Flocculation and rheology of kaolinite/quartz suspensions

Rheological properties of suspensions of Na-kaolinite and colloidal quartz (Min-U-Sil) at constant overall volume concentration of 2% are determined with a Weissenberg Rheogoniometer using a combined Couette and cone-and-plate geometry. The results are interpreted in terms of the flocculation behaviour of the constituent particles in the presence of high salt concentrations (0.1–0.75m NaCl) at pH 6, 7 and 8. In these chemical environments these suspensions are pseudoplastic for much of the range of mixture compositions becoming Newtonian for suspensions containing only quartz. These properties reflect the dominant influence of interactions between kaolinite particles on the flocculation behaviour of the mixture.Nomenclature a ₁, a₂ radii of spheres - A Hamaker constant - b radius of cylinder - C volumetric solids concentration - e ₀ electronic charge - H ₀ shortest distance between surfaces - I ionic strength - J collision frequency per unit volume - k Boltzmann constant - l length of cylinder - m number in eq. (2) - N particle number concentration - S (H ₀ + b)/b in eq. (4) - T absolute temperature - U electrophoretic mobility - V _A van der Waals attractive energy - V _R coulombic energy - V _T total energy of interaction - X H ₀/2a₁ in eq. (5) - Y a ₂/a₁ in eq. (5) - thickness of plate - shear rate - permittivity - zeta potential - k Debye-Hückel parameter - µ dynamic viscosity - µ _pl plastic viscosity - v valency of counter ion - shear stress - _B Bingham stress - ₁, ₂ dimensionless potentials - ₁, ₂ surface potentials     

The “Twin capillary” a simple device to separate shear- and slip-flow of fluids

When the flow behaviour of fluids is investigated with capillary-or rotational rheometers, adhesion of the fluid to the wall is normally one of the boundary conditions. For many fluids, especially for suspensions, this assumption is not valid. These fluids tend to slip at the wall. Therefore the normal evaluation of rheometer measurements leads to apparent but not compatible flow functions. The flow behaviour of these fluids can be characterized with two material functions which describe separately slipping in the boundary layer and shearing within the fluid. Only if both functions are known, correct predictions of flow processes are possible. A simple equipment to separate the shear function and the slip function is described.List of symbols Y^* apparent shear rate - Y _w ^* apparent wall shear rate - Y_w wall shear rate corrected with Rabinowitsch and Weissenberg correction - Y_s reduced shear rate (slip corrected) - Y_ws reduced wall shear rate (slip corrected) - * (r) velocity distribution in a capillary - _G slip velocity (at the wall) - * (r) velocity distribution in a capillary (without slip) - shear stress - _w wall shear stress - V_S total volume rate - V_G shear volume rate - V_G slip volume rate - p ₁ pressure in the reservoir channel of the capillary rheometer - p ₀ athmospheric pressure - L capillary length - R capillary radius     

Luikov equations applicable to sublimation-drying

Present paper presents a derivation of Luikov equations applicable to sublimation-drying. The physical situation and transfer mechanism are elucidated clearly. The coefficients appearing in Luikov equations are given in a more explicit way. Some formulation mistakes in recent publications are indicated.

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Anwendung der Luikov-Gleichungen auf die Sublimationstrocknung

Zusammenfassung Die Untersuchung bezieht sich auf eine Ableitung der Luikov-Gleichungen, mittels deren sich der Vorgang der Sublimationstrocknung analysieren läßt. Physikalische Anfangssituation und Austauschmechanismen werden klar herausgestellt und die in den Luikov-Gleichungen auftretenden Koeffizienten in expliziter Weise angegeben. Ferner erfolgt Hinweis auf Formulierungsfehler in jüngeren Veröffentlichungen.

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Nomenclature C M _v/V _f, concentration of vapor, kg/m³ - c _pv specific heat of vapor at constant pressure, J/kg K - c _pw specific heat of adsorbed water at constant pressure, J/kg K - c _s specific heat of solid skeleton, J/kg K - C _s M _s/V _f, concentration of solid skeleton, kg/m³ - C _w M _w/V _f, concentration of adsorbed water, kg/m³ - f V _w/V _f, volumetric fraction of adsorbed water - j _F mass flux of vapor by diffusion (Fick) transfer, kg/m² s - j _D mass flux of vapor by filtration (Darcy) transfer, kg/m² s - j _v total mass flux of vapor, kg/m² s - k permeability, m² - M _s mass of solid skeleton, kg - M _v mass of vapor in pores, kg - M _w mass of adsorbed water, kg - P pressure, Pa - q heat flux, W/m² - R gas constant, J/kg K - T temperature, K - V _f volume of the framework of porous medium, m³ - V _v volume of vapor in porous medium, m³ - V _w volume of the absorbed water, m³ Greek symbols /(c _p), effective thermal diffusivity, m²/s - _m effective vapor diffusivity in porous medium, m²/s - _p R T /, Luikov pressure diffusivity, m²/s - +f, porosity of the porous medium - effective thermal conductivity of porous body, W/m K - dynamic viscosity of vapor, kg/m s - kinematic viscosity, m²/s - Ck/=k/, Luikov filtration motion coefficient, s - V _v/V _f, volumetric fraction of vapor - density of absorbed water, kg/m³ - (c _p) M _v c _pv+M _s c _s+M _w c _pw /V _f=Cc _pv+C _s c _s+fc _pw, effective product of density and specific heat of humid porous body, J/m³K     

Evaporation of single liquid drops in an immiscible liquid at elevated pressures: equipment and preliminary results

This paper describes a piece of equipment which has been specially designed to permit the study of the evaporation of single liquid drops in an immiscible liquid at elevated pressures up to the order of 1 MPa. It is equipped with a piezoelectric discharger for yielding nucleation in each drop and with a dilatometer for detecting the change in volume of the drop in the course of evaporation succeeding the nucleation. Some preliminary results, obtained with the equipment, for n-pentane drops evaporating in a water medium at pressures up to 0.5 MPa (5 atm) are also presented.List of symbols C _D drag coefficient - D ₀ initial, equivalent spherical diameter of drop (= (6 V ₀/)^1/3) - H vertical position of two-phase bubble measured from the nozzle tip - H _t column height required for drop to vaporize completely - Nu Nusselt number related to instantaneous two-phase bubble diameter and thermal conductivity of the continuous phase - p ^* pressure at the nozzle tip - Pe Peclet number related to instantaneous diameter and rise velocity of two-phase bubble and to thermal diffusivity of the continuous phase - Re Reynolds number related to instantaneous diameter and rise velocity of two-phase bubble and to viscosity of the continuous phase - t time - t _v time required for drop to vaporize completely - T temperature in the bulk of the medium - T excess, assumed in theoretical model, of T above the temperature of two-phase bubble - T ₁ ^* excess of T above the saturation temperature of the dispersed-phase fluid corresponding to p ^* - T ₂ ^* excess of T above the temperature at which the sum of the saturated vapor pressures of the dispersed- and the continuous-phase fluids is equal to p ^* - V volume of two-phase bubble - V ₀ initial value of V     

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     

Dynamic fracture of a kane-mindlin plate

We study dynamic crack problems for an elastic plate by using Kane-Mindlin's kinematic assumptions. The general solutions of the Laplace transformed displacements and stresses are first derived. Path independent integrals for stationary cracks subjected to transient loads and steadily growing cracks are deduced. For a stationary crack in a very thin plate subjected to impact loads, the crack tip dynamic stress intensity factor (DSIF), K₁(t), is related to the far field plane stress one, K₁⁰(t), by where ν is Poisson's ratio. For a crack steadily growing with speed V, the crack tip DSIF, K₁(V), is given by where K₁⁰(V) is the plane stress DSIF and A(V) and B(V) are known functions of V. These results are applied to compute the DSIF for a semi-infinite stationary crack in an unbounded plate subjected to impact pressure on the crack faces. The results of DSIF for a finite crack in an infinite plate under uniform impact pressure on the crack surfaces show that for each plate thickness, the maximum DSIF is higher than that for the plane stress case.     

The site model theory and the standard linear solid

Summary The site model theory (SMT) is shown to lead to the same deformation behaviour as that displayed by the standard linear solid (SLS), group I, for all loading conditions. If a second deformation mechanism (inter-molecular slip) is introduced the result is the same as that obtained with the standard linear solid, group II, and models the behaviour of a polymer melt near to the solidification temperature.

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Zusammenfassung Es wird gezeigt, daß ein einfaches Platzwechsel-Modell (site model theory) bei allen Belastungsbedingungen das gleiche Deformationsverhalten voraussagt wie der lineare Drei-Parameter-Festkörper (standard linear solid, group I). Wenn ein weiterer Deformationsmechanismus (zwischenmolekulare Gleitung) eingeführt wird, entspricht das Verhalten dagegen demjenigen einer linearen Drei-Parameter-Flüssigkeit (standard linear solid, group II), welche das Verhalten einer Polymerschmelze in der Nähe der Schmelztemperatur beschreibt.

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a = ₁₂ ⁰ + ₂₁ ⁰ , see eq. [1] - b =N ₁ ⁰ ₁₂ ⁰ (V ₁₂ +V ₂₁), see eq. [1] - c = 2N _s ⁰ V _s see eq. [6] - k Boltzmann constant - t time - E,E ₁,E ₂ spring constants, see figures 1 and 3 - E _u unrelaxed modulus - N ₁ ⁰ site 1 equilibrium population in the unstressed state - N _s number of units available for slip - N(t) decrease in site 1 population - N _s (t) net number of slip jumps in the stressaided direction - T temperature (K) - V _i,j activation volume for jumps in directioni j - V _s activation volume for the slip process - strain - strain rate - incremental change in strain per unit change in site population - µ,µ ₁,µ ₂ dashpot constants, see figures 1 and 3 - applied stress - ₀ initial applied stress, (stress relaxation) =(t) (creep) - incremental change in stress per unit change in site population - ⁰ jump rate for slip in the unstressed state - _i,j ⁰ jump rate in the directioni j in the unstressed state With 3 figures and 3 tables     

The influence of free convection on soil salinization in arid regions

Evaporation of groundwater in a region with a shallow water table and small natural replenishment causes accumulation of salts near the ground surface. Water in the upper soil layer becomes denser than in the depth. This is a potentially unstable situation which may result in convective currents. When free convection takes place, estimates of the salinity profile, salt precipitation rate, etc., obtained within the framework of a 1-D (vertical) model fail.Very simplified model of the process is proposed, in which the unsaturated zone is represented by a horizontal soil layer at a constant water saturation, and temperature changes are neglected. The purpose of the model is to obtain a rough estimate of the role of natural convection in the salinization process.A linear stability analysis of a uniform vertical flow is given, and the stability limit is determined numerically as a function of evaporation rate, salt concentration in groundwater, and porous medium dispersivity. The loss of stability corresponds to quite realistic Rayleigh numbers. The stability limit depends in nonmonotonic way on the evaporation rate.The developed convective regime was simulated numerically for a 2-D vertical domain, using finite volume element discretization and FAS multigrid solver. The dependence of the average salt concentration in the upper layer on the Rayleigh number was obtained.List of Main Symbols horizontal wavenumber - _L , _T dispersivities (longitudinal and transversal) - D ^* diffusion coefficient (in a porous medium) - g acceleration of gravity - H thickness of the vadoze zone - k permeability - p pressure - Pe Péclet number - q mass flux - Ra Rayleigh number Greek _L , _T dimensionless dispersivities - coefficient of concentration expansion - coefficient of viscosity variation - volumetric fraction of the liquid phase - viscosity - density - stream function - mass fraction of salt in water Vectors and tensors D dispersion coefficient - e unit vector - I unit tensor - J nonadvective salt flux - V liquid phase velocity - x radius-vector     

Transport in ordered and disordered porous media V: Geometrical results for two-dimensional systems

In this paper we continue the geometrical studies of computer generated two-phase systems that were presented in Part IV. In order to reduce the computational time associated with the previous three-dimensional studies, the calculations presented in this work are restricted to two dimensions. This allows us to explore more thoroughly the influence of the size of the averaging volume and to learn something about the use of anon-representative region in the determination of averaged quantities.

Nomenclature

Roman Letters A interfacial area of the– interface associated with the local closure problem, m² - a _i i=1, 2, gaussian probability distribution used to locate the position of particles - l unit tensor - 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 a unit cell - n _i i=1, 2 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward 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 - V averaging volume, m³ - V volume of the-phase contained in the averaging volume,V, m³ - x position of the centroid of an averaging area, m - x ₀ reference position of the centroid of an averaging area, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters V /V, volume average porosity - _{a i} standard deviation ofa _i - _r standard deviation ofr - intrinsic phase average of      

Momentum and energy balances for dispersed two-phase flow

Summary The equations of motion and the mechanical energy balances for two-phase flow systems are derived by integration over a volume containing a large number of elements of the dispersed phase.List of symbols A, A boundary of volumes V, V - dA, dA surface element of A, A - A _s boundary of particles in V - dA _s surface element of A _s - F force per unit volume of the system - g –gz=gravity vector - g acceleration by gravity - I unit tensor - p pressure - Q dissipation in the continuous phase - Q _s dissipation in the dispersed phase - R compression work in the continuous phase - R _s compression work in the dispersed phase - t time - u velocity of continuous phase - u _s velocity of dispersed phase - u magnitude of u - u _s magnitude of u _s - V volume in the two-phase system - V part of V occupied by the continuous phase - W work done by F - z vertical coordinate - local volume fraction of the dispersed phase - pI–=stress tensor of the continuous phase - _s turbulent particle stress tensor - density of the continuous phase - _s density of the dispersed phase - shearing-stress tensor of the continuous phase - _s turbulent particle shearing-stress tensor - nabla operator - u, u _s velocity gradient tensor - substantial derivative (Shell Internationale Research Maatschappij N.V.)(Bataafse Internationale Petroleum Maatschappij N.V.)     

Dual optical fibre measurements of the particle concentration in gas/solid flows

Numerous optical probe designs to measure particle volume fraction have been proposed in the literature. Unfortunately, almost all of them suffer from an ill-defined measurement volume, poor sensitivity or require frequent and tedious calibration. We propose an improvement in the design of a dual optical fibre probe. It has a well-defined measuring volume and a near-linear sensitivity. A general calibration theory for optical fibre probes is also proposed. The design and the theory have been tested in a simple experimental set-up with encouraging results.

List of symbols

Latin letters a probe glass thickness (m) - b glare diameter (m) - d _p particle diameter (m) - dS surface element (m²) - D optical fibre diameter (m) - f maximum packing signal, function of distance to the probe (V) - g radial distribution, function of particle volume fraction - h probe sensitivity, function of distance to the probe (V) - I intensity of light reflected off a particle (V) - l light ray length (m) - N number of elements - n _p number of particles - p penetration depth of light distribution, function of distance and particle volume fraction (m^–1) - q glare correction, function of probe geometry and particle size - r distance to the probe surface (m) - r distance from the probe surface to the first particle in a given direction ( , ) in a given suspension (m) - R probe receptivity to light, function of angle of incidence and distance to particle - S probe response signal (V) - S ₀ probe response at zero volume fraction of particles (V) - S _dense probe response at maximum packing of particles (V) - V _cyl cylinder volume (m³) - V _p particle volume (m³) - z independent length variable (m) Greek letters _p volume fraction of particles - _p,max volume fraction of particles at maximum packing - angle between incident and receiving optical fibre (rad) - exponent - exponent - altitude angle in spherical coordinates (rad) - azimuthal angle in spherical coordinates (rad) - mean free path of light (m) - dimensionless probe response signal - dimensionless reciprocal mean free path of light - dimensionless distance to probe surface - independent dimensionless length Other symbols ~ instantaneous value - <> expectancy value     

Measurements of flow rates with a fiber-optic transducer in a suddenly opened chamber

A fiber-optic transducer is used in a circuit-breaker to measure instantaneous flow rates. The time response (<0.5 ms) is estimated by exposing the optical fiber to a very rapidly varying flow at the open end of a shock-tube. When the flow in the circuit-breaker is generated by an initial high pressure level, the transducer gives results in good agreement with a very simple model if compressible effects are taken into account.List of symbols p _i pressure of vessel at instant t - p ₀ i initial pressure of vessel - _i density of vessel at instant t - ₀ i initial density of vessel - p pressure in the measurement section at instant t - density in the measurement section at instant t - u velocity in the measurement section at instant t - D inner diameter of the electrodes and of the throat in the calibration wind tunnel - V volume of vessel - U velocity in the throat section in the calibration wind tunnel - m mass flow rate - ratio of specific heats - r gas constant - Y deviation of the fiber - C _d drag coefficient - E Young modulus - d diameter of the fiber - L length of the fiber cantilever - V ₁, V ₂ amplified signals given by the two cells - k coefficient of calibration - V ₀ ^* detector response without air flow - s detector lateral displacement     

Self-similar solution for deep-penetrating hydraulic fracture propagation

The propagation of a vertical hydraulic fracture of a constant height driven by a viscous fluid injected into a crack under constant pressure, is considered. The fracture is assumed to be rectangular, symmetric with respect to the well, and highly elongated in the horizontal direction (the Perkins and Kern model). The fracturing fluid viscosity is assumed to be different from the stratum saturating fluid viscosity, and the stratum fluid displacement by a fracturing fluid in a porous medium is assumed to be piston-like. The compressibility of the fracturing fluid is neglected. The stratum fluid motion is governed by the equation of transient seepage flow through a porous medium.A self-similar solution to the problem is constructed under the assumption of the quasi-steady character of the fracturing fluid flow in a crack and in a stratum and of a locally one-dimensional character of fluid-loss through the crack surfaces. Crack propagation under a constant injection pressure is characterized by a variation of the crack sizel in timet according to the lawl(t)=l _o (1+At)^1/4, where the constantA is the eigenvalue of the problem. In this case, the crack volume isVl, the seepage volume of fracturing fluidV _f l ³, and the flow rate of a fluid injected into a crack isQ ₀l ^–1.     

Landslide generated impulse waves. 2. Hydrodynamic impact craters

Landslide generated impulse waves were investigated in a two-dimensional physical laboratory model based on the generalized Froude similarity. Digital particle image velocimetry (PIV) was applied to the landslide impact and wave generation. Areas of interest up to 0.8 m by 0.8 m were investigated. PIV provided instantaneous velocity vector fields in a large area of interest and gave insight into the kinematics of the wave generation process. Differential estimates such as vorticity, divergence, and elongational and shear strain were extracted from the velocity vector fields. At high impact velocities flow separation occurred on the slide shoulder resulting in a hydrodynamic impact crater, whereas at low impact velocities no flow detachment was observed. The hydrodynamic impact craters may be distinguished into outward and backward collapsing impact craters. The maximum crater volume, which corresponds to the water displacement volume, exceeded the landslide volume by up to an order of magnitude. The water displacement caused by the landslide generated the first wave crest and the collapse of the air cavity followed by a run-up along the slide ramp issued the second wave crest. The extracted water displacement curves may replace the complex wave generation process in numerical models. The water displacement and displacement rate were described by multiple regressions of the following three dimensionless quantities: the slide Froude number, the relative slide volume, and the relative slide thickness. The slide Froude number was identified as the dominant parameter.List of symbols a wave amplitude (L) - b slide width (L) - c wave celerity (LT^–1) - d _g granulate grain diameter (L) - d _p seeding particle diameter (L) - F slide Froude number - g gravitational acceleration (LT^–2) - h stillwater depth (L) - H wave height (L) - l _s slide length (L) - L wave length (L) - M magnification - m _s slide mass (M) - n _por slide porosity - Q _d water displacement rate (L³) - Q _D maximum water displacement rate (L³) - Q _s maximum slide displacement rate - s slide thickness (L) - S relative slide thickness - t time after impact (T) - t _D time of maximum water displacement volume (L³) - t _qD time of maximum water displacement rate (L³) - t _si slide impact duration (T) - t _sd duration of subaqueous slide motion (T) - T wave period (T) - v velocity (LT^–1) - v _p particle velocity (LT^–1) - v _px streamwise horizontal component of particle velocity (LT^–1) - v _pz vertical component of particle velocity (LT^–1) - v _s slide centroid velocity at impact (LT^–1) - V dimensionless slide volume - V _d water displacement volume (L³) - V _D maximum water displacement volume (L³) - V _s slide volume (L³) - x streamwise coordinate (L) - z vertical coordinate (L) - slide impact angle (°) - bed friction angle (°) - x mean particle image x-displacement in interrogation window (L) - _x random displacement x error (L) - _tot total random velocity v error (LT^–1) - _xx streamwise horizontal elongational strain component (1/T) - _xz shear strain component (1/T) - _zx shear strain component (1/T) - _zz vertical elongational strain component (1/T) - water surface displacement (L) - density (ML^–3) - _g granulate density (ML^–3) - _p particle density (ML^–3) - _s mean slide density (ML^–3) - _w water density (ML^–3) - granulate internal friction angle (°) - _y vorticity vector component (out-of-plane) (1/T)     

Oscillating sedimentation of spheres in viscoplastic fluids

In applications of chemical engineering often the sedimentation is used to separate disperse particles from liquid phases. Some real liquids, e.g., polymer fluids, paints, and skin creams show viscoplastic flow behavior, i.e., they have a yield stress. In such fluids it is possible that suspended particles do not move under action of gravity although the density of the particles is greater than the fluid density. A possibility to sediment stuck spherical particles is shown. The fluid is set in sinusoidal vibration so that the particles undergo forced oscillations. This effect is investigated for single spheres. A model is given and several theoretical results are discussed. A criterion is presented that allows one to predict the combinations of the vibration parameters (amplitude and frequency) which are needed to sediment the spheres. The theoretical investigations are confirmed by experiments. The motion of several glass and steel spheres in an oscillating tube filled with aqueous carbopol solutions are detected. The comparison between theory and experiment shows good agreement.Nomenclature C Stokes drag coefficient - D strain rate tensor - E unit tensor - F, F _w external resp. drag force - f body force vector - G Green deformation tensor - G dimensionless (shear) modulus - g acceleration of gravity - K abbreviation (Eq. (15)) - p pressure - R, spherical coordinates - R ₀, R _a sphere resp. body radius - T extra stress tensor - t time - S stress tensor - U ₀, W ₀ displacement amplitude - u displacement vector - velocity vector - V viscoelastic number - V _k sphere volume - V steady sink velocity - Y yield stress parameter - Y _g limiting value - Z Stokes number - density ratio - second invariant of D - radii ratio - y differential viscosity - , Lamé constants - ^* = + i complex (shear) modulus - f, _k fluid resp. sphere density - second invariant of T - _f yield stress - phase angle - frequency     

Viscosity of suspensions modeled with a shear-dependent maximum packing fraction

A large amount of data from the literature on viscosity of concentrated suspensions of rigid spherical particles are analyzed to support the new concept that the maximum packing fraction ( _M ) is shear-dependent. Incorporation of this behavior in a rheological model for viscosity () as a function of particle volume fraction () succeeds in describing virtually all non-Newtonian effects over the entire concentration range and also accounts for a yield stress. The most successful model is one proposed by Krieger and Dougherty for Newtonian viscosities, (, _M ), but with _M varying from a low-shear limit _M0 to a high-shear limit _M. Microstructural interpretations of this behavior are advanced, with arguments suggesting that similar rheological models should apply to suspensions of nonspherical and irregular particles.Symbols a particle size scale (for spheres, the diameter) - A lumped kinetic parameter in eqs. (23) and (24) - BS butadiene-styrene - C coefficient in Arrhenius model, eq. (2) - D coefficient in Mooney model, eq. (3) - e _i parameter representing one of the three electroviscous effects (i = 1, 2, or 3) - f fraction of total particulates that exist in the dispersed phase, eq. (22) - h solution factor, in Arrhenius model, eq. (2) - k crowding factor, in Mooney model, eq. (3) - k _D ,k _F kinetic rate coefficient for producing particles of dispersed or flocculated type, respectively - K Einstein coefficient for particles of any shape, eq. (1); equal to [] - KD Krieger-Dougherty model, eq. (6) - m exponent to characterize shear-dependence in viscosity models of Cross, eq. (10), and eq. (23), and also in yield stress prediction eq. (24) - N number of monodisperse components in a blend of spheres with different diameters - PD polydispersity (in size) parameter - S generalized shape parameter - T temperature - V _c volume of chamber in figure 6, representing the entire volume of the sample - V _P total volume of particles in the sample - V _D ,V _F sample volumes in which dispersed particles or flocculated particles, respectively, prevail; volumes of the dispersed phase or flocculated phase, containing both particles and carrier fluid - V _PD ,V _PF particle volume within the phase volumeV _D orV _F , respectively Greek coefficient in definition of _c in eq. (8); of order unity - coefficient regulating -sensitivity in eq. (10) - shear rate,dv ₁/dx ₂ in simple shear - shear viscosity of the suspension - ₀, low-shear and high-shear limiting values of - _s viscosity of the suspending fluid - [] intrinsic viscosity, - _r reduced viscosity,/ _s - Boltzmann's constant; in _c - shear stress - _c parameter characterizing sensitivity of viscosity to stress, in eq. (8) - _B dynamic yield stress in the floc model - _y yield stress - volume fraction occupied by solids in a suspension - _M maximum value of attainable by a given collection of particles under given conditions of flow - _M0, _M limiting values of _M at the low- and high- conditions, respectively     

Laminar elastico-viscous flow through channels with porous walls of different permeability

Summary The two-dimensional steady flow of an incompressible elastico-viscous fluid through a porous channel that has the fluid sucked or injected with different normal velocities V ₁ and V ₂ at the walls is considered. For the viscous case Terrill and Shrestha [2] have given a series solution for small suction Reynolds number. In this paper the solution of [2] is extended to include the effect of the elasticity of the fluid. It is found that the elasticity of the fluid comes into effect only when suction or blowing is present. The general expressions for the pressure distribution and the friction coefficient are given and are found to increase as the elastic parameter increases. Finally, the resulting solution is confirmed by numerical results.Nomenclature p _ik stress tensor - p arbitrary isotropic pressure - g _ik metric tensor of a fixed coordinate system x ⁱ - ^ik stress components, defined in (2) - rate of stress tensor, defined in (3) - d ^ik rate of strain tensor - N() distribution function of relaxation times - limiting viscosity at small rate of shear - short memory coefficient - density - v ⁱ components of velocity vector - v _,j ⁱ components of the velocity gradient tensor - = ₀ / kinematic viscosity - k ₀ ^* k ₀/ - x, y distances parallel, perpendicular to channel walls - u, v velocity components in x, y directions - h distance between walls, channel width - V _i velocity of suction at walls - V ₁ velocity of suction at the wall y=0 - V ₂ velocity of suction at the wall y=h - U a constant velocity at x=0 - ₁ V ₂/V ₁–1 - ₂ –V ₁/V ₂+1 - =y/h nondimensional distance perpendicular to the walls - R _i =V _i h/ suction Reynolds number, i=1, 2 - R*=k ₀ ^* /h ² elastic number - K _i , c, d c ₀, c ₁, c ₂ d ₀, d ₁, d ₂ constants given in (19), (26), and (27) - R _e=4hU/ entrance Reynolds number - P _x pressure component along x-axis - C _f friction coefficient - C _f ⁰ friction coefficient at wall y=0 - C _f ¹ friction coefficient at wall y=h     

Some refinements concerning the boundary conditions at the macroscopic level

The study of boundary effects initiated in a previous paper is continued. New assumptions regarding the geometrical structure of the boundary surface are introduced. Under these assumptions, it is shown that macroscopic Neumann conditions do not generally affect the determination of the macroscopic field in the case of the transport process considered — heat conduction. For this type of boundary condition, the boundary effect is generally confined within a thin layer near the boundary. When heat sources are taken into account within the porous domain, the result is different. In this case, making use of a Neumann boundary condition, expressed in terms of macroscopic variables, amounts to introducing an extra flux. Under normal circumstances, however, this additional flux is negligible.Roman Letters A cross-sectional area of a unit cell - A _e cross-sectional area of a unit cell at the boundary surface - A _sf interfacial area of the s-f interface contained within the averaging volume - surface area per unit volume (A _sf/ ) - A _sf interfacial area of the s-f interface contained within the macroscopic system - g closure vector - h closure vector - k heat transfer coefficient at the s-f interface - K_eff effective thermal conductivity tensor - _x unit cell length - n unit vector - n_e outwardly directed unit normal vector at the boundary - n_sf outwardly directed unit normal vector for thes-phase at f-s interface - q heat flux density - T ^* macroscopic temperature defined by the macroscopic problem - s closure variable - V volume of the macroscopic system - V boundary surface of the macroscopic domain - V ₁ macroscopic sub-surface of the boundary surface - x local coordinate Greek Letters _s,_f volume fraction - _s, gl_f microscopic thermal conductivities - true microscopic temperature - ^* microscopic temperature corresponding toT ^* - microscopic error temperature - vector defined by Equation (34) - < > spatial average     

The equations of miscible flow with negligible molecular diffusion

A set of equations with generalized permeability functions has been proposed by de la Cruz and Spanos, Whitaker, and Kalaydjian to describe three-dimensional immiscible two-phase flow. We have employed the zero interfacial tension limit of these equations to model two phase miscible flow with negligible molecular diffusion. A solution to these equations is found; we find the generalized permeabilities to depend upon two empirically determined functions of saturation which we denote asA andB. This solution is also used to analyze how dispersion arises in miscible flow; in particular we show that the dispersion evolves at a constant rate. In turn this permits us to predict and understand the asymmetry and long tailing in breakthrough curves, and the scale and fluid velocity dependence of the longitudinal dispersion coefficient. Finally, we illustrate how an experimental breakthrough curve can be used to infer the saturation dependence of the underlying functionsA andB.Roman Letters A a surface area; cross-sectional area of a slim tube or core - A _1s pore scale area of interface between solid and fluid 1 - A ₁₂ pore scale area of interface between fluid 1 and fluid 2 - A(S ₁) fluid flow weighting function defined by Equation (3.21) - a _i ,b _a ,c _a ,d _i macro scale parameters,i=1...2 (Section 3); polynomial coefficients,i=1...N (Section 7) - B(S ₁) fluid flow weighting function defined by Equation (3.16) - c _e effluent concentration - c _i mass concentration fluidi=1...2 - c _fi fractional mass concentration of fluidi=1...2 - D dispersion tensor - D _m mechanical dispersion tensor - D ₀ molecular dispersion tensor - D _L longitudinal dispersion coefficient - D _T transverse dispersion coefficient - D _L ⁰ defined by Equation (6.21) - F(c _f2) defined by Equation (5.17) - f ₁(S ₁) fractional flow - g acceleration of gravity - j ₂ deviation mass flux of fluid 2 - K permeability of porous medium - K _ij generalized relative permeability function,i=1...2,j=1...2 - K _ri relative permeability functions,i=1...2 - L length of a slim tube or core - M _i total mass of fluidi=1...2 in volumeV - N number of points used to generate numerical curves - n unit normal to a surface - P pressure - P _i pressure in fluidi=1...2 - P _c capillary pressure - P ₁₂ macroscopic capillary pressure parameter - P(x) normal distribution function - q Darcy velocity of total fluid - q _i Darcy velocity of fluidi=1...2 - S _i saturation of fluidi=1...2 - S _L a low saturation value forS ₁ - S _H a high saturation value forS ₁ - u average intersitial fluid velocity - u _S isosaturation velocity - V volume used for volume averaging - V(c _f2) function defined by Equation (6.28) - V _e effluent volume - V _f fluid volume - V _i volume of fluidi=1...2 (Section 2); injected fluid volume - V _p pore volume of a slim tube or core - v macro scale fluid velocity - v _i macro scale velocity of fluidi=1...2 - _q (S ₁) isosaturation speed - _g (S ₁) component of isosaturation velocity due to gravity - w(S _L,S _H,t) width of a displacement front - w(t) overall width of a displacement front Greek Letters static interfacial tension - _ME macroscopic dispersivity - divergence operator - porosity - _i fraction of pore space occupied by fluidi=1...2 - (S ₁) effective viscosity of the fluid - _i viscosity of fluidi=1...2 - ₁₂ macroscopic fluid viscosity coupling parameter - macro scale fluid density - _i density of fluid i=1...2 - _q effective gravitational fluid density     

The response of a laminar separation bubble to ‘aircraft engine representative’ freestream disturbances

Abstract This paper investigates a large, laminar separation bubble that extends from near the leading edge and along the pressure surface of a low-pressure turbine blade. Whilst stability analyses are not performed, experiments suggest that the separation is convectively unstable at positive incidence. The separation then appears to exhibit absolute instability and, finally, global instability as the incidence of the blade becomes more negative. The conjectured onset of absolute and global instability is used to provide a possible explanation for the separations reduced sensitivity to aircraft engine representative disturbances as the incidence of the blade becomes more negative.Symbols C chord (m) - C _X axial chord (m) - C _P =( P ₀₁– P)/( P ₀₁– P ₂) static pressure coefficient - = fC / V ₂ reduced (nondimensional wake passing) frequency - h span (m) - i incidence (°) - P pressure (Pa) - R reattachment point - Re ₂= V ₂ C / cascade exit Reynolds number - Re _*= V */ Reynolds number based on displacement thickness - s blade pitch (m) - s entropy (J/kg.K) - S separation point - T temperature (K) - mean velocity (m/s) - u _RAW raw velocity (m/s) - u _RMS RMS velocity (m/s) - V = V ₂ C _P isentropic velocity (m/s) - V _REV reverse velocity (m/s) - x axial direction (m) - circumferential direction (rads) - _S entropy coefficient Subscipts 0 stagnation - 1 cascade inlet - 2 cascade exit - freestream