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).     

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.     

The role of shear stresses in brittle fracture

Using the three-dimensional model for brittle fracture developed earlier by S.A.F. Murrell and P.J. Digby (1970,1972) shear stress concentrations are calculated for brittle bodies and the relative roles of tensile and shear stresses in the fracture process are considered. It is found that the maximum shear stress and the maximum tensile stress occur at different places on a crack, and that there is a wide range of stress states for which they do not occur on the same crack. Furthermore, if the theoretical cleavage strength is σ_max and the theoretical shear strength is τ_max, then cleavage precedes inelastic shear and brittle fracture is possible, for suitable stress systems, when $\text{σ}{}_{\max }\text{<}\text{2τ}{}_{\mathrm{<mtext>max</mtext>}}\text{(1 ? ν)}$, where ν is the Poisson's ratio of the solid matrix. This appears to be in accordance with empirical observations.     

Asymptotic analysis of growing plane strain tensile cracks in elastic-ideally plastic solids

An exact asymptotic analysis is presented of the stress and deformation fields near the tip of a quasistatically advancing plane strain tensile crack in an elastic-ideally plastic solid. In contrast to previous approximate analyses, no assumptions which reduce the yield condition, a priori, to the form of constant in-plane principal shear stress near the crack tip are made, and the analysis is valid for general Poisson ratio ν. Specific results are given for ν = 0.3 and 0.5, the latter duplicating solutions in previous work by L.I. Slepyan, Y.-C. Gao and the present authors. The crack tip field is shown to divide into five angular sectors of four different types ; in the order in which these sweep across a point in the vicinity of the advancing crack, they are : two plastic sectors which can be described asymptotically (i.e., as r → 0, where r is distance from the crack tip) in slip-line terminology as ‘constant stress’ and ‘centered fan’ sectors, respectively ; a plastic sector of non-constant stress which cannot be described asymptotically in terms of slip lines; an elastic unloading sector; and a trailing plastic sector of the same type as that directly preceding the elastic sector. Further, these four different sector types constitute the full set of asymptotically possible solutions at the crack tip. As is known from prior work, the plastic strain accumulated by a material point passing through such a moving ‘centered fan’ sector is O(ln r) as r → 0 ; it is proved in the present work that the plastic strain accumulated by a material point passing through the ‘constant stress’ sector ahead of a growing crack must be less singular than In r as r → 0. As suggested also in earlier studies, the rate of increase of opening gap δ at a point currently at a distance r behind, but very near, the crack tip is given for crack advance under contained yielding by

$\text{δ}\text{?}\text{=}\text{α}\text{J}\text{?}\text{σ}{}_0\text{+β(}\text{σ}{}_0\text{E}\text{)}\text{a}\text{?}\text{ln(}\text{R}\text{r}\text{)}$

where a is crack length, σ0 is tensile yield strength, E is Young's modulus, J is the value of the J-integral taken in surrounding elastic material, and the parameters α and R are undetermined by the asymptotic analysis. The exact solution for ν = 0.3 gives β = 5.462, which agrees very closely with estimates obtained from finite element solutions. An approximate analysis based on use of slip line representations in all plastic sectors is outlined in the Appendix.     

A line plastic-zone model for steady mode III crack growth in an elastic-plastic material

Steady-state quasi-static growth of a crack in the anti-plane shear mode through an elastic-plastic material is analyzed. The material is non-hardening and small-scale yielding conditions are assumed. The essential feature of the model is that the active plastic-zone is assumed to be a pair of discrete lines emanating from the crack tip out of the crack plane on which a suitable yield condition is satisfied. An exact solution is obtained for the plastic strain left in the wake of this active line plastic-zone. The extent of the plastic zone from the tip is determined to be 0.071 $\text{(}\text{k}\text{τ}{}_0\text{)}{}^2$ where k and τ₀ are the remote elastic stress intensity factor and the shear flow stress, respectively, and it is found that 36% of the elastic energy flowing into the crack-tip region during growth is dissipated through plastic work and 64% is trapped as residual elastic energy in the plastic-zone wake.     

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.     

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).     

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.     

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.     

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.     

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     

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).     

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.