Asymptotics for the Expected Lifetime of Brownian Motion on Thin Domains in ℝ n

We derive a three-term asymptotic expansion for the expected lifetime of Brownian motion and for the torsional rigidity on thin domains in ? ⁿ , and a two-term expansion for the maximum (and corresponding maximizer) of the expected lifetime. The approach is similar to that which we used previously to study the eigenvalues of the Dirichlet Laplacian and consists of scaling the domain in one direction and deriving the corresponding asymptotic expansions as the scaling parameter goes to zero. Apart from being dominated by the one-dimensional Brownian motion along the direction of the scaling, we also see that the symmetry of the perturbation plays a role in the expansion. As in the case of eigenvalues, these expansions may also be used to approximate the exit time for domains where the scaling parameter is not necessarily close to zero.     

A parameter-uniform numerical method for time-dependent singularly perturbed differential-difference equations

A numerical study is made for solving a class of time-dependent singularly perturbed convection–diffusion problems with retarded terms which often arise in computational neuroscience. To approximate the retarded terms, a Taylor’s series expansion has been used and the resulting time-dependent singularly perturbed differential equation is approximated using parameter-uniform numerical methods comprised of a standard implicit finite difference scheme to discretize in the temporal direction on a uniform mesh by means of Rothe’s method and a B-spline collocation method in the spatial direction on a piecewise-uniform mesh of Shishkin type. The method is shown to be accurate of order O(M⁻¹ + N⁻² ln³N), where M and N are the number of mesh points used in the temporal direction and in the spatial direction respectively. An extensive amount of analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter. Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations. Comparisons of the numerical solutions are performed with an upwind and midpoint upwind finite difference scheme on a piecewise-uniform mesh to demonstrate the efficiency of the method.     

Properties of rigid polyurethane foams filled with milled carbon fibers

The effect of milled carbon fibers of two types (differing in length) on the properties of rigid polyurethane foams in the density range from 50 to 90 kg/m³ is investigated. The coefficient of thermal expansion and properties of the foams in tension and compression as functions of fiber content in them are determined. It is found that the long fibers are more efficient in improving the properties of the foams in their rise direction. The elongation at break of the foams decreases significantly with increasing fiber content.     

CFD simulation of precession in sudden pipe expansion flows with low inlet swirl

In this paper, three-dimensional, time-dependent calculations are carried out using the finite volume CFD code CFX4 and the VLES approach with standard k–ε model to simulate the turbulent swirl flow in an axisymmetric sudden expansion with an expansion ratio of 5.0 for a Reynolds number of 10⁵. This flow is unstable over the entire swirl number range considered between 0 and 0.48, and a large-scale coherent structure is found to precess about the centerline. Compared with the unswirled case, inclusion of a slight inlet swirl (swirl number below 0.23) can reduce the precession speed, cause the precession to be against the mean swirl and suppress the flapping motion. Several modes of precession are predicted as the swirl intensity increases, in which the precession, as well as the spiral structure, reverses direction. Accompanying the transition between different modes, abrupt changes in precession frequency are also experienced. Grid sensitivity and comparison with smaller expansion ratio data are also discussed.     

A counterexample to uniqueness and regularity for harmonic maps between hyperbolic spaces

We construct a harmonic diffeomorphism from the Poincaré ballH ⁿ⁼¹ to itself, whose boundary value is the identity on the sphereS ⁿ, and which is singular at a boundary point, as follows: The harmonic map equations between the corresponding upper-half-space models reduce to a nonlinear o.d.e. in the transverse direction, for which we prove the existence of a solution on the whole R₊ that grows exponentially near infinity and has an expansion near zero. A conjugation by the inversion brings the singularity at the origin, and a conjugation by the Cayley transform and an isometry of the ball moves the singularity at any point on the sphere.     

Large-scale structures as gradient lines: The case of the trkal flow

Based on expansion terms of the Beltrami-flow type, we use multiscale methods to effectively construct an asymptotic expansion at large Reynolds numbers R for the long-wavelength perturbation of the nonstationary anisotropic helical solution of the force-free Navier—Stokes equation (the Trkal solution). We prove that the systematic asymptotic procedure can be implemented only in the case where the scaling parameter is R ^1/2. Projections of quasistationary large-scale streamlines on a plane orthogonal to the anisotropy direction turn out to be the gradient lines of the energy density determined by the initial conditions for two modulated anisotropic Beltrami flows (modulated as a result of scaling) with the same eigenvalues of the curl operator. The three-dimensional streamlines and the curl lines, not coinciding, fill invariant vorticity tubes inside which the velocity and vorticity vectors are collinear up to terms of the order of 1/R.     

The coupling system of Navier–Stokes equations and elastic Navier–Lame equations in a blood vessel

In this paper, the blood flow problem is considered in a blood vessel, and a coupling system of Navier–Stokes equations and linear elastic equations, Navier–Lame equations, in a cylinder with cylindrical elastic shell is given as the governing equations of the problem. We provide two finite element models to simulating the three-dimensional Navier–Stokes equations in the cylinder while the asymptotic expansion method is used to solving the linearly elastic shell equations. Specifically, in order to discrete the Navier–Stokes equations, the dimensional splitting strategy is constructed under the cylinder coordinate system. The spectral method is adopted along the rotation direction while the finite element method is used along the other directions. By using the above strategy, we get a series of two-dimensional-three-components (2D-3C) fluid problems. By introduce the S-coordinate system in E³ and employ the thickness of blood vessel wall as the expanding parameter, the asymptotic expansion method can be established to approximate the solution of the 3D elastic problem. The interface contact conditions can be treated exactly based on the knowledge of tensor analysis. Finally, numerical test shows that our method is reasonable.     

On a nonlinear singular diffusion problem:existence and uniqueness

The singular diffusion equation u_t=(u^?1u_x)_x:arises in many areas of application, e.g. in the central limit approximation to Carleman's model of Boltzman equation, or, in the expansion of a thermalized electron cloud in plasma physics. This paper concerns the existence and uniqueness of solution of a mixed boundary value problem of equation u_t=(u^m=1u_x)_x for ?1 < m ≤0.     

Thermal expansion coefficients of laminar composites

The conclusion of [1], according to which the coefficient of linear expansion of a laminar composite in a direction orthogonal to the laminations may exceed the greater of the coefficients of linear expansion of the components, has been experimentally verified. The experiments were performed on laminated metal-plastics composed of alternating layers of thin sheet steel and epoxy-phenolic resin. The coefficients of linear expansion were determined in a direction normal to the laminations at temperatures of from 20 to 100°C and various component ratios. The experimental and theoretical results are compared.Moscow Power Engineering Institute. Translated from Mekhanika Polimerov, No. 3, pp. 567–568, May–June, 1969.     

On functions convex in the direction of the real axis with a fixed second coefficient

In the paper we consider the class Γ of analytic and univalent functions f in the unit disk Δ, normalized by f(0) = f′(0) − 1 = 0, having real coefficients and such that f(Δ) is convex in the direction of the real axis. We are especially interested in some subclasses of Γ. The most important of them is Γ(c) consisting of those functions which have the second coefficients of the Taylor expansion fixed and equal to c. We obtain the Koebe set for this class as well as for the classes Γ⁺(c) and Γ⁻(c) of functions which are in some sense convex in the direction of positive and negative axes respectively.     

Intersection Local Times of Independent Brownian Motions as Generalized White Noise Functionals

A 'chaos expansion' of the intersection local time functional of two independent Brownian motions in R ^d is given. The expansion is in terms of normal products of white noise (corresponding to multiple Wiener integrals). As a consequence of the local structure of the normal products, the kernel functions in the expansion are explicitly given and exhibit clearly the dimension dependent singularities of the local time functional. Their L ^p -properties are discussed. An important tool for deriving the chaos expansion is a computation of the 'S-transform' of the corresponding regularized intersection local times and a control about their singular limit.     

Un modèle d'expansion de plasma dans le videA model for plasma expansion in vacuum

In this paper, we propose a model describing the expansion of a plasma in vacuum. Our starting point consists of a 2-fluid Euler system (isentropic case) coupled with the Poisson equation. Since numerical simulations of this model are very expensive, we investigate a quasi-neutral limit of it. We show that electron emission happens at the plasma–vaccum interface. This emission is well modeled by a Child–Langmuir law. The difficulty consists in accounting for the motion of the plasma–vacuum interface. In this paper, we formally and numerically justify why electron emission produces a reaction pressure which slows down the plasma expansion. To cite this article: P. Degond et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 399–404.     

Modeling and simulation of plasma jet by lattice Boltzmann method

In order to find a simple and efficient simulation for plasma spray process, an attempt of modeling was made to calculate velocity and temperature field of the plasma jet by hexagonal 7-bit lattice Boltzmann method (LBM) in this paper. Utilizing the methods of Chapman–Enskog expansion and multi-scale expansion, the authors derived the macro equations of the plasma jet from the lattice Boltzmann evolution equations on the basis of selecting two opportune equilibrium distribution functions. The present model proved to be valid when the predictions of the current model were compared with both experimental and previous model results. It is found that the LBM is simpler and more efficient than the finite difference method (FDM). There is no big variation of the flow characteristics, and the isotherm distribution of the turbulent plasma jet is compared with the changed quantity of the inlet velocity. Compared with the velocity at the inlet, the temperature at the inlet has a less influence on the characteristics of plasma jet.     

An Asymptotic Expansion for the Distribution of Hotelling'sT-Statistic under Nonnormality

In this paper we obtain an asymptotic expansion for the distribution of Hotelling'sT²-statisticT²under nonnormality when the sample size is large. In the derivation we find an explicit Edgeworth expansion of the multivariatet-statistic. Our method is to use the Edgeworth expansion and to expand the characteristic function ofT².     

The Effects of Flexion and Torsion on a Fluid Flow Through a Curved Pipe

We consider an injection of incompressible viscous fluid in a curved pipe with a smooth central curve γ . The one-dimensional model is obtained via singular perturbation of the Navier—Stokes system as ɛ , the ratio between the cross-section area and the length of the pipe, tends to zero. An asymptotic expansion of the flow in powers of ɛ is computed. The first term in the expansion depends only on the tangential injection along the central curve γ of the pipe and the velocity as well as the pressure drop are in the tangential direction. The second term contains the effects of the curvature (flexion) of γ in the direction of the tangent while the effects of torsion appear in the direction of the normal and the binormal to γ . The boundary layers at the ends of the pipe are studied. The error estimate is proved. Accepted 21 March 2001. Online publication 9 August 2001.     

Some remarks on the Kähler geometry of the Taub-NUT metrics

In this article, we investigate the balanced condition and the existence of an Engliš expansion for the Taub-NUT metrics on \mathbbC²{\mathbb{C}^2} . Our first result shows that a Taub-NUT metric on \mathbbC²{\mathbb{C}^2} is never balanced unless it is the flat metric. The second one shows that an Engliš expansion of the Rawnsley’s function associated to a Taub-NUT metric always exists, while the coefficient a ₃ of the expansion vanishes if and only if the Taub-NUT metric is indeed the flat one.     

Genus expansion of HOMFLY polynomials

In the planar limit of the’ t Hooft expansion, the Wilson-loop vacuum average in the three-dimensional Chern-Simons theory (in other words, the HOMFLY polynomial) depends very simply on the representation (Young diagram), H_R(A|q)|_q=1 = (σ₁(A)^|R|. As a result, the (knot-dependent) Ooguri-Vafa partition function $\sum\nolimits_R {H_{R\chi R} \left\{ {\bar pk} \right\}}$ becomes a trivial τ -function of the Kadomtsev-Petviashvili hierarchy. We study higher-genus corrections to this formula for H_R in the form of an expansion in powers of z = q ? q^?1. The expansion coefficients are expressed in terms of the eigenvalues of cut-and-join operators, i.e., symmetric group characters. Moreover, the z-expansion is naturally written in a product form. The representation in terms of cut-and-join operators relates to the Hurwitz theory and its sophisticated integrability. The obtained relations describe the form of the genus expansion for the HOMFLY polynomials, which for the corresponding matrix model is usually given using Virasoro-like constraints and the topological recursion. The genus expansion differs from the better-studied weak-coupling expansion at a finite number N of colors, which is described in terms of Vassiliev invariants and the Kontsevich integral.     

Projecting the one-dimensional Sierpinski gasket

LetS⊂ℝ² be the Cantor set consisting of points (x,y) which have an expansion in negative powers of 3 using digits {(0,0), (1,0), (0,1)}. We show that the projection ofS in any irrational direction has Lebesgue measure 0. The projection in a rational directionp/q has Hausdorff dimension less than 1 unlessp+q ≡ 0 mod 3, in which case the projection has nonempty interior and measure 1/q. We compute bounds on the dimension of the projection for certain sequences of rational directions, and exhibit a residual set of directions for which the projection has dimension 1. This work was partially completed while the author was at the Institut Fourier, Grenoble, France.     

Second order asymptotics for the propagation speed of interfaces in the Allen-Cahn phase field model for elastic solids

For a phase field model, which consists of the elasticity equations coupled to the Allen-Cahn equation, we state an asymptotic expansion for the propagation speed of the diffusive interface. The error of the expansion is of order η², where η is the width of the interface. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Trois formulations d'un modèle de plasma quasi-neutre avec courant non nul

In this Note, we propose three formulations of a model describing a quasi-neutral plasma with non-vanishing current. In order to study and compare the numerical efficiency of each formulation, two test-problems are implemented in one dimension. The first is a periodic perturbation of a uniform stationary plasma. The second is a case of plasma expansion in vacuum between two electrodes. To cite this article: P. Crispel et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).