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 $\text{τ}{}^{\mathrm{+}}$.     

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.     

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.     

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.     

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).     

Mass and heat transfer by natural convection in a vertical cavity

This paper reports a fundamental study of laminar natural convection in a rectangular enclosure with heat and mass transfer from the side, when the bouyancy effect is due to density variations caused by either temperature or concentration variations. In the first part of the study scale analysis is used to determine the scales of the flow, temperature and concentration fields in boundary layer flow for all values of Prandtl and Lewis numbers. In particular, scale analysis shows that in the extreme case where the flow is driven by bouyancy due to temperature variations, the ratio of mass transfer rate divided by heat transfer rate scales as $\text{Le}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ only if (Pr > 1, Le < 1) or (Pr < 1, Sc < 1), and as $\text{Le}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$ if (Pr > 1, Le > 1) or (Pr < 1, Sc > 1). In the second part of the study, the boundary layer scales derived in the first part are used to determine the heat and mass transport characteristics of a vertical slot filled with fluid. Criteria for the existence of distinct thermal and concentration boundary layers in the slot are determined. Numerical solutions for the flow and concentration fields in a slot without distinct thermal boundary layers are reported. These solutions support further the method of scale analysis employed in the first part of the study     

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).     

Correlation of turbulent flow rate-pressure drop data for non-newtonian solutions and slurries in pipes

Isothermal and non-isothermal flow rate-pressure drop data in turbulent flow through smooth pipes have been obtained for non-Newtonian fluids, including aqueous solutions of polymers and aqueous suspensions of titanium dioxide. It has been found that the friction factor, f, is a function of a new form of Reynolds number, Re_B, based on the parameters A, x and w of Bowen's correlation, viz.

$\text{τ}{}_{\mathrm{w}}\text{D}{}^{\mathrm{x}}\text{=A}\text{u}{}^{\mathrm{w}}$

where τ_w is the wall shear strees, ?u the mean velocity, D the pipe diameter; A, x and w are experimentally derived parameters which characterise the fluid.     

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).     

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.     

Shear-layer acoustic radiation in an excited subsonic jet: experimental study

The subharmonic acoustic radiation of a tone excited subsonic jet shear-layer has been investigated experimentally. Two jet velocities $U_j=20\text{m}?{\text{s}}^{\mathrm{?1}}$ and $U_j=40\text{m}?{\text{s}}^{\mathrm{?1}}$ were studied. For $U_j=20\text{m}?{\text{s}}^{\mathrm{?1}}$, the natural boundary-layer at the nozzle exit is laminar. When the perturbation is applied, the fluctuations of the first and the second subharmonics of the excitation frequency are detected in the shear-layer. In addition, the first subharmonic near pressure field along the spreading jet is constituted of two strong maxima of sinusoidal shape. The far-field directivity pattern displays two lobes separated by an extinction angle ${\theta }^{?}$ at around 85° from the jet axis. These observations follow the results of Bridges about the vortex pairing noise. On the other hand, for $U_j=40\text{m}?{\text{s}}^{\mathrm{?1}}$, the initial boundary-layer is transitional and only the first subharmonic is observed in the presence of the excitation. The near pressure field is of Gaussian shape in the jet periphery and the acoustic far-field is superdirective as observed by Laufer and Yen. The state of the initial shear-layer seems to be the key feature to distinguish these two different radiation patterns. To cite this article: V. Fleury et al., C. R. Mecanique 333 (2005).     

Fractional vapour content of a liquid pool through which vapour is bubbled

Data from a large number of Russian, American and German sources are examined and found to be correlated in general by

$\text{α}\text{1?α)}\text{1}\text{2}\text{= K[F}{}_{\mathrm{D}}\text{P}{}^{\mathrm{m}}\text{]}{}^{\mathrm{n}}$

where α is voidage or fractional vapour content, K is a constant, F_D is a Froude number and P is a physical properties group. However, the exponent m is found to vary from 0 to 0.3 and the exponent n from $\text{2}\text{3}$ to 0.79, depending upon the sources of the data. The most probable value for n is $\text{2}\text{3}$ but a firm choice cannot be made for m, which is either 0.16 or 0.3. The different values of m depend chiefly upon the method of measurement of the voidage.     

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.     

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.     

The effect of hydrodynamic coupling on the axisymmetric drainage of thin films

The effect of hydrodynamic coupling of adjacent phases on the axisymmetric drainage of thin films is examined using a prototype model of coalescence. For long times, pressure forces in the film dominate flow in all three regions, and finally all move effectively as one, whereas for short times, profiles are sharp and initial flow differences in the three regions can dominate pressure effects. For intermediate times, temporal evolution of velocity profiles depends in a complicated way on the kinematic viscosity ratio and the parameter $\text{R = (?}{}_{\mathrm{A}}\text{μ}{}_{\mathrm{A}}\text{/?}{}_{\mathrm{B}}\text{/μ}{}_{\mathrm{B}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, as well as on initial conditions and pressure gradient. Generally speaking, the initial flows have less of an effect on overall drainage time than the presence of induced circulation in adjacent phases. Analytical solutions are plotted for a range of systems and representative initial conditions and pressure gradients. In a subsequent article, film-thinning equations are solved using this information.     

An experiment to measure the pressure dependence of the zero-shear-rate viscosity

A simple experimental method, based on Stokes' law for falling spheres, has been devised and used to measure the pressure-dependence of the zero-shear-rate viscosity of a polypropylene melt. The experiment was performed by maintaining three thick-walled test cylinders containing the polymer melt and the falling sphere at the same elevated temperature but different pressures for periods of time ranging from 20 to 48 hours.When compared with experiments using high-pressure capillary or rotational viscometers, this experimental method has the advantages that viscous heating is non-existent and the apparatus and data analysis are relatively simple. The principal disadvantage encountered here, thermal degradation at high temperatures, could probably be reduced by molding specimens under vacuum and by shortening the exposure time. Since the falling-sphere experiment provides data at very low shear rates and the capillary and rotational viscometers generate data at high shear rates, the two experimental methods are complementary.The pressure coefficient b [=d(In η₀/dp] was determined for Hercules Pro-fax 6523 polypropylene in two series of experiments at different temperatures. For seven experiments at 218.3°C and pressures up to 97.9 $\text{MN}\text{m}{}^2$ (14,200 psi), the average value of b ± 95% confidence limits was found to be 14.8 ($\text{GN}\text{m}{}^2$)^?1 ± 2.9.The average b was 12.6 ($\text{GN}\text{m}{}^2$)^?1 ± 1.4 in a series of eight experiments at 232.2°C and pressures up to 123 $\text{MN}\text{m}{}^2$ (17,800 psi).     

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.     

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.