Two-phase flow in heterogeneous porous media III: Laboratory experiments for flow parallel to a stratified system

Two-phase flow in stratified porous media is a problem of central importance in the study of oil recovery processes. In general, these flows are parallel to the stratifications, and it is this type of flow that we have investigated experimentally and theoretically in this study. The experiments were performed with a two-layer model of a stratified porous medium. The individual strata were composed of Aerolith-10, an artificial: sintered porous medium, and Berea sandstone, a natural porous medium reputed to be relatively homogeneous. Waterflooding experiments were performed in which the saturation field was measured by gamma-ray absorption. Data were obtained at 150 points distributed evenly over a flow domain of 0.1 × 0.6 m. The slabs of Aerolith-10 and Berea sandstone were of equal thickness, i.e. 5 centimeters thick. An intensive experimental study was carried out in order to accurately characterize the individual strata; however, this effort was hampered by both local heterogeneities and large-scale heterogeneities.The theoretical analysis of the waterflooding experiments was based on the method of large-scale averaging and the large-scale closure problem. The latter provides a precise method of discussing the crossflow phenomena, and it illustrates exactly how the crossflow influences the theoretical prediction of the large-scale permeability tensor. The theoretical analysis was restricted to the quasi-static theory of Quintard and Whitaker (1988), however, the dynamic effects described in Part I (Quintard and Whitaker 1990a) are discussed in terms of their influence on the crossflow.Roman Letters A interfacial area between the -region and the -region contained within V, m² - a vector that maps onto , m - b vector that maps onto , m - b vector that maps onto , m - B second order tensor that maps onto , m² - C second order tensor that maps onto , m² - E energy of the gamma emitter, keV - f fractional flow of the -phase - g gravitational vector, m/s² - h characteristic length of the large-scale averaging volume, m - H height of the stratified porous medium , m - i unit base vector in the x-direction - K local volume-averaged single-phase permeability, m² - K - {K}, large-scale spatial deviation permeability - { K} large-scale volume-averaged single-phase permeability, m² - K ^* large-scale single-phase permeability, m² - K ^** equivalent large-scale single-phase permeability, m² - K local volume-averaged -phase permeability in the -region, m² - K local volume-averaged -phase permeability in the -region, m² - K - {K } , large-scale spatial deviation for the -phase permeability, m² - K ^* large-scale permeability for the -phase, m² - l thickness of the porous medium, m - l characteristic length for the -region, m - l characteristic length for the -region, m - L length of the experimental porous medium, m - characteristic length for large-scale averaged quantities, m - n outward unit normal vector for the -region - n outward unit normal vector for the -region - n unit normal vector pointing from the -region toward the -region (n = - n ) - N number of photons - p pressure in the -phase, N/m² - p ₀ reference pressure in the -phase, N/m² - local volume-averaged intrinsic phase average pressure in the -phase, N/m² - large-scale volume-averaged pressure of the -phase, N/m² - large-scale intrinsic phase average pressure in the capillary region of the -phase, N/m² - - , large-scale spatial deviation for the -phase pressure, N/m² - p_c , capillary pressure, N/m² - p _c capillary pressure in the -region, N/m² - p capillary pressure in the -region, N/m² - {p _c } ^c large-scale capillary pressure, N/m² - q -phase velocity at the entrance of the porous medium, m/s - q -phase velocity at the entrance of the porous medium, m/s - S_wi irreducible water saturation - S /, local volume-averaged saturation for the -phase - S _i initial saturation for the -phase - S _r residual saturation for the -phase - S ^* { }^*/}^*, large-scale average saturation for the -phase - S saturation for the -phase in the -region - S saturation for the -phase in the -region - t time, s - v -phase velocity vector, m/s - v local volume-averaged phase average velocity for the -phase, m/s - {v } large-scale averaged velocity for the -phase, 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 -{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 large-scale averaging volume, m³ - y position vector relative to the centroid of the large-scale averaging volume, m - {y}^c large-scale average of y over the capillary region, m Greek Letters local porosity - local porosity in the -region - local porosity in the -region - local volume fraction for the -phase - local volume fraction for the -phase in the -region - local volume fraction for the -phase in the -region - {}^* { }^*+{ }^*, large-scale spatial average volume fraction - { }^* large-scale spatial average volume fraction for 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, Ns/m² - V /V , volume fraction of the -region ( + =1) - V /V , volume fraction of the -region ( + =1) - attenuation coefficient to gamma-rays, m^-1 - -      

Transport in ordered and disordered porous media III: Closure and comparison between theory and experiment

In this paper we examine the closure problem associated with the volume averaged form of the Stokes equations presented in Part II. For both ordered and disordered porous media, we make use of a spatially periodic model of a porous medium. Under these circumstances the closure problem, in terms of theclosure variables, is independent of the weighting functions used in the spatial smoothing process. Comparison between theory and experiment suggests that the geometrical characteristics of the unit cell dominate the calculated value of the Darcy's law permeability tensor, whereas the periodic conditions required for thelocal form of the closure problem play only a minor role.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 - A interfacial area of the- interface associated with the local closure problem, m² - A _p surface area of a particle, m² - b vector used to represent the pressure deviation, m^–1 - B ⁰ B+I, a second order tensor that maps v _m ontov - B second-order tensor used to represent the velocity deviation - d _p 6V _p/A_p, effective particle diameter, m - d a vector related to the pressure, m - D a second-order tensor related to the velocity, m² - g gravity vector, m/s² - I unit tensor - K traditional Darcy's law permeability tensor calculated on the basis of a spatially periodic model, m² - 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 characteristic length for the volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - characteristic length (pore scale) for the-phase - _i i=1, 2, 3 lattice vectors, m - weighting function - m(-y) , convolution product weighting function - 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 - 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 _m , spatial deviation pressure, N/m² - r ₀ radius of a spherical averaging volume, m - r _m support of the convolution product weighting function - r position vector, m - r position vector locating points in the-phase, m. - V averaging volume, m³ - B 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 _m superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - v traditional superficial volume averaged velocity, m/s - v – v _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     

Viscoelastic swirling flow with free surface in cylindrical chambers

The flow of a viscoelastic liquid driven by the steadily rotating bottom cover of a cylindrical cup is investigated. The flow field and the shape of the free surface are determined at the lowest significant orders of the regular domain perturbation in terms of the angular velocity of the bottom cap. The meridional field superposed on a primary azimuthal field shows a structure of multiple cells. The velocity field and the shape of the free surface are strongly effected by the cylinder aspect ratio and the elasticity of the liquid. The use of this flow configuration as a free surface rheometer to determine the first two Rivlin-Ericksen constants is shown to be promising.Nomenclature R, ,Z Coordinates in the physical domain D - , , Coordinates in the rest stateD ₀ - r, ,z Dimensionless coordinates in the rest stateD ₀ - Angular velocity - Zero shear viscosity - Surface tension coefficient - Density - Dimensionless surface tension parameter - ₁, ₂ The first two Rivlin-Ericksen constants - Stream function - Dimensionless second order meridional stream function - ^* Dimensionless second normal stress function - 2 Dimensionless sum of the first and second normal stress functions - N ₁,N ₂ The first and second normal stress functions - n Unit normal vector - D Stretching tensor - A _n nth order Rivlin-Ericksen tensor - S Extra-stress - u Velocity field - U Dimensionless second order meridional velocity field - V Dimensionless first order azimuthal velocity field - p Pressure - Modified pressure field - P Dimensionless second order pressure field - J Mean curvature - a Cylinder radius - d Liquid depth at rest - D Dimensionless liquid depth at rest - h Free surface height - H Dimensionless free surface height at the second order     

Thermal entrance region heat transfer for MHD laminar flow in parallel-plate channels with unequal wall temperatures

The problem of thermal entry heat transfer for Hartmann flow in parallel-plate channels with uniform but unequal wall temperatures considering viscous dissipation, Joule heating and axial conduction effects is approached by the eigenfunction expansion method. The series expansion coefficients for the nonorthogonal eigenfunctions are obtained by using a method for nonorthogonal series described by Kantorovich and Krylov [21]. Numerical results are obtained for the case with entrance condition parameter _o=1 and open circuit condition K=1. The parametric values of Ha=0, 2, 6, 10 and Br=0, –1 are considered for Hartmann and Brinkman numbers, respectively.

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Zusammenfassung Das Problem der Wärmeübertragung im thermischen Einlauf einer Hartmannströmung im ebenen Spalt mit einheitlichen, aber ungleichen Wandtemperaturen wurde unter Berücksichtigung viskoser Dissipation, Joulescher Heizung und axialer Wärmeleitung mit Hilfe einer Entwicklung nach Eigenfunktionen behandelt. Die Koeffizienten der Entwicklung für nichtorthogonale Eigenfunctionen wurde nach einer Methode für nichtorthogonale Reihen nach Kantorovicz und Krylow [21] berechnet. Numerische Ergebnisse werden für den Eintrittsparameter _o=1 und die Bedingung für den offenen Stromkreis K=1 erhalten. Die Parameterwerte Ha=0, 2, 6, 10 und Br=0, –1 werden für die jeweiligen Werte der Hartmann- und der Brinckman-Zahl betrachtet.

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Nomenclature a one-half of channel height - ¯B,B₀ magnetic field Induction vector and magnitude of applied magnetic field - Br Brinkman number, _f U_m ²/(k_c) - C_n,D_n coefficients in the series expansion of _e, see eq. 16 - c_p specific heat at constant pressure - ,E₀ electric field intensity vector and component - E_n,O_n even and odd eigenfunctions - Ha Hartmann number, (/_f)^1/2 B_o a - h₁,h₂ local heat transfer coefficients at lower and upper plates - ¯J,J_y electric current density vector and component - K external loading parameter, E_o/(B_o U_m) - k thermal conductivity - Nu₁, Nu₂ local Nusselt numbers, h₁,a/k and h₂a/k, respectively - P fluid pressure - Pe Peclet number, PrRe - Pr Prandtl number, C_{p f}/k - q₁,q₂ rates of heat transfer per unit area,^–k(T/Z)Z=–a'^–k(T/Z) Z=a respectively - Re Reynolds number, U_ma/u_f - T,T₀,T₁,T₂ fluid temperature, uniform entrance temperature, uniform but different lower and upper plate temperatures, respectively - T_b,T_m bulk temperature and (T₁+T₂)/2 - U,U_m,u axial, mean and dimensionless velocities, respectively - ¯V velocity vector - X,Z axial and transverse coordinates - x,z dimensionless coordinates - _n,_n even and odd eigenvalues - ,₀,_b dimensionless fluid, entrance and bulk temperatures, respectively - _c,_e,_f characteristic temperature difference (T₂-T_m), and dimensionless fluid temperatures, defined by eq. (10) - _e,_f magnetic permeability and viscosity of fluid - fluid density - electric conductivity - viscous dissipation function - (1-)/2     

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

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

Nomenclature

Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - a _i i=1, 2, 3 gaussian probability distribution used to locate the position of particles - I unit tensor - L general characteristic length for volume averaged quantities, m - L characteristic length for , m - L characteristic length for , m - characteristic length for the -phase particles, m - ⁰ reference characteristic length for the-phase particles, m - characteristic length for the-phase, m - _i i=1, 2, 3 lattice vectors, m - m convolution product weighting function - m _v special convolution product weighting function associated with the traditional volume average - n _i i=1, 2, 3 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward the-phase - n _e outwardly directed unit normal vector at the entrances and exits of the-phase - r _p position vector locating the centroid of a particle, m - r gaussian probability distribution used to determine the size of a particle, m - r ₀ characteristic length of an averaging region, m - r position vector, m - r _m support of the weighting functionm, m - averaging volume, m³ - V volume of the-phase contained in the averaging volume,, m³ - x positional vector locating the centroid of an averaging volume, m - x ₀ reference position vector associated with the centroid of an averaging volume, m - y position vector locating points relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - /L, small parameter in the method of spatial homogenization - standard deviation ofa _i - _r standard deviation ofr - _r intrinsic phase average of      

The chaotic response of a fluttering panel: The influence of maneuvering

The influence of maneuvering on the chaotic response of a fluttering buckled plate on an aircraft has been studied. The governing equations, derived using Lagrangian mechanics, include geometric non-linearities associated with the occurrence of tensile stresses, as well as coupling between the angular velocity of the maneuver and the elastic degrees of freedom. Numerical simulation for periodic and chaotic responses are conducted in order to analyze the influence of the pull-up maneuver on the dynamic behavior of the panel. Long-time histories phase-plane plots, and power spectra of the responses are presented. As the maneuver (load factor) increases, the system exhibits complicated dynamic behavior including a direct and inverse cascade of subharmonic bifurcations, intermittency, and chaos. Beside these classical routes of transition from a periodic state to chaos, our calculations suggest amplitude modulation as a possible new mode of transition to chaos. Consequently this research contributes to the understanding of the mechanisms through which the transition between periodic and strange attractors occurs in, dissipative mechanical systems. In the case of a prescribed time dependent maneuver, a remarkable transition between the different types of limit cycles is presented.Nomenclature a plate length - a _r u _r /h - D plate bending stiffness - E modulus of elasticity - g acceleration due to gravity - h plate thickness - j₁,j₂,j₃ base vectors of the body frame of reference - K spring constant - M Mach number - n 1 + ₀/g - N ₁ applied in-plane force - p–p aerodynamic pressure - P pa ⁴/Dh - q ₀/2 - Q _r generalized Lagrangian forces - R rotation matrix - R ₄ N, a ²/D - t time - kinetic energy - u plate deflection - u displacement of the structure - u _r modal amplitude - v₀ velocity - x coordinates in the inertial frame of reference - z coordinates in the body frame of reference - Ka/(Ka+Eh) - - elastic energy - 2qa ³/D - a/_mh - Poisson's ratio - material coordinates - air density - _m plate density - - _r prescribed functions - _r sin(r z/a) - angular velocity - a/v₀ - skew-symmetric matrix form of the angular velocity     

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     

On forced vibration of anisotropic cylinders

Summary The dynamic response of a circular cylinder with thick walls of transverse curvilinear isotropy subjected to a uniformly distributed pressure varying periodically with time is analyzed by means of the Laplace transformation, and the exact solution is obtained in closed form. The previously obtained solutions for forced vibrations with isotropy, and free vibrations with transverse curvilinear isotropy are included as special cases of the general results reported here.Nomenclature t time - r, , z cylindrical coordinates - _ii components of normal strain - _ii components of normal stress - u radial displacement - c _ij elastic constant - mass density - c ² c ₁₁/ - ² c ₂₂/c ₁₁ - a, b inner, outer radius of the cylinder - , A, B constants - forced angular frequency - function defined by (9) - p, real, complex variables - constant defined by (14) - real number - , Lamé elastic constants - J (x) Bessel function of first kind - Y (x) Bessel function of second kind - I (x) modified Bessel function of first kind - K (x) modified Bessel function of second kind     

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.     

Effect of measuring volume length on two-component laser velocimeter measurements in a turbulent boundary layer

The effects of finite measuring volume length on laser velocimetry measurements of turbulent boundary layers were studied. Four different effective measuring volume lengths, ranging in spanwise extent from 7 to 44 viscous units, were used in a low Reynolds number (Re=1440) turbulent boundary layer with high data density. Reynolds shear stress profiles in the near-wall region show that u v strongly depends on the measuring volume length; at a given y-position, u v decreases with increasing measuring volume length. This dependence was attributed to simultaneous validations on the U and V channels of Doppler bursts coming from different particles within the measuring volume. Moments of the streamwise velocity showed a slight dependence on measuring volume length, indicating that spatial averaging effects well known for hot-films and hot-wires can occur in laser velocimetry measurements when the data density is high.List of symbols time-averaged quantity - u wall friction velocity, ( _w /)^1/2 - v kinematic viscosity - d _p pinhole diameter - l _eff spanwise extent of LDV measuring volume viewed by photomultiplier - l ⁺ non-dimensional length of measuring volume, l _eff u /v - y ⁺ non-dimensional coordinate in spanwise direction, y u /v - z ⁺ non-dimensional coordinate in spanwise direction, z u /v - U ⁺ non-dimensional mean velocity, /u - u instantaneous streamwise velocity fluctuation, U &#x2329;U - v instantaneous normal velocity fluctuation, V–V - u RMS streamwise velocity fluctuation, u ^21/2 - v RMS normal velocity fluctuation, v ^21/2 - Re Reynolds number based on momentum thickness, U 0/v - R _uv cross-correlation coefficient, u v/u v - R₁₂(0, 0, z) two point correlation between u and v with z-separation, <u(0, 0, 0) v (0, 0, z)>/<u(0, 0, 0) v (0, 0, 0)> - N rate at which bursts are validated by counter processor - T Taylor time microscale, u (dv/dt²)^–1/2     

On the cross-effects of coupled macroscopic transport equations in porous media

In this paper, the derivation of macroscopic transport equations for this cases of simultaneous heat and water, chemical and water or electrical and water fluxes in porous media is presented. Based on themicro-macro passage using the method of homogenization of periodic structures, it is shown that the resulting macroscopic equations reveal zero-valued cross-coupling effects for the case of heat and water transport as well as chemical and water transport. In the case of electrical and water transport, a nonsymmetrical coupling was found.Notations b mobility - c concentration of a chemical - D rate of deformation tensor - D molecular diffusion coefficient - D _ij ^eff macroscopic (or effective) diffusion tensor - electric field - E ₀ initial electric field - k _ij molecular tensor - j, j ^*, current densities - K _ij macroscopic permeability tensor - l characteristic length of the ERV or the periodic cell - L characteristic macroscopic length - L _ijkl coupled flows coefficients - n _i unit outward vector normal to - p pressure - q _t ,q _t ⁺ , heat fluxes - q _c ,q _c ⁺ , chemical fluxes - s specific entropy or the entropy density - S entropy per unit volume - t time variable - t _ij local tensor - T absolute temperature - v _i velocity - V ₀ initial electric potential - V electric potential - x macroscopic (or slow) space variable - y microscopic (or fast) space variable - _i local vectorial field - _i local vectorial field - electric charge density on the solid surface - , bulk and shear viscosities of the fluid - _ij local tensor - _ij local tensor - _i local vector - _ij molecular conductivity tensor - _ij ^eff effective conductivity tensor - homogenization parameter - fluid density - ₀ ion-conductivity of fluid - _ij dielectric tensor - _i ¹ , _i ² , _i ³ local vectors - ⁴ local scalar - _S solid volume in the periodic cell - _L volume of pores in the periodic cell - boundary between _S and _L - ^s rate of entropy production per unit volume - total volume of the periodic cell - _l volume of pores in the cell On leave from the Politechnika Gdanska; ul. Majakowskiego 11/12, 80-952, Gdask, Poland.     

The stokes problems for a conducting fluid with a suspension of particles

The Stokes problems of an incompressible, viscous, conducting fluid with embedded small spherical particles over an infinite plate, set into motion in its plane by impulse and by oscillation, in the presence of a transverse magnetic field, are studied. The velocities of the fluid and of the particles and the wall shear stress are obtained. The stress is found to increase due to the particles and the magnetic field, with the effect of the particles diminishing as the field strength is increased.Nomenclature H ₀ strength of the imposed magnetic field - k density ratio of particles to fluid (per unit volume of flow field) - m _e ² H ₀ ² / - t time - y co-ordinate normal to the plate - u fluid velocity - v particle velocity - _e magnetic permeability of the fluid - kinematic viscosity of the fluid - electric conductivity of the fluid - fluid density - particle relaxation time - frequency of oscillation of the plate     

Saint-Venant's principle in linear two-dimensional elasticity for non-striplike domains

Let be a simply connected domain in the x ₁-x ₂ plane which lies within the strip 0<x ₂, is a simple closed piecewise smooth curve. Let l= [(x ₁, x ₂): (x ₁, x ₂) and x ₁>0], _l = [(x ₁ x ₂): (x ₁ ,x ₂) and x ₁>1>0].Suppose that a two-dimensional homogeneous isotropic elastic body occupies , that a self-equilibrated stress loading is applied to - l, and that l is stress-free. Knowles [2] and Flavin [6] showed that the elastic energy in _l decays exponentially with respect to l with an exponential decay constant of the form k/b, where k is a universal constant. It is shown here that a decay constant of the form c/ may be obtained where c is a universal constant and is a characteristic dimension of , which is more appropriate than b for general non-striplike domains. In addition, an appropriate decay theorem is obtained for coil-like domains.     

Experimental investigation of the creeping motion of a drop in a vertical tube

New experimental data regarding the motion of a drop along the axis of a vertical tube, filled with another highly viscous liquid, are obtained. The experiments are realised with sufficiently large drops for an internal circulation to develop and also for different pairs of fluids; the preponderant role of the gravity on the drop shape and consequently on its terminal velocity is pointed out. Moreover, by means of a visualization technique, details on the flow both inside and outside the drop are given.List of symbols g gravity acceleration - r distance from the drop center - R equivalent radius of the drop, i.e. the radius of the sphere having the same volume as the drop - R _EQ radius of the equatorial section of the drop - R _T tube radius - L _AX half length of the drop - U ₀ terminal velocity of the drop - P _s Poiseuille number= U _{0 e} /4 g R ² - Fr Fronde number = U ₀ ² _e /2 g R - Re Reynolds number = 2 U ₀ R _e / _e - E _o Eötvös number = 4g R ²/ - deformation parameter = _e U ₀/ - apparent density of the suspended liquid= | _i – _e | - _i viscosity of the suspended liquid - _e viscosity of the suspending liquid - drop-to-tube radius ratio = R/R _T - _EQ equatorial drop-to-tube radius ratio = R EQ/R _T - interfacial tension     

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     

On laminar flow through a uniformly porous pipe

Numerous investigations ([1] and [4–9]) have been made of laminar flow in a uniformly porous circular pipe with constant suction or injection applied at the wall. The object of this paper is to give a complete analysis of the numerical and theoretical solutions of this problem. It is shown that two solutions exist for all values of injection as well as the dual solutions for suction which had been noted by previous investigators. Analytical solutions are derived for large suction and injection; for large suction a viscous layer occurs at the wall while for large injection one solution has a viscous layer at the centre of the channel and the other has no viscous layer anywhere. Approximate analytic solutions are also given for small values of suction and injection.

Nomenclature

General r distance measured radially - z distance measured along axis of pipe - u velocity component in direction of z increasing - v velocity component in direction of r increasing - p pressure - density - coefficient of kinematic viscosity - a radius of pipe - V velocity of suction at the wall - r ²/a ² - R wall or suction Reynolds number, Va/ - f() similarity function defined in (6) - u ₀() eigensolution - U(0) a velocity at z=0 - K an arbitrary constant - B _K Bernoulli numbers Particular Section 5 perturbation parameter, –2/R - ² a constant, –K - x / - g(x) f()/ Section 6 perturbation parameter, –R/2 - ² a constant, –K - g() f() - g _c ()=g() near centre of pipe - * point where g()=0 Section 7 2/R - ² K - t (1–)/ - w(t, ) [1–f(t)]/ - ₀, ₁ constants - g() f()– ₀ - ₀/ - ₀ a constant - * point where f()=0     

Asymptotic Nusselt numbers for Newtonian flow through a parallel-plate channel with recycle

General expressions for evaluating the asymptotic Nusselt number for a Newtonian flow through a parallel-plate channel with recycle at the ends have been derived. Numerical results with the ratio of thicknesses as a parameter for various recycle ratios are obtained. A regression analysis shows that the results can be expressed by log Nur^0.83=0.3589 (log)² -0.2925 (log) + 0.3348 forR 3, 0.1 0.9; logNu=0.5982(log)² +0.3755 × 10^–2 (log) +0.8342 forR 10^–2, 0.1 0.9.

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Asymptotische Nusselt-Zahlen für die Newtonsche Strömung durch einen Kanal aus parallelen Platten mit Rückführung

Zusammenfassung In dieser Untersuchung wurden allgemeine Ausdrücke hergeleitet um die asymptotische Nusselt-Zahl für eine Newtonsche Strömung durch einen Kanal aus parallelen Platten mit Rückführung an den Enden berechnen zu können. Es wurden numerische Ergebnisse mit den Dicken-Verhältnissen, als Parameter für verschiedene Rückführungs-verhältnisse, erhalten. Eine Regressionsanalyse zeigt, daß die Ergebnisse wie folgt ausgedrückt werden können: log Nur^0,83=0,3589 (log)² -0,2925 (log) + 0,3348 fürR 3, 0,1 0,9; logNu=0,5982(log)² +0,3755 × 10^–2 (log) + 0,8342 fürR 10^–2, 0,1 0,9.

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Nomenclature A₁ shooting value,d(0)/d - A₂ shooting value,d(1)/d - B channel width - Gz Graetz number, U_bW²/L - h _m logarithmic average convective heat transfer coefficient - h _x average local convective heat transfer coefficient - k thermal conductivity - L channel length - Nu average local Nusselt number, 2 h_xW/k - Nu _m logarithmic average Nusselt number, 2h_mW/k - R recycle ratio, reverse volume flow rate divided by input volume flow rate - T temperature of fluid - T _m bulk temperature, Eq. (8) - T ₀ temperature of feed stream - T _s wall temperature - U velocity distribution - U _b reference velocity,V/BW - V input volume flow rate - v dimensionless velocity distribution, U/U_b - W channel thickness - x longitudinal coordinate - y transversal coordinate - Z₁-z₆ functions defined in Eq. (A1) - thermal diffusivity - least squares error, Eq. (A7) - weight, Eqs. (A8), (A9) - dimensionless coordinate,y/W - dimensionless coordinate,x/GzL - function, Eq. (7)     

Mean and turbulent velocity measurements of supersonic mixing layers

The behavior of supersonic mixing layers under three conditions has been examined by schlieren photography and laser Doppler velocimetry. In the schlieren photographs, some large-scale, repetitive patterns were observed within the mixing layer; however, these structures do not appear to dominate the mixing layer character under the present flow conditions. It was found that higher levels of secondary freestream turbulence did not increase the peak turbulence intensity observed within the mixing layer, but slightly increased the growth rate. Higher levels of freestream turbulence also reduced the axial distance required for development of the mean velocity. At higher convective Mach numbers, the mixing layer growth rate was found to be smaller than that of an incompressible mixing layer at the same velocity and freestream density ratio. The increase in convective Mach number also caused a decrease in the turbulence intensity ( _u/U).List of symbols a speed of sound - b total mixing layer thickness between U ₁ – 0.1 U and U ₂ + 0.1 U - f normalized third moment of u-velocity, f u³/(U)³ - g normalized triple product of u² , g u²/(U)³ - h normalized triple product of u ², h u ²/(U)³ - l _u axial distance for similarity in the mean velocity - l _u axial distance for similarity in the turbulence intensity - M Mach number - M _c convective Mach number (for ₁ = ₂), M _c (U ₁ – U ₂)/(a ₁ + a ₂) - P static pressure - r freestream velocity ratio, r U ₂/U ₁ - Re unit Reynolds number, Re U/ - s freestream density ratio, s ₂/₁ - T _t total temperature - u instantaneous streamwise velocity - u deviation of u-velocity, uu – U - U local mean streamwise velocity - U ₁ primary freestream velocity - U ₂ secondary freestream velocity - average of freestream velocities, (U ₁ + U ₂)/2 - U freestream velocity difference, U U ₁ – U ₂ - instantaneous transverse velocity - v deviation of -velocity, – V - V local mean transverse velocity - x streamwise coordinate - y transverse coordinate - y ₀ transverse location of the mixing layer centerline - ensemble average - ratio of specific heats - boundary layer thickness (y-location at 99.5% of free-stream velocity) - similarity coordinate, (y – y ₀)/b - compressible boundary layer momentum thickness - viscosity - density - standard deviation - dimensionless velocity, (U – U ₂)/U - 1 primary stream - 2 secondary stream A version of this paper was presented at the 11th Symposium on Turbulence, October 17–19, 1988, University of Missouri-Rolla     

Nonstationary vibration of a rotating shaft with nonlinear spring characteristics during acceleration through a critical speed (A critical speed of a summed-and-differential harmonic oscillation)

Nonstationary vibration of a flexible rotating shaft with nonlinear spring characteristics during acceleration through a critical speed of a summed-and-differential harmonic oscillation was investigated. In numerical simulations, we investigated the influence of the angular acceleration , the initial angular position of the unbalance _n and the initial rotating speed on the maximum amplitude. We also performed experiments with various angular accelerations. The following results were obtained: (1) the maximum amplitude depends not only on but also on _n and : (2) when the initial angular position _n changes. the maximum amplitude varies between two values. The upper and lower bounds of the maximum amplitude do not change monotonously for the angular acceleration: (3) In order to always pass the critical speed with finite amplitude during acceleration. the value of must exceed a certain critical value.Nomenclature O-xyz rectangular coordinate system - , ₁, ₁ inclination angle of rotor and its projections to thexy- andyz-planes - I _r polar moment of inertia of rotor - I diametral moment of inertia of rotor - i _r ratio ofI _r toI - dynamic unbalance of rotor - directional angle of fromx-axis - c damping coefficient - spring constant of shaft - N _nt ,N _nt nonlinear terms in restoring forees in ₁ and ₁ directions - ₄ representative angle - a small quantity - V. V _u .V _N potential energy and its components corresponding to linear and nonlinear terms in the restoring forees - directional angle - _n coefficients of asymmetrical nonlinear terms - _n coefficients of symmetrical nonlinear terms - coefficients of asymmetrical nonlinear terms experessed in polar coordinates - coefficients of symmetrical nonlinear terms expressed in polar coordinates - rotating speed of shaft - t time - _n initial angular position of att=0 - p natural frequency - p ₁.p _t natural frequencies of forward and backward precessions - , ₁, ₁ total phases of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation - , ₁, ₁ phases of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation - P, R _t ,R _b amplitudes of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation - difference between phases ( = _f – _u) - acceleration of rotor - initial rotating speed - t _t ,r _b amplitudes of nonstationary oscillation during acceleration - (r _t )_max, (r _b )_max maximum amplitudes of nonstationary oscillation during acceleration - (r ₁ ¹ )_max, (r _b ¹ )_max maximum value of angular acceleration of non-passable case - ₀ critical value over which the rotor can always pass the critical speed - p ₁,p ₂,p ₃,p ₄ natural frequencies of experimental apparatus     

The rheology of polymeric liquid crystals

The molecular theory of Doi has been used as a framework to characterize the rheological behavior of polymeric liquid crystals at the low deformation rates for which it was derived, and an appropriate extension for high deformation rates is presented. The essential physics behind the Doi formulation has, however, been retained in its entirety. The resulting four-parameter equation enables prediction of the shearing behavior at low and high deformation rates, of the stress in extensional flows, of the isotropic-anisotropic phase transition and of the molecular orientation. Extensional data over nearly three decades of elongation rate (10^–2–10¹) and shearing data over six decades of shear rate (10^–2–10⁴) have been correlated using this analysis. Experimental data are presented for both homogeneous and inhomogeneous shearing stress fields. For the latter, a 20-fold range of capillary tube diameters has been employed and no effects of system geometry or the inhomogeneity of the flow-field are observed. Such an independence of the rheological properties from these effects does not occur for low molecular weight liquid crystals and this is, perhaps, the first time this has been reported for polymeric lyotropic liquid crystals; the physical basis for this major difference is discussed briefly. A Semi-empirical constant in eq. (18), N/m² - c rod concentration, rods/m³ - c ^* critical rod concentration at which the isotropic phase becomes unstable, rods/m³ - C interaction potential in the Doi theory defined in eq. (3) - d rod diameter, m - D semi-empirical constant in eq. (19), s^–1 - D _r lumped rotational diffusivity defined in eq. (4), s^–1 - rotational diffusivity of rods in a concentrated (liquid crystalline) system, s^–1 - D _ro rotational diffusivity of a dilute solution of rods, s^–1 - f distribution function defining rod orientation - F tensorial term in the Doi theory defined in eq. (7) (or eq. (19)), s^–1 - G tensorial term in the Doi theory defined in eq. (8) - K _B Boltzmann constant, 1.38 × 10^–23 J/K-molecule - L rod length, m - S scalar order parameter - S tensor order parameter defined in eq. (5) - t time, s - T absolute temperature, K - u unit vector describing the orientation of an individual rod - rate of change ofu due to macroscopic flow, s^–1 - v fluid velocity vector, m/s - v velocity gradient tensor defined in eq. (9), s^–1 - V mean field (aligning) potential defined in eq. (2) - x coordinate direction, m - Kronecker delta (= 0 if = 1 if = ) - _r ratio of viscosity of suspension to that of the solvent at the same shear stress - _s solvent viscosity, Pa · s - ^* viscosity at the critical concentrationc ^*, Pa · s - v ₁, v₂ numerical factors in eqs. (3) and (4), respectively - deviatoric stress tensor, N/m² - volume fraction of rods - ⁰ constant in eq. (16) - ^* volume fraction of rods at the critical concentrationc ^* - average over the distribution functionf(u, t) (= d ²u f(u, t)) - gradient operator - d ²u integral over the surface of the sphere (|u| = 1)