Distribution of eddy viscosity and mixing length in a non-Newtonian flow of a solid particle-gas suspension

The present work concerns the study of the radial distribution of eddy viscosity and mixing length and their dependence upon the Reynolds number and the concentration of the solid phase in a non-Newtonian flow of a suspension of solid particles in a gas. The investigated systems have a pseudoplastic character and the deviation from Newtonian behaviour increases with an increase in the concentration of the dispersed phase. Relations are presented for the eddy viscosity and mixing length in the flow of pseudoplastic fluids. From the analysis of results it follows that the mixing length and eddy viscosity increases with an increase in the Reynolds number. In contrast, an increase in the concentration of the solid phase and consequently of the pseudoplasticity causes a decrease in the investigated quantities. The radial distribution of the mixing length and the eddy viscosity is characterized by a maximum, after which the investigated quantities vary only slightly. This enables the area of the core of the turbulent flow to be defined. E Non-dimensional eddy viscosity - K fluid consistency defined by Ostwald-De Waele's formula (power law) - K fluid consistency, eq. (12) - L mixing length - L _t non-dimensional mixing length - N ⁺ position parameter, eq. (3) - n power-law index - n ⁺ Reichardt's position parameter, eq. (4) - n slope of the dependence ln _w = f[ln (8w/D)] - R pipe radius - r radial distance from the pipe axis - Re=D W /µ Reynolds number - U ⁺ non-dimensional local mean axial velocity, eq. (2) - u ^* = w (/8)^0.5 friction velocity - non-dimensional local mean axial velocity - local mean axial velocity - u turbulence velocity component - mean axial velocity at pipe axis - w average velocity over cross-section of pipe - loading ratio of solid to air, i.e. ratio of mass flow rates of solids to air - Y _m ⁺ =Rⁿ u^2–n /K 8^n–1 non-dimensional distance of pipe centre from the wall - y distance from pipe wall - y ⁺ non-dimensional distance from pipe wall, eq. (5) - friction factor - µ _L laminar part of viscosity - µ _t eddy viscosity - density - shear stress - _w shear stress on the wall     

Analysis of slip and temperature jump coefficients in a binary gas mixture

The possibility of simplifying the formulas obtained by the Maxwell-Loyalka method for the velocity _u, temperature _T and diffusion _d slip coefficients and the temperature jump coefficient in a binary gas mixture with frozen internal degrees of freedom of the molecules is considered. Special attention is paid to gases not having sharply different physicochemical properties. The formulas are written in a form convenient for use without linearization in the thermal diffusion coefficient. They are systematically analyzed for mixtures of inert gases, N₂, O₂, CO₂, and H₂ at temperatures extending from room temperature to 2500°K. It is shown that for the molecular weight ratios m^* = m₂/m₁ considered the expressions for _u and can be radically simplified. With an error acceptable for practical purposes (up to 10%) it is possible to employ expressions of the same structural form as for a single-component gas: for _u if 1 m^* 6, and for if 1 m^* 3. When 1 m^* 2 the expression for _T can be simplified with a maximum error of 5%. Within the limits of accuracy of the method the expression for _t can be linearized in the thermal diffusion coefficient. An approximate expression convenient for practical calculations is proposed for _d Finally, the , _u, and _T for a single-component polyatomic gas with easy excitation of the internal degrees of freedom of the molecules are similarly analyzed; it is shown that these expressions can be considerably simplified.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 152–159, November–December, 1990.     

Momentum balance in highly distorted turbulent pipe flows

An in depth study into the development and decay of distorted turbulent pipe flows in incompressible flow has yielded a vast quantity of experimental data covering a wide range of initial conditions. Sufficient detail on the development of both mean flow and turbulence structure in these flows has been obtained to allow an implied radial static pressure distribution to be calculated. The static pressure distributions determined compare well both qualitatively and quantitatively with earlier experimental work. A strong correlation between static pressure coefficient Cp and axial turbulence intensity is demonstrated.List of symbols C _p static pressure coefficient = (pw-p)/1/2 - D pipe diameter - K turbulent kinetic energy - (r, , z) cylindrical polar co-ordinates. / 0 - R, y pipe radius, distance measured from the pipe wall - U, V axial and radial time mean velocity components - mean value of u - u, u/ , / - u, , w fluctuating velocity components - axial, radial turbulence intensity - turbulent shear stress - u friction velocity, (u ² = ₀/p) - ₀ wall shear stress - ^* boundary layer thickness A version of this paper was presented at the Ninth Symposium on Turbulence, University of Missouri-Rolla, October 1–3, 1984     

A technique for evaluation of three-dimensional behavior in turbulent boundary layers using computer augmented hydrogen bubble-wire flow visualization

A system is described which allows the recreation of the three-dimensional motion and deformation of a single hydrogen bubble time-line in time and space. By digitally interfacing dualview video sequences of a bubble time-line with a computer-aided display system, the Lagrangian motion of the bubble-line can be displayed in any viewing perspective desired. The u and v velocity history of the bubble-line can be rapidly established and displayed for any spanwise location on the recreated pattern. The application of the system to the study of turbulent boundary layer structure in the near-wall region is demonstrated.List of Symbols Reynolds number based on momentum thickness u /v - t⁺ nondimensional time - u shear velocity - u local streamwise velocity, x-direction - u ⁺ nondimensional streamwise velocity - v local normal velocity, -direction - x ⁺ nondimensional coordinate in streamwise direction - ⁺ nondimensional coordinate normal to wall - ₊ ^wire wire nondimensional location of hydrogen bubble-wire normal to wall - z ⁺ nondimensional spanwise coordinate - momentum thickness - v kinematic viscosity - _W wall shear stress     

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 dynamics of solid-solid phase transitions 2. Incoherent interfaces

Incoherent phase transitions are more difficult to treat than their coherent counterparts. The interface, which appears as a single surface in the deformed configuration, is represented in its undeformed state by a separate surface in each phase. This leads to a rich but detailed kinematics, one in which defects such as vacancies and dislocations are generated by the moving interface. In this paper we develop a complete theory of incoherent phase transitions in the presence of deformation and mass transport, with phase interface structured by energy and stress. The final results are a complete set of interface conditions for an evolving incoherent interface.Frequently used symbols A_i,C_i generic subsurface of S_t - B_i undeformed phase-i region - C configurational bulk stress, Eshelby tensor - F deformation gradient - G inverse deformation gradient - H relative deformation gradient - J bulk Jacobian of the deformation - ¯K, K_i total (twice the mean) curvature of and S_i - Lin (U, V) linear transformations from U into V - Lin⁺ linear transformations of ³ with positive determinant - Orth⁺ rotations of ³ - Q^a external bulk mass supply of species a - ¯S bulk Cauchy stress tensor - S bulk Piola-Kirchhoff stress tensor - S_i undeformed phase i interface - U_i relative velocity of S_i - Unim⁺ linear transformations of ³ with unit determinant - ¯V, V_i normal velocity of and S_i - intrinsic edge velocity of S and A _i S - W_i volume flow across the phase-i interface - X material point - b external body force - e internal bulk configurational force - f_i external interfacial force (configurational) - ¯g external interfacial force (deformational) - grad, div spatial gradient and divergence - gradient and divergence on - h relative deformation - h^a, diffusive mass flux of species a and list of mass fluxes - ¯m outward unit normal to a spatial control volume - ¯n, n_i unit normal to and S_i - n subspace of ³ orthogonal to n - ¯q^a external interfacial mass supply of species a - s ......... - ¯v, v_i compatible velocity fields of and S_i - ¯w, w_i compatible edge velocity fields for and A_i - x spatial point - y_i deformation or motion of phase i - y^. material velocity - generic subsurfaces of - , _i deformed body and deformed phase-i region - () energy supplied to by mass transport - symmetry group of the lattice - _i, surface jacobians - lattice - () power expended on - spatial control volume - S deformed phase interface - lattice point density - interfacial power density - , A total surface stress - C configurational surface stress for phase 1 (material) - ¯C_i configurational surface stress (spatial) - F_i tangential deformation gradient - G_i inverse tangential deformation gradient - H incoherency tensor - ¯1(x), 1_i(X) inclusions of ¯n(x) and n _i (X) into ³ - K configurational surface stress for phase 2 (material) - ¯L, l_i curvature tensor of and S_i - ¯P(x), P_i(X) projections of ³ onto ¯n(x) and n_i (X) - ¯S, S deformational surface stress (spatial and material) - ¯a, a normal part of total surface stress - c normal part of configurational surface stress for phase 1 (material) - e_i internal interfacial configurational force - ¯v, v_i unit normal to and A _i - (x),_i(X) projections of ³ onto ¯n(x) and n _i (X) - _i normal internal force (material) - bulk free energy - slip velocity - _i=(–1)ⁱ _i ......... - ^a, chemical potential of species a and list of potentials - ^a, bulk molar density of species a and list of molar densities - _i normal internal force (spatial) - surface tension - , _i effective shear - referential-to-spatial transform of field - interfacial energy - grand canonical potential - l unit tensor in ³ - x, vector and tensor product in ³ - (...)^., _t(...) material and spatial time derivative - , Div material gradient and divergence - gradient and divergence on S_i - (...), (...) normal time derivative following and S_i - (...) limit of a bulk field asx ,x_i - [...],...> jump and average of a bulk field across the interface - (...)_ext extension of a surface tensor to ³ - tangential part of a vector (tensor) on and S_i     

A rotating hot-wire technique for spatial sampling of disturbed and manipulated duct flows

The documentation and control of flow disturbances downstream of various open inlet contractions was the primary focus with which to evaluate a spatial sampling technique. An X-wire probe was rotated about the center of a cylindrical test section at a radius equal to one-half that of the test section. This provided quasi-instantaneous multi-point measurements of the streamwise and azimuthal components of the velocity to investigate the temporal and spatial characteristics of the flowfield downstream of various contractions. The extent to which a particular contraction is effective in controlling ingested flow disturbances was investigated by artificially introducing disturbances upstream of the contractions. Spatial as well as temporal mappings of various quantities are presented for the streamwise and azimuthal components of the velocity. It was found that the control of upstream disturbances is highly dependent on the inlet contraction; for example, reduction of blade passing frequency noise in the ground testing of jet engines should be achieved with the proper choice of inlet configurations.List of symbols K _uv correlation coefficient= - P percentage of time that an azimuthal fluctuating velocity derivative dv/d is found - U streamwise velocity component U=U (, t) - V azimuthal or tangential velocity component due to flow and probe rotation V=V (, t) - mean value of streamwise velocity component - U _m resultant velocity from and - mean value of azimuthal velocity component induced by rotation - u fluctuating streamwise component of velocity u=u(, t) - v fluctuating azimuthal component of velocity v = v (, t) - u phase-averaged fluctuating streamwise component of velocity u=u(0) - v phase-averaged fluctuating azimuthal component of velocity v=v() - û average of phase-averaged fluctuating streamwise component of velocity (u()) over cases I-1, II-1 and III-1 û = û() - average of phase-averaged fluctuating azimuthal component of velocity (v()) over cases I-1, II-1 and III-1 - u fluctuating streamwise component of velocity corrected for non-uniformity of probe rotation and/or phase-related vibration u = u(0, t) - v fluctuating azimuthal component of velocity corrected for non-uniformity or probe rotation and/or phase-related vibration v=v (, t) - u ² rms value of corrected fluctuating streamwise component of velocity - rms value of corrected fluctuating azimuthal component of velocity - phase or azimuthal position of X-probe     

Measurements of fully developed turbulent flow in a trapezoidal duct

The turbulence characteristics of fully developed isothermal air flows through a symmetric trapezoidal duct were examined experimentally using Pitot tube and hot-wire anemometry over a Reynolds number range of 3.7–11.6×10⁴. The measurements included local wall shear stress and the cross-sectional distributions of mean axial velocity, secondary velocities and Reynolds stresses. Four secondary flow cells were detected in a symmetric half of the duct. Although secondary velocity components were typically less than about 1% of the bulk axial velocity, their effect was especially pronounced on the distributions of turbulent kinetic energy and local wall shear stress.List of symbols a, b, c, d trapezoidal duct dimensions (Fig. 1) - A, B coefficients in log law (Table 1) - D _h equivalent hydraulic diameter - f Darcy friction factor, (2D _h /U _b ² ) (dP/dx) - k turbulent kinetic energy per unit mass, - k ⁺ dimensionless turbulent kinetic energy, k/( ^*)² - P static pressure - Re Reynolds number, U _b D _h / - s distance along inclined wall, measured from top corner (Fig. 1) - u, v, w fluctuating components of the velocities in the x, y, z directions - u ⁺, v ⁺, w ⁺ dimensionless turbulence intensities; u ²/ ^*, v ²/ ^*, w ²/ ^* - u ^* local friction velocity, ( _w /)^1/2 - ^* average friction velocity, (¯gt/)^1/2 - axial mean velocity (time-average) - U _b average mean axial velocity - U _sec resultant of ¯V and ¯W, (¯V ²+¯ ²)^1/2 - U ⁺ dimensionless velocity, /u ^* - ¯V, ¯W mean velocities in the y, z directions (secondary velocities) - x axial direction - y, 2 horizontal and vertical directions (Fig. 1) - z ⁺ dimensionless distance from (and normal to) a wall, zu ^*/v - distance from wall (at y=0) to location of maximum axial velocity - laminar dynamic viscosity - v kinematic viscosity - air density - _w local wall shear stress - ¯ _w average of local wall shear stresses over all walls - ¯ average wall shear stress, (dP/dx) (D _h /4) - corner angle of trapezoidal duct (Fig. 1) A version of this paper was presented at the 10th Symposium on Turbulence, University of Missouri-Rolla, Sept. 22–24, 1986     

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

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

Stabilität einer in zylindrischen Gleitlagern laufenden,unwuchtfreien Welle

Zusammenfassung Das Problem der sogenannten Lagerinstabilität wurde rechnerisch untersucht. Als Vorbereitung wurde der Schmiermitteldruck und die Feder- und Dämpfungskonstanten des Schmierfilms berechnet. Die lineare Stabilitätsrechnung ergab, daß die Größe des Breitenverhältnisses die Stabilitätsgrenzkurve nur wenig, die Anfahrkurve jedoch stark beeinflußt. Wie man aus dem Verlauf der Grenzkurven ersieht, muß man bei der Stabilisierung eines instabilen Gleichgewichtes zwei Fälle unterscheiden: Beim leicht belasteten Lager ist das Lagerspiel r zu verkleinern und die Eigenfrequenz _k der Welle zu erhöhen, beim schwer belasteten Lager ist die Zähigkeit und die Lagerbreite B zu verkleinern und das Lagerspiel r zu vergrößern. Die nichtlineare Bahnberechnung ergab: Auch wenn nach linearer Rechnung das Gleichgewicht stabil ist, kann bei höheren Drehzahlen eine hinreichend große Anfangsstörung Instabilität hervorrufen. Die Bahnkurven werden bei Instabilität mit wachsender Amplitude mehr und mehr kreisförmig. Dabei stellt sich bei Drehzahlen unter 2 _k die Halbdrehfrequenzschwingung mit einer endlichen Rotoramplitude ein, während bei Drehzahlen über 2 _k die Eigenfrequenzschwingung entsteht, deren Amplitude unaufhörlich anwächst. Dieses Anwachsen beruht auf einer durch den Schmiermitteldruck bedingten Energiezufuhr. Soll also in Wirklichkeit die Amplitude endlich bleiben, so muß eine äußere Dämpfung auf die Welle wirken. Für einen einfachen Dämpfungsansatz wurde daher abschließend die stationäre, kreisförmige Wellenschwingung berechnet, wobei sich wiederum die Halbdrehfrequenz- und die Eigenfrequenzschwingung ergaben. Für große Werte des Parameters / nimmt danach die Rotoramplitude mit der Drehzahl zu, für kleine /-Werte nimmt sie dagegen nach Überschreiten eines Maximums wieder ab. In diesem Falle läßt sich die Rotoramplitude auch durch ein Verkleinern des Parameters / vermindern. Die bisher bekannten Versuchsergebnisse scheinen die Rechenergebnisse zu bestätigen.Gekürzte Fassung der Dissertation an der T.H. Karlsruhe, 1962; Referent: Prof. Dr.-Ing. K. Kollmann; Korreferent: Prof. Dr. K. Nickel und Prof. Dr. F. Weidenhammer; insbesondere Herrn Professor Kollmann danke ich herzlich für die Anregung und großzügige Förderung der Arbeit.     

The effective permeability of a heterogeneous porous medium

The effective (single-phase) permeability of an (infinite) heterogeneous porous medium is studied using a formalism of Green's functions. We give formal expressions for it in the form of a series expansion involving the microscopic random-permeability field many-body correlation functions of higher and higher order.The particular case of a log-normal medium of infinite extent is studied using field-theoretical methods. Using partial series resummation techniques, we derivea formula up to all orders in the local correlations which was first reckoned by many authors by means of a first-order calculation. The formula — which remains an approximation — works whatever the dimensionality of the space, and gives the following simple estimate for the effective permeability in 3 D:K _eff=k ^1/33. The method is general and the approximations can be systematically improved on when more complex situations are studied.Roman Letters D number of dimensions of the space in which the flow takes place - f(r) body force field,N - f(q) Fourier-transformed body-force field, Nm³ - G ₀(r, r) Green's function of the Laplace operator, m^–1 - g(k,r, r) velocity propagator before averaging, m^–1 - G(r, r) velocity propagator after averaging, m^–1 - j(r) a scalar dimensionless field - k(r) local value of the permeability at point r, m² - K _eff effective permeability - K _g geometric average of the local permeability, m² - l typical size of the averaging volume, m - L characteristic length of the porous medium or of the reservoir, m - L(r, r) projection operator, m^–2 - M(r, r) scattering operator, m^–3 - p(r) local value of the pressure, Nm^–2 - p(k,r, r) pressure propagator before averaging, m^–1 - P(r, r) pressure propagator after averaging, m^–1 - r position vector, m - r modulus of vectorr, m - unit vector pointing in the direction ofr - q Fourier wave vector, m^–1 - q modulus of the Fourier wave-vectorq, m^–1 - unit vector pointing in the direction ofq - projector over vector - 1 unit tensor - X(r) a local random variable - ¯X(r) volume averaged local random variable - X (r) ensemble averaged local random variable - V large-scale averaging volume, m³ - Z(j) generating functional of a random field - Z(r,j) modified generating functional of a random field - Z normalization factor Greek Letters ₀ average value of the logarithm of the permeability - (r) fluctuation of the logarithm of permeability at pointr - viscosity of the fluid, Nt/m² - (r–r) two-point correlation function of the fluctuations of the logarithm of the permeability - _k correlation length of the permeability correlation function, m - _u correlation length of the velocity correlation function, m     

Laminar forced convection in elliptic ducts

The problem of laminar forced convection heat transfer in short elliptical ducts with (i) uniform wall temperature and (ii) prescribed wall heat flux is examined in detail with the well known Lévêque theory of linear velocity profile near the wall. Moreover, consideration is given to the variation of the slope of the linear velocity profile with the position on the duct wall. A correction factor for the temperature dependent viscosity is included. Expressions for the local and average Nusselt numbers and wall temperatures are obtained. For the case of constant heat flux the Nusselt numbers are higher than for constant wall temperature.The results corresponding to the classical Graetz and Purday problems are deduced as special cases.Nomenclature a, b semiaxes of ellipse, b Graetz number (average), Re Pr D _e/Z - h _i ^o local heat transfer coefficient - J _n(x) Bessel function of order n - K thermal conductivity of the fluid - [X] Laplace transform of X - N _u ^o local Nusselt number, h _i ^o D _e/K - perimeter average Nusselt number - overall average Nusselt number - Nu _w wall Nusselt number - Nu Nusselt number at large distance from the inlet - p Laplace transform parameter - Pr Prandtl number, C _a/K - Re Reynolds number, D _e / _a - T temperature of the fluid - T ₁, T _W inlet and wall temperatures, respectively - u _z local isothermal velocity along the axis of the duct - average fluid velocity - x, y, z Cartesian coordinates, z-axis parallel to the axis of the duct (z=0 at duct inlet) - Z length of the duct - thermal diffusivity, K/C - * correction factor for the temperature dependent viscosity - (x) gamma function - coordinate measured normal to the wall of the duct - _a, _w viscosity of fluid at average and wall temperatures - , , z elliptic cylindrical coordinates - density of fluid - (z) heat flux     

Slip flow in the hydrodynamic entrance region of a tube and a parallel plate channel

Summary The problem of slip flow in the entrance region of a tube and parallel plate channel is considered by solving a linearized momentum equation. The condition is imposed that the pressure drop from momentum considerations and from mechanical energy considerations should be equal. Results are obtained for Kn=0, 0.01, 0.03, 0.05, and 0.1 and the pressure drop in the entrance region is given in detail.Nomenclature A cross-sectional area of duct - c mean value of random molecular speed - d diameter of tube - f _p - f _t - h half height of parallel plate channel - Kn Knudsen number - L molecular mean free path - n directional normal of solid boundary - p pressure - p ₀ pressure at inlet - r radial co-ordinate - r _t radius of tube - R non-dimensional radial co-ordinate - Re _p 4hU/ - Re _t 2r _t U/ - s direction along solid boundary - T absolute temperature - u velocity in x direction - u* non-dimensional velocity - U bulk velocity = (1/A) _A u dA - v velocity in y direction - x axial co-ordinate - x* stretched axial co-ordinate - X non-dimensional axial co-ordinate - X* non-dimensional stretched axial co-ordinate - Y non-dimensional channel co-ordinate - eigenvalue in parallel plate channel - stretching factor - eigenvalue in tube - density - kinematic viscosity - i index - p parallel plate - t tube - v velocity vector - gradient operator - ² Laplacian operator     

A three-parameter model describing the experimental relation between shear stress and shear rate for laminar flow of aqueous polymer solutions

Summary A three-parameter model is introduced to describe the shear rate — shear stress relation for dilute aqueous solutions of polyacrylamide (Separan AP-30) or polyethylenoxide (Polyox WSR-301) in the concentration range 50 wppm – 10,000 wppm. Solutions of both polymers show for a similar rheological behaviour. This behaviour can be described by an equation having three parameters i.e. zero-shear viscosity ₀, infinite-shear viscosity , and yield stress ₀, each depending on the polymer concentration. A good agreement is found between the values calculated with this three-parameter model and the experimental results obtained with a cone-and-plate rheogoniometer and those determined with a capillary-tube rheometer.

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Zusammenfassung Der Zusammenhang zwischen Schubspannung und Schergeschwindigkeit von strukturviskosen Flüssigkeiten wird durch ein Modell mit drei Parametern beschrieben. Mit verdünnten wäßrigen Polyacrylamid-(Separan AP-30) sowie Polyäthylenoxidlösungen (Polyox WSR-301) wird das Modell experimentell geprüft. Beide Polymerlösungen zeigen im untersuchten Schergeschwindigkeitsbereich von ein ähnliches rheologisches Verhalten. Dieses Verhalten kann mit drei konzentrationsabhängigen Größen, nämlich einer Null-Viskosität ₀, einer Grenz-Viskosität und einer Fließgrenze ₀ beschrieben werden. Die Ergebnisse von Experimenten mit einem Kegel-Platte-Rheogoniometer sowie einem Kapillarviskosimeter sind in guter Übereinstimmung mit den Werten, die mit dem Drei-Parameter-Modell berechnet worden sind.

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a Pa^–1 physical quantity defined by:a = {1 – ( / ₀)}/ ₀ - c l concentration (wppm) - D m capillary diameter - L m length of capillary tube - P Pa pressure drop - R m radius of capillary tube - u m s^–1 average velocity - v _r m s^–1 local axial velocity at a distancer from the axis of the tube - shear rate (–dv _r /dr) - local shear rate in capillary flow - s^–1 wall shear rate in capillary flow - Pa s dynamic viscosity - _a Pa s apparent viscosity defined by eq. [2] - ( _a ) Pa s apparent viscosity in capillary tube at a distanceR from the axis - ₀ Pa s zero-shear viscosity defined by eq. [4] - Pa s infinite-shear viscosity defined by eq. [5] - l ratior/R - kg m^– density - Pa shear stress - ₀ Pa yield stress - _r Pa local shear stress in capillary flow - _R Pa wall shear stress in capillary flow _R = (PR/2L) - _v m³ s^–1 volume rate of flow With 8 figures and 1 table     

On the continuum modelling of porous media containing fluid: a molecular viewpoint with particular attention to scale

Mass conservation and linear momentum balance relations for a porous body and any fluid therein, valid at any given length scale in excess of nearest-neighbour molecular separations, are established in terms of local weighted averages of molecular quantities. The mass density field for the porous body at a given scale is used to identify its boundary at this scale, and a porosity field is defined for any pair of distinct length scales. Specific care is paid to the interpretation of the stress tensor associated with each of the body and fluid at macroscopic scales, and of the force per unit volume each exerts on the other. Consequences for the usual microscopic and macroscopic viewpoints are explored.Nomenclature material system; Section 2.1. - porous body (example of a material system); Sections 2.1, 3.1, 4.1 - fluid body (example of a material system); Sections 2.1, 3.1, 4.1 - weighting function; Sections 2.1, 2.3 - ,h weighting function corresponding to spherical averaging regions of radius and boundary mollifying layer of thicknessh; Section 3.2 - Euclidean space; Section 2.1 - V space of all displacements between pairs of points in; Section 2.1 - mass density field corresponding to; (2.3)₁ - ^P , ^f mass density fields for , ; (4.1) - P momentum density field corresponding to; (2.3)₂ - v velocity field corresponding to; (2.4) - S _r (X) interior of sphere of radiusr with centre at pointx; (3.3) - boundary ofany region - region in which ^p > 0 with = ,h; (3.1) - subset of whose points lie at least+h from boundary of ; (3.4) - abbreviated versions of ; Section 3.2, Remark 4 - strict interior of ; (3.7) - analogues of for fluid system ; Section 3.2 - general version of corresponding to any choice of weighting function; (4.6) - interfacial region at scale; (3.8) - ₀ scale of nearest-neighbour separations in ; Section 3.2. Remark 1 - porosity field at scales ( ₁; ₂); (3.9) - pore space at scales ( ₁; ₂); (3.12)     

Mean and turbulent velocity measurements of supersonic mixing layers

The behavior of supersonic mixing layers under three conditions has been examined by schlieren photography and laser Doppler velocimetry. In the schlieren photographs, some large-scale, repetitive patterns were observed within the mixing layer; however, these structures do not appear to dominate the mixing layer character under the present flow conditions. It was found that higher levels of secondary freestream turbulence did not increase the peak turbulence intensity observed within the mixing layer, but slightly increased the growth rate. Higher levels of freestream turbulence also reduced the axial distance required for development of the mean velocity. At higher convective Mach numbers, the mixing layer growth rate was found to be smaller than that of an incompressible mixing layer at the same velocity and freestream density ratio. The increase in convective Mach number also caused a decrease in the turbulence intensity ( _u/U).List of symbols a speed of sound - b total mixing layer thickness between U ₁ – 0.1 U and U ₂ + 0.1 U - f normalized third moment of u-velocity, f u³/(U)³ - g normalized triple product of u² , g u²/(U)³ - h normalized triple product of u ², h u ²/(U)³ - l _u axial distance for similarity in the mean velocity - l _u axial distance for similarity in the turbulence intensity - M Mach number - M _c convective Mach number (for ₁ = ₂), M _c (U ₁ – U ₂)/(a ₁ + a ₂) - P static pressure - r freestream velocity ratio, r U ₂/U ₁ - Re unit Reynolds number, Re U/ - s freestream density ratio, s ₂/₁ - T _t total temperature - u instantaneous streamwise velocity - u deviation of u-velocity, uu – U - U local mean streamwise velocity - U ₁ primary freestream velocity - U ₂ secondary freestream velocity - average of freestream velocities, (U ₁ + U ₂)/2 - U freestream velocity difference, U U ₁ – U ₂ - instantaneous transverse velocity - v deviation of -velocity, – V - V local mean transverse velocity - x streamwise coordinate - y transverse coordinate - y ₀ transverse location of the mixing layer centerline - ensemble average - ratio of specific heats - boundary layer thickness (y-location at 99.5% of free-stream velocity) - similarity coordinate, (y – y ₀)/b - compressible boundary layer momentum thickness - viscosity - density - standard deviation - dimensionless velocity, (U – U ₂)/U - 1 primary stream - 2 secondary stream A version of this paper was presented at the 11th Symposium on Turbulence, October 17–19, 1988, University of Missouri-Rolla     

Floquet bundles for scalar parabolic equations

For linear scalar parabolic equations such as on a finite interval 0x, with various boundary conditions, we obtain canonical Floquet solutions u _n (t, x). These solutions are characterized by the property that z(u _n (t, x))=n for all t, where z(·) denotes the zero crossing (lap) number of Matano. The coefficients a(t, x) and b(t, x) are not assumed to be periodic in t, but if they are, the solutions u _n (t, x) reduce to the standard Floquet solutions. Our results may naturally be expressed in the language of linear skew product flows. In this context, we obtain for each N1 an exponential dichotomy between the bundles span {u _n (·,·)} _n ^=1/N and .     

The theory of a vibrating-rod densimeter

The paper presents a theory of a device for the accurate determination of the density of fluids over a wide range of thermodynamic states. The instrument is based upon the measurement of the characteristics of the resonance of a circular section tube, or rod, performing steady, transverse oscillations in the fluid. The theory developed accounts for the fluid motion external to the rod as well as the mechanical motion of the rod and is valid over a defined range of conditions. A complete set of working equations and corrections is obtained for the instrument which, together with the limits of the validity of the theory, prescribe the parameters of a practical design capable of high accuracy.Nomenclature A, B, C, D constants in equation (60) - A _j , B _j constants in equation (18) - a _j ⁺ , a _j ^– wavenumbers given by equation (19) - C _f drag coefficient defined in equation (64) - C _f ^/0 , C _f ^/1 components of C _f in series expansion in powers of - c speed of sound - D _b drag force of fluid b - D ₀ coefficient of internal damping - E extensional modulus - force per unit length - F _j ⁺ , F _j ^– constants in equation (24) - f, g functions of defined in equations (56) - G modulus of rigidity - I second moment of area - K constant in equation (90) - k, k constants defined in equations (9) - L half-length of oscillator - Ma Mach number - m _a mass per unit length of fluid a - m _b added mass per unit length of fluid b - m _s mass per unit length of solid - n _j eigenvalue defined in equation (17) - P power (energy per cycle) - P _a , P _b power in fluids a and b - p pressure - R radius of rod or outer radius of tube - R _c radius of container - R _i inner radius of tube - r radial coordinate - T tension - T _visc temperature rise due to heat generation by viscous dissipation - t time - v _r , v radial and angular velocity components - y lateral displacement - z axial coordinate - dimensionless tension - _a dimensionless mass of fluid a - _b dimensionless added mass of fluid b - _b dimensionless drag of fluid b - dimensionless parameter associated with - ₀ dimensionless coefficient of internal damping - dimensionless half-width of resonance curve - dimensionless frequency difference defined in equation (87) - spatial resolution of amplitude - R, , , _s , increments in R, , , _s , - dimensionless amplitude of oscillation - dimensionless axial coordinate - ratio of to - _a , _b ratios of to for fluids a and b - angular coordinate - parameter arising from distortion of initially plane cross-sections - _f thermal conductivity of fluid - dimensionless parameter associated with - viscosity of fluid - _a , _b viscosity of fluids a and b - dimensionless displacement - _j jth component of - density of fluid - _a , _b density of fluids a and b - _s density of tube or rod material - density of fluid calculated on assumption that * - dimensionless radial coordinate - * dimensionless radius of container - dimensionless times - _{rr rr}, _r radial normal and shear stress components - spatial component of defined in equation (13) - _j jth component of - dimensionless streamfunction - ⁰, ¹ components of in series expansion in powers of - phase angle - _r phase difference - _ra , _rb phase difference for fluids a and b - streamfunction - _j jth component defined in equation (22) - dimensionless frequency (based on ) - _a , _b dimensionless frequency in fluids a and b - _s dimensionless frequency (based on _s ) - angular frequency - ₀ resonant frequency in absence of fluid and internal damping - _r resonant frequency in absence of internal fluid - _ra , _rb resonant frequencies in fluids a and b - dimensionless frequency - dimensionless frequency when _a vanishes - dimensionless frequencies when _a vanishes in fluids a and b - dimensionless resonant frequency when _a , _b, _b and ₀ vanish - dimensionless resonant frequency when _a , _b and _b vanish - dimensionless resonant frequency when _b and _b vanish - dimensionless frequencies at which amplitude is half that at resonance     

The unsteady aerodynamics of a first stage stator vane row

The fundamental unsteady aerodynamics on a vane row of an axial flow research compressor stage are experimentally investigated, demonstrating the effects of airfoil camber and steady loading. In particular, the rotor wake generated unsteady surface pressure distributions on the first stage vane row are quantified over a range of operating conditions. These cambered airfoil unsteady data are correlated with predictions from a flat plate cascade inviscid flow model. At the design point, the unsteady pressure difference coefficient data exhibit good correlation with the nonseparated predictions, with the aerodynamic phase lag data exhibiting fair trendwise correlation. The quantitative phase lag differences are associated with the camber of the airfoil. An aft suction surface flow separation region is indicated by the steady state surface static pressure data as the aerodynamic loading is increased. This separation affects the increased incidence angle unsteady pressure data.List of symbols b airfoil semi-chord - C airfoil chord - C _p dynamic pressure coefficient, - _p static pressure coefficient, - i incidence angle - k reduced frequency, - N number of rotor revolutions - p dynamic pressure difference - static pressure difference, - S stator vane circumferential spacing - U _t rotor blade tip speed - u longitudinal perturbation velocity - V absolute velocity - V _axial absolute axial velocity - v transverse perturbation velocity - x _sep location of separation point - inlet angle - inlet air density - blade passing angular frequency     

Rayleigh problem in slip flow with transverse magnetic field

An analytical continuum solution of the Rayleigh problem in slip flow with applied magnetic field is obtained using a modified initial condition and slip boundary conditions. The results are uniformly valid for all times and show that the velocity slip and the local skin friction coefficient remain almost unaffected by the imposition of the magnetic field for small times. They increase however with the magnetic field for large times. The present results reduce to the corresponding results of the hydrodynamic case when there is no magnetic field.Nomenclature A constant - b characteristic length - B magnetic field vector - B ₀ magntidue of the applied magnetic field normale to the plate - B _x magnitude of the induced magnetic field parallel to the plate - C slip coefficient, (2–f)/f - C _f skin friction coefficient, - C _D average drag coefficient - erfc(_x) complementary error function, - E electric field vector - f Maxwell's reflection coefficient - H _a Hartmann number, (B ₀ ² b ²/)^1/2 - nondimensional magnetic parameter - J current vector - Kn=L/b Knudsen number - L mean free path - M Mach number - p constant parameter - P _m magnetic Prandtl number, Re _m/Re= ₀ - q velocity vector - Re Reynolds number, Ub/ - Re _m magnetic Reynolds number, ₀ Ub - t time - nondimensional time, tU/b - u velocity of the fluid parallel to the plate - nondimensional velocity, u/U - U velocity of the plate - Laplace transform of - x, y coordinates along and normal to the plate respectively - y nondimensional distance, y/b - Z nondimensional parameter, 1/Re ^1/2 Kn - ratio of specific heats - boundary layer thickness - velocity slip - viscosity - ₀ magnetic permeability - kinematic viscosity - nondimensional time parameter, ( /Re)^1/2/Kn - density - electrical conductivity