The Einstein viscosity correction in n dimensions

The effective viscosity of a dilute suspension of rigid $\text{n-}\text{dimensional}$ hyperspheres in a viscous fluid at small particle Reynolds numbers is determined; the result being

$\text{μ}{}_{\mathrm{eff}}\text{=μ}\text{1+}\text{n+2}\text{2}\text{φ}$

. Expressions are also given for the $\text{n-}\text{dimensional}$ Stokes velocity and pressure fields for a hypersphere in a pure straining flow.     

Calculating liquid droplets settling on a cylinder in gas-liquid spray flow

Adding atomized liquid to air flowing around a cylinder gives an appreciable increase in heat transfer by forming a liquid film on the cylinder surface. The heat transfer coefficient depends upon the amount of liquid forming the film, which is limited by two phenomena: droplet deflection from the liquid film on the surface and droplets not striking the cylinder. This paper presents a method of calculating the quantity of liquid droplets settling on a cylinder surface in a gas-liquid spray flow. A coefficient $\text{k}$, the volume ratio of the liquid entering the film to the amount of liquid directed at the cylinder, is introduced. $\text{k}$ values were calculated by means of numerical computation and the theory verified experimentally. The calculation method permits estimation of the dependence of the amount of liquid settling on a cylinder on the droplet diameter distribution parameters and on the linear gas velocity     

Phenomenology of inertia effects in a dispersed solid-fluid mixture

Combining single particle results, average equations and thermodynamic considerations, we propose a way to build the equations describing a suspension of rigid spherical particles in a carrier fluid, with emphasis on inertia effects including virtual mass. The spatial fluctuations of the fluid velocity field are depicted by two phenomenological functions $\text{?(α}{}_{\mathrm{s}}\text{)}\text{and}\text{g(α}{}_{\mathrm{s}}\text{)}$ of the particle volume fraction, and a third function h(α_s) is necessary to describe the intensity of the particles internal stress. It is shown that all inertia effects occurring in the relative translational motion can be derived from the two functions $\text{?}\text{and}\text{g–h}$ only. The conditions under which the above system of equations is hyperbolic are determined and comparison is made with what is presently known about $\text{?, g}\text{and}\text{h}$ in the dilute limit.     

Investigation of droplet deposition from a turbulent gas stream

Turbulent deposition of particles from two-phase flow onto the smooth wall of a tube has been studied theoretically and experimentally. A model is proposed for the deposition motion of large particles based on turbulent diffusion in the core followed by a free flight towards the wall. The theory shows that within the Stokes regime, the dimensionless deposition velocity $\text{k-}{}_{\mathrm{<mtext>d</mtext>}}\text{/}\text{u}$* depends on Re and $\text{τ}{}^{\mathrm{+}}$ only, where u* is the friction velocity, Re is the tube Reynolds number and $\text{τ}{}^{\mathrm{+}}$ is the dimensionless particle relaxation time. Deposition data are obtained for air-water droplet flow through a 12.7-mm i.d. acrylic tubing at $\text{Re}\text{= 52,500}$ and 94,600. The proposed theory satisfactorily describes the existing deposition data as well as present measurements, covering a wide range of Re and $\text{τ}{}^{\mathrm{+}}$.     

An experimental study of choked foam flows in a convergent-divergent' nozzle

Choked flow of a foam in a convergent-divergent nozzle has been investigated. The foam consisted of air and a solution of a surface active agent in water. The upstream gas-liquid volume ratio $\text{δ}{}_0$ was in the range 0.053–1.57. The experimental results are in very good agreement with a homogeneous frictionless nozzle flow theory, assuming isothermal behaviour of the gas and no relative motion between the phases, for throat gas-liquid volume ratios $\text{δ}{}_1$ as high as 0.8; for ratios in the range $\text{0.8 < δ}{}_{\mathrm{t}}\text{< 2.98}$ the agreement, while only approximate, is still quite close. Departures from the homogeneous theory are explained in terms of (a) the failure of the assumption of the isothermal behaviour and (b) the existence of relative velocity between the phases. The latter effect predominates at low values of $\text{δ}{}_1$ but at large values, it appears that both contribute to errors in the predictions.     

Two-phase flow in a vertical pipe and the phenomenon of choking: Homogeneous diffusion model—I. Homogeneous flow models

The paper examines the topological structure of all possible solutions which can exist in flows through adiabatic constant-area ducts for which the homogeneous diffusion model has been assumed. The conservation equations are one-dimensional with the single space variable $\text{z.}$ but gravity effects are included. The conservation equations are coupled with three equations of state: a pure substance, a perfect gas with constant specific heats, and a homogeneous two-phase system in thermodynamic equilibrium. The preferred state variables are pressure $\text{P.}$ enthalpy $\text{h.}$ and mass flux $\text{G}{}^2$.The three conservation equations are first-order but nonlinear. They induce a family of solutions which are interpreted as curves in a four-dimensional phase space conceived as a union of three-dimensional spaces ($\text{P, h, G}{}^2\text{, z}$) with $\text{G}{}^2\text{=}\text{const}$ treated as a parameter. It is shown that all points in these spaces are regular, so that no singular solutions need to be considered. The existence and uniqueness theorem leads to the conclusion that through every point in phase space there passes one and only one solution-curve.The set of differential equations, treated as a system of algebraic equations of each point of the phase space, determines the components of a rate-of-change vector which are obtained explicitly by Cramer's rule. This vector is tangent to the solution curve. Each solution curve turns downward in $\text{z}$ at some specific elevation $\text{z}{}^1$, and this determines the condition for choking. Choking occurs always when the exit flow velocity at $\text{L = z}{}^1$ is equal to the local velocity of propagation of small plane disturbances of sufficiently large wavelength, that is when the flow rate $\text{G}$ becomes equal to a specified, critical flow rate, $\text{G}{}^1$. (The possible dependence of the sonic velocity on frequency in a real flow is ignored, because it has not been allowed for in the equations of the model under study.) A criterion, analogous to the Mach number, which indicates the presence or absence of choking in a cross section is the ratio $\text{K = G/G}{}^7$ of the mass-flow rate $\text{G}$ to the local critical mass flow rate. $\text{G}{}^7$, $\text{K = 1}$ denoting choking. The critical parameters depend only on the thermodynamic properties of the fluid and are independent of the gravitational acceleration and shearing stress at the wall.The topological characteristics of the solutions allow us to study all flow patterns which can, and which cannot, occur in a pipe of given length $\text{L}$ into which fluid is discharged through a rounded entrance from a stagnation reservoir and whose back-pressure is slowly lowered. The set of flow patterns is analogous to that which occurs with a perfect gas, except that the characteristic numerical values are different. They must be obtained by numerical integration and the influence of gravity must be allowed for.The preceding conclusions are valid for all assumptions concerning the shearing stress at the wall which make if dependent on the state parameters only, but not on their derivatives with respect to $\text{z}$. However, the study is limited to upward flows for which the shearing stress at the wall and the gravitational acceleration are codirectional.     

Viscous dissipation effects on thermophoretically augmented aerosol particle transport across laminar boundary layers

The sensitivity of aerosol particle motion to local temperature gradients has motivated this investigation of viscous dissipation effects on mass transport rates across nonisthermal, low mass-loading ‘dusty gas’ laminar boundary layers (lbl). From numerical lbl transfer calculations, including ‘ash’ particle thermophoresis and variable thermophysical properties, it has been found that for a specified wall temperature, $\text{T}{}_{\mathrm{<mtext>w</mtext>}}$, and mainstream static temperature, $\text{T}{}_{\mathrm{<mtext>e</mtext>}}$ cous dissipation within the boundary layer increases total particle deposition rates, its relative importance being dependent on $\text{T}{}_{\mathrm{<mtext>w</mtext>}}\text{/T}{}_{\mathrm{<mtext>e</mtext>}}$. For combustion turbine blades which operate at near-unity Mach number, neglect of viscous dissipation is found to cause about a 25% underestimate of the fouling rate at $\text{T}{}_{\mathrm{<mtext>w</mtext>}}\text{/T}{}_{\mathrm{<mtext>e</mtext>}}\text{= 0.8}$ for particle diameters between $\text{0.6 × 10}{}^{\mathrm{?2}}\text{μ}\text{m}$ and 0.3 μm. Alternatively, for conditions of fixed adiabatic wall temperature, $\text{T}{}_{\mathrm{<mtext>aw</mtext>}}$, or fixed stagnation (reservoir) temperature, $\text{T}{}_0$, dusty gas acceleration to appreciable Mach numbers is associated with reduced particle arrival rates due, in part, to the associated reduction in mainstream gas temperature. Recently developed mass transfer rate correlations are extended and found to be successful when tested against the present numerical calculations.     

Experimental and numerical investigation a bubble-driven laminar flow in a axisymmetric vessel

Measurements have been obtained, by laser-Doppler anemometry (LDA), of the axisymetric, recirculating liquid flow caused by a column of air bubbles ($\text{5–6}\text{1}\text{2}\text{mm dia.}$) rising through caster oil in a cylindrical enclosure (100 mm dia.). The liquid velocities correspond to creeping flow. Axial and radial liquid velocity profiles are reported at eight axial stations and, close to within the bubble column, as a function of time. The maximum liquid velocity found outside the bubble column is about 0.5 of that of the bubbles and a very rapid radical decay from this value is noted. The temporal variation of the velocity field, due to the passage of the air bubbles, is undetectable at radial locations greater than about $\text{1}\text{1}\text{2}$ bubble radii from the centreline.The variation of bubble velocity with axial distance was aise measured by LDA for liquid height to enclosure diámeter ratios of 0.98 and 2.78. The maximum bubble velocities were about 0.1–0.2 higher than the Strokes law terminal velocity. The increase is due to the convection of the bubble column by the liquid flow. The maximum bubble velocity is established within approximately three bubble diameters of the air inlet.The motion of the liquid has been calculated by the numerical solution of the steady form of the equations of motion, with the inner boundary of the area of integration lying 1.3 bubble radii from the centerline. The boundary conditions at this surface are assumed to be steady and are taken from measurements of the time-averaged velocity components. The assumption of steady flow at this boundary is supported by experimental observation and results in calculations which are generally in close agreement with the measurements. Discrepancies are confined to the immediate vicinity of the bubble column near to the top and bottom of the enclosure. These are ascribed to a combination of small asymmetries in the experiment and inadequate numerical resolution in these regions.     

Arbitrary motion of an elliptic disc at a fluid interface

The problem of calculating the disturbance due to finite elliptic discs at the interface ($\text{x}{}_3\text{= 0}$) of two incompressible immiscible fluids of different coefficients of viscosity is solved, assuming that body and inertia forces are negligible. When the direction of motion is parallel to the interface, our solution, which is based on potential functions analogous to the Papkovitch-Neuber functions of linear elasticity, satisfies not only the interface conditions of continuity of fluid velocity and stresses but also that of zero normal velocity at the interface. It is also remarkable that this solution produces in each of the fluids a flow field that is totally independent of the properties of the other fluid. These results are not peculiar to elliptic discs, but also hold for finite discs of other shapes. The method of solution presented here can be readily applied to the more general cases where the two-phase fluid, in the absence of the disc, moves with an arbitrarily directed velocity which is a general polynomial function of the coordinates $\text{x}{}_1$ and $\text{x}{}_2$ at the interface. The procedure for carrying this out is demonstrated by treating the case of an elliptic disc in linear shear flow.     

Dynamics of dual-particles settling under gravity

As a first step towards understanding particle–particle interaction in fluid flows, the motion of two spherical particles settling in close proximity under gravity in Newtonian fluids was investigated experimentally for particle Reynolds numbers ranging from 0.01 to 2000. It was observed that particles repel each other for Re>0.1 and that the separation distance of settling particles is Reynolds number dependent. At lower Reynolds numbers, i.e. for Re<0.1, particles settling under gravity do not separate.The orientation preference of two spherical particles was found to be Reynolds number dependent. At higher Reynolds numbers, the line connecting the centres of the two particles is always horizontal, regardless of the way the two particles are launched. At lower Reynolds numbers, however, the particle centreline tends to tilt to an arbitrary angle, even of the two particles are launched in the horizontal plane. Because of the tilt, a side migration of the two particles was found to exist. A linear theory was developed to estimate the side migration velocity. It was found that the maximum side migration velocity is approximately 6% of the vertical settling velocity, in good agreement with the experimental results.Counter-rotating spinning of the two particles was observed and measured in the range of Re=0–10. Using the linear model, it is possible to estimate the influence of the tilt angle on the rate of rotation at low Reynolds numbers. Dual particles settle faster than a single particle at small Reynolds numbers but not at higher Reynolds numbers, because of particle separation. The variation of particle settling velocity with Reynolds number is presented. An equation which can be used to estimate the influence of tilt angle on particle settling velocity at low Reynolds number is also derived.     

Sedimentation of bidisperse suspensions

The average settling velocity of a suspension of identical particles through otherwise quiescent fluid is smaller than the settling velocity of a single particle in an unbounded fluid. When a suspension settles out to form a deposit, this hindered settling effect may lead to complicated sedimentation behaviour, even if the initial suspension is uniformly distributed. This study analyses the bulk sedimentation of bidisperse suspensions and calculates the evolution of the volume fraction of each species from an initially vertically uniform state through to the final steady state where both species have fully settled out of suspension and have formed a deposit. These calculations are analytical and employ the method of characteristics to reveal how both particle species evolve. The profiles often include ‘shocks’, across which discontinuous changes in volume fraction occur. Rarefaction fans may also be found across which the gradients of volume fraction are discontinuous. These new analytical solutions reveal the evolving composition of the suspension and the deposit and may be compared to experimental observations. They also provide test cases that can be used to verify recent numerical techniques for computing the bulk sedimentation behaviour of polydisperse suspensions.     

On unimodular constitutive expressions

We consider constitutive expressions which the stress σ(X, t) at a particle X at time t is given by σ (X, t) = $\text{F}$[F[X, τ)] where $\text{F}$[F(X, τ)] denotes a functional of the history of the deformation gradient matrix [F(X, τ)] from time τ = 0 unti τ = t. This expression is restricted by the requirement of invariance under a superposed rotation of the physical system and by the further requirement that the constitutive expression shall be invariant under the group of unimodular transformations, i.e. $\text{F}$[F(X, τ)] = $\text{F}$[F(X, τ) H] must hold for all matrices H such that det H - 1. We employ results from the classical theory of invariants in order to determine the general form of the expression $\text{F}$[F(X, τ)] which is consistent with these restrictions. Special cases are considered where the functional is replaced by a function of the strain, rate of strain, ? matrices. The case of shear flow is briefly discussed.     

Local properties of vertical mercury-argon two-phase flow in a circular tube under transverse magnetic field

Experimental data are presented in this paper on the profiles of local void fraction, bubble impaction rate, bubble velocity and its spectrum, and also bubble length and its spectrum, of mercury-argon two-phase slug flow flowing upwards in a vertical circular tube in the presence of a transverse magnetic field. Decrease in void fraction and increase in bubble velocity are significant when the magnetic flux density is larger than $\text{0.3～0.4}$$\text{T}\text{(Ha ? 100)}$. This effect is discussed by analyzing the bubble size distribution. Recovery of local void fraction profile in the downstream of an obstacle and diffusion of void injected from only one nozzle in the presence of magnetic field are also discussed.     

On centrifugal separation of particles of two different sizes

We consider flow in a centrifugal force field of a non-dilute suspension with particles or droplets of two sizes. The volume fraction and the velocity fields are determined assuming small convection and shear terms. The resulting flow field is quite different from that in a gravitational settling of a similar mixture. In particular, the volume fraction is a function of time and radius in the sectors separated by kinematic shocks and the settling velocity is a non-monotonic function of the particle size.     

Size segregation and particle velocity fluctuations in settling concentrated suspensions

We investigate the sedimentation of concentrated suspensions at low Reynolds numbers to study collective particle effects on local particle velocity fluctuations and size segregation effects. Experiments are carried out with polymethylmetacrylate (PMMA) spheres of two different mean diameters (190 and 25 μm) suspended in a hydrophobic index-matched fluid. Spatial repartitions of both small and large spheres and velocity fluctuations of particles are measured using fluorescently labelled PMMA spheres and a particle image velocimetry method. We also report measurements of the interstitial fluid pressure during settling. Experiments show that size segregation effects can occur during the sedimentation of concentrated suspensions of either quasi-monodisperse or bidisperse spheres. Size segregation is correlated to the organisation of the sedimentation velocity field into vortex-like structures of finite size. A loss of size segregation together with a significant decrease of the fluid pressure gradient in the bulk suspension is observed when the size of vortex-like structures gets on the order of the container size. However, the emergence of channels through the settling zone prevents a complete loss of size segregation in very concentrated suspensions.     

Burn-out,circumferential film flow distribution and pressure drop for an eccentric annulus with heated rod

Measurements of (1) burn-out, (2) circumferential film flow distribution, and (3) pressure drop in a $\text{17 × 27.2 × 3500}\text{mm}$ concentric and eccentric annulus geometry are presented. The eccentric displacement was varied between 0 and 3 mm. The working fluid was water. Burn-out curves at 70 bar are presented for mass velocities between 500 and 1500 kg/m²s and for inlet subcoolings of 10°C and 100°C. The film flow measurements correspond to the steam qualities $\text{χ = 19 \%}$ and 24 % for the mass velocity $\text{G = 602}$ kg/m²s and $\text{χ = 20 \%}$ and 23 % for $\text{G = 1200}$ kg/m²s. The influence of the circumferential rod film flow variation on burn-out is discussed.     

The stratified flow of gas and non-newtonian liquid in horizontal pipes

Predictions of pressure drop and holdup are presented for the stratified flow of gas and non-Newtonian liquid obeying the Ostwald-de Waele power law model. The model of Taitel & Dukler (1976) for gas/Newtonian liquid flow is extended to liquids possessing either shear-thinning or shear-thickening laminar flow behaviour and computed results are given for flow behaviour indices in the range $\text{0.1 ≤ n ≤ 2}$. In particular, conditions are defined for drag reduction of the liquid flow by the presence of the gas. It is concluded that drag reduction occurs over the largest ranges of liquid and gas flow rates at the lowest $\text{n}$ values, provided that liquid flow remains laminar, but that maximum drag reduction may be expected for shear-thickening liquids with $\text{n}$ values of 2 or greater. Ratios of the liquid flow rate in the presence of gas to that for liquid flow alone under a constant pressure gradient are also presented. These ratios frequently exceed unity and are greatest for highly shear-thinning liquids.Although the Taitel & Dukler approach is consistent with experiments on gas/Newtonian liquid flow, and, in addition, appears to be valid for immiscible Newtonian liquid-liquid systems, provided that the viscosity ratio of the two phases is at least five, experiments are required to confirm its applicability for gas/non-Newtonian systems.     

Espaces d'écoulements dits « universels »Spaces of universal flows

An isochoric motion can be performed both in perfect fluid, in Newtonian fluid, in Maxwell fluid (slow motions) and in Rivlin–Ericksen fluid of second grade whatever be viscosities and viscometric coefficients, iff the motion is universal. Every universal motion with steady vorticity is a generalised Belrami flow, and fulfils the Stokes equation. If the velocity $\text{u}$ of an universal motion complies with $\text{rot}\text{[(?}{}^{\mathrm{<mtext>t</mtext>}}\text{(}\text{Δ}\text{u}\text{))}\text{u}\text{]=}\text{0}$, the motion stands for feasible motion in every second order fluid. Brothers of the potential flows, all the sets of universal motions make up bundles of linear or cono??d spaces with various dimensions, finite or infinite, issued from the rest $\text{u}\text{≡}\text{0}$. The structures appear by scanning parallel to the potential flows. To cite this article: M. Bouthier, C. R. Mecanique 331 (2003).     

Hydromagnetic flow due to torsional oscillations of an infinite disk about a steady non-zero mean

The present investigation is the study of the laminar hydromagnetic flow due to torsional oscillations of an infinite disk about a steady non-zero mean in an electrically conducting fluid. Separate solutions have been obtained for the limiting cases of low and high frequency oscillations. The low frequency solution is obtained by expressing the flow functions in powers (ik) while for high frequencies, the flow functions are expressed in powers of $\text{(ik)}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$, where $\text{k =}\text{ω}\text{Ω}$ being the ratio of the frequency of oscillations ω to the mean disk-angular velocity Ω and i² = ?1. It is found that the oscillating part of the transverse shearing stress has a phase-lead while that of the radial shearing stress has a phase lag behind the disk oscillations. The phase-lead in the former case and phase-lag in the latter case decrease with the increase in the strength of the applied magnetic field.