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     

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     

Convective heat transfer within fibrous insulation slabs

A numerical study of convective heat flow within a fibrous insulating slab is presented. The material is treated as an anisotropic porous medium and the variation of properties with temperature is taken into account. Good agreement is obtained with available experimental data for the same geometry.

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Zusammenfassung Für den konvektiven Wärmestrom in einem faserförmigen Isolierstoff wird eine numerische Berechnung angegeben. Der Stoff wird als anisotropes poröses Medium mit temperaturabhängigen Stoffwerten angesehen. Die Übereinstimmung mit verfügbaren Versuchswerten ist gut.

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Nomenclature C_p specific heat of the gas at the mean temperature - Da Darcy number=k_y/H² - Gr^* modified Grashof number=gTHk_y/²= (Grashof number) × (Darcy number) - H thickness of the specimen - P gas pressure - Pr^* modified Prandtl number= C_p/_x - Ra^* modified Rayleigh number=Gr^* Pr^* - R_p ratio of permeabilities=k_y/k_x - R_k ratio of conductivities= _y/_x - T absolute temperature of the gas - t₁ absolute temperature of the hot face - T₂ absolute temperature of the cold face - T_m mean temperature of the gas=(T₁+T₂)/2 - k_x specific permeability of the porous medium along the x-direction - k_y specific permeability of the porous medium along the y-direction - p T/T_m - q exponent - r exponent - u gas velocity along the x-direction - v gas velocity along the y-direction - X_* distance along the x-direction - y_* distance along the y-direction - T temperature difference=t₁–T₂ - thermal coefficient of expansion of the gas - _m thermal coefficient of expansion of the gas at the mean temperature - _* T–T_m - dimensionless temperature= ^*/T - _a apparent thermal conductivity of the porous medium along the x-direction - _al local apparent thermal conductivity of the porous medium along the x-direction - _x thermal conductivity of the porous medium along the x-direction in the absence of convection - _y thermal conductivity of the porous medium along the y-direction in the absence of convection - dynamic viscosity of the gas - _m dynamic viscosity of the gas at the mean temperature - kinematic viscosity of the gas - _m kinematic viscosity of the gas at the mean temperature - density of the gas - _m density of the gas at the mean temperature - ^* stream function at any point - dimensionless stream function= ^*/( _m/_m)     

New concepts in relative permeabilities at high capillary numbers for surfactant flooding

Flooding oil reservoirs with surfactant solutions can increase the amount of oil that can be recovered. Macroscopic modelling of the process requires relative permeabilities to be functions of saturation and capillary number. With only limited experimental data, relative permeabilities have usually been assumed to be linear functions of saturation at high capillary numbers. The experimental data is reviewed, some of which suggest that this assumption is not necessarily correct. The basis for the assumption is therefore reviewed and it is concluded that the linear model corresponds to microscopically segregated flow in the porous medium. Based on new but equally plausible complementary assumptions about the flow pattern, a mixed flow model is derived. These models are then shown to be limiting cases of a droplet model which represents the mixing scale within the porous medium and gives a physical basis for interpolating between the models. The models are based on physical concepts of flow in a porous medium and so the approach described here represents a significant improvement in the understanding of high capillary number flow. This is shown by the fact that fewer parameters are needed to describe experimental data.Notation A total cross-sectional area assigned to capillary bundle - A ⁽ⁱ⁾ physical cross-sectional area of tube i - c ⁽ⁱ⁾ ordered configurational label for droplets in tube i - c configuration label for tube i (order not considered) - D defined by Equation (26) - E(...) expectation value with respect to the trinomial distribution - S _r ⁽⁾ fractional flow of phase - k absolute permeability - k _r relative permeability of phase - k _r ⁰ endpoint relative permeability of phase - L capillary tube length in bundle model - m ⁽ⁱ⁾ number of droplets of phase a occupying tube i - n exponent for phase a in Equation (2) - N number of droplets in bundle model - N _c capillary number - p pressure - p(c') probability of configuration c - Q ⁽ⁱ⁾ total volume flow rate in tube i - S saturation of phase - S flowing saturation of phase - S _r residual saturation of phase - S _r ⁽⁾ saturations when fractional flow of phase is 1 in the case of varying residual saturations for three-phase flow ( ) - t _c residence time for droplet configuration c - v ⁽ⁱ⁾ total fluid velocity in bundle tube i - , phase label - p pressure differential across capillary bundle - ⁽ⁱ⁾ tube conductivity defined by Equation (7) - viscosity of phase - interfacial tension - gradient operator - ... average over tube droplet configurations     

On shock layer method for the hypersonic flow problems

Chernyi’s series method[1] is not proper for the case that(γ-l)/(γ+l)<<2/(γ+1)×M²sin²β (γ=c_p/c_v-adiabatic index number, M-Much number, β-shock incidence). In this paper, we only suppose that in the neighbour of the shock, there exists a shock layer in which the density of the gas is very big, but we do not remove the case that (γ-1)/(γ+1)<<2/(γ+1)M²sin²β.     

On the closure problem for Darcy's law

In a previous derivation of Darcy's law, the closure problem was presented in terms of an integro-differential equation for a second-order tensor. In this paper, we show that the closure problem can be transformed to a set of Stokes-like equations and we compare solutions of these equations with experimental data. The computational advantages of the transformed closure problem are considerable.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 _e area of entrances and exits for the-phase contained within the averaging volume, m² - B second-order tensor used to respresent the velocity deviation - b vector used to represent the pressure deviation, m^–1 - C second-order tensor related to the permeability tensor, m^–2 - D second-order tensor used to represent the velocity deviation, m² - d vector used to represent the pressure deviation, m - g gravity vector, m/s² - I unit tensor - K C ^–1,–D, Darcy's law permeability tensor, m² - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - l _i i=1, 2, 3, lattice vectors, m - 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 - p pressure in the-phase, N/m ² - p intrinsic phase average pressure, N/m² - p –p , spatial deviation of the pressure in the-phase, N/m² - r position vector locating points in the-phase, m - r ₀ radius of the averaging volume, m - t time, s - v velocity vector in the-phase, m/s - v intrinsic phase average velocity in the-phase, m/s - v phase average or Darcy velocity in the \-phase, m/s - v –v , spatial deviation of the velocity in the-phase m/s - V averaging volume, m³ - V volume of the-phase contained in 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     

Injection of ideal fluid from a slot into a free stream

A cold gas is injected from a slot into a free stream of hot gas. In a simple model this leads to a two-fluid free boundary problem with the jump relation |u^-|²–|u⁺|² = ( constant) on the free boundary {u=0}, where u is the stream function. We prove that for any (–1, ) there exists a unique solution (Q, u) where Q is the flux of the injected fluid. Various properties of the solution u and of the free boundary are established.     

The effect of background particulates on the delayed transition of a heated 9:1 ellipsoid

An experimental study was done to quantify the effects of a variety of background particulates on the delayed laminar-turbulent transition of a thermally stabilized boundary layer in water. A Laser-Doppler Velocimeter system was used to measure the location of boundary layer transition on a 50 mm diameter, 9:1 fineness ratio ellipsoid. The ellipsoid had a 0.15 m RMS surface finish. Boundary layer transition locations were determined for length Reynolds numbers ranging from 3.0 × 10⁶ to 7.5 × 106. The ellipsoid was tested in three different heating conditions in water seeded with particles of four distinct size ranges. For each level of boundary layer heating, measurements of transition were made for clean water and subsequently, water seeded with 12.5 m, 38.9 m, 85.5 m and 123.2 m particles, alternately. The three surface heating conditions tested were no heating, T = 10°C and T = 15°C where T is the difference between the inlet model heating water temperature, T _i, and free stream water temperature, T . The effects of particle concentration were studied for 85.5 m and 123.2 m particulates.The results of the study can be summarized as follows. The 12.5 m and 38.9 m particles has no measurable effect on transition for any of the test conditions. However, transition was significantly affected by the 85.5 m and 123.2 m particles. Above a length Reynolds number of 4 × 10⁶ the boundary layer transition location moved forward on the body due to the effect of the 85.5 m particles for all heating conditions. The largest percentage changes in transition location from clean water, were observed for 85.5 m particles seeded water.Transition measurements made with varied concentrations of background particulates indicated that the effect of the 85.5 m particles on the transition of the model reached a plateau between 2.65 particulates/ml concentration and 4.2 particles/ml. Measurements made with 123.3 m particles at concentrations up to 0.3 part/ml indicated no similar plateau.     

Flow through a helically coiled annulus

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

Non-isothermal masstransport in a circular tube for flow of Non-Newtonian liquid

The non-isothermal mass transport in a circular tube has been investigated for an Ostwald-de-Waele liquid. The influence of the velocity profile parameter, the flow parameter, the thermal diffusion, the diffusive heat transfer as well as the ratio of the Schmidt- to the Prandtl number upon the concentration- and temperature profile along the tube have been determined. The results are compared with the plug flow case and also with the isothermal mass transport. It was found that considerable differences in the mass transport and heat transport occur through their coupling effect.

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Nichtisothermer Stofftransport in einem Rohr für die Strömung einer nicht-Newtonschen Flüssigkeit

Zusammenfassung Für eine Ostwald-de-Waele Flüssigkeit wurde der nichtisotherme Stofftransport in einem Kreiszylinderrohr untersucht. Dabei wurde der Einfluß des Geschwindigkeitsparameters, des Strömungsparameters, der Wärmediffusion, des diffusen Wärmetransportes, sowie der Einfluß des Verhältnisses von Schmidt- zur Prandtlzahl auf das Konzentrations- und Temperaturprofil entlang der Rohrlauflänge untersucht. Die Ergebnisse wurden mit dem Fall eines Kolbenprofiles und mit denen des isothermen Stofftransportes verglichen. Dabei konnten erhebliche Differenzen in der Konzentrations- und Wärmeverteilung durch den Koppelungseffekt festgestellt werden.

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Nomenclature a radius of tube - c concentration - c _i initial concentration at inlet z=0 - c _w wall concentration - D diffusion coefficient - J _m Besselfunction of first kind andm ^th order - j diffusive mass flux - K material constant - p liquid pressure - dp/dz pressure gradient along tube - q heat flux - r, ,z cylindrical coordinates - S entropy - T temperature { _{Tw wall temperature} ^{Ti initial temperature at inlet} - w axial velocity of liquid - k _T thermal diffusion ratio - N thermodynamic parameter - .=k _T/T parameter - _n zeros ofJ ₀ (_n)=0 - parameter (=0 plug flow, 1 Ostwald-de-Waele flow) - material constant { _{< 1 dilatant} ^{> 1 pseudoplastics} } - effective chemical potential - thermal conductivity - thermal diffusivity of mixture - liquid density - dynamic viscosity of Newtonian liquid - shear stress - eigenvalues     

The value of the critical exponent for reaction-diffusion equations in cones

Let D R ^N be a cone with vertex at the origin i.e., D = (0, )x where S ^N–1 and x D if and only if x = (r, ) with r=¦x¦, . We consider the initial boundary value problem: u _t = u+u ^p in D×(0, T), u=0 on Dx(0, T) with u(x, 0)=u ₀(x) 0. Let ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on and let ₊ denote the positive root of (+N–2) = ₁. Let p ^* = 1 + 2/(N + ₊). If 1 < p < p ^*, no positive global solution exists. If p>p ^*, positive global solutions do exist. Extensions are given to the same problem for u _t=+¦x¦ u ^p .This research was supported in part by the Air Force Office of Scientific Research under Grant # AFOSR 88-0031 and in part by NSF Grant DMS-8 822 788. The United States Government is authorized to reproduce and distribute reprints for governmental purposes not withstanding any copyright notation therein.     

The Conjugacy of Stochastic and Random Differential Equations and the Existence of Global Attractors

We consider stochastic differential equations in d-dimensional Euclidean space driven by an m-dimensional Wiener process, determined by the drift vector field f₀ and the diffusion vector fields f₁,...,f_m, and investigate the existence of global random attractors for the associated flows . For this purpose is decomposed into a stationary diffeomorphism given by the stochastic differential equation on the space of smooth flows on R^d driven by m independent stationary Ornstein Uhlenbeck processes z¹,...,z^m and the vector fields f₁,...,f_m, and a flow generated by the nonautonomous ordinary differential equation given by the vector field (_t/x)^–1[f₀(_t)+ _i=1 ¹ f_i(_t)z _t ⁱ ]. In this setting, attractors of are canonically related with attractors of . For , the problem of existence of attractors is then considered as a perturbation problem. Conditions on the vector fields are derived under which a Lyapunov function for the deterministic differential equation determined by the vector field f₀ is still a Lyapunov function for , yielding an attractor this way. The criterion is finally tested in various prominent examples.     

Instationäre Messung der Wärmeleitfähigkeit mit optischer Registrierung

Zusammenfassung Die Wärmeleitfähigkeit von Wasser wird im Temperaturbereich von 20 bis 90 °C und bei 1 bar mit einem neuen instationären Absolutverfahren bestimmt. Zur Aufzeichnung des instationären Feldes des Brechungsindex und der Temperatur werden zwei interferometrische Anordnungen benutzt: Die Methode vonMach-Zehnder und das Biprismaverfahren. Die Ergebnisse stehen in guter Übereinstimmung zu den Messungen anderer Autoren, die stationäre Methoden benutzten. Die Unsicherheit der 374 Einzelmessungen wird auf höchstens±1% geschätzt. Damit ist nachgewiesen, daß ein instationäres Meßverfahren mit optischer Registrierung mit den klassischen stationären Verfahren hinsichtlich der Meßunsicherheit konkurrieren kann. Das instationäre Verfahren kommt ohne kalorische Messungen aus und besteht bei optischer Registrierung im wesentlichen aus Längenmessungen.

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Unsteady-state measurements of the thermal conductivity with optical recording

The thermal conductivity of water in the temperature region from 20 to 90°C and at 1 bar was measured by means of a new unsteady-state absolute method. To record the unsteady-state field of the index of refraction and of the temperature, two interferometric arrangements were used: TheMach-Zehnder and the biprisma methods. The results are in good agreement with measurements of other authors, who had used steady-state methods. The maximum degree of uncertainty of the 374 measurements is estimated to be±1%. Thus it is shown that unsteady-state methods with optical recording can well be compared with classical steady-state methods regarding uncertainties. The method does not require caloric measurements and uses primarly determinations of lengths.

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Bezeichnungen A=2 a in den Auswerteverfahren gebrauchte Abkürzung, m - B=q _x=0/ in den Auswerteverfahren gebrauchte Abkürzung, grd/m - F Fläche, m² - Fo=a/x ² Fourierzahl - I Strom, A - R Widerstand, - U Spannung, V - T Kelvintemperatur, °K - a=/c _p Temperaturleitfähigkeit, m²/s - Wärmeeindringzahl, Ws^1/2/m²grd - c _p isobare spezifische Wärmekapazität, kJ/kggrd - l Modellänge, m - n Brechungsindex - q Wärmestromdichte, W/m² - t Celsiustemperatur, °C - =1/ spezifisches Volumen, m³/kg - x Wandabstand, Ortskoordinate, m - z Ordnungszahl der Interferenzstreifen - =/_x=0 dimensionslose Übertemperatur - n=n -n _x=0 Brechungsindexdifferenz - Verhältnis der Wärmeeindringzahlen - Übertemperatur, grd - Wärmeleitfähigkeit, W/mgrd - Lichtwellenlänge, m - =1/ Dichte, kg/msu3 - Zeit, s Indices Zustand des Bades, Umgebung - x=0 Wand - z Stelle der z-ten Ordnung - i Laufparameter - err errechneter Wert - mess Meßwert - Bez Bezug Auszug aus der von der Fakultät für Maschinenwesen, und Elektrotechnik der Technischen Hochschule München genehmigten Dissertation von J.Bach.     

Theoretical investigation of the effect of an internal heat source on turbulent liquid metal heat transfer in a channel with uniform heat flux

Summary The effect of an internal heat source on the heat transfer characteristics for turbulent liquid metal flow between parallel plates is studied analytically. The analysis is carried out for the conditions of uniform internal heat generation, uniform wall heat flux, and fully established temperature and velocity profiles. Consideration is given both to the uniform or slug flow approximation and the power law approximation for the turbulent velocity profile. Allowance is made for turbulent eddying within the liquid metal through the use of an idealized eddy diffusivity function. It is found that the Nusselt number is unaffected by the heat source strength when the velocity profile is assumed to be uniform over the channel cross section. In the case of a 1/7-power velocity expression, the Nusselt numbers are lower than those in the absence of internal heat generation, and decrease with diminishing eddy conduction. Nusselt numbers, in the absence of an internal heat source, are compared with existing calculations, and indications are that the present results are adequate for preliminary design purposes.Nomenclature A hydrodynamic parameter - a half height of channel - a ₁ a constant, 1+0.01 Pr Re ^0.9 - a ₂ a constant, 0.01 Pr Re ^0.9 - C _p specific heat at constant pressure - D _h hydraulic diameter of channel, 4a - h heat transfer coefficient, q _w/(t _w–t _b) - I ₁ integral defined by (17) - I ₂ integral defined by (18) - k diffusivity parameter, (1+0.01 Pr Re ^0.9)^1/2 - m exponent in power velocity expression - Nu Nusselt number, hD _h/ - Nu ₀ Nusselt number in absence of internal heat generation - Pr Prandtl number, / - Q heat generation rate per volume - q _w wall heat flux - Re Reynolds number for channel, 2/ - s ratio of heat generation rate to wall heat flux, Qa/q _w - T dimensionless temperature, (t _w–t)/(t _w–t _b) - t fluid temperature, t _w wall temperature, t _b fluid bulk temperature - u fluid velocity in x direction, , fluid mean velocity - x longitudinal coordinate measured from channel entrance - x ⁺ dimensionless longitudinal coordinate, 2(x/a)/Pr Re - y transverse coordinate measured from channel centerline - z transverse coordinate measured from channel wall, a–y - molecular diffusivity of heat, /C _p - dummy variable of integration - dummy variable of integration - _H eddy diffusivity of heat - _M eddy diffusivity of momentum - dummy variable of integration - fluid thermal conductivity - _T dimensionless diffusivity, Pr ( _H/) - fluid kinematic viscosity - dummy variable of integration - fluid density - dummy variable of integration - ratio of eddy diffusivity for heat transfer to that for momentum transfer, _H/ _M - average value of - dimensionless velocity distribution, u/     

The Effect of Wettability on Three-Phase Relative Permeability

We study three-phase flow in water-wet, oil-wet, and fractionally-wet sandpacks. We use CT scanning to measure directly the oil and water relative permeabilites for three-phase gravity drainage. In an analogue experiment, we measure pressure gradients in the gas phase to determine the gas relative permeability. Thus we find all three relative permeabilities as a function of saturation. We find that the gas relative permeability is approximately half as much in a oil-wet medium than in an water-wet medium at the same gas saturation. The water relative permeability in the water-wet medium and the oil relative permeability in the oil-wet medium are similar. In the water-wet medium the oil relative permeability scales as k _roS _o ⁴ for S _o>S _or, where S _or is the waterflood residual oil saturation. With octane as the oil phase, k _roS _o ² for S _o<S _or, while with decane as the oil phase, k _ro falls sharply for S _o<S _or. The water relative permeability in the oil-wet medium resembles the oil relative permeability in the water-wet medium for a non-spreading oil such as decane. These observations can be explained in terms of wetting, spreading, and the pore scale configurations of fluid.     

Testing viscometric predictions of the internal viscosity model,using dilute viscous theta solutions

A stress-symmetrized internal viscosity (I.V.) model for flexible polymer chains, proposed by Bazua and Williams, is scrutinized for its theoretical predictions of complex viscosity ^* () = – i and non-Newtonian viscosity (), where is frequency and is shear stress. Parameters varied are the number of submolecules,N (i.e., molecular weightM = NM _s ); the hydrodynamic interaction,h ^*; and/f, where andf are the I.V. and friction coefficients of the submolecule. Detailed examination is made of the eigenvalues _p (N, h ^*) and how they can be estimated by various approximations, and property predictions are made for these approximations.Comparisons are made with data from our preceding companion paper, representing intrinsic properties [], [], [] in very viscous theta solutions, so that theoretical foundations of the model are fulfilled. It is found that [ ()] data can be predicted well, but that [ ()] data cannot be matched at high. The latter deficiency is attributed in part to unrealistic predictions of coil deformation in shear.     

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 method for measuring time constants of a thermocouple wire in varying flow states

A new measuring method is suggested for determining the time constant of a thermocouple wire to be applied for the measurement of the true fluid temperatures in varying flow states. Based on the techniques of internal heating which are commonly used to measure mean time constants, we extend the existing method to measure instantaneous time constants continuously. A method of measurement and analysis is presented and verified experimentally.List of Symbols A _s surface area [m²] - c specific heat [J/kg K] - D diameter [m] - h heat transfer coefficient [W/m² K] - I current [A] - k thermal conductivity [W/m K] - L length [m] - r resistance per unit length [/m] - T temperature [°C] - t time [s] - t _c characteristic time to reach uniform state [s] - u velocity of stream [m/s] - V volume [m³] - x axial coordinate [m] - thermal diffusivity [m²/s] - normalized temperature (TT )/(T _RT )) - density [kg/m³] - time constant [s] - angular velocity [rad/s] - a amplitude - i initial condition - j junction of thermocouple - R reference point - surrounding The work was supported by Turbo and Power Machinery Research Center at Seoul National University and the authors are grateful to Mr. M. H. Yang for his assistance in the experiment.     

Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions

In this paper, we show that the maximum principle holds for quasilinear elliptic equations with quadratic growth under general structure conditions.Two typical particular cases of our results are the following. On one hand, we prove that the equation (1) {ie77-01} where {ie77-02} and {ie77-03} satisfies the maximum principle for solutions in H ¹()L(), i.e., that two solutions u ₁, u ₂H¹() L() of (1) such that u ₁u₂ on , satisfy u ₁u₂ in . This implies in particular the uniqueness of the solution of (1) in H ₀ ¹ ()L().On the other hand, we prove that the equation (2) {ie77-04} where fH^–1() and g(u)>0, g(0)=0, satisfies the maximum principle for solutions uH¹() such that g(u)¦Du|{²L¹(). Again this implies the uniqueness of the solution of (2) in the class uH ₀ ¹ () with g(u)¦Du|{²L¹().In both cases, the method of proof consists in making a certain change of function u=(v) in equation (1) or (2), and in proving that the transformed equation, which is of the form (3) {ie77-05}satisfies a certain structure condition, which using ((v₁ -v ₂)⁺)ⁿ for some n>0 as a test function, allows us to prove the maximum principle.     

Flow in porous media III: Deformable media

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