On surface waves in generalized thermoelasticity

Knowles' representation theorem for harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space is extended to include harmonically time-dependent free processes for thermoelastic surface waves in generalized thermoelasticity of Lord and Shulman and of Green and Lindsay.r , , r , , .This work was done when author was unemployed.     

Peristaltic pumping of a second-order fluid in a planar channel

The peristaltic motion of a non-Newtonian fluid represented by the constitutive equation for a second-order fluid was studied for the case of a planar channel with harmonically undulating extensible walls. A perturbation series for the parameter ( half-width of channel/wave length) obtained explicit terms of 0(²), 0(²Re²) and 0(₁Re²) respectively representing curvature, inertia and the non-Newtonian character of the fluid. Numerical computations were performed and compared to the perturbation analysis in order to determine the range of validity of the terms.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada     

Effect of the Nature of the Interaction between a Gas and a Surface on the Relations for the Knudsen Layer in the Case of Strong Evaporation

The effect of the temperature accommodation coefficient _T on the relations at the Knudsen layer edge is investigated for strong evaporation using the moment method. An explicit expression for the dimensionless density as a function of the temperature and the Mach number M is obtained for 0 < _T < 1. For _T = 0 the entire solution is obtained in explicit form. It is shown that for = 0 and a condensation coefficient << 1 the temperature outside the Knudsen layer changes sharply as M varies from 0 to a certain value much less than unity after which the temperature ceases to depend on . For the model of specular reflection of the molecules from the surface the density and the temperature outside the Knudsen layer are found in explicit form as functions of the Mach number.     

Waves in a plasma with Hall dispersion

In a magnetohydrodynamic approximation, an investigation is made of the propagation of waves in a plasma, whose characteristic frequency is much less than the collision frequency of the electrons _e ^–1. It is assumed that the magnetic field is sufficiently strong so that the equality _ee1 will be satisfied, where _e is the cyclotron frequency of the rotation of the electrons. With large magnetic Reynolds numbers (R_m1), which are characteristic for many astrophysical problems, this latter condition leads to a need to take account of dispersion effects connected with Hall currents, in the absence of Joule dissipation. The dispersion equation for the propagation of small perturbations is analyzed in the limiting cases of weak dispersion and of a wave propagating along the magnetic field. In the case of weak dispersion, an equation is derived for nonlinear waves. The solutions are found in the form of stationary solitons. The region of such solutions is analyzed. A typical example of a medium with Hall dispersion is an interplanetary plasma, in which the parameter _ee is generally great.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 108–113, May–June, 1974.     

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

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

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

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

Modellvorstellung zur turbulenten Filmkondensation strömenden Dampfes

Zusammenfassung Der Wärmeübergang bei turbulenter Film kondensation strömenden Dampfes an einer waagerechten ebenen Platte wurde mit Hilfe der Analogie zwischen Impuls-und Wärmeaustausch untersucht. Zur Beschreibung des Impulsaustausches im Film wurde ein Vierbereichmodell vorgestellt. Nach diesem Modell wird die wellige Phasengrenze als starre rauhe Wand angesehen. Die Abhängigkeit einer Schubspannungs-Nusseltzahl von der Film-Reynoldszahl und Prandtlzahl wurde berechnet und dargestellt.

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A model for turbulent film condensation of flowing vapour

The heat transfer in turbulent film condensation of flowing vapour on a horizontal flat plate was investigated by means of the analogy between momentum and heat transfer. To describe the momentum transfer in the film a four-region model was presented. With this model the wavy interfacial surface is treated as a stiff rough wall. A shear Nusselt number has been calculated and represented as a function of film Reynolds number and Prandtl number.

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Formelzeichen a Temperaturleitkoeffizient - k Mischungswegkonstante - k _s äquivalente Sandkornrauhigkeit - Nu _x lokale Schubspannungs-Nusseltzahl,Nu _x=x_xv/uw - Pr Prandtlzahl,Pr=v/a - Pr _t turbulente Prandtlzahl,Pr _t =_m/_q - q Wärmestromdichte _q - R Wärmeübergangswiderstand - R_f Wärmeübergangswiderstand des Films - Re _F Reynoldszahl der Filmströmung - T Temperatur - U, V Geschwindigkeitskomponenten des Dampfes in waagerechter und senkrechter Richtung - u, Geschwindigkeitskomponenten des Kondensats in waagerechter und senkrechter Richtung - V Querschwankungsgeschwindigkeit des Kondensats und des Dampfes - u _/gtD Schubspannungsgeschwindigkeit an der Phasengrenze für die Dampfgrenzschicht, uD =(/)^1/2 - u _F Schubspannungsgeschwindigkeit an der Phasengrenze für den Kondensatfilm,u _F =(/)^1/2 - u _w Schubspannungsgeschwindigkeit an der Wand der Kühlplatte,u _w =(w/)^1/2 - y Wandabstand - x Wärmeübergangskoeffizient - gemittelte Kondensatfilmdicke - _s Dicke der zähen Schicht der Filmströmung an der welligen Phasengrenze - ₄ Dicke der zähen Schicht der Filmströmung an der gemittelten glatten Phasengrenze - Wärmeleitzahl - dynamische Viskosität - v kinematische Viskosität - Dichte - Oberflächenspannung - _w Wandschubspannung - Schubspannung an der Phasengrenzfläche - _m turbulente Impulsaustauschgröße - _q turbulente Wärmeaustauschgröße Indizes d Wert des Dampfes - w Wert an der Wand - x lokaler Wert inx - Wert an der Phasengrenze Stoffgrößen ohne Index gelten für das Kondensat     

A non-dualistic unified field theory of gravitation,electromagnetism and spin

The wisdom of classicalunified field theories in the conceptual framework of Weyl, Eddington, Einstein and Schrödinger has often been doubted and in particular there does not appear to be any empirical reason why the Einstein-Maxwell (E-M) theory needs to be geometrized. The crux of the matter is, however not whether the E-M theory is aesthetically satisfactory but whether it answers all the modern questions within the classical context. In particular, the E-M theory does not provide a classical platform from which the Dirac equation can be derived in the way Schrödinger's equation is derived from classical mechanics via the energy equation and the Correspondence Principle. The present paper presents a non-dualistic unified field theory (UFT) in the said conceptual framework as propounded by M. A. Tonnelat. By allowing the metric formds ²=g dx ^v x ^v and the non-degenerate two-formF=(1/2> _l) dx ^vdx ^vto enter symmetrically into the theory we obtain a UFT which contains Einstein's General Relativity and the Born-Infeld electrodynamics as special cases. Above all, it is shown that the Dirac equation describing the electron in an external gravito-electromagnetic field can be derived from the non-dualistic Einstein equation by a simple factorization if the Correspondence Principle is assumed.     

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     

Zum Impuls-, Wärme- und Stoffaustausch in turbulenten Strömungen reagierender Binärgemische

Zusammenfassung Für ein reagierendes Binärgemisch konstanter Dichte werden die Reynolds'schen Gleichungen angegeben. Die Transportkoeffizienten sowie die Wärmekapazitäten der beiden Komponenten werden als konstant angenommen. Für die unbekannten Reynolds'schen Terme werden Transportgleichungen angegeben. Weiter wird der Einfluß der Turbulenz auf die chemische Produktionsdichte diskutiert.

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About the transfer of momentum, heat and mass in turbulent flows of binary mixturesPart I: The reynolds equations and the transport equations

The Reynolds equations for a reacting binary mixture of constant density are given. The transport coefficients as well as the specific heats of the components are assumed to be constant. Transport equations for the unknown Reynolds-terms are given. The influence of turbulence on the chemical production of species in discussed.

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Formelzeichen c Massenkonzentration - c_p soezifische Wärme bei konstantern Druck - D binärer Diffussionskoeffizient - h spezifische Enthalpie - h_o=h + v _k ² /2 totale spezifische Enthalpie - h^o Reaktionsenthalpie - j_k Massendiffusionsstromvektor - k Reaktionsgeschwindigkeitskonstante - p Druck - Pr= c_p/ Prandtl-Zahl - q²/2 kinetische Energie der Schwankungsbewegung - q_k Energiestromvektor - R universelle Gaskonstante - Sc=/D Schmidt-Zahl - t Zeit - T absolute Temperatur - v_k Geschwindigkeitsvektor - x_k Ortsvektor Griechische Symbole Dissipationsfunktion - Wärmeleitfähigkeit - dynamische Viskosität - =/ kinematische Viskosität - Dichte - Produktionsdichte - _jk viskoser Spannungstensor Indizes auf die Komponente bezogen - 1 auf die Komponente 1 bezogen - 2 auf die Komponente 2 bezogen - mol molekularer Anteil - tur turbulenter Anteil - res resultierender Anteil     

ON THE UNIFICATION OF THE HAMILTON PRINCIPLES IN NONHOLONOMIC SYSTEM AND IN HOLONOMIC SYSTEM

ＯＮＴＨＥＵＮＩＦＩＣＡＴＩＯＮＯＦＴＨＥＨＡＭＩＬＴＯＮＰＲＩＮＣＩＰＬＥＳＩＮＮＯＮＨＯＬＯＮＯＭＩＣＳＹＳＴＥＭＡＮＤＩＮＨＯＬＯＮＯＭＩＣＳＹＳＴＥＭ（梁立孚）（韦扬）ＯＮＴＨＥＵＮＩＦＩＣＡＴＩＯＮＯＦＴＨＥＨＡＭＩＬＴＯＮＰＲＩＮＣＩＰＬ...     

A particle transport problem with a quadratic nonlinearity and general boundary conditions: The stationary solution

Summary We study a stationary, nonlinear, particle transport problem in slab geometry with general boundary conditions. The existence and uniqueness of the solution is proved by means of fixed point techniques, provided that the source term is sufficiently small.

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Sommario Si studia un problema stazionario nonlineare di particelle in geometria piana con condizioni al contorno generali. L'esistenza e unicitá della soluzione è dimostrata con tecniche di punto fisso purchè il termine di sorgente sia sufficientemente piccolo.

     

Almost analytic solutions and their tests of the horizontal diffusion equation for the movement of water in unsaturated soil

Itisalwaysdifficulttofindthesolutionsoftheequationforthemovementofwaterinunsaturatedsoi1.Theprimar}'reasonisthatthehydraulicconductivityK(T)orthediffusivityofsoiIwaterD(o)isfunctionofwaterpotential(W)orwatercontent'(o)'Atpresent,thegeneralwaystofindthesol…     

Symmetry groups of attractors

For maps equivariant under the action of a finite group on ⁿ, the possible symmetries of fixed points are known and correspond to the isotropy subgroups. This paper investigates the possible symmetries of arbitrary, possibly chaotic, attractors and finds that the necessary conditions of Melbourne, Dellnitz & Golubitsky [15] are sufficient, at least for continuous maps.This result shows that the reflection hyperplanes are important in determining those groups which are admissible; more precisely, a subgroup of is admissible as the symmetry group of an attractor if there exists a with / cyclic such that fixes a connected component of the complement of the set of reflection hyperplanes of reflections in but not in . For finite reflection groups this condition on reduces to the condition that is an isotropy subgroup. Our results are illustrated for finite subgroups of O(3).     

Rheological models for gypsum plaster pastes

Shear stress and shear rate data obtained for gypsum plaster pastes were correlated by means of different rheological models. The pastes were prepared from a commercial calcium sulfate hemihydrate at various water/plaster ratios ranging from 100/150 to 100/190. The tests were performed at 25°C using a rotating coaxial cylinder viscosimeter. The measurements were accomplished by applying a step-wise decreasing shear rate sequence. Discrimination among the models was made: (1) on the basis of the fitting goodness; (2) by checking the physical meaning of the calculated parameters; (3) on the basis of the stability of the parameters and of their prediction capacity beyond the limits of the experimental data. In the light of above, the Casson model seemed to be most effective for application to gypsum plaster pastes. K Consistency - n Power-law index - N Number of experimental data - P Number of parameters - Shear rate (s^–1) - ₀ Viscosity (Pa · s) - _d Dispersing medium viscosity (Pa · s) - _p Plastic viscosity (Pa · s) - Viscosity at zero shear rate (Pa · s) - Viscosity at infinite shear rate (Pa · s) - [] Intrinsic viscosity - ² Variance - Shear stress (Pa) - ₀ Yield stress (Pa) - Solid volume fraction - _m Maximum solid volume fraction     

Flow in vicinity of blunt body stagnation point in diverging hypersonic stream

In the hypersonic thin shock layer approximation for a small ratio k of the densities before and after the normal shock wave the solution of [1] for the vicinity of the stagnation point of a smooth blunt body is extended to the case of nonuniform outer flow. It is shown that the effect of this nonuniformity can be taken into account with the aid of the effective shock wave radius of curvature R_*, whose introduction makes it possible to reduce to universal relations the data for different nonuniform outer flows with practically the same similarity criterion k. The results of the study are compared with numerical calculations of highly underexpanded jet flow past a sphere.Notations x, y a curvilinear coordinate system with axes directed respectively along and normal to the body surface with origin at the forward stagnation point - R radius of curvature of the meridional plane of the body surface - uV, vV., , p V ² respectively the velocity projections on the x, y axes, density, and pressure - and V freestream density and velocity The indices =0 and=1 apply to plane and axisymmetric flows Izv. AN SSSR, Mekhanika Zhidkosti i Gaza, Vol. 5, No. 3, pp. 102–105, 1970.     

The Thermodynamic Significance of the Local Volume Averaged Temperature

Since the temperature is not an additive function, the traditional thermodynamic point of view suggests that the volume integral of the temperature has no precise physical meaning. This observation conflicts with the customary analysis of non-isothermal catalytic reactors, heat pipes, driers, geothermal processes, etc., in which the volume averaged temperature plays a crucial role. In this paper we identify the thermodynamic significance of the volume averaged temperature in terms of a simple two-phase heat transfer process. Given the internal energy as a function of the point temperature and the density

Thixotropic behaviour of concentrated bauxite residue suspensions

An experimental study on the time-dependent rheology of highly concentrated and flocculated suspensions of bauxite residue (red mud) is presented. Both the thixotropic breakdown with shear and recovery at rest have been quantitatively examined using a vane-shear instrument and a capillary rheometer. It is demonstrated that both the yield stress and the apparent viscosity of the material can be drastically reduced, by orders of magnitude, by shear-induced agitation with a simple anchor impeller. The rate of thixotropic decay is a function of solids concentration, agitation time and speed. With prolonged agitation, the flow properties are brought to an equilibrium level characterized by a finite yield stress and a shear rate-dependent (shear-thinning) viscosity. In the absence of shear, the yield stress gradually increases with time but at a rate considerably slower than the rate of decay in the shear field. Even after an extended rest period of several months, only a fraction of the initial yield stress can be regained. The observed behaviour has been interpreted in terms of a time-dependent but non-reversible transformation of a network structure of flocculated particles initially developed in the red mud. Using a structural kinetic approach, models have been developed for correlating the experimental kinetic data. The paper concludes with a discussion on the practical consequences of the work. D Diameter of capillary tube - k ₁ Breakdown rate constant in eq. (2) - k ₁ Modified rate constant in eq. (7) - k ₂ Recovery rate constant in eq. (8) - N Impeller rotational speed - P Pressure - t Agitation (mixing time) - t Resting (aging) time - V Average velocity - Average mixing shear rate - Apparent viscosity ( _w/(8V/D)) - _w Wall shear stress - _y Yield stress - _y ^E Yield stress at infinite resting time - Dimensionless structural parameter - e Equilibrium condition (t ) - o Initial condition (t = 0) - Conditions att and      

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.     

Magnetohydrodynamic laminar flow between porous disks

The steady two-dimensional laminar flow of an incompressible conducting fluid between two parallel circular disks in the presence of a transverse magnetic field is investigated. A solution is obtained by perturbing the creeping flow solution and it is valid only for small suction or injection Reynolds numbers. Expressions for velocity, induced magnetic field, pressure, and shear stress distribution are determined and are compared with the creeping flow and hydrodynamic solutions. It is found that the overall effect of the magnetic field on the flow is the same as that in the Hartmann flow.Nomenclature stream function - 2h channel width - z, r axial and radial coordinates - radius of the disk - U _r radial component of velocity - U _r average velocity in the radial direction, U _r d - U _z axial component of velocity - U ₀ injection or suction velocity - dimensionless axial coordinate, z/h - f() function defined in (8) - density - coefficient of kinematic viscosity - electrical conductivity - magnetic permeability - H ₀ impressed magnetic field - h _r induced magnetic field, H _r /H ₀ - M Hartmann number, H ₀ h(/)^1/2 - R Reynolds number, U ₀ h/ - R _m magnetic Reynolds number, U ₀ r - A constant defined in (15) - K constant defined in (27) - C ₂ constant defined in (26) - p pressure - C _p pressure coefficient - C _f skin friction coefficient     

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

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