Flow in porous media III: Deformable media

Stokes flow in a deformable medium is considered in terms of an isotropic, linearly elastic solid matrix. The analysis is restricted to steady forms of the momentum equations and small deformation of the solid phase. Darcy's law can be used to determine the motion of the fluid phase; however, the determination of the Darcy's law permeability tensor represents part of the closure problem in which the position of the fluid-solid interface must be determined.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, 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 a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - E Young's modulus for the-phase, N/m² - e i unit base vectors (i = 1, 2, 3) - g gravity vector, m²/s - H height of elastic, porous bed, m - k unit base vector (=e ₃) - characteristic length scale for the-phase, m - L characteristic length scale for volume-averaged quantities, m - n unit normal vector pointing from the-phase toward the-phase (n = -n ) - p pressure in the-phase, N/m² - P p – g·r, N/m² - r ₀ radius of the averaging volume, m - r position vector, m - t time, s - T total stress tensor in the-phase, N/m² - T ⁰ hydrostatic stress tensor for the-phase, N/m² - u displacement vector for the-phase, m - V averaging volume, m₃ - V volume of the-phase contained within the averaging volume, m³ - v velocity vector for the-phase, m/s Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - shear coefficient of viscosity for the-phase, Nt/m² - first Lamé coefficient for the-phase, N/m² - second Lamé coefficient for the-phase, N/m² - bulk coefficient of viscosity for the-phase, Nt/m² - T –T ⁰ , a deviatoric stress tensor for the-phase, N/m²     

The contribution of internal degrees of freedom to the non-Newtonian rheology of model polymer fluids

We report non-equilibrium molecular dynamics simulations of rigid and non-rigid dumbbell fluids to determine the contribution of internal degrees of freedom to strain-rate-dependent shear viscosity. The model adopted for non-rigid molecules is a modification of the finitely extensible nonlinear elastic (FENE) dumbbell commonly used in kinetic theories of polymer solutions. We consider model polymer melts — that is, fluids composed of rigid dumbbells and of FENE dumbbells. We report the steady-state stress tensor and the transient stress response to an applied Couerte strain field for several strain rates. We find that the rheological properties of the rigid and FENE dumbbells are qualitatively and quantitatively similar. (The only exception to this is the zero strain rate shear viscosity.) Except at high strain rates, the average conformation of the FENE dumbbells in a Couette strain field is found to be very similar to that of FENE dumbbells in the absence of strain. The theological properties of the two dumbbell fluids are compared to those of a corresponding fluid of spheres which is shown to be the most non-Newtonian of the three fluids considered.Symbol Definition b dimensionless time constant relating vibration to other forms of motion - F force on center of mass of dumbbell - F _i force on bead i of dumbbell - F force between center of masses of dumbbells and - F _ij force between beads i and j - h vector connecting bead to center of mass of dumbbell - H dimensionless spring constant for dumbbells, in units of / ² - I moment of inertia of dumbbell - J general current induced by applied field - k _B Boltzmann's constant - L angular momentum - m mass of bead, (= m/2) - M mass of dumbbell, g - N number of dumbbells in simulation cell - P translational momentum of center of mass of dumbbell - P pressure tensor - P _xy xy component of pressure tensor - Q separation of beads in dumbbell - Q _eq equilibrium extension of FENE dumbbell and fixed extension of rigid dumbbell - Q ₀ maximum extension of dumbbell - r _ij vector connecting beads i and j - r position vector of center of mass dumbbell - R vector connecting centers of mass of two dumbbells - t time - t ^* dimensionless time, in units of m/ - T ^* dimensionless temperature, in units of /k - u potential energy - u velocity vector of flow field - u _x x component of velocity vector - V volume of simulation cell - X general applied field - strain rate, s^–1 - ^* dimensionless shear rate, in units of /m ² - general transport property - Lennard-Jones potential well depth - friction factor for Gaussian thermostat - shear viscosity, g/cms - ^* dimensionless shear viscosity, in units of m/ ² - ^* dimensionless number density, in units of ^–3 - Lennard-Jones separation of minimum energy - relaxation time of a fluid - angular velocity of dumbbell - orientation angle of dumbbell      

Unsteady convective diffusion in a rotating parallel plate channel

The paper presents an exact analysis of the dispersion of a passive contaminant in a viscous fluid flowing in a parallel plate channel driven by a uniform pressure gradient. The channel rotates about an axis perpendicular to its walls with a uniform angular velocity resulting in a secondary flow. Using a generalized dispersion model which is valid for all time, we evaluate the longitudinal dispersion coefficientsK _i (i=1, 2, ...) as functions of time. It is shown thatK ₁=0 andK ₃,K ₄, ... decay rapidly in comparison withK ₂. ButK ₂ decreases with increasing (the dimensionless rotation parameter) for values of upto approximately =2.2. ThereafterK ₂ increases with further increase in and its value gets saturated for large values of (say, 500) and does not change any further with increase in . A physical explanation of this anomalous behaviour ofK ₂ is given.

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Instationäre konvektive Diffusion in einem rotierenden Parallelplattenkanal

Zusammenfassung In dieser Untersuchung wird eine exakte Analyse der Ausbreitung eines passiven Kontaminierungsstoffes in einer zähen Flüssigkeit gegeben, die, befördert durch einen gleichförmigen Druckgradienten, in einem Parallelplattenkanal strömt. Der Kanal rotiert mit gleichförmiger Winkelgeschwindigkeit um eine zu seinen Wänden senkrechte Achse, wodurch sich eine Sekundärströmung ausbildet. Unter Verwendung eines generalisierten, für alle Zeiten gültigen Dispersionsmodells werden die longitudinalen DispersionskoeffizientenK _i (i=1, 2, ...) als Funktionen der Zeit ermittelt. Es wird gezeigt, daßK ₁=0 gilt und dieK ₃,K ₄, ... gegenüberK ₂ schnell abnehmen.K ₂ nimmt ab, wenn , der dimensionslose Rotationsparameter, bis etwa zum Wert 2,2 ansteigt. Danach wächstK ₂ mit bis auf einem Endwert an, der etwa ab =500 erreicht wird. Dieses anomale Verhalten vonK ₂ findet eine physikalische Erklärung.

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List of symbols C solute concentration - D molecular diffusivity - K _i longitudinal dispersion coefficients - 2L depth of the channel - P ₀ dimensionless pressure gradient along main flow - Pe Péclet number - q velocity vector - Q _x,Q _y mass flux along the main flow and the secondary flow directions - dimensionless average velocity along the main flow direction - (x, y, z) Cartesian co-ordinates Greek symbols dimensionless rotation parameter - the inclination of side walls withx-axis - kinematic viscosity - fluid density - dimensionless time - angular velocity of the channel - dimensionless distance along the main flow direction - dimensionless distance along the vertical direction - dimensionless solute concentration - integral of the dispersion coefficientK ₂() over a time interval     

Optical co-ordinates in general relativity 2 — the problem of distance

Summary Let denote the congruence of null geodesics associated with a given optical observer inV ₄. We prove that determines a unique collection of vector fieldsM₍₎ ( =1, 2, 3) and ₍₀₎ overV ₄, satisfying a weak version of Killing's conditions.This allows a natural interpretation of these fields as the infinitesimal generators of spatial rotations and temporal translation relative to the given observer. We prove also that the definition of the fieldsM₍₎ and ₍₀₎ is mathematically equivalent to the choice of a distinguished affine parameter f along the curves of, playing the role of a retarded distance from the observer.The relation between f and other possible definitions of distance is discussed.

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Sommario Sia la congruenza di geodetiche nulle associata ad un osservatore ottico assegnato nello spazio-tempoV ₄. Dimostriamo che determina un'unica collezione di campi vettorialiM₍₎ ( =1, 2, 3) e ₍₀₎ inV ₄ che soddisfano una versione in forma debole delle equazioni di Killing. Ciò suggerisce una naturale interpretazione di questi campi come generatori infinitesimi di rotazioni spaziali e traslazioni temporali relative all'osservatore assegnato. Dimostriamo anche che la definizione dei campiM₍₎, ₍₀₎ è matematicamente equivalente alla scelta di un parametro affine privilegiato f lungo le curve di, che gioca il ruolo di distanza ritardata dall'osservatore. Successivamente si esaminano i legami tra f ed altre possibili definizioni di distanza in grande.

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Work performed in the sphere of activity of: Gruppo Nazionale per la Fisica Matematica del CNR.     

Theoretical derivation of tangential velocity profiles in a flat vortex chamber-influence of turbulence and wall friction

Summary As part of a study on the hydrodynamics of a cyclone separator, a theoretical investigation of the flow pattern in a flat box cyclone (vortex chamber) has been carried out. Expressions have been derived for the tangential velocity profile as influenced by internal friction (eddy viscosity) and wall friction. The most important parameter controlling the tangential velocity profile is = –u ₀ R/(v+ ), where u ₀ is the radial velocity at the outer radius R of the cyclone, the kinematic liquid viscosity and is the kinematic eddy viscosity. For values of greater than about 10 the tangential velocity profile is nearly hyperbolic, for smaller than 1 the tangential velocity even decreases towards the centre. It is shown how and also the wall friction coefficient may be obtained from experimental velocity profiles with the aid of suitable graphs. Because of the close relation between eddy viscosity and eddy diffusion, measurements of velocity profiles in flat box cyclones will also provide information on the eddy motion of particles in a cyclone, a motion reducing its separation efficiency.List of symbols A cross-sectional area of cyclone inlet - h height of cyclone - p static pressure in cyclone - p static pressure difference in cyclone between two points on different radius - r radius in cyclone - r ₁ radius of cyclone outlet - R radius of cyclone circumference - u radial velocity in cyclone - u ₀ radial velocity at circumference of flat box cyclone - v tangential velocity - v ₀ tangential velocity at circumference of flat box cyclone - w axial velocity - z axial co-ordinate in cyclone - friction coefficient in flat box cyclone (for definition see § 5) - ₁ value of friction coefficient for ₁<< ₂ - ₂ value of friction coefficient for ₂<<1 - = - ₁ value of for ₁<< ₂ - ₂ value of for ₂<<1 - thickness of laminar boundary layer - =/h - turbulent kinematic viscosity - ratio of z to h - k ratio of height of cyclone to radius R of cyclone - parameter describing velocity profile in cyclone =–u ₀ R/(+) - kinematic viscosity of fluid - density of fluid - ratio of r to R - ₁ value of at outlet of cyclone - ₂ value of at inner radius of cyclone inlet - _w shear stress at cyclone wall - angular momentum in cyclone/angular momentum in cyclone inlet - ₁ value of at = ₁ - ₂ value of at = ₂     

The rheology of polymeric liquid crystals

The molecular theory of Doi has been used as a framework to characterize the rheological behavior of polymeric liquid crystals at the low deformation rates for which it was derived, and an appropriate extension for high deformation rates is presented. The essential physics behind the Doi formulation has, however, been retained in its entirety. The resulting four-parameter equation enables prediction of the shearing behavior at low and high deformation rates, of the stress in extensional flows, of the isotropic-anisotropic phase transition and of the molecular orientation. Extensional data over nearly three decades of elongation rate (10^–2–10¹) and shearing data over six decades of shear rate (10^–2–10⁴) have been correlated using this analysis. Experimental data are presented for both homogeneous and inhomogeneous shearing stress fields. For the latter, a 20-fold range of capillary tube diameters has been employed and no effects of system geometry or the inhomogeneity of the flow-field are observed. Such an independence of the rheological properties from these effects does not occur for low molecular weight liquid crystals and this is, perhaps, the first time this has been reported for polymeric lyotropic liquid crystals; the physical basis for this major difference is discussed briefly. A Semi-empirical constant in eq. (18), N/m² - c rod concentration, rods/m³ - c ^* critical rod concentration at which the isotropic phase becomes unstable, rods/m³ - C interaction potential in the Doi theory defined in eq. (3) - d rod diameter, m - D semi-empirical constant in eq. (19), s^–1 - D _r lumped rotational diffusivity defined in eq. (4), s^–1 - rotational diffusivity of rods in a concentrated (liquid crystalline) system, s^–1 - D _ro rotational diffusivity of a dilute solution of rods, s^–1 - f distribution function defining rod orientation - F tensorial term in the Doi theory defined in eq. (7) (or eq. (19)), s^–1 - G tensorial term in the Doi theory defined in eq. (8) - K _B Boltzmann constant, 1.38 × 10^–23 J/K-molecule - L rod length, m - S scalar order parameter - S tensor order parameter defined in eq. (5) - t time, s - T absolute temperature, K - u unit vector describing the orientation of an individual rod - rate of change ofu due to macroscopic flow, s^–1 - v fluid velocity vector, m/s - v velocity gradient tensor defined in eq. (9), s^–1 - V mean field (aligning) potential defined in eq. (2) - x coordinate direction, m - Kronecker delta (= 0 if = 1 if = ) - _r ratio of viscosity of suspension to that of the solvent at the same shear stress - _s solvent viscosity, Pa · s - ^* viscosity at the critical concentrationc ^*, Pa · s - v ₁, v₂ numerical factors in eqs. (3) and (4), respectively - deviatoric stress tensor, N/m² - volume fraction of rods - ⁰ constant in eq. (16) - ^* volume fraction of rods at the critical concentrationc ^* - average over the distribution functionf(u, t) (= d ²u f(u, t)) - gradient operator - d ²u integral over the surface of the sphere (|u| = 1)     

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

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

Darcyan flow with relaxation effect

The equation for the flow of a liquid through a porous medium is either elliptic or parabolic which implies that a disturbance in pressure or head is transmitted with infinite velocity. This is unsatisfactory from a physical viewpoint, although not necessarily from a practical one. If Darcy's law is completed with an inertial term the flow equation becomes a strongly damped wave equation. The proposed additional term can be identified from experiments with confined flow and free surface flow when the pressure or head varies harmonically with time.Symbols c ₀ characteristic speed - c ₁ characteristic speed - c _w characteristic speed - C auxiliary parameter - d diameter of a bead - E modulus of elasticity - F auxiliary function - h piezometric height - H height of groundwater table - I ₀(x) modified Bessel function - I ₁(x) modified Bessel function - k permeability - K hydraulic conductivity - L tube length or channel length - L ₀ reference length - m auxiliary parameter - n porosity - p pressure - Q mass flux - r=(x, y, z) Cartesian coordinates - dimensionless Cartesian coordinates - r(p) tube radius - r ₀ tube radius - R ₀ tube radius - r _p equivalent pore radius - t time - velocity vector - V volume - coefficient of compressibility - coefficient of compressibility - coefficient of compressibility - expansibility - relaxation time - dynamic viscosity - Poisson's ratio - density - dimensionless time - angular frequency     

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.     

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

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

Free convection from a vertical plate with concentrated and distributed thermal sources

An analysis is developed for the laminar free convection from a vertical plate with uniformly distributed wall heat flux and a concentrated line thermal source embedded at the leading edge. We introduce a parameter=(1 +Q _L/Q_w)^–1=(1 + Ra_L/Ra_w)^–1 to describe the relative strength of the two thermal sources; and propose a unified buoyancy parameter=( Ra_L+ Ra_w)^1/5 with=1/(1 +Pr ^–1) to properly scale the dependent and independent variables. The variables are so defined that the resulting nonsimilar boundary-layer equations can describe exactly the buoyancy-induced flow from the dual sources with any relative strength to fluids of any Prandtl number from very small values to infinity. These nonsimilar equations are readily reducible to the self-similar equations of an adiabatic wall plume for=0, and to those of free convection from uniform flux plate for=1. Rigorous finite-difference solutions for fluids of Pr from 0.001 to are obtained over the entire range of from 0 to 1. The effects of both relative source strength and Prandtl number on the velocity profiles, temperature profiles, and the variations of wall temperature, are clearly illustrated.

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Freie Konvektion an einer vertikalen Platte mit einer konzentrierten und einer gleichmäßig verteilten Wärmequelle

Zusammenfassung Für die freie Konvektion an einer vertikalen Platte mit einer gleichmäßig verteilten Wandwärmestromdichte und einer in der Vorderkante eingebetteten linienförmigen Wärmequelle wird eine Berechnungsmethode entwickelt. Zur Beschreibung der relativen Stärke der beiden Wärmequellen führen wir einen Parameter=(1 + Q_L/Q_w)^–1=(1 + Ra_L/Ra_w)^–1 ein und schlagen einen vereinheitlichten Auftriebsparameter=( Ra _L+ Ra _w)^1/5 mit=1/(1 +Pr ^–1 für die Skalierung der abhängigen und unabhängigen Variablen vor. Die Variablen werden so definiert, daß mit den sich ergebenden unabhängigen Grenzschichtgleichungen die von den beiden Wärmequellen beliebiger Stärke verursachte Auftriebsströmung von Fluiden beliebiger Prandtl-Zahl genau beschrieben werden kann. Diese unabhängigen Gleichungen können ohne weiteres auf die selbstähnlichen Gleichungen für den Fall einer lokalen Wärmezufuhr an einer sonst adiabatischen Wand für=0 und jenen der freien konvektion an einer Platte mit einheitlichem Wärmestrom für=1 zurückgeführt werden. Für Fluide mit der Prandtl-Zahl zwischen 0,001 und Unendlich werden nach der strengen finite Differenzen-Methode Lösungen im Bereich von zwischen 0 und 1 erhalten. Der jeweilige Einfluß der relativen Quellenstärke und der Prandtl-Zahl auf die Geschwindigkeits- und Temperaturprofile sowie die Veränderung der Wandtemperatur werden deutlich dargestellt.

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Nomenclature C _f friction coefficient - C _p specific heat - f reduced stream function - g gravitational acceleration - k thermal conductivity - L width of the plate - Nu local Nusselt number - Pr Prandtl number - q _w wall heat flux - Q _L heat generated by the line source - Q _w heat released by the uniform-flux wall from 0 tox, q _w Lx - Ra _L local Rayleigh number, g T _L ^* x ³/( ) - Ra _w local Rayleigh number,g T _w ^* w ³/( ) - T fluid temperature - T temperature of ambient fluid - T _L ^* characteristic temperature of the line source,Q _{L/(C p} L) - T _w ^* characteristic temperature of the uniform flux wall, =q _w x/k=Q _w /(C _p L) - u velocity component in then-direction - U₀ dimensionless velocity,u/(/x) Ra _L ^2/5 - U ₁ dimensionless velocity,u/(/x) Ra _w ^2/5 - velocity component in they-direction - x coordinate parallel to the plate - y coordinate normal to the plate - thermal diffusivity - thermal expansion coefficient - pseudo-similarity variable,(y/x) - dimensionless temperature, (T–T )/(T _L ^* +T _w ^* ) - ₀ dimensionless temperature, (Ra_l)^1/5 (T–T )/T _L ^* - ₁ dimensionless temperature, (Ra_w)Ra_w)^1/5 (T–T )/T _w ^* - (Ra _L+Ra_w)^1/5 - kinematic viscosity - (1 +Ra _L/Ra_w)^–1=(1 +T _L ^* /T _w ^* )^–1=(1 + Q_L/Q_w)^–1 - density - Pr/(1 +Pr) - _w wall shear stress - stream function     

Flow through a helically coiled annulus

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

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     

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     

Interaction of a vortex couple with a free surface

The characteristics of three-dimensional flow structures (scars and striations) resulting from the interaction between a heterostrophic vortex pair in vertical ascent and a clean free surface are described. The flow features at the scar-striation interface (a constellation of whirls or coherent vortical structures) are investigated through the use of flow visualization, a motion analysis system, and the vortex-element method. The results suggest that the striations are a consequence of the short wavelength instability of the vortex pair and the helical instability of the tightly spiralled regions of vorticity. The whirls result from the interaction of striations with the surface vorticity. The whirl-merging is responsible for the reverse energy cascade leading to the formation and longevity of larger vortical structures amidst a rapidly decaying turbulent field.List of symbols A _c Area of a vortex core (Fig. 6b) - AR Aspect ratio of the delta wing model - B base width of delta wing - b ₀ initial separation of the vortex couple - d ₀ depth at which the vortex pair is generated - c average whirl spacing in the x-direction - E energy density - Fr Froude number ( ) - g gravitational acceleration - L length of the scar band - L _ko length of the Kelvin oval - N _w number of whirls in each scar band - P _c Perimeter of a vortex core - q surface velocity vector - r _c core size of the whirl ( = 2A _c/P _c) - Re Reynolds number ( = ) - Rnd a random number - s inboard edge of the scar front (Fig. 6 a) - t time - u velocity in the x-direction - velocity in the y-direction - V _b velocity imposed on a scar by the vortex couple (Fig. 6 a) - V ₀ initial mutual-induction velocity of the vortex couple (=₀/2b ₀) - V _t tangential velocity at the edge of the whirl core - w width of the scar front (Fig. 6 a) - z complex variable - z _k position of the whirl center - half included angle of V-shaped scar band - wave number - _m initial mean circulation of the whirls - ₀ initial circulation of the vortex pair - _w circulation of a whirl - _min minimum survival strength of a whirl - t time step - gDz increment of z - gD change in vorticity - cut-off distance in velocity calculations - critical merging distance - curvature of the surface - wavelength - kinematic viscosity - angular velocity of a whirl core     

Impacts of local dispersion and first-order decay on solute transport in randomly heterogeneous porous media

Stochastic subsurface transport theories either disregard local dispersion or take it to be constant. We offer an alternative Eulerian-Lagrangian formalism to account for both local dispersion and first-order mass removal (due to radioactive decay or biodegradation). It rests on a decomposition of the velocityv into a field-scale componentv , which is defined on the scale of measurement support, and a zero mean sub-field-scale componentv _s , which fluctuates randomly on scales smaller than. Without loss of generality, we work formally with unconditional statistics ofv _s and conditional statistics ofv . We then require that, within this (or other selected) working framework,v _s andv be mutually uncorrelated. This holds whenever the correlation scale ofv is large in comparison to that ofv _s . The formalism leads to an integro-differential equation for the conditional mean total concentration c which includes two dispersion terms, one field-scale and one sub-field-scale. It also leads to explicit expressions for conditional second moments of concentration cc. We solve the former, and evaluate the latter, for mildly fluctuatingv by means of an analytical-numerical method developed earlier by Zhang and Neuman. We present results in two-dimensional flow fields of unconditional (prior) mean uniformv . These show that the relative effect of local dispersion on first and second moments of concentration dies out locally as the corresponding dispersion tensor tends to zero. The effect also diminishes with time and source size. Our results thus do not support claims in the literature that local dispersion must always be accounted for, no matter how small it is. First-order decay reduces dispersion. This effect increases with time. However, these concentration moments c and cc of total concentrationc, which are associated with the scale below, cannot be used to estimate the field-scale concentrationc directly. To do so, a spatial average over the field measurement scale is needed. Nevertheless, our numerical results show that differences between the ensemble moments ofc and those ofc are negligible, especially for nonpoint sources, because the ensemble moments ofc are already smooth enough.     

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).     

Effects of MHD free convection and mass transfer on the flow past an oscillating infinite coaxial vertical circular cylinder

The effects of MHD free convection and mass transfer are taken into account on the flow past oscillating infinite coaxial vertical circular cylinder. The analytical expressions for velocity, temperature and concentration of the fluid are obtained by using perturbation technique.

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Einwirkungen von freier MHD-Konvektion und Stoffübertragung auf eine Strömung nach einem schwingenden unendlichen koaxialen vertikalen Zylinder

Zusammenfassung Die Einwirkungen der freien MHD-Konvektion und Stoffübertragung auf eine Strömung nach einem schwingenden, unendlichen, koaxialen, vertikalen Zylinder wurden untersucht. Die analytischen Ausdrücke der Geschwindigkeit, Temperatur und Fluidkonzentration sind durch die Perturbationstechnik erhalten worden.

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Nomenclature C _p Specific heat at constant temperature - C the species concentration near the circular cylinder - C _w the species concentration of the circular cylinder - C the species concentration of the fluid at infinite - * dimensionless species concentration - D chemical molecular diffusivity - g acceleration due to gravity - Gr Grashof number - Gm modified Grashof number - K thermal conductivity - Pr Prandtl number - r _a ,r _b radius of inner and outer cylinder - a, b dimensionless inner and outer radius - r,r coordinate and dimensionless coordinate normal to the circular cylinder - Sc Schmidt number - t time - t dimensionless time - T temperature of the fluid near the circular cylinder - T _w temperature of the circular cylinder - T temperature of the fluid at infinite - u velocity of the fluid - u dimensionless velocity of the fluid - U ₀ reference velocity - z,z coordinate and dimensionless coordinate along the circular cylinder - coefficient of volume expansion - * coefficient of thermal expansion with concentration - dimensionless temperature - H ₀ magnetic field intensity - coefficient of viscosity - _e permeability (magnetic) - kinematic viscosity - electric conductivity - density - M Hartmann number - dimensionless skin-friction - frequency - dimensionless frequency     

Laminar flow in the entrance region of a porous-wall channel

Summary The effect of fluid injection at the walls of a two-dimensional channel on the development of flow in the entrance region of the channel has been investigated. The integral forms of the boundary layer equations for flow in the channel were set up for an injection velocity uniformly distributed along the channel walls.With an assumed polynomial of the n-th degree for the one-parameter velocity profile a solution of the above boundary layer equations was obtained by an iteration method. A closed form solution was also obtained for the case when a similar velocity profile was assumed. The agreement between the entrance region velocity profiles of the present analysis for an impermeable-walled channel and of Schlichting¹) and Bodoia and Osterle²) is found to be very good.The results of the analysis show that fluid injection at the channel walls increases the rate of the growth of the boundary layer thickness, and hence reduces considerably the entrance length required for a fully developed flow.Nomenclature h half channel thickness - L entrance length with wall-injection - L ₀ entrance length without wall-injection - p static pressure - p=p/U ₀ ² dimensionless pressure - Re=U ₀ h/ Reynolds number at inlet cross-section - u velocity in the x direction at any point in the channel - =u/U ₀ dimensionless velocity in the x direction at any point in the channel - U _av average velocity at a channel cross-section - U _c center line velocity - U ₀ inlet cross-section velocity - _c =U _c /U ₀ dimensionless center line velocity - v velocity in the y direction at any point in the channel - v ₀ constant injection velocity of fluid at the wall - v=v/v ₀ dimensionless velocity in the y direction at any point in the channel - x distance along the channel wall measured from the inlet cross-section - x=x/hRe dimensionless distance in the x direction - y distance perpendicular to the channel wall - y=y/h dimensionless distance in the y direction - thickness of the boundary layer - =/h dimensionless boundary layer thickness - =/ dimensionless distance within the boundary layer region - =v ₀ h/ injection parameter or injection Reynolds number - kinematic viscosity - 1+ie - mass density of the fluid - parameter defined in (14)     

Analyse und Berechnung der Strömungsvorgänge im zylindrischen Scherspalt von Plastifizieraggregaten

Zusammenfassung Dieser Aufsatz zeigt eine Möglichkeit auf, zylindrische Scherteile einer Plastifiziereinheit, auf der strukturviskose Materialien verarbeitet werden, approximativ zu berechnen. Es ist möglich, den Volumenstrom und Druckabfall, die mittlere Schergeschwindigkeit, Scherdeformation und Schubspannung im Scherspalt zu approximieren. Durch diese Gleichungen wird eine Abschätzung der Verteil- und Zerteilvorgänge im Scherelement möglich.

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A method is described for approximating the flow in cylindrical shearing gaps of plasticating extruder, which is applicable to shear thinning materials. It is possible to calculate the through-put and pressure drop as well as the shear rate, strain and shear stress in the gap. With these equations the distribution and separation process in shearing gaps can be evaluated.

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D Zylinderdurchmesser - d ₁ Schnecken-Kerndurchmesser der Meteringzone - d _s Durchmesser des zylindrischen Scherteils - K Konstante im Potenzfließgesetz - K _0T Koeffizient des Potenzfließgesetzes - L ₁ Länge der Anlaufschräge - L _s Länge des zylindrischen Scherteils - n Fließindex - n ₀ Drehzahl - p Druckabfall über der Scherteillänge - s Scherspalthöhe - T _M Massetemperatur - ₀ Umfangsgeschwindigkeit - _0x Geschwindigkeitskomponente inx-Richtung - _x, _z Geschwindigkeit inx- bzw.z-Richtung als Funktion der Koordinatey - Volumenstrom - x, z Ortskoordinaten - Exponent des Potenzfließgesetzes - Schergeschwindigkeit - mittlere Schergeschwindigkeit - Viskosität - dimensionslose Höhe - Dichte der Schmelze - Schubspannung - _yx, _yz Schubspannungskomponenten - _xx, _zz Normalspannungskomponenten - _ps dimensionsloser Druckgradient - dimensionsloser Volumenstrom - _x, _z dimensionslose Geschwindigkeit inx- bzw.z-Richtung