilistic model of the geometric imperfections of a real structure is proposed, in order to provide a general theory of the stochastic response of structures in presence of small random deviations from the perfect scheme. The main statistical measures of the stochastic response are derived and an application to the study of a particular conservative elastic system is developed.

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Sommario Si propone una teoria generale della risposta probabilistica di strutture, in presenza di piccole deviazioni aleatorie dei dati iniziali rispetto allo schema geometrico perfetto. Si deducono le principali proprietà statistiche della risposta della struttura a sollecitazioni esterne deterministiche, e si sviluppa una applicazione riguardante il comportamento aleatorio di un particolare sistema elastico conservativo.

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List of symbols element of the sample space of events - _kn random variables modelling the structural imperfections - P(o) probability density of random variables - random imperfection of the unloaded structure - u additional displacement of the loaded structure - u_o deterministic fundamental solution for the perfect structure - difference between the additional displacement of the loaded structure and the deterministic fundamental solution for the perfect structure - V₁=u₁ buckling mode of the perfect structure - _i intrinsic coordinates of the structure - suitable measure of the magnitude of the random imperfections - scalar geometric variable representing the internal product - random imperfection divided by - single scalar variable denoting the magnitude of the prescribed loads - potential energy of the structure - potential energy of the perfect structure - difference between and - _c lowest critical load - _s real local maximum for the magnitude of the prescribed loads - _c divided by _S - E{} expected value of a random variable - ² variance of a random variable - , random variables defined by Eq. (21)     

An example of a quasiconvex function that is not polyconvex in two dimensions

We study the different notions of convexity for the function f () = ||² (||² – 2 det ) where ^2×2, introduced by Dacorogna & Marcellini. We show that f is convex, polyconvex, quasiconvex, rank-one convex, if and only if ¦¦ 2/3 2, 1, 1+ (for some >0), 2/3, respectively.     

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]).     

Similarity Solutions for Thawing Processes with a Heat Flux Condition at the Fixed Boundary

Similarity solutions for a mathematical model for thawing in a saturated semi-infinite porous medium is considered when change of phase induces a density jump and a heat flux condition of the type is imposed on the fixed face x=0. Different cases depending on physical parameters are analysed and the explicit solution is obtained if and only if an inequality for the thermal coefficient q ₀ is verified. An improvement for the existence of a similarity solution for the same free boundary problem with a constant temperature on the fixed face x=0 is also obtained. Sommario. Vengono considerate soluzioni di similarità per un modello matematico di disgelo di un mezzo poroso saturo semi-infinito allorquando il cambiamento di fase induce un salto di densità ed una condizione di flusso di calore del tipo viene imposta sulla faccia fissa x=0. Si analizzano differenti casi dipendenti da parametri fisici e la soluzione esplicita viene ottenuta se e solo se risulta verificata una diseguaglianzo per il coefficiente termico q ₀. Si ottiene altresi un miglioramento della condizione di esistenza di una soluzione di similarità per lo stesso problema al contorno libero con temperatura costante sulla faccia fissa x=0.     

Wärmeübertragung in einem axial rotierenden,durchströmten Rohr im Bereich des thermischen Einlaufs

Zusammenfassung Der Einfluß der Rotation auf das Temperaturprofil und die Wärmeübergangszahl einer turbulenten Rohrströmung im Bereich des thermischen Einlaufs wird theoretisch untersucht und mit Meßwerten verglichen. Es wird angenommen, daß das Geschwindigkeitsprofil voll ausgebildet ist. Die Rotation hat aufgrund der radial ansteigenden Zentrifugalkräfte einen ausgeprägten Einfluß auf die Unterdrückung der turbulenten Bewegung. Dadurch verschlechtert sich die Wärmeübertragung mit steigender Rotations-Reynoldszahl und die thermische Einlauflänge nimmt beträchtlich zu.

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Heat transfer in an axially rotating pipe in the thermal entrance region. Part 1: Effect of rotation on turbulent pipe flow

The effects of rotation on the temperature distribution and the heat transfer to a fluid flowing inside a tube are examined by analysis in the thermal entrance region. The theoretical results are compared with experimental findings. The flow is assumed to have a fully developed velocity profile. Rotation was found to have a very marked influence on the suppression of the turbulent motion because of radially growing centrifugal forces. Therefore, a remarkable decrease in heat transfer with increasing rotational Reynolds number can be observed. The thermal entrance length increases remarkably with growing rotational Reynolds number.

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Formelzeichen a Temperaturleitzahl - C _n , ,C ₁,C ₃ Konstanten - c _p spezifische Wärme bei konstantem Druck - D Rohrdurchmesser - E Funktion nach Gl. (30) - H _n Eigenfunktionen - l hydrodynamischer Mischungsweg - l _q thermischer Mischungsweg - Massenstrom - N=Re /Re Reynoldszahlenverhältnis - Nu Nusseltzahl - Nu Nusseltzahl für die thermisch voll ausgebildete Strömung - Pr Prandtlzahl - Pr _t turbulente Prandtlzahl - Wärmestromdichte - Re _* Schubspannungsreynoldszahl - R _n Eigenfunktionen - Durchfluß-Reynoldszahl - Re v =D/ Rotations-Reynoldszahl - Ri Richardsonzahl - R Rohrradius - r Koordinate in radialer Richtung - dimensionslose Koordinate in radialer Richtung - T Temperatur - T Temperaturschwankung - T _b bulk temperature - mittlere Axialgeschwindigkeit - v Geschwindigkeit - v Geschwindigkeitsschwankung - turbulenter Wärmestrom - dimensionsloser Wandabstand - =1/6 Konstante - Integrationsvariable - Integrationsvariable - , ₁, ₂, dimensionslose Temperaturen - Wärmeleitzahl - _n Eigenwerte - kinematische Viskosität - Dichte - tangentiale Koordinate - , Hilfsfunktionen Indizes m in der Rohrmitte - r radial - w an der Rohrwand - z axial - 0 am Rohreintritt - 0 ohne Rotation - tangential     

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     

Darcy-Forchheimer natural,forced and mixed convection heat transfer in non-Newtonian power-law fluid-saturated porous media

The governing equation for Darcy-Forchheimer flow of non-Newtonian inelastic power-law fluid through porous media has been derived from first principles. Using this equation, the problem of Darcy-Forchheimer natural, forced, and mixed convection within the porous media saturated with a power-law fluid has been solved using the approximate integral method. It is observed that a similarity solution exists specifically for only the case of an isothermal vertical flat plate embedded in the porous media. The results based on the approximate method, when compared with existing exact solutions show an agreement of within a maximum error bound of 2.5%.Nomenclature A cross-sectional area - b _i coefficient in the chosen temperature profile - B ₁ coefficient in the profile for the dimensionless boundary layer thickness - C coefficient in the modified Forchheimer term for power-law fluids - C ₁ coefficient in the Oseen approximation which depends essentially on pore geometry - C _i coefficient depending essentially on pore geometry - C _D drag coefficient - C _t coefficient in the expression forK ^* - d particle diameter (for irregular shaped particles, it is characteristic length for average-size particle) - f _p resistance or drag on a single particle - F _R total resistance to flow offered byN particles in the porous media - g acceleration due to gravity - g _x component of the acceleration due to gravity in thex-direction - Grashof number based on permeability for power-law fluids - K intrinsic permeability of the porous media - K ^* modified permeability of the porous media for flow of power-law fluids - l _c characteristic length - m exponent in the gravity field - n power-law index of the inelastic non-Newtonian fluid - N total number of particles - Nux,_D,F local Nusselt number for Darcy forced convection flow - Nux,_D-F,F local Nusselt number for Darcy-Forchheimer forced convection flow - Nux,_D,M local Nusselt number for Darcy mixed convection flow - Nux,_D-F,M local Nusselt number for Darcy-Forchheimer mixed convection flow - Nux,_D,N local Nusselt number for Darcy natural convection flow - Nux,_D-F,N local Nusselt number for Darcy-Forchheimer natural convection flow - pressure - p exponent in the wall temperature variation - _Pe c characteristic Péclet number - _Pe x local Péclet number for forced convection flow - _Pe x modified local Péclet number for mixed convection flow - _Ra c characteristic Rayleigh number - _Ra x local Rayleigh number for Darcy natural convection flow - _Ra x local Rayleigh number for Darcy-Forchheimer natural convection flow - Re convectional Reynolds number for power-law fluids - Reynolds number based on permeability for power-law fluids - T temperature - T _e ambient constant temperature - T w,_ref constant reference wall surface temperature - T _w(X) variable wall surface temperature - T _w temperature difference equal toT w,_ref–T _e - T ₁ term in the Darcy-Forchheimer natural convection regime for Newtonian fluids - T ₂ term in the Darcy-Forchheimer natural convection regime for non-Newtonian fluids (first approximation) - T _N term in the Darcy/Forchheimer natural convection regime for non-Newtonian fluids (second approximation) - u Darcian or superficial velocity - u ₁ dimensionless velocity profile - u _e external forced convection flow velocity - u _s seepage velocity (local average velocity of flow around the particle) - u _w wall slip velocity - U c _M characteristic velocity for mixed convection - U c _N characteristic velocity for natural convection - x, y boundary-layer coordinates - x ₁,y ₁ dimensionless boundary layer coordinates - X coefficient which is a function of flow behaviour indexn for power-law fluids - effective thermal diffusivity of the porous medium - shape factor which takes a value of/4 for spheres - shape factor which takes a value of/6 for spheres - ₀ expansion coefficient of the fluid - _T boundary-layer thickness - T ₁ dimensionless boundary layer thickness - porosity of the medium - similarity variable - dimensionless temperature difference - coefficient which is a function of the geometry of the porous media (it takes a value of 3 for a single sphere in an infinite fluid) - ₀ viscosity of Newtonian fluid - ^* fluid consistency of the inelastic non-Newtonian power-law fluid - constant equal toX(2 ^2–n )/ - density of the fluid - dimensionless wall temperature difference     

Laminar source flow between parallel porous disks with equal suction and injection

An analysis is presented for laminar source flow between parallel stationary porous disks with suction at one of the disks and equal injection at the other. The solution is in the form of an infinite series expansion about the solution at infinite radius, and is valid for all suction and injection rates. Expressions for the velocity, pressure, and shear stress are presented and the effect of the cross flow is discussed.Nomenclature a distance between disks - A, B, ..., J functions of R _w only - F static pressure - p dimensionless static pressure, p(a ²/ ²) - Q volumetric flow rate of the source - r radial coordinate - r dimensionless radial coordinate, r/a - R radial coordinate of a point in the flow region - R dimensionless radial coordinate of a point in the flow region, R - Re source Reynolds number, Q/2a - R _w wall Reynolds number, Va/ - reduced Reynolds number, Re/r ² - critical Reynolds number - velocity component in radial direction - u dimensionless velocity component in radial direction, a/ - average radial velocity, Q/2a - u dimensionless average radial velocity, Re/r - ratio of radial velocity to average radial velocity, u/u - velocity component in axial direction - v dimensionless velocity component in axial direction, v - V magnitude of suction or injection velocity - z axial coordinate - z dimensionless axial coordinate, z a - viscosity - density - kinematic viscosity, / - shear stress at lower disk - shear stress at upper disk - ₀ dimensionless shear stress at lower disk, - ₁ dimensionless shear stress at upper disk, - dimensionless stream function     

Material functions generated by the complex viscosity

Based on the complex viscosity model various steady-state and transient material functions have been completed. The model is investigated in terms of a corotational frame reference. Also, BKZ-type integral constitutive equations have been studied. Some relations between material functions have been derived. C ^–1 Finger tensor - F[], (F ^–1[]) Fourier (inverse) transform - rate of deformation tensor in corotating frame - h(I, II) Wagner's damping function - J (x) Bessel function - m parameter inh (I, II) - m(s) memory function - m _k, n_k integers (powers in complex viscosity model) - P principal value of the integral - parameter in the complex viscosity model - rate of deformation tensor - shear rates - [], [] incomplete gamma function - (a) gamma function - steady-shear viscosity - ^* complex viscosity - , real and imaginary parts of ^* - ₀ zero shear viscosity - ⁺, ₁ ⁺ stress growth functions - ^–, ₁ ^- stress relaxation functions - (s) relaxation modulus - ₁(s) primary normal-stress coefficient - ø(a, b; z) degenerate hypergeometric function - ₁, ₂ time constants (parameters of ^*) - frequency - extra stress tensor     

Slip flow of non-plasticized PVC compounds

Measurements on seven rigid PVC compounds were carried out with a slit rheometer working in combination with an injection moulding machine. Plastication of the compounds occurred in the screw of the plastication unit, which also forced the melt through the die with a controlled forward velocity. The rectangular slit had a length of 90 mm and a widthB of 20 mm. The heightH could be varied between 0.8 and 3.3 mm. Pressures and temperatures were recorded at several positions in and before the die. Measurements were carried out at shear rates from 10 to 2000 s^–1.When the reduced volume output was plotted against the wall shear stress _W , only four compounds showed master curves independent ofH, which is indicative of wall adhesion. In the other cases this plot did not produce such a master curve, but the plot of the mean velocity against _W was independent ofH (slip curve). This indicated that slip flow prevailed with a slip velocityv _G When, in the case of wall slip, the smooth inner surfaces of the die were replaced by surfaces with grooves perpendicular to the direction of flow, slip flow was prevented and the flow curves were shifted to much higher values of _Wc Above a critical value of the wall shear stress ( _Wc ) at which slip flow began, the output became nearly independent of _W . From the measurements made below _Wc a vs. relation for the shear flow could be derived, which was used to calculate the superimposed shear flow . Exact values of the slip velocity were then given by . Wall slip only occurred for compounds with a high shear viscosity, which corresponds to a high molecular weight (K-value).Dedicated to Professor H. Janeschitz-Kriegl on the occasion of his 60th birthday.     

Apparent thickening behavior of dilute polystyrene solutions in extensional flows

An experimental investigation was undertaken to study the apparent thickening behavior of dilute polystyrene solutions in extensional flow. Among the parameters investigated were molecular weight, molecular weight distribution, concentration, thermodynamic solvent quality, and solvent viscosity. Apparent relative viscosity was measured as a function of wall shear rate for solutions flowing from a reservoir through a 0.1 mm I.D. tube. As increased, slight shear thinning behavior was observed up until a critical wall shear rate was exceeded, whereupon either a large increase in or small-scale thickening was observed depending on the particular solution under study. As molecular weight or concentration increased, decreased and, the jump in above , increased. increased as thermodynamic solvent quality improved. These results have been interpreted in terms of the polymer chains undergoing a coil-stretch transition at . The observation of a drop-off in at high (above ) was shown to be associated with inertial effects and not with chain fracture due to high extensional rates.     

Rheological models for gypsum plaster pastes

Shear stress and shear rate data obtained for gypsum plaster pastes were correlated by means of different rheological models. The pastes were prepared from a commercial calcium sulfate hemihydrate at various water/plaster ratios ranging from 100/150 to 100/190. The tests were performed at 25°C using a rotating coaxial cylinder viscosimeter. The measurements were accomplished by applying a step-wise decreasing shear rate sequence. Discrimination among the models was made: (1) on the basis of the fitting goodness; (2) by checking the physical meaning of the calculated parameters; (3) on the basis of the stability of the parameters and of their prediction capacity beyond the limits of the experimental data. In the light of above, the Casson model seemed to be most effective for application to gypsum plaster pastes. K Consistency - n Power-law index - N Number of experimental data - P Number of parameters - Shear rate (s^–1) - ₀ Viscosity (Pa · s) - _d Dispersing medium viscosity (Pa · s) - _p Plastic viscosity (Pa · s) - Viscosity at zero shear rate (Pa · s) - Viscosity at infinite shear rate (Pa · s) - [] Intrinsic viscosity - ² Variance - Shear stress (Pa) - ₀ Yield stress (Pa) - Solid volume fraction - _m Maximum solid volume fraction     

Transport in ordered and disordered porous media I: The cellular average and the use of weighting functions

In this work we consider transport in ordered and disordered porous media using singlephase flow in rigid porous mediaas an example. We defineorder anddisorder in terms of geometrical integrals that arise naturally in the method of volume averaging, and we show that dependent variables for ordered media must generally be defined in terms of thecellular average. The cellular average can be constructed by means of a weighting function, thus transport processes in both ordered and disordered media can be treated with a single theory based on weighted averages. Part I provides some basic ideas associated with ordered and disordered media, weighted averages, and the theory of distributions. In Part II a generalized averaging procedure is presented and in Part III the closure problem is developed and the theory is compared with experiment. Parts IV and V provide some geometrical results for computer generated porous media.Roman Letters A interfacial area of the- interface contained within the macroscopic region, m² - A_e area of entrances and exits for the-phase contained within the macroscopic system, m² - g gravity vector, m/s² - I unit tensor - K traditional Darcy's law permeability tensor, m² - L general characteristic length for volume averaged quantities, m - characteristic length (pore scale) for the-phase - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - N unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p0 reference pressure in the-phase, N/m² - p traditional intrinsic volume averaged pressure, N/m² - r0 radius of a spherical averaging volume, m - r position vector, m - r position vector locating points in the-phase, m - averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - v velocity vector in the-phase, m/s - v traditional superficial volume averaged velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V/V, volume average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m²     

On the universality of the velocity profiles of a turbulent flow in an axially rotating pipe

If a fluid enters an axially rotating pipe, it receives a tangential component of velocity from the moving wall, and the flow pattern change according to the rotational speed. A flow relaminarization is set up by an increase in the rotational speed of the pipe. It will be shown that the tangential- and the axial velocity distribution adopt a quite universal shape in the case of fully developed flow for a fixed value of a new defined rotation parameter. By taking into account the universal character of the velocity profiles, a formula is derived for describing the velocity distribution in an axially rotating pipe. The resulting velocity profiles are compared with measurements of Reich [10] and generally good agreement is found.Nomenclature b constant, equation (34) - D pipe diameter - l mixing length - l ₀ mixing length in a non-rotating pipe - N rotation rate,N=Re /Re _D - p pressure - R pipe radius - Re _D flow-rate Reynolds number, - Re rotational Reynolds number, Re =v _w D/ - Re* Reynolds number based on the friction velocity, Re*=v*R/ - (Re*)₀ Reynolds number based on the friction velocity in a non-rotating pipe - Ri Richardson number, equation (10) - r coordinate in radial direction - dimensionless coordinate in radial direction, - v _r ,v ,v _z time mean velocity components - v _r ,v ,v _z velocity fluctations - v _w tangential velocity of the pipe wall - v* friction velocity, - axial mean velocity - v _ZM maximum axial velocity - dimensionless radial distance from pipe wall, - y ⁺ dimensionless radial distance from pipe wall - y ₁ ⁺ constant - Z rotation parameter,Z =v _w/v _* =N Re _D /2Re* - _m eddy viscosity - ( _m )₀ eddy viscosity in a non-rotating pipe - coefficient of friction loss - von Karman constant - ₁ constant, equation (31) - density - dynamic viscosity - kinematic viscosity     

Flow visualization of compressible vortex structures using density gradient techniques

Mathematical results are derived for the schlieren and shadowgraph contrast variation due to the refraction of light rays passing through two-dimensional compressible vortices with viscous cores. Both standard and small-disturbance solutions are obtained. It is shown that schlieren and shadowgraph produce substantially different contrast profiles. Further, the shadowgraph contrast variation is shown to be very sensitive to the vortex velocity profile and is also dependent on the location of the peak peripheral velocity (viscous core radius). The computed results are compared to actual contrast measurements made for rotor tip vortices using the shadowgraph flow visualization technique. The work helps to clarify the relationships between the observed contrast and the structure of vortical structures in density gradient based flow visualization experiments.Nomenclature a Unobstructed height of schlieren light source in cutoff plane, m - c Blade chord, m - f Focal length of schlieren focusing mirror, m - C _T Rotor thrust coefficient, T/( ² R ⁴) - I Image screen illumination, Lm/m ² - l Distance from vortex to shadowgraph screen, m - n _b Number of blades - p Pressure,N/m ² - p Ambient pressure, N/m ² - r, , z Cylindrical coordinate system - r _c Vortex core radius, m - Non-dimensional radial coordinate, (r/r _c ) - R Rotor radius, m - Tangential velocity, m/s - Specific heat ratio of air - Circulation (strength of vortex), m ²/s - Non-dimensional quantity, ² 8²p r _c ² - Refractive index of fluid medium - ₀ Refractive index of fluid medium at reference conditions - Gladstone-Dale constant, m ³/kg - Density, kg/m ³ - Density at ambient conditions, kg/m ³ - Non-dimensional density, (/ ) - Rotor solidity, (n _b c/ R) - Rotor rotational frequency, rad/s     

A general model for the effective viscosity of pseudoplastic and dilatant fluids

Summary A new and very general expression is proposed for correlation of data for the effective viscosity of pseudoplastic and dilatant fluids as a function of the shear stress. Most of the models which have been proposed previously are shown to be special cases of this expression. A straightforward procedure is outlined for evaluation of the arbitrary constants.

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Zusammenfassung Eine neue und sehr allgemeine Formel wird für die Korrelation der Werte der effektiven Viskosität von strukturviskosen und dilatanten Flüssigkeiten in Abhängigkeit von der Schubspannung vorgeschlagen. Die meisten schon früher vorgeschlagenen Methoden werden hier als Spezialfälle dieser Gleichung gezeigt. Ein einfaches Verfahren für die Auswertung der willkürlichen Konstanten wird beschrieben.

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Nomenclature b arbitrary constant inSisko model (eq. [5]) - n arbitrary exponent in eq. [1] - x independent variable - y(x) dependent variable - y ₀(x) limiting behavior of dependent variable asx 0 - y(x) limiting behavior of dependent variable asx - z original dependent variable - arbitrary constant inSisko model (eq. [5]) andBird-Sisko model (eq. [6]) - arbitrary exponent in eqs. [2] and [8] - effective viscosity = shear stress/rate of shear - _A effective viscosity at = _A - _B empirical constant in eqs. [2] and [8] - ₀ limiting value of effective viscosity as 0 - ₀() limiting behavior of effective viscosity as 0 - limiting value of effective viscosity as - () limiting behavior of effective viscosity as - rate of shear - arbitrary constant inBird-Sisko model (eq.[6]) - shear stress - _A arbitrary constant in eqs. [2] and [8] - ₀ shear stress at inBingham model - _1/2 shear stress at = ( ₀ + )/2 With 8 figures     

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.

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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.

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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     

Analytical solution for transient laminar fully developed free convection in vertical channels

Using Green's function method, analytical solutions for transient fully developed natural convection in open-ended vertical circular and two-parallel-plate channels are presented. Different fundamental boundary conditions for these two configurations have been investigated and the corresponding fundamental solutions are obtained. These fundamental solutions may be used to obtain solutions satisfying more general thermal boundary conditions. In terms of the obtained unsteady temperature and velocity profiles, the transient volumetric flow rate, mixing cup emperature and local nusselt number are estimated.Zusammenfassung Für oben und unten offene vertikale Kanäle mit Kreisquerschnitt bzw. als Parallelplattenanordnung werden unter Verwendung der Methode der Greenschen Funktionen analytische Lösungen für die nichtstationäre, vollausgebildete, natürliche Konvektion gefunden und zwar unter Zugrundelegung verschiedener Fundamental-Randbedingungen bezüglich beider Konfigurationen. Die so ermittelten Fundamentallösungen können zur Gewinnung von Lösungen für allgemeine Randbedingungen dienen. Der zeitlich veränderliche Volumenstrom, die Mischtemperatur und die Nusselt-Zahl werden mit Bezug auf die erhaltenen nichtstationären Profile für Temperatur und Geschwindigkeit näher analysiert.

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Analytische Lösung für die nichtstationäre vollausgebildete laminare freie Konvektion invertikalen Kanälen

Nomenclature a local heat transfer coefficient based on the area of the heat transfer surface,q/(T _w –T ₀)=±(T/y)_w/(T_w–T₀), minus and plus signs apply respectively for heating and cooling in case of parallel-plate channel and vice versa in case of a tube - average heat transfer coefficient over the channel - c _p specific heat of fluid at constant pressure - f volumetric flow rate, for circular channels and or two-parallel-plate channels - F dimensionless volumetric flow rate,f/(2lvGr^*) for circular channels forfw/(lvGr ^*) for two-parallel-plate channels - g gravitational body force per unit mass (acceleration) - G Green's function - Gr Grashof number,±g(T _w–T₀)w³/v² in case of an isothermal boundary of±gqw ⁴/2kv² in case of a uniform heat flux (UHF) on the heat transfer boundary, the plus and minus signs apply to upward (heating) and downward (cooling) flows, respectively. ThusGr is a positive number in both cases. - Gr ^* modified Grashof number,wGr/l - h heat gained or lost by fluid from the entrance up to a particular elevation in the channel, ₀ fc _p(T _m–T ₀) for all cases - J ₀ Bessel function of zero order - k thermal conductivity of fluid - l height of channel - L dimensionless height of channel,1/Gr ^* - Nu local Nusselt number,|a| w/k - average Nusselt number, - p pressure of fluid inside the channel at any cross-section - p pressure defect at any point,p–p _s - p ₀ pressure of fluid at the channel entrance - p _s hydrostatic pressure, ₀ gz where the minus and plus signs are for upward (heating) and downward (cooling) flows, respectively - p dimensionless pressure defect at any point(pw ⁴)/(₀ l ²² Gr ²) - Pr Prandtl number,c _p/k - q heat flux at the heat transfer surface,q=±k(T/y)_w where the minus and plus signs are, respectively, for cooling and heating in case of circular pipe and vice versa in case of a parallel-plate channel - Ra Rayleigh number,GrPr - Ra ^* modified Rayleigh number,Gr ^*Pr - t time - T fluid temperature at any point - T _m mixing-cup (mixed-mean) temperature over any cross section, for circular channels, and for two-parallelplate channels - T ₀ initial and channel-inlet fluid temperature - T _w temperature of the heat-transfer wall - u axial velocity component at any point - U dimensionless axial velocity,uw ²/(lvGr^*) - w radius of circular tube or width (between plates) of parallel-plate channel - y radial or transverse coordinate - y dimensionless radial or transverse coordinate,y/w - z axial coordinate - Z dimensional axial coordinate,z/(lGr ^*) Greek symbols constant appears in Eq. (8) - parameter appears in Eq. (9) which equals the integration of with respect to or volumetric coefficient of thermal expansion - _n eigenvalues - parameter appears in Eq. (7) - _n eigenvalues - parameter appears in Eq. (12) - _n eigenvalues - parameter appears in Eq. (9) - dimensionless temperature,(T–T ₀)/(T_w–T₀) in case of an isothermal heat transfer boundary and(T–T ₀)/(qw/2k) for UHF boundary - _m dimensionless mixing cup temperature,(T _m–T₀)/(T_w–T₀) in case of an isothermal heat transfer boundary and(T _m–T₀)/(qw/2k) for UHF boundary - _w dimensionless temperature of the heat-transfer wall, equals unity in case of an isothermal heat transfer boundary and(T _w–T₀)/(qw/2k) for a UHF boundary - _n eigenvalues - dynamic viscosity of fluid - kinematic viscosity of fluid, /₀ - fluid density at temperatureT,₀[1–(T–T ₀)] - ₀ fluid density atT ₀ - demensionless time,tk/(cw²)     

Heat transfer between gas-liquid spray stream flowing perpendicularly to the row of the cylinders

Experimental investigation and analysis of heat transfer process between a gas-liquid spray flow and the row of smooth cylinders placed in the surface perpendicular to the flow has been performed. Among others, there was taken into account in the analysis the phenomenon of droplets bouncing and omitting the cylinder as well as the phenomenon of the evaporation process from the liquid film surface.In the experiments test cylinders were used, which were placed between two other cylinders standing in the row.From the experiments and the analysis the conclusion can be drawn that the heat transfer coefficients values for a row of the cylinders are higher than for a single cylinder placed in the gasliquid spray flow.

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Wärmeübergang an eine senkrecht anf eine Zylinderreihe auftreffende Gas-Flüssigkeits-Sprüh-Strömung

Zusammenfassung Es wurden Messungen und theoretische Analysen des Wärmeübergangs zwischen einer Gas-FlüssigkeitsSprüh-Strömung und den glatten Oberflächen einer Zylinderreihe durchgeführt, die senkrecht zum Sprühstrahl angeordnet waren. Dabei wurde in der Analyse unter anderem das Phänomen betrachtet, daß die Tropfen die Zylinderwand treffen und verfehlen können und daß sich ein Verdampfungsprozeß aus dem flüssigen Film an der Zylinderoberfläche einstellt.Gemessen wurde an einem zwischen zwei Randzylindern befindlichen Zylinder.Die Experimente und die Analyse gestatten die Schlußfolgerung, daß der Wärmeübergangskoeffizient für eine Zylinderreihe höher ist als für einen einzelnen Zylinder in der Sprühströmung.

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Nomenclature a distance between axes of cylinders, m - c _l specific heat capacity of liquid, J/kg K - c _g specific heat capacity of gas, J/kg K - D cylinder diameter, m - g _l mass velocity of liquid, kg/m²s - ¯k average volume ratio of liquid entering film to amount of liquid directed at the cylinder in gas-liquid spray flow, dimensionless - k() local volume ratio of liquid entering film to amount of liquid directed at the cylinder in gas-liquid spray flow, dimensionless - L specific latent heat of vaporisation, J/kg - m mass fraction of water in gas-liquid spray flow, dimensionless - M constant in Eq. (9) - p pressure, Pa - p _g statical pressure of gas, Pa - p _w pressure of gas on the cylinder surface, Pa - p external pressure on the liquid film surface, Pa - r cylindrical coordinate, m - R radius of cylinder, m - T temperature, K, °C - T _l, t_l liquid temperature in the gas-liquid spray, K, °C - T _w,t_w temperature of cylinder surface, K, °C - T temperature of gas-liquid film interface, K - U liquid film velocity, m/s - w gas velocity on cylinder surface, m/s - w _g gas velocity in free stream, m/s - W _l liquid vapour mass ratio in free stream, dimensionless - W liquid vapour mass ratio at the edge of a liquid film, dimensionless - x coordinate, m - y coordinate, m - z complex variable, dimensionless - average heat transfer coefficient, W/m²K - local heat transfer coefficient, W/m² K - average heat transfer coefficient between cylinder surface and gas, W/m² K - _g, local heat transfer coefficient between cylinder surface and gas, W/m² K - mass transfer coefficient, kg/m²s - liquid film thickness, m - _lg dynamic diffusion coefficient of liquid vapour in gas, kg/m s - pressure distribution function on a cylinder surface - function defined by Eq. (3) - _l liquid dynamic viscosity, kg/m s - _g gas dynamic viscosity, kg/m s - cylindrical coordinate, rad, deg - _l thermal conductivity of liquid, W/m K - _g thermal conductivity of gas, W/m K - mass transfer driving force, dimensionless - _l density of liquid, kg/m³ - _g density of gas, kg/m³ - _w shear stress on the cylinder surface, N/m² - _w shear stress exerted by gas at the liquid film surface, N/m² - air relative humidity, dimensionless - T -T _w - _w =T _wT_l Dimensionless parameters I= enhancement factor of heat transfer - m ^*=M _l/M_g molar mass of liquid to the molar mass of gas ratio - Nu _g= D/ _g gas Nusselt number - Pr _g=c _{g g}/_g gas Prandtl number - Pr _l=c_lll liquid Prandtl number - ¯r=(r–R)/ dimensionless coordinate - Re _g=w_gD _g/_g gas Reynolds number - Re _g,max=w_g,max D _g/_g gas Reynolds number calculated for the maximal gas velocity between the cylinders - Sc=m ^* _g/_l–g Schmidt number =/R dimensionless film thickness     

Certain yawed magnetogasdynamic flows

Certain steady yawed magnetogasdynamic flows, in which the magnetic field is everywhere parallel to the velocity field, are related to certain reduced three-dimensional compressible gas flows having zero magnetic field. Under a restriction, the reduced flows are linked, by certain reciprocal relations, to a four parameter class of plane gas flows. In the instance of constant entropy an approximation method is suggested for obtaining magnetogasdynamic flows from the corresponding plane, irrotational gasdynamic flows and examples are given.

Nomenclature

magnetogasdynamic flow variables H magnetic intensity - q fluid velocity - fluid density - p pressure - s entropy - Q _t, H _t component of q, H in the x–y plane - w , h component of q, H perpendicular to the x–y plane reduced gasdynamic flow factor of proportionality - q* fluid velocity - * fluid density - p* pressure - Q _t ^* =u*î+v*, w* components of q* - l arbitrary constant - A _v Alfvén speed - Q _t, , p fluid velocity, density, pressure of the reciprocal gas dynamic flow - L, n, k, arbitrary constants - , velocity potential, stream function - angle made by Q _t, Q _t ^* , and V with the x-axis - adiabatic gas constant - a ²=(–1)/2 constant - M Mach number - W constant value of w* - E approximate constant value of g(p) - * modified potential function - modified velocity coordinate - +i - complex potential of the irrotational flow - B arbitrary constant - V incompressible flow velocity - V modified fluid velocity - X _p, Y _p points on the profile