Filament stretching equation in Lagrangian coordinates

The general integral of the very simple equation ^21/n/() was found to describe the cross sectional area of filaments of isothermal power law fluids while in transient stretching where is time and is the initial location of fluid molecules at time = 0 given as the distance from a reference point fixed in space. Any such stretching transient given as a solution of the above equation is physically realizable subject to the restrictions > 0 and/ < 0.     

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     

Transformation and some solutions of the equations of motion of an ideal gas

The equations of one-dimensional and plane steady adiabatic motion of an ideal gas are transformed to a new form in which the role of the independent variables are played by the stream function and the function introduced by Martin [1, 2], It is shown that the function retains a constant value on a strong shock wave (and on a strong shock for plane flows). For one-dimensional isentropic motions the resulting transformation permits new exact solutions to be obtained from the exact solutions of the equations of motion. It is shown also that the one-dimensional motions of an ideal gas with the equation of state p=f(t) and the one-dimensional adiabatic motions of a gas for which p=f() are equivalent (t is time, is the stream function). It is shown that if k=s=–1, m and n are arbitrary (m+n0) and =1, the general solution of the system of equations which is fundamental in the theory of one-dimensional adiabatic self-similar motions [3] is found in parametric form with the aid of quadratures. Plane adiabatic motions of an ideal gas having the property that the pressure depends only on a single geometric coordinate are studied.     

Uniaxial extension of an isothermal fluid cylinder in the presence of inertia,surface tension and gravity

Effects of inertia, surface tension and gravity in the constant force stretching of isothermal cylindrical filaments of Newtonian, power-law and Maxwell-type fluids were analysed in Lagrangian coordinates. Solution for the purely gravitational extension of Newtonian fluid cylinder was found to be as simple as = 1 – C ₃ (1 – ) where designates the cross sectional area, the Lagrangian distance and the time. Analytical solutions were also available for the case of inertialess Newtonian and power-law fluids.A first-order backward differencing scheme and minimal computer time were sufficient to numerically analyse the constant force extension of Maxwell-type fluids in the presence of inertia, gravity and surface tension. Effects of inertia, surface tension and gravity on the severity of neck down occurring at either end of the filament are summarized in diagrams. The present approach is valid on any other constitutive model as far as there is a numerical scheme to analyse thehomogeneous extension of a cylinder of that particular fluid.     

Free convection heat transfer from an isothermal plate with arbitrary inclination

This paper presents a new formulation for the laminar free convection from an arbitrarily inclined isothermal plate to fluids of any Prandtl number between 0.001 and infinity. A novel inclination parameter is proposed such that all cases of the horizontal, inclined and vertical plates can be described by a single set of transformed equations. Moreover, the self-similar equations for the limiting cases of the horizontal and vertical plates are recovered from the transformed equations by setting=0 and=1, respectively. Heated upward-facing plates with positive and negative inclination angles are investigated. A very accurate correlation equation of the local Nusselt number is developed for arbitrary inclination angle and for 0.001 Pr .

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Wärmeübertragung bei freier Konvektion an einer isothermen Platte mit beliebiger Neigung

Zusammenfasssung Diese Untersuchung stellt eine neue Formulierung der laminaren freien Konvektion von Flüssigkeiten mit einer Prandtl-Zahl zwischen 0,001 und unendlich an einer beliebig schräggestellten isothermen Platte dar. Ein neuer Neigungsparameter wird eingeführt, so daß alle Fälle der horizontalen, geneigten oder vertikalen Platte von einem einzigen Satz transformierter Gleichungen beschrieben werden können. Die unabhängigen Gleichungen für die beiden Fälle der horizontalen and vertikalen Platte wurden für=0 und=1 aus den transformierten Gleichungen wieder abgeleitet. Es wurden erwärmte aufwärtsgerichtete Platten mit positiven und negativen Neigungswinkeln untersucht. Eine sehr genaue Gleichung wurde für die lokale Nusselt-Zahl bei beliebigen Neigungswinkeln und für 0,001 Pr entwickelt.

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Nomenclature C _p specific heat - f reduced stream function - g gravitational acceleration - Gr local Grashof number,g(T _w –T _w ) x³/v² - h local heat transfer coefficient - k thermal conductivity - n constant exponent - Nu local Nusselt number,hx/k - p pressure - Pr Prandtl number, v/ - Ra local Rayleigh number,g(T _w –T )J x³/v - T fluid temperature - T _w wall temperature - T temperature of ambient fluid - u velocity component in x-direction - v velocity component in y-direction - x coordinate parallel to the plate - y coordinate normal to the plate Greek symbols thermal diffusivity - thermal expansion coefficient - (Ra¦sin¦)^1/4/( Ra cos()^1/5 - pseudo-similarity variable, (y/) - dimensionless temperature, (T–T )/(T _w–T ) - ( Ra cos)^1/5+(Rasin)^1/4 - v kinematic viscosity - 1/[1 +(Ra cos)^1/5/( Ra¦sin)^1/4] - density of fluid - Pr/(1+Pr) - _w wall shear stress - angle of plate inclination measured from the horizontal - stream function - dimensionless dynamic pressure     

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     

The swirling radial free jet

Summary A general similarity solution suggested by Watson for the problem of the laminar, radial, free-jet with swirl has been previously discussed by Riley who also calculated the order to which the solution was valid. That problem is considered in more detail here and higher order terms are given. It is shown that a perturbation scheme for the stream function consisting of a series of inverse powers of and which uses the asymptotic similarity solution as the basic solution is inadequate, and a modification to the series so as to include terms like ⁿ (ln ) ^m must be adopted in order to satisfy the boundary conditions. It is also shown that the general similarity solution may be obtained from the asymptotic series representing the general case with swirl for certain special values of the free constants and also for the no-swirl or free-jet problem. The asymptotic series is given to order ^–13 for the case of swirl and to order ^–29 when there is no swirl.     

Contributions to the theory of the method of steepest descent

Applying the method of steepest descent to F(x ₁,..., x _n ) one obtains a sequence of points _v . To obtain conditions for convergence of _v , the derived set H of the _v in the case of divergence is studied. In this case H is a continuum on which not only grad F vanishes everywhere, but also the rank of the Hessian of F is everywhere less than n-1.     

An example of a quasiconvex function that is not polyconvex in two dimensions

We study the different notions of convexity for the function f () = ||² (||² – 2 det ) where ^2×2, introduced by Dacorogna & Marcellini. We show that f is convex, polyconvex, quasiconvex, rank-one convex, if and only if ¦¦ 2/3 2, 1, 1+ (for some >0), 2/3, respectively.     

Squeezing flows of Newtonian liquid films an analysis including fluid inertia

A theoretical study is made of the flow behavior of thin Newtonian liquid films being squeezed between two flat plates. Solutions to the problem are obtained by using a numerical method, which is found to be stable for all Reynolds numbers, aspect ratios, and grid sizes tested. Particular emphasis is placed on including in the analysis the inertial terms in the Navier-Stokes equations.Comparison of results from the numerical calculation with those from Ishizawa's perturbation solution is made. For the conditions considered here, it is found that the perturbation series is divergent, and that in general one must use a numerical technique to solve this problem.Nomenclature a half of the distance, or gap, between the two plates - a ₀ the value of a at time t=0 - adot da/dt - ä d² a/dt ² - d³ a/dt ³ - a ⁱ components of a contravariant acceleration vector - f unknown function of z ₀ and t defined in (6) - f ⁱ function defined in (9) f ¹=r ₀ g(z ₀, t) f ²= ₀ f ³=f(z ₀, t) - F force applied to the plates - g unknown function of z ₀ and t defined in (6) - g g/z ₀ - h grid dimension in the z ₀ direction (see Fig. 5) - Christoffel symbol - i, j, k, l indices - k grid dimension in the t direction (see Fig. 5) - r radial coordinate direction defined in Fig. 1 - r ₀ radial convected coordinate - R radius of the circular plates - t time - v _r fluid velocity in the r direction - v _z fluid velocity in the z direction - v fluid velocity in the direction - x ⁱ cylindrical coordinate x ¹=r x²= x³=z - z vertical coordinate direction defined in Fig. 1 - z ₀ vertical convected coordinate - tangential coordinate direction - ₀ tangential convected coordinate - viscosity - kinematic viscosity, / - ⁱ convected coordinate ¹=r₀ ²=₀ ³=z₀ - density     

On the Ellipticity of the Middle Surface of a Shell and its Application to the Asymptotic Analysis of ‘Membrane’ Shells

We consider a surface S = (), where ² is a bounded, connected, open set with a smooth boundary and : ³ is a smooth map; let () denote the components of the two-dimensional linearized strain tensor of S and let 0 with length 0 > 0. We assume the the norm ,|| ()||0, in the space V0() = { H¹() × H¹() × L²(); = 0 on 0 } is equivalent to the usual product norm on this space. We then establish that this assumption implies that the surface S is uniformly elliptic and that we necessarily have 0 = .     

Analysis of slip and temperature jump coefficients in a binary gas mixture

The possibility of simplifying the formulas obtained by the Maxwell-Loyalka method for the velocity _u, temperature _T and diffusion _d slip coefficients and the temperature jump coefficient in a binary gas mixture with frozen internal degrees of freedom of the molecules is considered. Special attention is paid to gases not having sharply different physicochemical properties. The formulas are written in a form convenient for use without linearization in the thermal diffusion coefficient. They are systematically analyzed for mixtures of inert gases, N₂, O₂, CO₂, and H₂ at temperatures extending from room temperature to 2500°K. It is shown that for the molecular weight ratios m^* = m₂/m₁ considered the expressions for _u and can be radically simplified. With an error acceptable for practical purposes (up to 10%) it is possible to employ expressions of the same structural form as for a single-component gas: for _u if 1 m^* 6, and for if 1 m^* 3. When 1 m^* 2 the expression for _T can be simplified with a maximum error of 5%. Within the limits of accuracy of the method the expression for _t can be linearized in the thermal diffusion coefficient. An approximate expression convenient for practical calculations is proposed for _d Finally, the , _u, and _T for a single-component polyatomic gas with easy excitation of the internal degrees of freedom of the molecules are similarly analyzed; it is shown that these expressions can be considerably simplified.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 152–159, November–December, 1990.     

Heat transport in a parabolic tube of circular cross-section

The local temperature has been determined for a viscous liquid flowing through a paraboloidal tube. Wall temperature and inlet temperature have been considered constant. The liquid flow was considered as creeping flow and its velocity distribution was determined by solving the biharmonic differential equation of the stream function. The local temperature was evaluated numerically from the analytical results.

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Wärmetransport im Paraboloidrohr

Zusammenfassung Es wird die lokale Temperatur in einer viskosen Strömung durch ein Paraboloidrohr bestimmt. Dabei wird konstante Wand- und Einlauftemperatur angenommen. Die kriechende Strömungsgeschwindigkeit wurde aus der Lösung der biharmonischen Differentialgleichung der Stromfunktion bestimmt. Die lokale Temperatur wurde aus den analytischen Ergebnissen für einige Paraboloidrohre numerisch bestimmt.

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Nomenclature ₁ F ₁ confluent hypergeometric function - diffusivity - T(, , ) temperature - T _w temperature at the paraboloidal wall - T _i temperature at the inlet - u(, ) flow velocity of viscous liquid in -direction - volumetric flow - eigenvalues of confluent hypergeometric function - streamfunction - _o wall of paraboloidal tube - _o inlet of paraboloidal tube - , , paraboloidal coordinates     

An iteration method for integral equations arising from axisymmetric loading problems

Let the concentrated forces and the centers of pressure with unknown density functions x(ξ) and y(ξ) respectively be distributed along the axis z outside the solid, then one can reduce an axismmetric loading problem of solids of revolution to two simultaneous Fredholm integral equations. An iteration method for solving such equations is duscussed. A lemma equivalent to E. Rakotch’s contractive mapping theorem and a theorem concerning the convergent proof of the iteration method are presented.     

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

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

Diffuse approximation method for solving natural convection in porous media

The diffuse approximation is presented and applied to natural convection problems in porous media. A comparison with the control volume-based finite-element method shows that, overall, the diffuse approximation appears to be fairly attractive.Nomenclature H height of the cavities - I functional - K permeability - p(M _i ,M) line vector of monomials - p ^T p-transpose - M current point - Nu Nusselt number - Ri inner radius - Ro outer radius - Ra Rayleigh number - x, y cartesian coordinates - u, v velocity components - T temperature - M vector of estimated derivatives - _t thermal diffusivity - coefficient of thermal expansion - practical aperture of the weighting function - scalar field - (M, M _i ) weighting function - streamfunction - kinematic viscosity     

Two-phase flow in heterogeneous porous media II: Numerical experiments for flow perpendicular to a stratified system

In this work, we make use of numerical experiments to explore our original theoretical analysis of two-phase flow in heterogeneous porous media (Quintard and Whitaker, 1988). The calculations were carried out with a two-region model of a stratified system, and the parameters were chosen be consistent with practical problems associated with groundwater flows and petroleum reservoir recovery processes. The comparison between theory (the large-scaled averaged equations) and experiment (numerical solution of the local volume averaged equations) has allowed us to identify conditions for which the quasi-static theory is acceptable and conditions for which a dynamic theory must be used. Byquasi-static we mean the following: (1) The local capillary pressure,everywhere in the averaging volume, can be set equal to the large-scale capillary pressure evaluated at the centroid of the averaging volume and (2) the large-scale capillary pressure is given by the difference between the large-scale pressures in the two immiscible phases, and is therefore independent of gravitational effects, flow effects and transient effects. Bydynamic, we simply mean a significant departure from the quasi-static condition, thus dynamic effects can be associated with gravitational effects, flow effects and transient effects. To be more precise about the quasi-static condition we need to refer to the relation between the local capillary pressure and the large-scale capillary pressure derived in Part I (Quintard and Whitaker, 1990). Herep _c ¦_y represents the local capillary pressure evaluated at a positiony relative to the centroid of the large-scale averaging volume, and {p _c }¦_x represents the large-scale capillary pressure evaluated at the centroid.In addition to{p _c } ^c being evaluated at the centroid, all averaged terms on the right-hand side of Equation (1) are evaluated at the centroid. We can now write the equations describing the quasi-static condition as , , This means that the fluids within an averaging volume are distributed according to the capillary pressure-saturation relationwith the capillary pressure held constant. It also means that the large-scale capillary pressure is devoid of any dynamic effects. Both of these conditions represent approximations (see Section 6 in Part I) and one of our main objectives in this paper is to learn something about the efficacy of these approximations. As a secondary objective we want to explore the influence of dynamic effects in terms of our original theory. In that development only the first four terms on the right hand side of Equation (1) appeared in the representation for the local capillary pressure. However, those terms will provide an indication of the influence of dynamic effects on the large-scale capillary pressure and the large-scale permeability tensor, and that information provides valuable guidance for future studies based on the theory presented in Part I.Roman Letters A scalar that maps {}*/t onto - A scalar that maps {}*/t onto - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - a vector that maps ({}*/t) onto , m - a vector that maps ({}*/t) onto , m - b vector that maps ({p}– g) onto , m - b vector that maps ({p}– g) onto , m - B second order tensor that maps ({p}– g) onto , m² - B second order tensor that maps ({p}– g) onto , m² - c vector that maps ({}*/t) onto , m - c vector that maps ({}*/t) onto , m - C second order tensor that maps ({}*/t) onto , m² - C second order tensor that maps ({}*/t) onto . m² - D third order tensor that maps ( ) onto , m - D third order tensor that maps ( ) onto , m - D second order tensor that maps ( ) onto , m² - D second order tensor that maps ( ) onto , m² - E third order tensor that maps () onto , m - E third order tensor that maps () onto , m - E second order tensor that maps () onto - E second order tensor that maps () onto - p _c =(), capillary pressure relationship in the-region - p _c =(), capillary pressure relationship in the-region - g gravitational vector, m/s² - largest of either or - - - i unit base vector in thex-direction - I unit tensor - K local volume-averaged-phase permeability, m² - K local volume-averaged-phase permeability in the-region, m² - K local volume-averaged-phase permeability in the-region, m² - {K } large-scale intrinsic phase average permeability for the-phase, m² - K –{K }, large-scale spatial deviation for the-phase permeability, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K ^* large-scale permeability for the-phase, m² - L characteristic length associated with local volume-averaged quantities, m - characteristic length associated with large-scale averaged quantities, m - I _i i = 1, 2, 3, lattice vectors for a unit cell, m - l characteristic length associated with the-region, m - ; characteristic length associated with the-region, m - l _H characteristic length associated with a local heterogeneity, m - - n unit normal vector pointing from the-region toward the-region (n =–n ) - n unit normal vector pointing from the-region toward the-region (n =–n ) - p pressure in the-phase, N/m² - p local volume-averaged intrinsic phase average pressure in the-phase, N/m² - {p } large-scale intrinsic phase average pressure in the capillary region of the-phase, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - P _c p –{p }, capillary pressure, N/m² - {p_c}^c large-scale capillary pressure, N/m² - r ₀ radius of the local averaging volume, m - R ₀ radius of the large-scale averaging volume, m - r position vector, m - , m - S /, local volume-averaged saturation for the-phase - S ^* {}*{}*, large-scale average saturation for the-phaset time, s - t time, s - u , m - U , m² - v -phase velocity vector, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - {v } large-scale intrinsic phase average velocity for the-phase in the capillary region of the-phase, m/s - {v } large-scale phase average velocity for the-phase in the capillary region of the-phase, m/s - v –{v }, large-scale spatial deviation for the-phase velocity, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - V local averaging volume, m³ - V volume of the-phase in, m³ - V large-scale averaging volume, m³ - V capillary region for the-phase within, m³ - V capillary region for the-phase within, m³ - V _c intersection of m³ - V volume of the-region within, m³ - V volume of the-region within, m³ - V () capillary region for the-phase within the-region, m³ - V () capillary region for the-phase within the-region, m³ - V () , region in which the-phase is trapped at the irreducible saturation, m³ - y position vector relative to the centroid of the large-scale averaging volume, m Greek Letters local volume-averaged porosity - local volume-averaged volume fraction for the-phase - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region (This is directly related to the irreducible saturation.) - {} large-scale intrinsic phase average volume fraction for the-phase - {} large-scale phase average volume fraction for the-phase - {}* large-scale spatial average volume fraction for the-phase - –{}, large-scale spatial deviation for the-phase volume fraction - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - a generic local volume-averaged quantity associated with the-phase - mass density of the-phase, kg/m³ - mass density of the-phase, kg/m³ - viscosity of the-phase, N s/m² - viscosity of the-phase, N s/m² - interfacial tension of the - phase system, N/m - , N/m - , volume fraction of the-phase capillary (active) region - , volume fraction of the-phase capillary (active) region - , volume fraction of the-region ( + =1) - , volume fraction of the-region ( + =1) - {p }– g, N/m³ - {p }– g, N/m³     

Determination of permeability tensors for two-phase flow in homogeneous porous media: Theory

In this paper we continue previous studies of the closure problem for two-phase flow in homogeneous porous media, and we show how the closure problem can be transformed to a pair of Stokes-like boundary-value problems in terms of pressures that have units of length and velocities that have units of length squared. These are essentially geometrical boundary value problems that are used to calculate the four permeability tensors that appear in the volume averaged Stokes' equations. To determine the geometry associated with the closure problem, one needs to solve the physical problem; however, the closure problem can be solved using the same algorithm used to solve the physical problem, thus the entire procedure can be accomplished with a single numerical code.Nomenclature a a vector that maps V onto , m^-1. - A a tensor that maps V onto . - A area of the - interface contained within the macroscopic region, m². - A area of the -phase entrances and exits contained within the macroscopic region, m². - A area of the - interface contained within the averaging volume, m². - A area of the -phase entrances and exits contained within the averaging volume, m². - Bo Bond number (= (=(–)g²/). - Ca capillary number (= v/). - g gravitational acceleration, m/s². - H mean curvature, m^-1. - I unit tensor. - permeability tensor for the -phase, m². - viscous drag tensor that maps V onto V. - ^* dominant permeability tensor that maps onto v , m². - ^* coupling permeability tensor that maps onto v , m². - characteristic length scale for the -phase, m. - l characteristic length scale representing both and , m. - L characteristic length scale for volume averaged quantities, m. - n unit normal vector directed from the -phase toward the -phase. - n unit normal vector representing both n and n . - n unit normal vector representing both n and n . - P pressure in the -phase, N/m². - p superficial average pressure in the -phase, N/m². - p intrinsic average pressure in the -phase, N/m². - p –p , spatial deviation pressure for the -phase, N/m². - r ₀ radius of the averaging volume, m. - r position vector, m. - t time, s. - v fluid velocity in the -phase, m/s. - v superficial average velocity in the -phase, m/s. - v intrinsic average velocity in the -phase, m/s. - v –v , spatial deviation velocity in the -phase, m/s. - V volume of the -phase contained within the averaging volmue, m³. - averaging volume, m³. Greek Symbols V /, volume fraction of the -phase. - viscosity of the -phase, Ns/m². - density of the -phase, kg/m³. - surface tension, N/m. - (v +v ^T ), viscous stress tensor for the -phase, N/m².     

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.     

Measurement of the Coefficient of Transverse Dispersion in Flow Through Packed Beds for a Wide Range of Values of the Schmidt Number

Experimental values of the coefficient of transverse dispersion (D _T) were measured with the system 2-naphthol/water, over a range of temperatures between 293K and 373K, which corresponds to a range of values of viscosity () between 2.83×10^–4 Ns/m² and 1.01×10^–3 Ns/m² and of molecular diffusion coefficient (D _m) between 1.03×10^–9 m²/s and 5.49×10^–9 m²/s. Since the density () of water is close to 10³ kg/m³, the corresponding variation of the Schmidt number (Sc=/D _m) was in the range 1000 – 50. More than 200 experimental values of the transverse dispersion coefficient were obtained using beds of silica sand with average particle sizes (d) of 0.297 and 0.496mm, operated over a range of interstitial liquid velocities (u) between 0.1mm/s and 14mm/s and this gave a variation of the Reynolds number (Re=du/) between 0.01 and 3.5.Plots of the dimensionless coefficient of transverse dispersion (D _T/D _m) vs. the Peclet number (Pe_m=ud/D _m) based on molecular diffusion bring into evidence the influence of Sc on transverse dispersion. As the temperature is increased, the value of Sc decreases and the values of D _T/D _m gradually approach the line corresponding to gas behaviour (i.e. Sc 1), which is known to be well approximated by the equation D _T/D _m=1/+ud/12D _m, where is the tortuosity with regard to diffusion.