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.     

Diffusion and wave behaviour in linear Voigt modelDiffusion et comportement ondulatoire dans le modèle linéaire de Voigt

A boundary value problem $\text{P}{}_{\mathrm{\epsilon }}$ related to a third order parabolic equation with a small parameter ε is analized. This equation models the one-dimensional evolution of many dissipative media as viscoelastic fluids or solids, viscous gases, superconducting materials, incompressible and electrically conducting fluids. Moreover, the third order parabolic operator regularizes various nonlinear second order wave equations. In this paper, the hyperbolic and parabolic behaviour of the solution of $\text{P}{}_{\mathrm{\epsilon }}$ is estimated by means of slow timeτ=εt and fast timeθ=t/ε. As consequence, a rigorous asymptotic approximation for the solution of $\text{P}{}_{\mathrm{\epsilon }}$ is established. To cite this article: M. De Angelis, P. Renno, C. R. Mecanique 330 (2002) 21–26     

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.     

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.     

Boundary layer in compressible flow of wet water vapour

In this paper the influence of small droplets, with radius $\text{10}{}^{\mathrm{?8}}\text{m}\text{< r < 10}{}^{\mathrm{?6}}\text{m}$, on laminar and turbulent boundary layer behavior is considered. It is found that the laminar boundary layer in a two-phase flow with strongly dispersed liquid retains dissipation energy and that the recovery factor of enthalpy is greater than unity. In turbulent boundary layers small droplets are transported by turbulent diffusion and this leads to the recovery factor being less than unity. Its value in both cases depends mainly on the nondimensional number $\text{Ds}\text{= C}{}_{\mathrm{Le}}\text{L/(U}{}_{\mathrm{e}}{}^2\text{/2)}$. The laminar boundary layer solution for non-equilibrium two-phase flow is obtained. Profiles of the droplet mass fraction, vapour and droplets temperatures and droplet radius are computed for the case of a steady two-dimensional flow. The turbulent boundary layer is treated using a semi-empirical theory assuming thermodynamic equilibrium.     

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.     

Bifurcations et solutions multiples en cavité 3D différentiellement chaufféeBifurcations and multiple solutions in a differentially heated cubic cavity

In this paper, a differentially heated square/cubic cavity is studied by performing three-dimensional direct numerical simulations. The first bifurcation observed at $\text{R}{}_{\mathrm{a}}\text{≈3.2×10}{}^7$ is due to the 3D vortex structures generated at the end regions of vertical boundary layers near the median plane. The main results of this Note are that the flow returns to a steady state for higher values of the Rayleigh number $\text{R}{}_{\mathrm{a}}$ (7×10⁷ and 10⁸ for example) still exhibiting these 3D vortex structures, and that multiple steady flows which differ by their symmetry properties, are obtained for $\text{R}{}_{\mathrm{a}}\text{=10}{}^8$. However, the flow reverts to unsteadiness for $\text{R}{}_{\mathrm{a}}\text{=3×10}{}^8$. In this latter case, the instability is due to the vertical boundary layers. To cite this article: G. de Gassowski et al., C. R. Mecanique 331 (2003).     

Homogenization limit for electrical conduction in biological tissues in the radio-frequency rangeLimite d'homogénéisation pour la conduction électrique dans les tissus biologiques dans le domaine des radiofréquences

We study an evolutive model for electrical conduction in biological tissues, where the conductive intra-cellular and extracellular spaces are separated by insulating cell membranes. The mathematical scheme is an elliptic problem, with dynamical boundary conditions on the cell membranes. The problem is set in a finely mixed periodic medium. We show that the homogenization limit u₀ of the electric potential, obtained as the period of the microscopic structure approaches zero, solves the equation $\text{?}\text{div}\text{(σ}{}_0\text{?}{}_{\mathrm{x}}\text{u}{}_0\text{+A}{}^0\text{?}{}_{\mathrm{x}}\text{u}{}_0\text{+∫}{}_0{}^{\mathrm{t}}\text{A}{}^1\text{(t?τ)?}{}_{\mathrm{x}}\text{u}{}_0\text{(x,τ)}\text{d}\text{τ?}\text{F}\text{(x,t))=0}$ where σ₀>0 and the matrices A⁰, A¹ depend on geometric and material properties, while the vector function $\text{F}$ keeps trace of the initial data of the original problem. Memory effects explicitly appear here, making this elliptic equation of non standard type. To cite this article: M. Amar et al., C. R. Mecanique 331 (2003).     

Linear methods in problems of non-linear differential equations with a small parameter

Methods have been considered for deriving asymptotical formulas for the systems of the type

$\text{ε}{}^{\mathrm{p}}\text{d}\text{x}{}_{\mathrm{k}}\text{d}\text{t}\text{= f}{}_{\mathrm{k}}\text{(x) + ε}\text{f}\text{?}{}_{\mathrm{k}}\text{(x) + …}$

by constructing an analog of the Schrödinger perturbation theory of the linear operator

$\text{∑}\text{k}\text{[f}{}_{\mathrm{k}}\text{(x) + ε}\text{f}\text{?}{}_{\mathrm{k}}\text{(x)]}\text{?F}\text{?x}{}_{\mathrm{k}}\text{= A}{}_{\mathrm{o}}\text{F + εA}{}_1\text{F.}$

These methods can be extended to some classes of partial differential equations, in particular, to Whitham's non-linear theory.     

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.     

Time-average local thickness measurement in falling liquid film flow

A method for monitoring time-varying local film thickness variation through the detection of laser scattering from suspended latex particles is briefly described. This method was used in conjunction with the Jeffreys theory of drainage from a flat plate to determine time-average local film thickness.Measurements were made at Reynolds numbers (equal to ($\text{4Q/ν}$)) from 145 to 4030 at varying distances along the direction of flow. At the bottom of the flow, 134 cm from the top, average film thickness is given by the expression: where $\text{a}{}_{\mathrm{i}}$ and $\text{n}{}_{\mathrm{i}}$ are constants unique to each of the three Reynolds number regions, wavy laminar, transitional and turbulent.     

Homogenized model of reaction–diffusion in a porous mediumUn modèle homogénéisé de réaction–diffusion dans un milieu poreux

We study the initial boundary value problem for the reaction–diffusion equation,

$\text{?}{}_{\mathrm{t}}\text{u}{}^{\mathrm{\epsilon }}\text{??·(a}{}^{\mathrm{\epsilon }}\text{?u}{}^{\mathrm{\epsilon }}\text{)+g(u}{}^{\mathrm{\epsilon }}\text{)=h}{}^{\mathrm{\epsilon }}$

in a bounded domain $\text{Ω}$ with periodic microstructure $\text{F}{}^{\mathrm{(\epsilon )}}\text{∪}\text{M}{}^{\mathrm{(\epsilon )}}$, where a^ε(x) is of order 1 in $\text{F}{}^{\mathrm{(\epsilon )}}$ and κ(ε) in $\text{M}{}^{\mathrm{(\epsilon )}}$ with κ(ε)→0 as ε→0. Combining the method of two-scale convergence and the variational homogenization we obtain effective models which depend on the parameter θ=lim_ε→0κ(ε)/ε². In the case of strictly positive finite θ the effective problem is nonlocal in time that corresponds to the memory effect. To cite this article: L. Pankratov et al., C. R. Mecanique 331 (2003).     

Non-linear boundary and eigenvalue problems for the emden-fowler equations by group methods

Two pragmatic boundary value and eigenvalue problems of the Emden-Fowler equation $\text{(t}{}^{\mathrm{\alpha }}\text{u′)′ + λt}{}^{\mathrm{\beta }}\text{?(u) = 0,?(u) = u}{}^{\mathrm{\gamma }}$ and e^u are studied using the simple one parameter group properties. In all cases boundary value problems are converted into initial value problems using the property of the invariance group. With $\text{?(u) = u}{}^{\mathrm{\gamma }}$ an eigenvalue problem is detailed and calculations presented.     

Effect of particulate thermophoresis in reducing the fouling rate advantages of effusion-cooling

To predict small particle diffusional mass transfer (deposition), including particle thermophoresis, transpiration cooling and variable properties, the coupled ordinary differential equations governing self-similar laminar boundary layers are solved numerically. Under typical combustion turbine conditions, although diffusional deposition rates can be dramatically reduced by transpiration cooling (eg by some 5-decades for mainstream submicron particles corresponding to a Schmidt number of about 10² (or d_p ≈ 0.7 × 10^?2μm) and a wall transpiration-cooled to $\text{T}{}_{\mathrm{<mtext>w</mtext>}}\text{T}{}_{\mathrm{<mtext>e</mtext>}}\text{= 0.8}$) actual deposition rate reductions will be smaller than previously expected (by about 1 decade for particles with Sc ≈ 10²), owing to thermophoretic particle drift ‘caused’ by the colder wall. Such micro-droplets, small enough to behave like ‘heavy molecules’ in combustion systems, are often important because they can cause adherence of the much larger, supermicron, ash particles which inertially impact on the same surface     

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 $\text{τ}{}^{\mathrm{+}}$.     

A simple stochastic model of two phase flow pressure drop accompanied by boiling inside circular tubes with in-line static mixers

A simple stochastic model has been developed for boiling pressure drop inside a circular tube with in-line static mixers. This model gives rise to the dimensionless correlating equation of the form:

$\text{[f] = a[Pr}{}_{\mathrm{m}}\text{]}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{[Re}{}_{\mathrm{L}}\text{]}{}^{\mathrm{?1}}\text{μm}\text{μ}{}_{\mathrm{L}}\text{ρ}\text{ρ}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{H}{}_{\mathrm{se}}\text{+ xH}{}_{\mathrm{LG}}\text{C}{}_{\mathrm{pm}}\text{ΔT}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$

This correlation shows good agreement with the experimental data.     

Holdup in two phase flow

A wide range of experimental holdup data have been analysed on the basis of the general correlations of Chen & Spedding (1983). For upward inclined flow, holdup data in the range $\text{(}\text{R}{}_{\mathrm{G}}\text{/}\text{R}{}_{\mathrm{L}}\text{) = 4.0}$ to 275 were handled using a modification of the Chen & Spedding method, and for the case of $\text{(}\text{R}{}_{\mathrm{G}}\text{/}\text{R}{}_{\mathrm{L}}\text{) ? 4.0}$, the modified Armand equation was found to be suitable. Horizontal stratified flow was examined using the Bernoulli equation, and shown to be a limiting case of the free draining of a tube initially filled with liquid. For downward inclined stratified flow, the Manning equation predicted the holdup accurately for low liquid rates and small angles of inclination. In addition, for these two cases of horizontal and downward stratified flow, the holdup also was examined in terms of the critical depth of flow as determined using the total energy relation.     

Dispersion in annular climbing film flow

The dispersion of a tracer injected as a pulse into a climbing liquid film is investigated for a series of water and air flow rates, and for a number of different electrolyte tracers. It is found that at all flow rates the observed concentration distribution depends on the nature of the tracer. This observation is explained in terms of two effects: molecular diffusion in a viscous sub-layer and ion fractionation associated with droplet formation at the gas-liquid interface. The overall dispersive characteristics of the system are described in terms of a mathematical model assuming dispersed plug flow in both the film and entrained droplets with interchange between these phases. This model is fitted to experimental tracer concentration distributions using a non-linear least-squares regression procedure. The parameter values obtained from the fitting procedure are studied to determine trends with flow rates and tracer properties. Values for a film dispersion parameter, $\text{P}{}_{\mathrm{f}}$, are found to correlate significantly with the molecular diffusion coefficients of the tracers. Consistent values for an ion fractionation coefficient, $\text{k}{}_{\mathrm{if}}$, are also obtained.