Transport in ordered and disordered porous media II: Generalized volume averaging

In this paper we develop the averaged form of the Stokes equations in terms of weighting functions. The analysis clearly indicates at what point one must choose a media-specific weighting function in order to achieve spatially smoothed transport equations. The form of the weighting function that produces the cellular average is derived, and some important geometrical theorems are presented.Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - A _p surface area of a particle, m² - d _p 6V _p/A_p, effective particle diameter, m - g gravity vector, m/s² - I unit tensor - 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 general characteristic length for volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - l characteristic length (pore scale) for the-phase - l _i i=1, 2, 3 lattice vectors, m - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - 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 - m _D special convolution product weighting function for disordered media - m _M master convolution product weighting function for ordered and disordered media - 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 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, m - r position vector, m - r position vector locating points in the-phase, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - V velocity vector in the-phase, m/s - vm superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - V volume of the-phase contained in the averaging volume, m³ - V _p volume of a particle, m³ - v traditional superficial volume averaged velocity, m/s - v v p _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 - V /V, volume fraction of the-phase     

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].     

Shear viscosity behavior of highly concentrated suspensions at low and high shear-rates

A method is proposed for determining the shear viscosity behavior of highly concentrated suspensions at low and high shear-rates through the use of a formulation that is a function of three parameters signifying the effects of particle size distribution. These parameters are the intrinsic viscosity [], a parametern that reflects the level of particle association at the initiation of motion and the maximum packing concentration _m. The formulation reduces to the modified Eilers equation withn = 2 for high shear rates. An analytical method was used for the calculation of maximum packing concentration which was subsequently correlated with the experimental values to account for the surface induced interaction of particles with the fluid. The calculated values of viscosities at low and high shear-rates were found to be in good agreement with various experimental data reported in literature. A brief discussion is also offered on the reliability of the methods of measuring the maximum packing concentration. _r = /₀ relative viscosity of the suspension - volumetric concentration of solids - k _n coefficient which characterizes a specific effect of particle interactions - _m maximum packing concentration - _r,0 relative viscosity at low shear-rates - [] intrinsic viscosity - n, n parameter that reflects the level of particle interactions at low and high shear-rates, respectively - _r, relative viscosity at high shear-rates - (_m)_s, (_m)_i, (_m)_l packing factors for small, intermediate and large diameter classes - v _s, v_i, v_l volume fractions of small, intermediate and large diameter classes, respectively - _si, _sl coefficient to be used in relating a smaller to an intermediate and larger particle group, respectively - _is, _il coefficient to be used in relating an intermediate to a smaller and larger particle group, respectively - _ls, _li coefficient to be used in relating a larger to a smaller and intermediate particle group, respectively - _m0 maximum packing concentration for binary mixtures - _m,e measured maximum packing concentration - _m,c calculated maximum packing concentration     

On an inversion theorem for conical regions in elasticity theory

Summary Two-dimensional stress singularities in wedges have already drawn attention since a long time. An inverse square-root stress singularity (in a 360° wedge) plays an important role in fracture mechanics.Recently some similar three-dimensional singularities in conical regions have been investigated, from which one may be also important in fracture mechanics.Spherical coordinates are r, , . The conical region occupied by the elastic homogeneous body (and possible anisotropic) has its vertex at r=0. The mantle of the cone is described by an arbitrary function f(, )=0. The displacement components be u. For special values of (eigenvalues) there exist states of displacements (eigenstates) % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakabbaaa6daaahjxzL5gapeqa% aiaadwhadaWgaaWcbaGaeqOVdGhabeaakiabg2da9iaadkhadaahaa% WcbeqaaiabeU7aSbaakiaadAgadaWgaaWcbaGaeqOVdGhabeaakiaa% cIcacqaH7oaBcaGGSaGaeqiUdeNaaiilaiabfA6agjaacMcaaaa!582B!\[u_\xi = r^\lambda f_\xi (\lambda ,\theta ,\Phi )\],which may satisfy rather arbitrary homogeneous boundary conditions along the generators.The paper brings a theorem which expresses that if is an eigenvalue, then also-1- is an eigenvalue. Though the theorem is related to a known theorem in Potential Theory (Kelvin's theorem), the proof has to be given along quite another line.

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Zusammenfassung Zwei-dimensionale Spannungssingularitäten in keilförmigen Gebieten sind schon längere Zeit untersucht worden und neuerdings auch ähnliche drei-dimensionale Singularitäten in konischen Gebieten.Kugelkoordinaten sind r, , . Das konische Gebiet hat seine Spitze in r=0. Der Mantel des Kegels lässt sich beschreiben mittels einer willkürlichen Funktion f(, )=0. Die Verschiebungskomponenten seien u. Für spezielle Werte von (Eigenwerte) bestehen Verschiebunszustände % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakabbaaa6daaahjxzL5gapeqa% aiaadwhadaWgaaWcbaGaeqOVdGhabeaakiabg2da9iaadkhadaahaa% WcbeqaaiabeU7aSbaakiaadAgadaWgaaWcbaGaeqOVdGhabeaakiaa% cIcacqaH7oaBcaGGSaGaeqiUdeNaaiilaiabfA6agjaacMcaaaa!582B!\[u_\xi = r^\lambda f_\xi (\lambda ,\theta ,\Phi )\],welche homogene Randwerte der Beschreibenden des Kegels entlang genügen.Das Bericht bringt ein Theorem, welches aussagt, das und =–1– beide Eigenwerte sind.

     

Flow in porous media II: The governing equations for immiscible,two-phase flow

The Stokes flow of two immiscible fluids through a rigid porous medium is analyzed using the method of volume averaging. The volume-averaged momentum equations, in terms of averaged quantities and spatial deviations, are identical in form to that obtained for single phase flow; however, the solution of the closure problem gives rise to additional terms not found in the traditional treatment of two-phase flow. Qualitative arguments suggest that the nontraditional terms may be important when / is of order one, and order of magnitude analysis indicates that they may be significant in terms of the motion of a fluid at very low volume fractions. The theory contains features that could give rise to hysteresis effects, but in the present form it is restricted to static contact line phenomena.Roman Letters (, = , , and ) 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 - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - g gravity vector, m²/s - H mean curvature of the- interface, m^–1 - H area average of the mean curvature, m^–1 - H – H , deviation of the mean curvature, m^–1 - I unit tensor - K Darcy's law permeability tensor, m² - K permeability tensor for the-phase, m² - K viscous drag tensor for the-phase equation of motion - K viscous drag tensor for the-phase equation of motion - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - n unit normal vector pointing from the-phase toward the-phase (n = –n ) - p _c p – P , capillary pressure, N/m² - 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, m - t time, s - 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 - surface tension of the- interface, N/m - viscous stress tensor for the-phase, N/m² - / kinematic viscosity, m²/s     

Resonant generation of a solitary wave in a thermocline

The resonant generation of a second-mode internal solitary wave, resulting from a ship internal waves system damping in a thermocline, is studied experimentally. The source of the stationary internal waves was provided by an oblong ellipsoid of revolution towed horizontally and uniformly at the depth of the thermocline center. The ranges of the Reynolds and Froude numbers were 500Re=Ul/v 15000 and 0.3Fi=U/N _max D1.0, respectively. When the body's speed and the linear long-wave second-mode phase speed were equal, an internal solitary wave of the bulge type was observed. The shape of the wave satisfied the Korteweg-de Vries equation. The Urcell parameter was equal to 10.2.List of Symbols L, B, H towing tank length, breadth and height respectively - z vertical coordinate - D characteristic vertical dimension of the body - a minor semiaxis of an ellipsoid - b major semiaxis of an ellipsoid (maximum ellipsoid diameter D=2a) - l length of the body ( =2b) - U velocity of the body - t temperature - g acceleration due to gravity - i fresh water density at ith level - _4° fresh water density for temperature t=4°C - o water density at the center of the thermocline - _i density variation due to the temperature variation at the ith horizon - N Brunt-Väisälä frequency - N _max maximum value of Brunt-Väisälä frequency - Re Reynolds number - Fi internal Froude number - f _n eigenfunction of the boundary-value problem for the nth mode - _n nth mode frequency - k _{n n}th mode horizontal wavenumber - C _n limiting phase speed of a linear nth mode interval wave (= _n/k_n;k_n 0) - Ur Urcell parameter - v fresh water kinematic viscosity - conventional density - half-length of a solitary wave - ₀ solitary wave height - time This work was partially supported by the INTAS (grant no. 94-4057) and by the Russian Foundation of Basic Research under grant no. 94-05-17004-a.A version of this paper was presented at the Second International Conference on Experimental Fluid Mechanics, Torino, Italy, 4–8 July, 1994.     

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     

Fractal-time stochastic processes and dynamic functions of polymeric systems

Dynamic material functions of polymeric systems are calculated via a defect-diffusion model. The random motion of defects is modelled by a fractaltime stochastic process. It is shown that the dynamic functions of polymeric solutions can be approximated by the defect-diffusion process of the mixed type. The relaxation modulus of Kohlrausch type is obtained for a fractal-time defect-diffusion process, and it is shown that this modulus is capable of portraying the dynamic behavior of typical viscoelastic solutions.The Fourier transforms of the Kohlrausch function are calculated to obtain and. A three-parameter model for and is compared with the previous calculations. Experimental measurements for five polymer solutions are compared with model predictions. D rate of deformation tensor - G(t) mechanical relaxation modulus - H relaxation spectrum - I(t) flux of defects - P _n (s) probability of finding a walker ats aftern-steps - P generating function ofP _n (s) - s(t) fraction of surviving defects - , () gamma function (incomplete) - ₀ zero shear viscosity - ^* () complex viscosity - frequency - t _n n-th moment - F[] Fourier transform - f ^* (u) Laplace transform off(t) - , components of ^* - G _f, _f ^* fractional model - G ₃, ₃ ^* three parameter model - complex conjugate ofz - material time derivative ofD     

Approximate solutions for drag coefficient of bubbles moving in shear-thinning elastic fluids

The drag coefficient for bubbles with mobile or immobile interface rising in shear-thinning elastic fluids described by an Ellis or a Carreau model is discussed. Approximate solutions based on linearization of the equations of motion are presented for the highly elastic region of flow. These solutions are in reasonably good agreement with the theoretical predictions based on variational principles and with published experimental data. C _D Drag coefficient - E ^* Differential operator [E ^{* 2} = ²/² + (sin/ ²)/(1/sin /)] - El Ellis number - F _D Drag force - K Consistency index in the power-law model for non-Newtonian fluid - n Flow behaviour index in the Carreau and power-law models - P Dimensionless pressure [=(p – p ₀)/₀ (U /R)] - p Pressure - R Bubble radius - Re ₀ Reynolds number [= 2R U /₀] - Re Reynolds number defined for the power-law fluid [= (2R) ⁿ U ^2–n /K] - r Spherical coordinate - t Time - U Terminal velocity of a bubble - u Velocity - Wi Weissenberg number - Ellis model parameter - Rate of deformation - Apparent viscosity - ₀ Zero shear rate viscosity - Infinite shear rate viscosity - Spherical coordinate - Parameter in the Carreau model - ^* Dimensionless time [=/(U /R)] - Dimensionless length [=r/R] - Second invariant of rate of deformation tensors - ^* Dimensionless second invariant of rate of deformation tensors [=/(U /R)²] - Second invariant of stress tensors - ^* Dimensionless second invariant of second invariant of stress tensor [= / ₀ ² (U /R)²] - Fluid density - Shear stress - ^* Dimensionless shear stress [=/ ₀ (U /R)] - _1/2 Ellis model parameter - ₁ ^2/* Dimensionless Ellis model parameter [= _1/2/ ₀(U /R)] - Stream function - ^* Dimensionless stream function [=/U R ²]     

An extension of Reiner's “Deborah Number” concept to a Wide field of rheological investigations

Summary Reiner defined a numeric, which he called theDeborah Number to represent the ratio of a relaxation time to a natural (observation) time. This implies aMaxwell model but is readily extended to complete relaxation spectra. Similar Numbers are proposed for retardation times and also for some conditions of coagulation thixotropy and for data from certain psychophysical experiments.     

Development and evaluation of a new turbulence generator for atomization research

Planar Mie scattering visualizations in compressible mixing layers are used to compute the probability density function of a passive scalar. Mixing layer flows with relative Mach numbers of 0.63 and 1.49 are studied. Ethanol condensation is used to generate both scalar transport seeding and product formation seeding. All PDFs exhibit a marching behavior. The condensation process in the product formation seeding is modeled to provide an estimate of the error embedded in the scalar transport PDFs. The mixing efficiency is found to be 0.56 in the product formation experiments, and the overprediction of mixing efficiency by the scalar PDFs is estimated to be 11% based on results from the ethanol condensation model.List of Symbols _291-01 Damköhler number based on - J droplet nucleation rate - k Boltzmann constant - m _c molecular mass of ethanol - M _r relative Mach number, M _r = 2U/(a₁ + a₂) - N ^* number of nucleated droplets - p(,) probability density function - P _d internal droplet pressure - P _m total mixed fluid probability - P _sat ethanol saturation partial pressure - P _v ethanol vapor partial pressure - r freestream velocity ratio, r=U ₂/U₁; droplet radius - r ^* critical nucleation radius - R gas constant for air - _291-2 Reynolds number based on - s freestream density ratio, s = ₂/₁ - T local static temperature - U ₁ high speed freestream velocity - U ₂ low speed freestream velocity - U _c large structure convection velocity, - U freestream velocity difference, U=U ₁–U₂ - x streamwise coordinate - y transverse coordinate - mixing layer thickness - _i incompressible mixing layer thickness - mixture fraction - similarity variable, = (y–y ₀)/ - _c condensed phase ethanol density - droplet surface tension     

The virtual origin of riblets in an open channel flow

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

Response of an elastic Bingham fluid to oscillatory shear

The response of an elastic Bingham fluid to oscillatory strain has been modeled and compared with experiments on an oil-in-water emulsion. The newly developed model includes elastic solid deformation below the yield stress (or strain), and Newtonian flow above the yield stress. In sinusoidal oscillatory deformations at low strain amplitudes the stress response is sinusoidal and in phase with the strain. At large strain amplitudes, above the yield stress, the stress response is non-linear and is out of phase with strain because of the storage and release of elastic recoverable strain. In oscillatory deformation between parallel disks the non-uniform strain in the radial direction causes the location of the yield surface to move in-and-out during each oscillation. The radial location of the yield surface is calculated and the resulting torque on the stationary disk is determined. Torque waveforms are calculated for various strains and frequencies and compared to experiments on a model oil-in-water emulsion. Model parameters are evaluated independently: the elastic modulus of the emulsion is determined from data at low strains, the yield strain is determined from the phase shift between torque and strain, and the Bingham viscosity is determined from the frequency dependence of the torque at high strains. Using these parameters the torque waveforms are predicted quantitatively for all strains and frequencies. In accord with the model predictions the phase shift is found to depend on strain but to be independent of frequency.Notation A plate strain amplitude (parallel plates) - A _R plate strain amplitude at disk edge (parallel disks) - G elastic modulus - m torque (parallel disks) - M normalized torque (parallel disks) = 2m/R ³₀ - N ratio of viscous to elastic stresses (parallel plates) =µ A/ ₀ ratio of viscous to elastic stresses (parallel disks) =µ A _R/₀ - r normalized radial position (parallel disks) =r/R - r radial position (parallel disks) - R disk radius (parallel disks) - t normalized time = t — /2 - t time - _E elastic strain - _P plate strain (displacement of top plate or disk divided by distance between plates or disks) - _PR plate strain at disk edge (parallel disks) - ₀ yield strain - _E normalized elastic strain = _E/₀ - _P normalized plate strain = _P/₀ - _PR normalized plate strain at disk edge (parallel disks) = _PR/₀ - ₀ normalized plate strain amplitude (parallel plates) =A/ ₀ — normalized plate strain amplitude at disk edge (parallel disks) =A _R/₀ - phase shift between _P andT (parallel plates) — phase shift between _PR andM (parallel disks) - µ Bingham viscosity - stress - ₀ yield stress - T normalized stress =/ ₀ - frequency     

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

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

Flow of second-order fluids over an enclosed rotating disc

Summary This paper is devoted to a study of the flow of a second-order fluid (flowing with a small mass rate of symmetrical radial outflow m, taken negative for a net radial inflow) over a finite rotating disc enclosed within a coaxial cylinderical casing. The effects of the second-order terms are observed to depend upon two dimensionless parameters ₁ and ₂. Maximum values ₁ and ₂ of the dimensionless radial distances at which there is no recirculation, for the cases of net radial outflow (m>0) and net radial inflow (m<0) respectively, decrease with an increase in the second-order effects [represented by T(=₁+₂)]. The velocities at ₁ and ₂ as well as at some other fixed radii have been calculated for different T and the associated phenomena of no-recirculation/recirculation discussed. The change in flow phenomena due to a reversal of the direction of net radial flow has also been studied. The moment on the rotating disc increases with T.Nomenclature , , z coordinates in a cylindrical polar system - z ₀ distance between rotor and stator (gap length) - =/z ₀, dimensionless radial distance - =z/z ₀, dimensionless axial distance - _s = _s/z₀, dimensionless disc radius - V =(u, v, w), velocity vector - dimensionless velocity components - uniform angular velocity of the rotor - , p fluid density and pressure - P =p/(₂ z ₀₂ ² , dimensionless pressure - ₁, ₂, ₃ kinematic coefficients of Newtonian viscosity, elastico-viscosity and cross-viscosity respectively - ₁, _{2 2}/z ₀ ² , resp. ₃/z ₀ ² , dimensionless parameters representing the ratio of second-order and inertial effects - m = , mass rate of symmetrical radial outflow - l a number associated with induced circulatory flow - Rm =m/(z ₀₁), Reynolds number of radial outflow - R _l =l/(z ₀₁), Reynolds number of induced circulatory flow - Rz =z ₀ ² /₁, Reynolds number based on the gap - ₁, ₂ maximum radii at which there is no recirculation for the cases Rm>0 and Rm<0 respectively - ₁(T), ₂(T) ₁ and ₂ for different T - U _1(T) ⁽⁺⁾ = dimensionless radial velocity, Rm>0 - V _1(T) ⁽⁺⁾ = , dimensionless transverse velocity, Rm>0 - U _2(T) ^(–) = , dimensionless radial velocity, Rm=–Rn<0, m=–n - V _2(T) ^(–) = , dimensionless transverse velocity, Rm<0 - C _m moment coefficient     

Material functions generated by the complex viscosity

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

Messung und Berechnung des örtlichen und mittleren Stoffübergangs an stumpf angeströmten Kreisscheiben bei unterschiedlicher Turbulenz

Zusammenfassung In einer vergleichenden Literaturübersicht zu Umströmung, Druck- bzw. Geschwindigkeitsverteilung sowie Wärme- und Stoffübergang werden bislang vorliegende Angaben zu stumpf angeströmten Kreisscheiben und -Zylindern zusammengefaßt. Wenige und zudem divergierende Ergebnisse zum Wärme- und Stoffübergang machen grundlegende experimentelle und theoretische Untersuchungen notwendig, wie sie in [l, 2] für die Eichung von Stoffübergangsmeßmethoden benötigt werden.Unter Einbeziehung des quer angeströmten Kreiszylinders wird gezeigt, daß genaue Angaben zum Wärme- und Stoffübergang bei zwei- wie dreidimensionalen Staupunktströmungen bislang nur über die Messung möglich sind. Über gemessene Geschwindigkeitsverteilungen berechnete Stoffübergangskoeffizienten werden von der Messung nicht bestätigt. Sie liegen gegenüber dem Experiment zu niedrig.Die Messungen wurden bei Turbulenzintensiten 0,8%Tu6%, Reynolds-Zahlen 2·10³5 und Scheibendurchmessern 9,3mmd73,7mm durchgeführt. Der Einfluß der Turbulenz auf den Stoffübergang im Staupunkt von Kreisscheiben kann nur näherungsweise über den Smith-Kuethe-Parameter Tu · Re/100 erfaßt werden. Differenzen zwischen Theorie nach Smith und Kuethe für Tu· Re<5 und Messung lassen sich über die Stabilitätstheorie erklären. Für eine genauere Erfassung des Stoffübergangs muß den unterschiedlichen Transportvorgängen über Turbulenzballen oder Längswirbeln sowie der Struktur der Turbulenz Rechnung getragen werden.

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Measuring and computation of local and average mass transfer to disks in cross flow at different turbulence intensities

The results of different publications concerning the flow, pressure and velocity distributions as well as the heat and mass transfer of disks and cylinders in cross flow are compared by a literature review. A few diverging results for heat and mass transfer require new experimental and theoretical approaches. The calibration of recently developed techniques for the determination of mass transfer rates as published in [1, 2] make these investigations expecially necessary. Including the cylinder in cross flow the authors show, that up to now exact data of heat and mass transfer for two- or three-dimensional flow at a forward stagnation region can be obtained by direct measuring only.Mass transfer coefficients computed from measured velocity distributions are not confirmed by the experimental results. Compared to the experimental data they are too low. The measurements were accomplished for turbulence intensities 0.8%Tu6%, Reynolds-numbers 2· 10³5 and disk diameters 9.3 mm d 73.7 mm.The influence of the turbulence on the stagnation point mass transfer of disks can be obtained only approximately by the Smith-Kuethe-parameter Tu·Re/100. Differences between theoretical results of Smith and Kuethe and experimental ones for Tu·Re/100<5 may be explained by the stability theory. For a more accurate determination of the mass transfer the different transport mechanisms of the scale of turbulence or the tree-dimensional flow pattern like Taylor-Görtler-vortices as well as the structure of the turbulence itself have to be regarded.

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Bezeichnungen a Temperaturleitkoeffizient - C_p Beiwert für den statischen Druck - C₂, C₃ Gradient der bezogenen Geschwindigkeit U⁺ am Staupunkt bei ebener, räumlicher Strömung - D_A Diffusionskoeffizient von Ammoniak in Luft - d Durchmesser - Fr=Sh/Re Frössling-Zahl für den Stoffübergang - Fr=Nu/Re Frössling-Zahl für den Wärmeübergang - Le=a/D_A Lewis-Zahl - L Bezugslänge - M Maschenweite von Turbulenzgittern - Nu=·d/ Nußbelt-Zahl - n Exponent der Prandtl-bzw. Schmidt-Zahl - Pr=/a Prandtl-Zahl - p Druck, Partialdruck - p_x statischer Druck an der Stelle x am Rand der Grenzschicht - Re=U · d/ Reynolds-Zahl - r Radius - r(x) radiale Distanz von der Rotationsachse eines Körpers zu einem Oberflächenelement - Sc=/D_A Schmidt-Zahl - Sh= _A ·d/D_A Sherwood-Zahl - T absolute Temperatur - Tu Turbulenzintensität (Turbulenzgrad) in% - U Strömungsgeschwindigkeit in x-Richtung am Rand der Grenzschicht - U Hauptströmungsgeschwindigkeit im freien Kanalquerschnitt - U⁺=U/U bezogene Geschwindigkeit in x-Richtung am Rand der Grenzschicht - u Strömungsgeschwindigkeit in x-Richtung, tangential zur Oberfläche - mittlere turbulente Geschwindig-keitsschwankung in x-Richtung - v Strömungsgeschwindigkeit in y-Richtung, normal zur Oberfläche - x Koordinate in Strömungsrichtung, tangential zur Oberfläche - x_G Entfernung vom Turbulenzgitter in Strömungsrichtung - x⁺ bezogene Länge x/r - y Koordinate normal zur Oberfläche - Wärmeübergangskoeffizient - _A Stoffübergangskoeffizient (Ammoniak) - dimensionsloses Temperaturgefälle an der Wand - Keilvariable - Wärmeleitkoeffizient - Wirbelweilenlänge (mm) - kinematische Zähigkeit - transformierte bezogene Länge - _A Partialdichte von Ammoniak Indices B mit Korrektur aufgrund der Verengung - m mittel - S bezogen auf die Kreisscheibe - Z bezogen auf den Kreiszylinder Herrn Prof. Dr.-Ing. habil. Josef Ipfelkofer zum 70. Geburtstag am 7. April 1977 gewidmet.     

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)     

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

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

Das erweiterte Korrespondenzgesetz für die dynamische Idealviskosität und die Idealwärmeleitfähigkeit reiner Stoffe

Zusammenfassung Zur Berechnung der dynamischen Idealviskosität Ideal (T) und der Idealwärmeleitfähigkeit ideal (T) benötigt man die kritische TemperaturT _kr, das kritische spezifische Volum _kr, die MolmasseM, den kritischen Parameter _kr und die molare isochore WärmekapazitätC _v(T). Sowohl das theoretisch, als auch das empirisch abgeleitete erweiterte Korrespondenzgesetz ergeben eine für praktische Zwecke ausreichende Genauigkeit für die Meßwertwiedergabe, die bei den assoziierenden Stoffen und den Quantenstoffen jedoch geringer ist als bei den Normalstoffen.

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The extended correspondence law for the ideal dynamic viscosity and the ideal thermal conductivity of pure substances

For the calculation of the ideal dynamic viscosity Ideal (T) and the ideal thermal conductivity ideal (T) the critical temperatureT _kr, the critical specific volumev _kr, the molecular massM, the critical parameter _kr, and the molar isochoric heat capacityC _v(T) is needed. Not only the theoretically determined but also the empirically determined extended correspondence law gives for practical use a good representation of the measured data, which for the associating substances and the quantum substances is not so good as for the normal substances.