Exact solution of the non-linear differential equation RR̈ + 32R2 − AR−4 + B = 0

The non-linear equation $\text{R}\text{R}\text{?}\text{+}\text{3}\text{2}\text{R}{}^2\text{- AR}{}^{\mathrm{?4}}\text{+ B = 0}$ is shown to represent simply periodic motion with a minimum at R₁ and a maximum at R₁R₀ or a maximum at R₁ and a minimum at R₁R₀^?1. R₀ is a function of the ratio $\text{A}\text{B}$ and is greater than 1 for $\text{A}\text{B}$ > 1 and less than 1 for $\text{A}\text{B}$ > 1. The period of the motion satisfies the simple relation T(R₀^?1) = R₀^?1T(R₀). The exact solution to the above equation is represented in terms of elliptic integrals of the first and second kinds and a simple algebraic function.     

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.     

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.     

Plastic flow and dislocation structure at small strains in OFHC copper deformed in tension,torsion and combined tension-torsion

OFHC copper specimens of 39 μm grain size were deformed to small strains (up to 8%) in tension, torsion and combined tension-torsion at 300 K and the resulting dislocation structures, distributions and densities were determined using transmission electron microscopy. Employing the von Mises yield criterion and the plastic-work hypothesis good agreement was obtained for the three testing conditions for (i) equivalent stress $\text{}\text{?}$gs vs equivalent strain $\text{}\text{?}$g3^p curves, (ii) the dislocation structure, distribution and density ρ as a function of $\text{}\text{?}$g3^p, and (iii) $\text{}\text{?}$gs as a function of $\text{ρ}\text{1}\text{2}$. Furthermore, upon comparing the $\text{}\text{?}$gs vs $\text{ρ}\text{1}\text{2}$ curve for polycrystalline copper with the τ_RSS vs $\text{ρ}\text{1}\text{2}$ curve for single crystals, an average Taylor factor M= (σ/τ_RSS) of approximately 3.2 was obtained, which is in good accord with that predicted theoretically for FCC metals. Almost equally good correlations for the stressstrain curves and for the dislocation density were obtained on the basis of maximum shear stress τ_max and maximum shear strain γ^p_max as on the basis of $\text{}\text{?}$gs and $\text{}\text{?}$g3^P. Therefore, the present results do not permit a positive decision on the question whether the dislocation density correlates better with $\text{}\text{?}$gs and $\text{}\text{?}$g3^P or with τ_max and γ^P_max.A single test in which the direction of straining in torsion was reversed yielded a density and distribution of dislocations (and a corresponding value of $\text{}\text{?}$gs) equivalent to those that developed at a smaller strain in unidirectional straining.     

The critical separation reynolds number for streaming flow past a circular cylinder

Employing a complex variable approach to the equations of motion for an incompressible viscous fluid and a more general approximation to the convecting stream function it is possible to calculate an approximation to the vorticity on the boundary for streaming flow past a circular cylinder without solving first for the complete flow field. In particular it is found that separation at the rear stagnation point first occurs when $\text{R1 = 2.78}$, where $\text{R1}$ is the critical Reynolds number. This result is in good agreement with the value of $\text{R1}$ obtained by experiment and the value obtained numerically. The convecting stream function satisfies the conditions of no slip and vorticity is not convected through the cylinder as it is in small Reynolds number Oseen theory.     

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.     

Some remarks on “Drag coefficients of a slowly moving sphere in Non-Newtonian fluids”

Viscoelastic solutions were ejected vertically downwards into air and various Newtonian fluids. The measured swell increased significantly when ejected into a liquid rather than air. The observed increase is considered a result of both bouyancy and drag forces on the solution. The following dimensions expression relating the ratio of the swell diameter in liquid and air D_L/D_A to the elastic shear compliance of the ejected solution J_e was experimentally observed.(D_L/D_A)⁶-1=30(Δ?/?_s)^{?$\text{1}\text{2}$}([g²η²_N?_s]$\text{1}\text{3}$J_e)^{$\text{3}\text{5}$}, where Δ? is the density difference between the extruded and Newtonian fluid, ?_s is the solution density, g is the gravitational constant, and η_N is the Newtonian fluid viscosity. Thus with this expression a simple extrudate swell technique exists to estimate the elastic shear compliance of a viscoelastic solution.     

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 $\text{τ}{}^{\mathrm{+}}$.     

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.     

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).     

Effects of incompressible surfactant on thermocapillary interactions of spherical drops

Collision efficiencies are determined for two surfactant-covered spherical drops in the limit of nearly uniform surface coverage in thermocapillary motion. The problem is linearized by assuming dilute surfactant concentration, with the effect of surfactant controlled by a single retardation parameter A  . The mobility function _LA$L_A$ along the drops’ line of centers is much less than zero over a wide range of parameters, so that the smaller drop can move faster than the larger one at moderate to large separations. At surface Péclet numbers less than 10, the incompressible surfactant model agrees well with solution of the full convective-diffusion equation for the minimum separation between drops. With the exception of non-conducting drops, the collision efficiencies become zero at moderate values of A. A model system of contaminated ethyl salicylate (ES) drops in diethylene glycol (DEG) is studied in thermocapillary motion. Population dynamics simulations confirm the coalescence-inhibiting effect of incompressible surfactant on the evolution of the ES/DEG drop-size distribution.     

Rheology of suspensions of spherical particles in a newtonian and a non-newtonian fluid

Properties of suspensions of spherical glass beads (25–38 μm dia.) in a Newtonian fluid and a non-Newtonian (NBS Fluid 40) fluid were measured at volume fractions, φ, of 0%, 10%, 20% and 30%. Measurements were made using a modified and computerized Weissenberg Rheogoniometer. Properties measured included steady shear viscosity, η($\text{γ}\text{.}$), first normal stress difference, N₁($\text{γ}\text{.}$), linear viscoelastic properties, η′(ω) and G′(ω), shear stress relaxation, σ^? ($\text{γ}\text{.}$, t), and growth, σ⁺($\text{γ}\text{.}$, t) and normal stress relaxation, N₁^?($\text{γ}\text{.}$, t).For a the Newtonian fluid, increasing φ causes both η and η′ to increase, with η′ showing a slight frequency dependence. Both N₁ and G′ are zero and stress relaxation and growth occur essentially instantaneously. For the NBS fluid, both η and η′ increse with φ at all $\text{γ}\text{.}$ and ω, respectively, the increase being greater as $\text{γ}\text{.}$ and ω approach zero. N₁ and G′ are less affected by the presence of the particles than η and η′ with the effect on G′ being more pronounced than on N₁. For fixed $\text{γ}\text{.}$, stress relaxation and growth exhibit greater non-linear effects as φ is increased. A model for predicting a priori the linear viscoelastic properties for suspensions was found to yeild reasonable estimates up to φ = 20%.     

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).     

Slow flow through a periodic array of spheres

Slow flow through a periodic array of spheres is studied theoretically, and the drag force by the fluid on a sphere forming the periodic array is calculated using a modification of the method developed by Hashimoto (1959). Results for the complete range of volume fraction $\text{c}$ of spheres are given for simple cubic, body-centered cubic, and face-centered cubic arrays and these agree well with the corresponding values reported by previous investigators. Also, series expansions for the drag force to 0($\text{c}{}^{10}$) are derived for each of these cubic arrays. The method is also applied to determine the drag force to 0($\text{c}{}^3$) on infinitely long cylinders in square and hexagonal arrays.     

Two new methods for the approximate determination of the initial coefficient of the first normal stress difference

Two methods for determining the initial coefficient of the first normal stress difference are presented. They are based on the evaluation of the steady viscosity function η($\text{γ}\text{.}$) and the viscosity function η⁺($\text{γ}\text{.}$, t) at the start-up of a flow with a very small rate of deformation $\text{γ}\text{.}$ < $\text{γ}\text{.}$₀. For the functions η($\text{γ}\text{.}$) and η⁺($\text{γ}\text{.}$), equations are given which can be used for a simple evaluation of the integral relationships obtaiend for ψ₁₀. The values for ψ₁₀ calculated by the two methods are compared with values obtained by the well-known methods via measurement of the ψ₁($\text{γ}\text{.}$) or η″(ω)/ω functions and extrapolation to zero). Both methods give values which are in satisfactory agreement with the experimental values.     

Flow patterns in polyethylene and polystyrene melts during extrusion through a die entry region: measurement and interpretation

Flow visualization experiments have been carried out on these melts flowing from a reservoir into a capillary die. The existence and magnitude of vortices at the die entrance have been determined over a range of extrusion conditions. The vortex size is interpreted in terms of the theory of viscoelastic fluid mechanics. It is found that the second-order fluid-perturbation solution cannot represent the observed experimental results. The data are correlated with (i) a Weissenberg number τ_ch$\text{V}\text{L}$ ≡ $\text{}\text{?}$gt($\text{γ}\text{?}$_w)$\text{γ}\text{?}$_w ≡ Ψ₁$\text{γ}\text{?}$_w/2η  (N₁)_w/ 2(σ₁₂)_w measured at the die wall and (ii) with the deformation-rate dependence of relaxation time. Interpretation of vortex formation and size in terms of elongational viscosity is offered.Several polystyrene and polyethylene melts have been rheologically characterized as part of this study with measurements of viscosity η and principal normal stress difference N₁. The zero shear viscosity η₀ of the polystyrenes varies with the 3.5 power of the weight-average molecular weight M_w while the principal normal stress difference coefficient Ψ₁ varies with the sixth power of M_w when evaluated at a shear rate of 1 sec^?1.     

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.     

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.