Effect of radial and rotational lubricant inertia on the pressure distribution in externally pressurised thrust bearings

Expressions are obtained for the pressure distribution in an externally pressurised thrust bearing for the condition when one bearing surface is rotated. The influence of centripetal acceleration and the combined effect of rotational and radial inertia terms are included in the analysis. Rotation of the bearing causes the lubricant to have a velocity component in an axial direction towards the rotating surface as it spirals radially outwards between the bearing surfaces. This results in an increase in the pumping losses and a decrease in the load capacity of the bearing. A further loss in the performance of the bearing is found when the radial inertia term, in addition to the rotational inertia term is included in the analysis.Nomenclature r, z, cylindrical co-ordinates - V _r, V , V _z velocity components in the r, and z directions respectively - U, X, W representative velocities - coefficient of viscosity - p static pressure at radius r - p mean static pressure at radius r - Q volume flow per unit time - 2h lubricant film thickness - density of the lubricant - r ₂ outside radius of bearing = D/2 - angular velocity of bearing - R dimensionless radius = r/h - P dimensionless pressure = h ³ p/Q - Re channel Reynolds number = Q/h     

Temperament Development Modeled as a Nonlinear Complex Adaptive System

Using approach-withdrawal (AW) as a specific instance of temperament, a theoretical model of temperament as a complex dynamic system is proposed. Developmental contextualism (Lerner, 1998) serves as a guiding theory in determining the structural components of the system and Kauffman's (1993) Boolean models of self-organization are adapted to estimate the parameter functions. In this model P(AW) = f(, ) where P(AW) is the probability density function of an approach or a withdrawal response, ( is a standardized parameter estimate of the biological sensitivity to stimulation, and is a standardized parameter estimate of the contextual response to an approach or withdrawal response. It is theorized that the functions of ( and follow a Hill function of the forms: d /dt = (²/c² + ²) – K₁ d /dt = ( ²/c² + ²) – K₂, where K₁, K₂, and c are system constants. This results in a double sigmoid function in which at extreme values of and the system stabilizes on a steady state of either approach or withdrawal response patterns. At intermediate parameter values the probability density functions of approach and withdrawal responses are wider. Thus, AW can be modeled as representing two basins of attraction. In addition, considerations are given to the systems sensitivity to initial conditions.     

Flow of viscoelastic fluid between confocal elliptic cylinders

Summary Using elliptic coordinates, the flow pattern of a fluid of grade four between two elliptic tubes is determined. A comparison between the position of the maximum of the axial velocity in the present case and in the case of two concentric circular tubes shows a basic difference. In the elliptic case the maximum is shifted towards the external wall, while in the case of concentric circular tubes the shift is in the direction of the internal wall. The secondary flow shows dissymmetry with reference to the intermediate line , which itself lies nearer to the external wall.

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Zusammenfassung Unter Benutzung elliptischer Koordinaten wird die Strömung zwischen zwei elliptischen Rohren bestimmt. Ein Vergleich zwischen der Lage des axialen Geschwindigkeitsmaximums im vorliegenden Fall und im Fall zweier konzentrischer Kreisrohre ergibt einen grundsätzlichen Unterschied: Das Maximum ist im elliptischen Fall zur äußeren Wand hin verschoben, während die Verschiebung im Fall der konzentrischen Kreisrohre zur inneren Wand hin erfolgt. Die Sekundärströmung ist unsymmetrisch relativ zur mittleren Stromlinie , die selbst näher zur äußeren Wand liegt.

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A planar domain representing the annular region - vector inx ₁,x ₂-plane - x _i rectangular coordinates - rectangular unit vectors - , elliptic coordinates - ₁, ₂ ellipses representing respectively the internal and external tubes - = ₂ – ₁ annular widthy = ( – ₁)/ - µ 1^st grade material constant - _i 2^nd grade material constants - _i 3^rd grade material constants - _i 4^th grade material constants - I unit tensor - T _E extra stress (T + pI) - V potential of body forces - material density = (p/) + V = –ax ₃ + () - a specific driving force - arbitrary scalar function - A _k Rivlin-Eriksen tensors - S stress scalar defined onA - t stress vector defined onA - P stress tensor defined onA - v axial velocity - v _i i ^th term in the approximation ofv - u velocity vector perpendicular to the axis 4( ₃ + ₄ + ₅ + 1/2₆) –2/µ(2 ₁ + ₂)( ₂ + ₃) - T stress tensor - p arbitrary hydrostatic pressure - u _i i ^th term in the approximation ofu - stream function definingu - _i i ^th term in the approximation of With 8 figures and 1 table     

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     

Viscous properties of calcium carbonate filled polymer melts

Summary The viscous properties of calcium carbonate filled polyethylene and polystyrene melts were examined. The relative vircosity _r defined in the previous paper gave an asymtptotic value( _r)_l in the range of the shear stress below 10⁵ dyne/cm².( _r)_l of the calcium carbonate filled system was higher than that of the glass beads or glass balloons filled system at the same volume fraction of the filler. Maron-Pierce equation with ₀ = 0.44 was able to approximate the( _r)_l — relationship. However, it was deduced here that the high value of( _r)_l of calcium carbonyl filled system was due to the apparent increase of and this increase was attributed to the fixed polymer layer formed on the powder particle. By assuming the particle as a sphere with a diameter of 2 µm, the thickness of the fixed polymer layer was estimated as about 0.17 µm. The yield stress estimated from the Casson's plots increased exponentially with.

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Zusammenfassung Es wurden die viskosen Eigenschaften von Polyäthylen-und Polystyrol-Schmelzen untersucht, die mit Kalziumkarbonat-Teilchen gefüllt waren. Für die relative Viskosität _r, wie sie in einer vorangegangenen Veröffentlichung definiert worden war, ergab sich bei Schubspannungen unterhalb 10⁵ dyn/cm² ein asymptotischer Wert( _r)_l. Dieser war bei den mit Kalziumkarbonat gefüllten Schmelzen höher als bei Schmelzen, die bis zur gleichen Volumenkonzentration mit Glaskugeln oder Glasballons gefüllt waren. Die ( _r) _l —-Abhängigkeit ließ sich durch eine Gleichung nachMaron und Pierce mit ₀ = 0,44 beschreiben. Es wurde jedoch geschlossen, daß der hohe( _r)_l-Wert der mit Kalziumkarbonat gefüllten Schmelzen auf eine scheinbare Zunahme von zurückzuführen ist, verursacht durch eine feste Polymerschicht auf der Teilchenoberfläche. Unter Annahme kugelförmiger Teilchen mit einem Durchmesser von 2 µm ließ sich die zugeordnete Schichtdicke zu 0,17 µm abschätzen. Die mittels der Casson-Beziehung geschätzte Fließspannung ergab eine exponentielle-Abhängigkeit.

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Laser velocimetry measurements in a horizontal gas-solid pipe flow

This paper presents laser measurements of particle velocities in a horizontal turbulent two-phase pipe flow. A phase Doppler particle analyzer, (PDPA), was used to obtain particle size, velocity, and rms values of velocity fluctuations. The particulate phase consisted of glass spheres 50 m in diameter with the volume fraction of the suspension in the range _p=10^-4 to _p=10^-3. The results show that the turbulence increases with particle loading.List of symbols a particle diameter - C _va velocity diameter cross-correlation - d pipe diameter - Fr ² Froude number - g gravitational constant - p(a) Probability density of the particle diameter - Re pipe Reynolds number based on the friction velocity - T characteristic time scale of the energy containing eddies - T _L integral scale of the turbulence sampled along the particle path - u, U, u characteristic fluid velocities: fluctuating, mean and friction - v characteristic velocity of the paricle fluctuations - f expected value of any random variable f - f¦g expected value of f given a value of the random variable g - _p particle volume fraction - _p particle response time - absolute fluid viscosity - v kinematic fluid viscosity - _p, _f densities, particle and fluid - _a ² particle diameter variance - _va ² velocity variance due to the particle diameter variance - _vT ² total particle velocity variance - _vt ² particle velocity variance due to the response to the turbulent field     

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     

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     

A new interpretation of viscosity and yield stress in dense slurries: Coal and other irregular particles

Illinois coal was ground and wet-sieved to prepare three powder stocks whose particle-size distributions were characterized. Three suspending fluids were used (glycerin, bromonaphthalene, Aroclor), with viscosities _s that differed by a factor of 100 and with very different chemistries, but whose densities matched that of the coal. Suspensions were prepared under vacuum, with coal volume fractions that ranged up to 0.46. Viscosities were measured in a cone-and-plate over a shear rate range 10^–3–10² s^–1. Reduced viscosity _r = /_s is correlated in the high-shear limit ( ) with/ _M, where _M is the maximum packing fraction for the high-shear microstructure, to reveal the roles of size distribution and suspending fluid character. A new model that invokes the stress-dependence of _M is found to correlate _r well under non-Newtonian conditions with simultaneous prediction of yield stress at sufficiently high; a critical result is that stress and not governs the microstructure and rheology. Numerous experimental anomalies provide insight into suspension behavior.     

The rheology of pigment dispersions

Summary A kinetic model is developed to relate the measured shear stress in a dispersion with the rate of deformation, and with the level of structure caused by the competing effects of flocculation and deflocculation.The model parameters are determined from experimental data obtained from equilibrium and transient oscillatory shear, using dispersions of a pigment in three different oil-based media. It is found that the model can successfully describe the flow behaviour of the dispersions under all three types of deformation, and account for different concentrations and temperatures.

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Zusammenfassung Es wird ein kinetisches Modell entwickelt, das die in einer Dispersion gemessene Schubspannung mit der Deformationsgeschwindigkeit in Beziehung setzt unter Berücksichtigung der im Wettbewerb stehenden Flokkulations-und Deflokkulationseffekte.Die Modellparameter werden mit Hilfe experimenteller Daten bestimmt, die zu stationären und oszillatorischen Scherströmungen unter Einbeziehung wechselnder Beanspruchungsarten erhalten wurden. Dabei wurden Pigment-Dispersionen in drei verschiedenen Medien auf Öl-Basis verwendet. Man findet, daß das Modell das Fließverhalten der Dispersionen unter allen betrachteten Deformationstypen sowie bei den verschiedenen angewandten Konzentrationen und Temperaturen erfolgreich zu beschreiben vermag.

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a, b experimental constants - c dispersion concentration by weight - d ₃ mean volume to surface diameter of a floc. - f (·,·) function defined by eq. [20] - g(·) a function of volume fraction - k Boltzmann's constant - n _s number of floccules containings flocs, per unit volume - ratio of the number of floccules containings flocs, per unit volume, to the total number of flocs per unit volume - t present time - B, D derived constants - E ₀,E ₁ activation energies for viscous flow of a dispersion at low and high shear rates - E _m activation energy for viscous flow of the medium - E _f ,E _d activation energies for flocculation and deflocculation of a dispersion - G shear rate - K ₁,K ₂ model constants - M parameter related to the total number of flow units - N number of flocs per unit volume - R, R ₀ the ratio of the rate function for deflocculation to the rate function for flocculation, and its value in the absence of shear - T absolute temperature - ₀, ₀ constants - (·,·),(·) rate functions for flocculation and deflocculation - , , v model parameters - , ₀, ₁ viscosity of a dispersion, and its value at low and high shear rates - _r the viscosity of a dispersion of floccules each containingr flocs - dynamic viscosity of a dispersion - µ, µ ₀ viscosity of a medium at temperatureT, and in the limit of high temperature - _p , _f , _F volume fraction of pigment, of flocs, of floccules - measured shear stress - non-dimensional time - a characteristic time for flocculation With 8 figures and 7 tables     

Similar solutions and the asymptotics of filtration equations

In this paper we consider the asymptotic behavior of solutions of the quasilinear equation of filtration as t. We prove that similar solutions of the equation u _t = (u )_xx asymptotically represent solutions of the Cauchy problem for the full equation u _t = [(u)]_xx if (u) is close to u for small u.     

Some remarks on differential equations of quadratic type

In this paper we study differential equations of the formx(t) + x(t)=f(x(t)), x(0)=x ₀ C HereC is a closed, bounded convex subset of a Banach spaceX,f(C) C, and it is often assumed thatf(x) is a quadratic map. We study the differential equation by using the general theory of nonexpansive maps and nonexpansive, non-linear semigroups, and we obtain sharp results in a number of cases of interest. We give a formula for the Lipschitz constant off: C C, and we derive a precise explicit formula for the Lipschitz constant whenf is quadratic,C is the unit simplex inR ⁿ, and thel ₁ norm is used. We give a new proof of a theorem about nonexpansive semigroups; and we show that if the Lipschitz constant off: CC is less than or equal to one, then lim_tf(x(t))–x(t)=0 and, if {x(t):t 0} is precompact, then lim_tx(t) exists. Iff¦C=L¦C, whereL is a bounded linear operator, we apply the nonlinear theory to prove that (under mild further conditions on C) lim_t f(x(t))–x(t)=0 and that lim_t x(t) exists if {x(t):t 0} is precompact. However, forn 3 we give examples of quadratic mapsf of the unit simplex ofR ⁿ into itself such that lim_t x(t) fails to exist for mostx ₀ C andx(t) may be periodic. Our theorems answer several questions recently raised by J. Herod in connection with so-called model Boltzmann equations.     

A Strongly Degenerate Quasilinear Equation: the Parabolic Case

We prove the existence and uniqueness of entropy solutions of the Neumann problem for the quasilinear parabolic equation u_t=÷ a(u, Du), where a(z,)=f(z,), and f is a convex function of with linear growth as ||||, satisfying other additional assumptions. In particular, this class includes the case where f(z,)=(z)(), >0, and is a convex function with linear growth as ||||.     

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–)     

Saint-Venant's principle in nonlinear plane elasticity with sufficiently small strains

A homogeneous, isotropic cylinder in an equilibrium state of plane strain, whose cross-section is a rectangle R : [0 < y ₁ < 2L; 0 < y ₂ < h] with h/L 1, is considered. There are no body forces and the long sides are stress free. At y ₁ = 0 and y ₁ = 2L, there are arbitrary loadings, each statically equivalent to a uniformly distributed tensile or compressive stress c. Within the theory of nonlinear elasticity and with the strains and strain gradients assumed to be sufficiently small (but with no such assumptions on the displacement gradients), it is proved that if (,=1,2) represents the Cauchy stress tensor and the Kronecker delta, then |–c₁₁| decays exponentially to zero in R with distance from the nearer end, and the decay constant depends only upon the material but is independent of L.     

Positively invariant regions for a problem in phase transitions

Positively invariant regions for the system v _t + p(W) _x = V _xx , W _t –V _x = W _xx are constructed where p < 0, w < , w > , p(w) = 0, w , > 0. Such a choice of p is motivated by the Maxwell construction for a van der Waals fluid. The method of an analysis is a modification of earlier ideas of Chueh, Conley, & Smoller [1]. The results given here provide independent L bounds on the solution (w, v).Dedicated to Professor James Serrin on the occasion of his sixtieth birthday     

Supersonic air flow past a sphere with account for vibrational relaxation

A numerical solution is obtained for the problem of air flow past a sphere under conditions when nonequilibrium excitation of the vibrational degrees of freedom of the molecular components takes place in the shock layer. The problem is solved using the method of [1]. In calculating the relaxation rates account was taken of two processes: 1) transition of the molecular translational energy into vibrational energy during collision; 2) exchange of vibrational energy between the air components. Expressions for the relaxation rates were computed in [2]. The solution indicates that in the state far from equilibrium a relaxation layer is formed near the sphere surface. A comparison is made of the calculated values of the shock standoff with the experimental data of [3].Notation uV_max, vV_max velocity components normal and tangential to the sphere surface - V_max maximal velocity - P V _max ² pressure - density - TT temperature - e_viRT vibrational energy of the i-th component per mole (i=–O₂, N₂) - =rb–1 shock wave shape - a _f the frozen speed of sound - HRT/m gas total enthalpy     

Transport in ordered and disordered porous media IV: Computer generated porous media for three-dimensional systems

In the method of volume averaging, the difference between ordered and disordered porous media appears at two distinct points in the analysis, i.e. in the process of spatial smoothing and in the closure problem. In theclosure problem, the use of spatially periodic boundary conditions isconsistent with ordered porous media and the fields under consideration when the length-scale constraint,r ₀L is satisfied. For disordered porous media, spatially periodic boundary conditions are an approximation in need of further study.In theprocess of spatial smoothing, average quantities must be removed from area and volume integrals in order to extractlocal transport equations fromnonlocal equations. This leads to a series of geometrical integrals that need to be evaluated. In Part II we indicated that these integrals were constants for ordered porous media provided that the weighting function used in the averaging process contained thecellular average. We also indicated that these integrals were constrained by certain order of magnitude estimates for disordered porous media. In this paper we verify these characteristics of the geometrical integrals, and we examine their values for pseudo-periodic and uniformly random systems through the use of computer generated porous media.

Nomenclature

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 _i i=1, 2, 3 gaussian probability distribution used to locate the position of particles - I unit tensor - L general characteristic length for volume averaged quantities, m - L characteristic length for , m - L characteristic length for , m - 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 the traditional volume average - n _i i=1, 2, 3 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward the-phase - n _e outwardly directed unit normal vector at the entrances and exits of 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 - r position vector, m - r _m support of the weighting functionm, m - averaging volume, m³ - V volume of the-phase contained in the averaging volume,, m³ - x positional vector locating the centroid of an averaging volume, m - x ₀ reference position vector associated with the centroid of an averaging volume, m - y position vector locating points 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 - /L, small parameter in the method of spatial homogenization - standard deviation ofa _i - _r standard deviation ofr - _r intrinsic phase average of      

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