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.     

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).     

A class of complete integrals of plane eikonal equations

A class of complete integrals of the plane eikonal equation

$\text{|}\text{grad}\text{u|}{}^2\text{= f(x,y)}$

for harmonic u(x,y) is determined by using complex variables. The case in which z = 0 is a singular point of the analytic function whose real part is log f is also treated. Illustrative examples are given.     

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.     

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).     

Dynamics of a system exhibiting the global bifurcation of a limit cycle at infinity

This paper concerns the dynamics of a class of non-linear oscillators of the form:

$\text{x″ + x ? ?x′(1?ax}{}^2\text{?bx′}{}^2\text{) = 0}$

. The non-linear term contains two parameters a and b which may be varied to give the Rayleigh and Van der Pol differential equations as special cases.The existence and approximation of limit cycles in this system are investigated using the Poincare-Bendixson theorem and the Lindstedt perturbation method. Analysis of the system at infinity is used to study the global bifurcation through which the limit cycle is created from four saddle-saddle connections between equilibrium points at infinity. Center manifold theory is used to determine the stability of the equilibrium points at infinity. Numerical integration is used to verify the analytical results.It is shown that an arbitrarily small perturbation to the damping term of the Rayleigh equation results in points close to the stable limit cycle escaping to infinity.     

An asymptotic solution of an initial value problem for a non-linear viscoelastic rod

Stress wave propagation in a one-dimensional materai, described by the non-linear constitutive law

$\text{?σ}\text{?t}\text{? E}\text{?ε}\text{?t}\text{+ k[σ ? E'g3 + βε}{}^2\text{] = 0}$

, is analyzed by a two-time variable perturbation method. Secular terms are “cast out” by employing special orthogonality conditions. The resulting expansion reveals a “mode coupling”, due to the non-linearity, and displays the significance of the three parameters present in the constitutive law. Numerical values for the parameters are assumed and the resulting displacements computed.     

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.     

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.     

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.     

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 $\text{τ}{}^{\mathrm{+}}$.