Flow in porous media I: A theoretical derivation of Darcy's law

Stokes flow through a rigid porous medium is analyzed in terms of the method of volume averaging. The traditional averaging procedure leads to an equation of motion and a continuity equation expressed in terms of the volume-averaged pressure and velocity. The equation of motion contains integrals involving spatial deviations of the pressure and velocity, the Brinkman correction, and other lower-order terms. The analysis clearly indicates why the Brinkman correction should not be used to accommodate ano slip condition at an interface between a porous medium and a bounding solid surface.The presence of spatial deviations of the pressure and velocity in the volume-averaged equations of motion gives rise to aclosure problem, and representations for the spatial deviations are derived that lead to Darcy's law. The theoretical development is not restricted to either homogeneous or spatially periodic porous media; however, the problem ofabrupt changes in the structure of a porous medium is not considered.Roman Letters A interfacial area of the - interface contained within the macroscopic system, m² - A _e area of entrances and exits for the -phase contained within the macroscopic system, m² - A interfacial area of the - interface contained within the averaging volume, m² - A ^* interfacial area of the - interface contained within a unit cell, m² - Ae area of entrances and exits for the -phase contained within a unit cell, m² - B second order tensor used to represent the velocity deviation (see Equation (3.30)) - b vector used to represent the pressure deviation (see Equation (3.31)), m^–1 - d distance between two points at which the pressure is measured, m - g gravity vector, m/s² - K Darcy's law permeability tensor, m² - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the -phase (see Figure 2), m - characteristic length scale for the -phase (see Figure 2), m - n unit normal vector pointing from the -phase toward the -phase (n =–n ) - n _e unit normal vector for the entrances and exits of the -phase contained within a unit cell - p pressure in the -phase, N/m² - p intrinsic phase average pressure for the -phase, N/m² - p –p , spatial deviation of the pressure in the -phase, N/m² - r ₀ radius of the averaging volume and radius of a capillary tube, m - v velocity vector for the -phase, m/s - v phase average velocity vector for the -phase, m/s - v intrinsic phase average velocity vector for the -phase, m/s - v –v , spatial deviation of the velocity vector for the -phase, m/s - V averaging volume, m³ - V volume of the -phase contained within the averaging volume, m³ Greek Letters V/V, volume fraction of the -phase - mass density of the -phase, kg/m³ - viscosity of the -phase, Nt/m² - arbitrary function used in the representation of the velocity deviation (see Equations (3.11) and (B1)), m/s - arbitrary function used in the representation of the pressure deviation (see Equations (3.12) and (B2)), s^–1     

Transport in ordered and disordered porous media V: Geometrical results for two-dimensional systems

In this paper we continue the geometrical studies of computer generated two-phase systems that were presented in Part IV. In order to reduce the computational time associated with the previous three-dimensional studies, the calculations presented in this work are restricted to two dimensions. This allows us to explore more thoroughly the influence of the size of the averaging volume and to learn something about the use of anon-representative region in the determination of averaged quantities.

Nomenclature

Roman Letters A interfacial area of the– interface associated with the local closure problem, m² - a _i i=1, 2, gaussian probability distribution used to locate the position of particles - l unit tensor - characteristic length for the-phase particles, m - ⁰ reference characteristic length for the-phase particles, m - characteristic length for the-phase, m - _i i=1,2,3 lattice vectors, m - m convolution product weighting function - m _V special convolution product weighting function associated with a unit cell - n _i i=1, 2 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward the-phase - r _p position vector locating the centroid of a particle, m - r gaussian probability distribution used to determine the size of a particle, m - r ₀ characteristic length of an averaging region, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume,V, m³ - x position of the centroid of an averaging area, m - x ₀ reference position of the centroid of an averaging area, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters V /V, volume average porosity - _{a i} standard deviation ofa _i - _r standard deviation ofr - intrinsic phase average of      

The chaotic response of a fluttering panel: The influence of maneuvering

The influence of maneuvering on the chaotic response of a fluttering buckled plate on an aircraft has been studied. The governing equations, derived using Lagrangian mechanics, include geometric non-linearities associated with the occurrence of tensile stresses, as well as coupling between the angular velocity of the maneuver and the elastic degrees of freedom. Numerical simulation for periodic and chaotic responses are conducted in order to analyze the influence of the pull-up maneuver on the dynamic behavior of the panel. Long-time histories phase-plane plots, and power spectra of the responses are presented. As the maneuver (load factor) increases, the system exhibits complicated dynamic behavior including a direct and inverse cascade of subharmonic bifurcations, intermittency, and chaos. Beside these classical routes of transition from a periodic state to chaos, our calculations suggest amplitude modulation as a possible new mode of transition to chaos. Consequently this research contributes to the understanding of the mechanisms through which the transition between periodic and strange attractors occurs in, dissipative mechanical systems. In the case of a prescribed time dependent maneuver, a remarkable transition between the different types of limit cycles is presented.Nomenclature a plate length - a _r u _r /h - D plate bending stiffness - E modulus of elasticity - g acceleration due to gravity - h plate thickness - j₁,j₂,j₃ base vectors of the body frame of reference - K spring constant - M Mach number - n 1 + ₀/g - N ₁ applied in-plane force - p–p aerodynamic pressure - P pa ⁴/Dh - q ₀/2 - Q _r generalized Lagrangian forces - R rotation matrix - R ₄ N, a ²/D - t time - kinetic energy - u plate deflection - u displacement of the structure - u _r modal amplitude - v₀ velocity - x coordinates in the inertial frame of reference - z coordinates in the body frame of reference - Ka/(Ka+Eh) - - elastic energy - 2qa ³/D - a/_mh - Poisson's ratio - material coordinates - air density - _m plate density - - _r prescribed functions - _r sin(r z/a) - angular velocity - a/v₀ - skew-symmetric matrix form of the angular velocity     

Experimental investigation of the creeping motion of a drop in a vertical tube

New experimental data regarding the motion of a drop along the axis of a vertical tube, filled with another highly viscous liquid, are obtained. The experiments are realised with sufficiently large drops for an internal circulation to develop and also for different pairs of fluids; the preponderant role of the gravity on the drop shape and consequently on its terminal velocity is pointed out. Moreover, by means of a visualization technique, details on the flow both inside and outside the drop are given.List of symbols g gravity acceleration - r distance from the drop center - R equivalent radius of the drop, i.e. the radius of the sphere having the same volume as the drop - R _EQ radius of the equatorial section of the drop - R _T tube radius - L _AX half length of the drop - U ₀ terminal velocity of the drop - P _s Poiseuille number= U _{0 e} /4 g R ² - Fr Fronde number = U ₀ ² _e /2 g R - Re Reynolds number = 2 U ₀ R _e / _e - E _o Eötvös number = 4g R ²/ - deformation parameter = _e U ₀/ - apparent density of the suspended liquid= | _i – _e | - _i viscosity of the suspended liquid - _e viscosity of the suspending liquid - drop-to-tube radius ratio = R/R _T - _EQ equatorial drop-to-tube radius ratio = R EQ/R _T - interfacial tension     

Asymptotic Nusselt numbers for Newtonian flow through a parallel-plate channel with recycle

General expressions for evaluating the asymptotic Nusselt number for a Newtonian flow through a parallel-plate channel with recycle at the ends have been derived. Numerical results with the ratio of thicknesses as a parameter for various recycle ratios are obtained. A regression analysis shows that the results can be expressed by log Nur^0.83=0.3589 (log)² -0.2925 (log) + 0.3348 forR 3, 0.1 0.9; logNu=0.5982(log)² +0.3755 × 10^–2 (log) +0.8342 forR 10^–2, 0.1 0.9.

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Asymptotische Nusselt-Zahlen für die Newtonsche Strömung durch einen Kanal aus parallelen Platten mit Rückführung

Zusammenfassung In dieser Untersuchung wurden allgemeine Ausdrücke hergeleitet um die asymptotische Nusselt-Zahl für eine Newtonsche Strömung durch einen Kanal aus parallelen Platten mit Rückführung an den Enden berechnen zu können. Es wurden numerische Ergebnisse mit den Dicken-Verhältnissen, als Parameter für verschiedene Rückführungs-verhältnisse, erhalten. Eine Regressionsanalyse zeigt, daß die Ergebnisse wie folgt ausgedrückt werden können: log Nur^0,83=0,3589 (log)² -0,2925 (log) + 0,3348 fürR 3, 0,1 0,9; logNu=0,5982(log)² +0,3755 × 10^–2 (log) + 0,8342 fürR 10^–2, 0,1 0,9.

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Nomenclature A₁ shooting value,d(0)/d - A₂ shooting value,d(1)/d - B channel width - Gz Graetz number, U_bW²/L - h _m logarithmic average convective heat transfer coefficient - h _x average local convective heat transfer coefficient - k thermal conductivity - L channel length - Nu average local Nusselt number, 2 h_xW/k - Nu _m logarithmic average Nusselt number, 2h_mW/k - R recycle ratio, reverse volume flow rate divided by input volume flow rate - T temperature of fluid - T _m bulk temperature, Eq. (8) - T ₀ temperature of feed stream - T _s wall temperature - U velocity distribution - U _b reference velocity,V/BW - V input volume flow rate - v dimensionless velocity distribution, U/U_b - W channel thickness - x longitudinal coordinate - y transversal coordinate - Z₁-z₆ functions defined in Eq. (A1) - thermal diffusivity - least squares error, Eq. (A7) - weight, Eqs. (A8), (A9) - dimensionless coordinate,y/W - dimensionless coordinate,x/GzL - function, Eq. (7)     

Nonstationary vibration of a rotating shaft with nonlinear spring characteristics during acceleration through a critical speed (A critical speed of a summed-and-differential harmonic oscillation)

Nonstationary vibration of a flexible rotating shaft with nonlinear spring characteristics during acceleration through a critical speed of a summed-and-differential harmonic oscillation was investigated. In numerical simulations, we investigated the influence of the angular acceleration , the initial angular position of the unbalance _n and the initial rotating speed on the maximum amplitude. We also performed experiments with various angular accelerations. The following results were obtained: (1) the maximum amplitude depends not only on but also on _n and : (2) when the initial angular position _n changes. the maximum amplitude varies between two values. The upper and lower bounds of the maximum amplitude do not change monotonously for the angular acceleration: (3) In order to always pass the critical speed with finite amplitude during acceleration. the value of must exceed a certain critical value.Nomenclature O-xyz rectangular coordinate system - , ₁, ₁ inclination angle of rotor and its projections to thexy- andyz-planes - I _r polar moment of inertia of rotor - I diametral moment of inertia of rotor - i _r ratio ofI _r toI - dynamic unbalance of rotor - directional angle of fromx-axis - c damping coefficient - spring constant of shaft - N _nt ,N _nt nonlinear terms in restoring forees in ₁ and ₁ directions - ₄ representative angle - a small quantity - V. V _u .V _N potential energy and its components corresponding to linear and nonlinear terms in the restoring forees - directional angle - _n coefficients of asymmetrical nonlinear terms - _n coefficients of symmetrical nonlinear terms - coefficients of asymmetrical nonlinear terms experessed in polar coordinates - coefficients of symmetrical nonlinear terms expressed in polar coordinates - rotating speed of shaft - t time - _n initial angular position of att=0 - p natural frequency - p ₁.p _t natural frequencies of forward and backward precessions - , ₁, ₁ total phases of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation - , ₁, ₁ phases of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation - P, R _t ,R _b amplitudes of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation - difference between phases ( = _f – _u) - acceleration of rotor - initial rotating speed - t _t ,r _b amplitudes of nonstationary oscillation during acceleration - (r _t )_max, (r _b )_max maximum amplitudes of nonstationary oscillation during acceleration - (r ₁ ¹ )_max, (r _b ¹ )_max maximum value of angular acceleration of non-passable case - ₀ critical value over which the rotor can always pass the critical speed - p ₁,p ₂,p ₃,p ₄ natural frequencies of experimental apparatus     

The theory of a vibrating-rod viscometer

The paper presents a complete theory for a viscometer based upon the principle of a circular-section rod, immersed in a fluid, performing transverse oscillations perpendicular to its axis. The theory is established as a result of a detailed analysis of the fluid flow around the rod and is subject to a number of criteria which subsequently constrain the design of an instrument. Using water as an example it is shown that a practical instrument can be designed so as to enable viscosity measurement with an accuracy of ±0.1%, although it is noted that many earlier instruments failed to satisfy one or more of the newly-established constraints.Nomenclature A, D constants in equation (46) - A _m , B _m , C _m , D _m constants in equations (50) and (51) - A _j , B _j constants in equation (14) - a _j ⁺ , a _j ^– wavenumbers given by equation (15) - C _f drag coefficient defined in equation (53) - c speed of sound - D _b drag force of fluid b - D ₀ coefficient of internal damping - E extensional modulus - f(z) initial deformation of rod - f(), F _m () functions of defined in equation (41) - F force in the rod - force per unit length near t=0 - F dimensionless force per unit length near t=0 - g _m amplitude of transient force - G modulus of rigidity - h, h* functions defined by equations (71) and (72) - H functions defined by equation (69) and (70) - I second moment of area - I _0,1, J _0,1, K _0,1 modified Bessel functions - k, k functions defined in equations (2) - L half-length of oscillator - Ma Mach number - m _b added mass per unit length of fluid b - m _s mass per unit length of solid - n _j eigenvalue defined in equations (15) and (16) - R radius of rod - R _c radius of container - 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 - y ₀ initial lateral displacement - y ₁, y ₂ successive maximum lateral displacement - z axial coordinate - dimensionless tension - dimensionless mass of fluid - dimensionless drag of fluid - amplification factor - logarithmic decrement in a fluid - _a , _b logarithmic decrement in fluids a and b - ₀ logarithmic decrement in vacuo - _j logarithmic decrement in mode j in a fluid - spatial resolution of amplitude - _v voltage resolution - r, , , _s, , increments in R, , , _s , , - dimensionless amplitude of oscillation - dimensionless axial coordinate - angular coordinate - _f thermal conductivity of fluid - viscosity of fluid - viscosity of fluid calculated on assumption that * - _a , _b viscosity of fluids a and b - _m constants in equation (10) - dimensionless displacement - _j j the component of - density of fluid - _a , _b density of fluids a and b - _s density of tube or rod material - dimensionless radial coordinate - * dimensionless radius of container - dimensionless times - spatial component of defined in equation (11) - _j , _tm jth, mth component of - dimensionless streamfunction - ⁰, ¹ components of in series expansion in powers of - streamfunction - dimensionless frequency (based on ) - angular frequency - ₀ angular frequency in absence of fluid and internal damping - _j angular frequency in mode j in a fluid - _a , _b frequencies in fluids a and b     

Stability characteristics of condensate films

The linear stability theory is used to study stability characteristics of laminar condensate film flow down an arbitrarily inclined wall. A critical Reynolds number exists above which disturbances will be amplified. The magnitude of the critical Reynolds number is in all practical situations so small that a laminar gravity-induced condensate film can be expected to be unstable. Several stabilizing effects are acting on the film flow; at an inclined wall these effects are due to surface tension, gravity and condensation mass transfer.

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Zusammenfassung Mit Hilfe der linearen Stabilitätstheorie werden die Stabilitätseigenschaften laminarer Kondensatfilme an einer geneigten Wand untersucht. Es zeigt sich, daß Kondensatfilme in jedem praktischen Fall ein unstabiles Verhalten aufweisen. Der stabilisierende Einfluß von Oberflächenspannung, Schwerkraft und Stoffübertragung durch Kondensation bewkkt jedoch, daß Störungen in bestimmten Wellenlängenbereichen gedämpft werden.

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Nomenclature c=c^*/u₀ complex wave velocity, celerity, dimensionless - c^*=c _r ^* + i c _i ^* complex wave velocity, celerity, dimensional - c_p specific heat at constant pressure - g gravitational acceleration - h_fg latent heat - k thermal conductivity of liquid - p^* pressure - p=p^*/u₀ ² dimensionless pressure - Pe=Pr Re^* Peclet number - Pr Prandtl number - Re^*=u₀ / Reynolds number (defined with surface velocity) - S temperature perturbation amplitude - t^* time - t=t^* u₀/ dimensionless time - T temperature - T_s saturation temperature - T_w wall temperature - T=T_s-T_w temperature drop across liquid film - u^*, v^* velocity components - u=u^*/u₀ dimensionless velocity components - v=v^*/u₀ dimensionless velocity components - u₀ surface velocity of undisturbed film flow - v _g ^* vapor velocity - x^*, y^* coordinates - x=x^*/ dimensionless coordinates - y=y^*/ dimensionless coordinates Greek Symbols =^* wave number, dimensionless - ^*=2 /^* wave number dimensional - ^* wave length, dimensional - =^*/ wave length, dimensionless - local thickness of undisturbed condensate film - kinematic viscosity, liquid - density, liquid - _g density vapor - surface tension - = (1 +) film thickness of disturbed film, Fig. 1 - stream function perturbation amplitude - angle of inclination Base flow quantities are denoted by^–, disturbance quantities are denoted by.     

Buoyancy- and pressure-driven motion in a vertical porous layer: Effects of quadratic drag

The seepage velocity arising from pressure and buoyancy driving forces in a slender vertical layer of fluid-saturated porous media is considered. Quadratic drag (Forcheimer effects) and Brinkman viscous forces are included in the analysis. Parameters are identified which characterize the influence of matrix permeability, quadratic drag and buoyancy. An explicit solution is obtained for pressure-driven flow which illustrates the influence of quadratic drag and the strong boundary layer behavior expected for low permeability media. The experimental data of Givler and Altobelli [2] for water seepage through a high porosity foam is found to yield good agreement with the present analysis. For the case of buoyancy-driven flow, a uniformly valid approximate solution is found for low permeability media. Comparison with the pressure-driven case shows strong similarities in the near-wall region.Nomenclature B function of - d layer thickness - D discriminant defined by Equation (9) - modified Darcy number - F Forcheimer constant - g gravitational acceleration - k porous matrix permeability - m parameter defined by Equation (11) - p pressure - p modified pressure - pressure gradient - R buoyancy parameter - T ₀ nominal layer temperature - u seepage velocity - dimensionless seepage velocity - _c composite approximation - _i boundary layer velocity - _o outer or core flow approximation - _m midplane velocity - U matching velocity - V cross-sectional average velocity - w variable defined by Equation (12) - x, z Cartesian coordinates - , dimensionless Cartesian coordinates - inertia parameter - T layer temperature difference - larger root of cubic given by Equation (8) - fluid dynamic viscosity - _e effective viscosity of fluid saturated medium - variable defined by Equation (18) - 0 fluid density - smaller root of cubic given by Equation (8) - variable defined by Equation (18) - stretched inner coordinate - porosity - function of      

Thermal effects in the capillary rheometer

Summary The effect of viscous heating in a capillary rheometer is analysed for a power-law fluid by means of a perturbation expansion based upon a boundary-layer-core structure. This expansion is found to complement the eigenfunction series solution obtained by earlier investigators. A similar analysis is presented for the work-of-expansion effect. These two thermal effects are superimposed together with a third perturbation effect due to the pressure dependence of viscosity.On the basis of the present theory, earlier work in this area is discussed and, in some cases, apparent inaccuracies or inconsistencies are pointed out. A means is indicated for correcting data on the basis of the present theory.

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Zusammenfassung Es wird der Effekt der Erwärmung einer Potenzflüssigkeit infolge viskoser Reibung in einem Kapillar-Rheometer mittels einer Störungsrechnung untersucht, die auf der Unterteilung der Strömung in eine Grenzschicht und einen Kern basiert. Diese Störungsentwicklung ergänzt eine früher von anderen Autoren gefundene Reihenentwicklung mit Hilfe von Eigenfunktionen. Eine ähnliche Untersuchung wird für die thermische Ausdehnungsarbeit durchgeführt. Diese beiden thermischen Effekte sind zusammen einem dritten Störeffekt superponiert, der von der Druckabhängigkeit der Viskosität herrührt.Aufgrund der vorgelegten Theorie werden verschiedene auf diesem Gebiet früher durchgeführte Arbeiten diskutiert, und es werden in einigen Fällen offensichtliche Ungenauigkeiten und Folgewidrigkeiten aufgedeckt. Schließlich wird eine Methode zur Korrektur von Meßdaten mit Hilfe der vorliegenden Theorie angegeben.

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Nomenclature a tube radius - b ; evaluated atT ₀ andp = 0 when used in perturbation expansion - C _p specific heat - f - f ^* - h defined by eq. [15] - k thermal conductivity - L tube length - m defined by eq. [8] - m ₀ m(T₀, 0) - n power-law index - p pressure - Pe C _p W a/k Peclet number - Pr C _pa/k Prandtl number - Q volumetric flow rate - Q ₀ unperturbed value ofQ in specified-p formulation - r radial coordinate - Re W a/ _a Reynolds number - T temperature - T ₀ inlet temperature - u radial velocity component - u ₀ 0 unperturbed radial velocity - w axial velocity component - w ₀ /W(1 – ) unperturbed axial velocity - W Q/(a ²) average axial velocity - W ₀ Q ₀/(a ²) - z axial coordinate - (3n + 1)/n - ^* ; evaluated atT ₀ andp = 0 when used in perturbation expansion - 4^1-n - ^* - (n + 1)/n - ... shear rate - 4W/a apparent shear rate - p total pressure drop - T _a W ²/k characteristic temperature difference - T _b total bulk-temperature rise - ^* T - r/a - shear viscosity - _a m₀ - (1 –)/ ^1/3 - —p/z - ₀ ... unperturbed value of - z-averaged value of - µ ^{n + 1}/n - z/(a Pe) - _L L/(a Pe) - mass density - _w shear stress at wall - streamfunction - ^*T₀ (absolute temperature scale) - ( )₁ leading-order effect due to viscous heating - ( ) ₁ ^* leading-order effect due to work-of-expansion Note: in specified-p formulation,W gets replaced byW ₀ in definition of Pe, Re, and. With 7 figures and 7 tables     

Rheological behaviour of two-phase flow of solid particles in a gas

The present investigation was concerned with the rheological behaviour of dilute suspensions of solid particles in a gas in a vertical cocurrent flow moving upwards. Starting from the experimentally determined dependence of the pressure drop on the concentration of solid particles and the Reynolds number of the carrier medium in the steady flow region, the rheological parameters were estimated using pseudo-shear diagrams. Air was the carrier medium and the dispersed phase was one of six fractions of polypropylene powder and five fractions of glass ballotini. The results show that the investigated two-phase systems have pseudoplastic character which becomes more pronounced with increases in concentration, equivalent diameter and density of solid particles in the flowing suspension. C _d coefficient of particle resistance - d _e equivalent diameter of particles - D column diameter - Fr Froude number - g gravitational acceleration - K rheological parameter - L length - n rheological parameter - p _t pressure drop due to friction - p _m total pressure drop - p _ag pressure drop due to acceleration of the gas phase - p _as pressure drop due to acceleration of the solid phase - p _g hydrostatic pressure of the gas phase - p _s specific effective weight of the dispersed phase - r radius - Re Reynolds number - Re _p Reynolds number of a particle - Re _G generalized Reynolds number - Re _G1 generalized Reynolds number relating to the end of the laminar flow region - Re _G2 generalized Reynolds number relating to the beginning of the turbulent flow region - w _z axial component of velocity - u _t steady free-fall velocity of a single particle - w average velocity - w _g average velocity of the gas phase - w _s average velocity of the dispersed phase of solid particles - relative mass fraction of solid particles - x _s volume fraction of solid particles - _g coefficient of pressure drop due to friction - µ dynamic viscosity - _g density of the gas phase - _m density of the suspension - _s density of solid particles - _ds density of the dispersed phase - _w shear stress at the wall     

The very slow flow of a Powell-Eyring fluid around a sphere

Summary The very slow flow of a Powell-Eyring type non-Newtonian fluid around a sphere is investigated by a variational technique. The result, a correction factor that is applied to the Stokes' equation, is given as a plot and as an equation which is empirically fit to the plot. Also, a comparison of the very slow flows of a simplified viscoelastic Oldroyd fluid and the Powell-Eyring fluid is made which indicates that in a certain restricted region of the very slow flows, both models give essentially the same results. The Oldroyd and Powell-Eyring model parameters are interrelated by forcing both models to fit the same tube flow viscosity data.Nomenclature B dimensionless quantity, v /R - C dimensionless second invariant - c ₁ constant determined by variational method - D dimensionless variational integral - D _2j , D _j+k position-independent variables used in specification of trial functions - E _2j , E _j+k position-independent variables used in specification of trial functions - f friction factor - f _corr friction factor correction - F _drag drag force on sphere - g, g ₀, g ₁ general trial function; first and second terms in the general trial function - G, H terms in the expression for C - j index - J variational integral - k index - K term in the expression for C - p, q integers - r integer, radial coordinate - R radius of sphere - Re Reynolds number - Re ₀ Reynolds number at point of zero shear rate - Re Reynolds number at infinite distance from sphere - Re _NN Reynolds number based on variable part of viscosity - u, v dimensionless position coordinates - V volume considered - v _i ith velocity component - v _r , v , v _z velocity components in the r, , and z-directions - v approach velocity of the fluid - x/ parameter in Powell-Eyring model - x _i i-position coordinate - parameter in Powell-Eyring model - rate of deformation - , _c , _N , ₀ coefficient of viscosity; cross viscosity; parameter in Powell-Eyring model; viscosity in limit of zero shear rate - spherical coordinate - , _ij rate of deformation tensor; ij-component of rate of deformation tensor - ₁, ₂ parameters in Oldroyd model - Newtonian viscosity - ₁, ₂ parameters in Oldroyd model - dimensionless radial coordinate, r/R - second invariant - fluid density - spherical coordinate - stream function     

On natural convection from a vertical plate with a prescribed surface heat flux in porous media

This paper presents a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 +x ²)), where is a constant andx is the distance along the surface. It is shown that for > -1/2 the solution develops from a similarity solution which is valid for small values ofx to one which is valid for large values ofx. However, when -1/2 no similarity solutions exist for large values ofx and it is found that there are two cases to consider, namely < -1/2 and = -1/2. The wall temperature and the velocity at large distances along the plate are determined for a range of values of .Notation g Gravitational acceleration - k Thermal conductivity of the saturated porous medium - K Permeability of the porous medium - l Typical streamwise length - q _w Uniform heat flux on the wall - Ra Rayleigh number, =gK(q _w /k)l/(v) - T Temperature - Too Temperature far from the plate - u, v Components of seepage velocity in the x and y directions - x, y Cartesian coordinates - Thermal diffusivity of the fluid saturated porous medium - The coefficient of thermal expansion - An undetermined constant - Porosity of the porous medium - Similarity variable, =y(1+x ) ^/3/x ^1/3 - A preassigned constant - Kinematic viscosity - Nondimensional temperature, =(T – T )Ra^1/3 k/qw - Similarity variable, = =y(log_e x)^1/3/x ^2/3 - Similarity variable, =y/x ^2/3 - Stream function     

An experimental study of the behavior of transient isolated convection in a rigidly rotating fluid

The temperature field of starting thermal plumes were measured in a rotating annulus with various rotation rates and buoyancies. The experiments revealed many details of the internal structure of these convective phenomena and also significant horizontal displacements from their source. Measurements show an increase in the maximum temperature observed in the thermal caps with increasing rotation and a more rapid cooling of the buoyancy source.List of symbols D angle relating inward centripetal acceleration to buoyant acceleration, defined by tan D = R/g - g gravitational acceleration - P total pressure of ambient fluid - R radial coordinate measured from rotation axis - R ₀ distance from rotation axis to buoyancy source - u velocity of fluid parcel along the radial direction - velocity of fluid parcel along the azimuthal direction - w velocity of fluid parcel along the axial direction - z axial coordinate, measured upward from the plane containing the buoyancy source - density of a buoyant parcel of fluid - ₀ density of the ambient fluid - azimuthal angle measured from the radial line passing through the buoyancy source - rotation rate of the R––z coordinate system in radians/second     

Transport in ordered and disordered porous media III: Closure and comparison between theory and experiment

In this paper we examine the closure problem associated with the volume averaged form of the Stokes equations presented in Part II. For both ordered and disordered porous media, we make use of a spatially periodic model of a porous medium. Under these circumstances the closure problem, in terms of theclosure variables, is independent of the weighting functions used in the spatial smoothing process. Comparison between theory and experiment suggests that the geometrical characteristics of the unit cell dominate the calculated value of the Darcy's law permeability tensor, whereas the periodic conditions required for thelocal form of the closure problem play only a minor role.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 - A interfacial area of the- interface associated with the local closure problem, m² - A _p surface area of a particle, m² - b vector used to represent the pressure deviation, m^–1 - B ⁰ B+I, a second order tensor that maps v _m ontov - B second-order tensor used to represent the velocity deviation - d _p 6V _p/A_p, effective particle diameter, m - d a vector related to the pressure, m - D a second-order tensor related to the velocity, m² - g gravity vector, m/s² - I unit tensor - K traditional Darcy's law permeability tensor calculated on the basis of a spatially periodic model, m² - K _m permeability tensor for the weighted average form of Darcy's law, m² - L general characteristic length for volume averaged quantities, m - L _p characteristic length for the volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - characteristic length (pore scale) for the-phase - _i i=1, 2, 3 lattice vectors, m - weighting function - m(-y) , convolution product weighting function - m _v special convolution product weighting function associated with the traditional averaging volume - m _g general convolution product weighting function - m _V unit cell convolution product weighting function - m _C special convolution product weighting function for ordered media which produces the cellular average - n unit normal vector pointing from the-phase toward the -phase - p pressure in the-phase, N/m² - p _m superficial weighted average pressure, N/m² - p _m intrinsic weighted average pressure, N/m² - p traditional intrinsic volume averaged pressure, N/m² - p – p _m , spatial deviation pressure, N/m² - r ₀ radius of a spherical averaging volume, m - r _m support of the convolution product weighting function - r position vector, m - r position vector locating points in the-phase, m. - V averaging volume, m³ - B 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 _m superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - v traditional superficial volume averaged velocity, m/s - v – v _m , spatial deviation 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 - _m m * , weighted average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m²     

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

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

Thermal entrance region heat transfer for MHD laminar flow in parallel-plate channels with unequal wall temperatures

The problem of thermal entry heat transfer for Hartmann flow in parallel-plate channels with uniform but unequal wall temperatures considering viscous dissipation, Joule heating and axial conduction effects is approached by the eigenfunction expansion method. The series expansion coefficients for the nonorthogonal eigenfunctions are obtained by using a method for nonorthogonal series described by Kantorovich and Krylov [21]. Numerical results are obtained for the case with entrance condition parameter _o=1 and open circuit condition K=1. The parametric values of Ha=0, 2, 6, 10 and Br=0, –1 are considered for Hartmann and Brinkman numbers, respectively.

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Zusammenfassung Das Problem der Wärmeübertragung im thermischen Einlauf einer Hartmannströmung im ebenen Spalt mit einheitlichen, aber ungleichen Wandtemperaturen wurde unter Berücksichtigung viskoser Dissipation, Joulescher Heizung und axialer Wärmeleitung mit Hilfe einer Entwicklung nach Eigenfunktionen behandelt. Die Koeffizienten der Entwicklung für nichtorthogonale Eigenfunctionen wurde nach einer Methode für nichtorthogonale Reihen nach Kantorovicz und Krylow [21] berechnet. Numerische Ergebnisse werden für den Eintrittsparameter _o=1 und die Bedingung für den offenen Stromkreis K=1 erhalten. Die Parameterwerte Ha=0, 2, 6, 10 und Br=0, –1 werden für die jeweiligen Werte der Hartmann- und der Brinckman-Zahl betrachtet.

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Nomenclature a one-half of channel height - ¯B,B₀ magnetic field Induction vector and magnitude of applied magnetic field - Br Brinkman number, _f U_m ²/(k_c) - C_n,D_n coefficients in the series expansion of _e, see eq. 16 - c_p specific heat at constant pressure - ,E₀ electric field intensity vector and component - E_n,O_n even and odd eigenfunctions - Ha Hartmann number, (/_f)^1/2 B_o a - h₁,h₂ local heat transfer coefficients at lower and upper plates - ¯J,J_y electric current density vector and component - K external loading parameter, E_o/(B_o U_m) - k thermal conductivity - Nu₁, Nu₂ local Nusselt numbers, h₁,a/k and h₂a/k, respectively - P fluid pressure - Pe Peclet number, PrRe - Pr Prandtl number, C_{p f}/k - q₁,q₂ rates of heat transfer per unit area,^–k(T/Z)Z=–a'^–k(T/Z) Z=a respectively - Re Reynolds number, U_ma/u_f - T,T₀,T₁,T₂ fluid temperature, uniform entrance temperature, uniform but different lower and upper plate temperatures, respectively - T_b,T_m bulk temperature and (T₁+T₂)/2 - U,U_m,u axial, mean and dimensionless velocities, respectively - ¯V velocity vector - X,Z axial and transverse coordinates - x,z dimensionless coordinates - _n,_n even and odd eigenvalues - ,₀,_b dimensionless fluid, entrance and bulk temperatures, respectively - _c,_e,_f characteristic temperature difference (T₂-T_m), and dimensionless fluid temperatures, defined by eq. (10) - _e,_f magnetic permeability and viscosity of fluid - fluid density - electric conductivity - viscous dissipation function - (1-)/2     

Natural convection heat transfer through an enclosed horizontal layer of supercritical carbon dioxide

In natural convection heat transfer through a thin horizontal layer of carbon dioxide, maxima in the equivalent thermal conductivities are obtained in the vicinity of the respective pseudocritical temperatures at pressures of 75.8, 89.6 and 103.4 bar. The maxima are the more pronounced, the closer the critical point is approached.Comparison of experimental results with Nusselt equations shows good agreement except for the immediate vicinity of the pseudocritical temperature.In visual observations a distinct change in flow structure appears in the immediate vicinity of the pseudocritical temperature. A steady state polygon pattern and a boiling-like action could not be observed in this geometry.

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Zusammenfassung Beim Wärmetransport durch freie Konvektion in einer dünnen waagerechten Schicht von Kohlendioxid ergaben sich Maxima der scheinbaren Wärmeleitfähigkeit in der Nähe der pseudokritischen Temperaturen bei Drükken von 75,8, 89,6 und 103,4 bar. Die Maxima sind um so ausgeprägter, je mehr man sich dem kritischen Punkt nähert.Ein Vergleich der Versuchsergebnisse mit Nusseltbeziehungen ergibt gute Übereinstimmung außer in unmittelbarer Umgebung der pseudokritischen Temperatur. Direkte Beobachtungen der Konvektionsmuster zeigen in unmittelbarer Umgebung der pseudokritischen Temperatur eine deutliche Strukturänderung. Ein stationäres Zellmuster und siedeähnliche Vorgänge konnten in dieser Anordnung nicht beobachtet werden.

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Nomenclature A area of the heating or cooling plate - C constant in the correlation - g acceleration of gravity - h heat transfer coefficient - k thermal conductivity of fluid in the gap - k _e equivalent thermal conductivity - m, n exponents of dimensionless numbers - q heat flux - T _C,PC absolute temperature; critical C, pseudocritical PC - Gr Grashof numberg ( _h– _c) ³/ ² - Nu Nusselt numberh/k - Pr Prandtl number/ - thermal diffusivity - coefficient of volume expansion - width of gap - _c,h temperature of cooling (c)-, heating (h)-plate - _m arithmetic mean temperature ( _c+ _h)/2 - kinematic viscosity - _c,h fluid density at the temperature of the cooling (c)- or heating (h)-plate - heat flow rate through the gap     

LDA measurements in low Reynolds number turbulent boundary layer

LDA measurements of the mean velocity in a low Reynolds number turbulent boundary layer allow a direct estimate of the friction velocity U from the value of /y at the wall. The trend of the Reynolds number dependence of / is similar to the direct numerical simulations of Spalart (1988).     

Heat transfer from laminar flow in ducts with elliptic cross-section

Summary The cooling of a hot fluid in laminar Newtonian flow through cooled elliptic tubes has been calculated theoretically. Numerical data have been computed for the two values 1.25 and 4 of the axial ratio of the elliptic cross-section . For =1.25 the influence of non-zero thermal resistance between outmost fluid layer and isothermal surroundings has also been investigated. Special attention has been given to the distribution of heat flux around the perimeter; when increases the flux varies more with the position at the circumference. This positional dependence becomes less pronounced, however, as the (position-independent) thermal resistance of the wall increases.Flattening of the conduit, while maintaining its cross-sectional area constant, improves the cooling. Comparison with rectangular pipes shows that this improvement is not as marked with elliptic as with rectangular pipes.Nomenclature A _k =A _{m, n} coefficients of expansion (6) - a, b half-axes of ellipse, b<a - a _p =a _{r, s} coefficients of representation (V) - D hydraulic diameter, = 4S/P; S = cross-sectional area, P = perimeter - D _e equivalent diameter, according to (13) - n coordinate (outward) normal to the tube wall - T temperature of fluid - T _i temperature of fluid at the inlet - T _s temperature of surroundings - v ₀ mean velocity of fluid - v _z longitudinal velocity of fluid - x, y carthesian coordinates coinciding with axes of ellipse - z coordinate in flow direction - , dimensionless half-axes of ellipse, =a/D and =b/D - _t heat transfer coefficient from fluid at bulk temperature to surroundings; equation (11) - _w heat transfer coefficient at the wall; equation (3) - axial ratio of ellipse, = a/b = / - , , , dimensionless coordinates; =x/D, =y/D, =z/D, =n/D - dimensionless temperature, = (T–T _s)/(T _i–T _s) - ₀ cup-mixing mean value of ; equation (10) - thermal conductivity of fluid - _m,n = _k eigenvalue - c volumetric heat capacity of fluid - _{m, n} = _k = _k eigenfunction; equations (6) and (I) - Nu total Nusselt number, = _t D/ - Nusselt number at large distance from the inlet - Nu _w wall Nusselt number, = _w D/, based on _w - Pé Péclet number, = ₀ Dc/