Nonlocal Effects in Two-Dimensional Conductivity

The paper deals with the asymptotic behaviour as ε → 0 of a two-dimensional conduction problem whose matrix-valued conductivity a _ε is ε-periodic and not uniformly bounded with respect to ε. We prove that only under the assumptions of equi-coerciveness and L ¹-boundedness of the sequence a _ε , the limit problem is a conduction problem of same nature. This new result points out a fundamental difference between the two-dimensional conductivity and the three-dimensional one. Indeed, under the same assumptions of periodicity, equi-coerciveness and L ¹-boundedness, it is known that the high-conductivity regions can induce nonlocal effects in three (or greater) dimensions.     

Asymptotic approximation for the solution of a boundary-value problem with varying type of boundary conditions in a thick two-level junction

We consider a mixed boundary-value problem for a Poisson equation in a plane two-level junction Ω_ε that is the union of a domain Ω₀ and a large number 3N of thin rods with thickness of order . The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. The homogeneous Dirichlet conditions and inhomogeneous Neumann conditions are given on the sides of the thin rods from the first level and the second level, respectively. Using the method of matched asymptotic expansions and special junction-layer solutions, we construct an asymptotic approximation for the solution and prove the corresponding estimates in the Sobolev space H ¹(Ω_ε) as ε → 0 (N → +∞). Published in Neliniini Kolyvannya, Vol. 9, No. 3, pp. 336–355, July–September, 2006.     

Nonlinear stability theory of channel flow with heat transfer – an asymptotic approach

A fully developed laminar Poiseuille flow subject to constant heat flux across the wall is analysed with respect to its stability behavior by applying a weakly nonlinear stability theory. It is based on an expansion of the disturbance control equations with respect to a perturbation parameter ε. This parameter is the small initial amplitude of the fundamental wave. This fundamental wave which is the solution of the linear (Orr-Sommerfeld) first order equation triggers all higher order effects with respect to ε. Heat transfer is accounted for asymptotically through an expansion with respect to a small heat transfer parameter ε _T . Both perturbation parameters, ε and ε _T , are linked by the assumption ε _T =O(ε²) by which a certain distinguished limit is assumed. The results for a fluid with temperature dependent viscosity show that heat transfer effects in the nonlinear range continue to act in the same way as in the initial linear range. Received on 11 August 1997     

The modulational theory of nonlinear gravity-wave in fluid with varying depth and nonuniform current

By means of WKB expansions, new fourth order evolution equations are derived for two-dimensional Stokes waves over the bottom with arbitrary depth. The effects of slowly varying depthh=h(ε ²x) and currentU=U(ε ²x,ε²t,ε⁴z) on the evolution of a packet of Stokes waves are considered as well. In addition, numerical simulation is performed for the evolution of single envelope by finite-difference method. Project supported by National Natural Science Foundation of China and Centre of Advanced Academic Research of Zhongshan University.     

Numerical Modeling of Combustion of Fuel-Droplet-Vapour Releases in the Atmosphere

The paper is concerned with a numerical simulation of fuel cloud behaviour which follows releases of a liquid fuel. The main aim of the work is to develop further a mathematical model to simulate such releases into the atmosphere. The model is validated by a comparison with experimental results. The influence of boundary conditions for turbulent kinetic energy k and its dissipation rate ε on the solution is investigated. It is concluded that the solution depends mainly on the combination of k and ε in the form k ^3/2/ε rather than each of these values separately. A way to define the boundary conditions for k and ε is suggested. The KIVA-II code has been used as the base of the code used. The original code has been modified to simulate low Mach number atmospheric flows, radiation, soot formation and turbulent combustion. This revised version was published online in July 2006 with corrections to the Cover Date.     

Phase Transition with the Line‐Tension Effect

We make the connection between the geometric model for capillarity with line tension and the Cahn‐Hilliard model of two‐phase fluids. To this aim we consider the energies where u is a scalar density function and W and V are double‐well potentials. We show that the behaviour of F _ε in the limit ε→0 and λ→∞ depends on the limit of ε log λ. If this limit is finite and strictly positive, then the singular limit of the energies F _ε leads to a coupled problem of bulk and surface phase transitions, and under certain assumptions agrees with the relaxation of the capillary energy with line tension. These results were announced in [ABS1] and [ABS2]. (Accepted November 5, 1997)     

Deduction of Saint-Venant"s Principle for the Asymptotic Residue of Elastic Rods

Let u(ε) be a rescaled 3-dimensional displacement field solution of the linear elastic model for a free prismatic rod Ω^ε having cross section with diameter of order ε, and let u ⁽⁰⁾ –Bernoulli–Navier displacement – and u ⁽²⁾ be the two first terms derived from the asymptotic method. We analyze the residue r(ε) = u(ε) − (u ⁽⁰⁾ + ε² u ⁽²⁾) and if the cross section is star-shaped, we prove such residue presents a Saint-Venant"s phenomenon near the ends of the rod. This revised version was published online in August 2006 with corrections to the Cover Date.     

Qualitative Behavior of Local Minimizers of Singular Perturbed Variational Problems

We consider a non-convex variational problem (P) and the corresponding singular perturbed problem (P _ε ). The qualitative behavior of stable critical points of (P _ε ) depending on ε and a lower order term is discussed and we prove compactness of a sequence of stable critical points as ε ↘ 0. Moreover we show whether this limit is the global minimizer of (P). Furthermore uniform convergence is considered as well as the convergence rate depending on ε.      

A Quasi-Realizable Cubic Low-Reynolds Eddy-Viscosity Turbulence Model with a New Dissipation Rate Equation

A non-linear relationship of the Reynolds stresses in function of the strain rate and vorticity tensors, with terms up to third order, is developed. Anisotropies in the normal stresses, influence from streamline curvature or rotation of the reference frame, and swirl effects are accounted for. The relationship is linked to ak–ε model with a modified transport equation for the dissipation rate. A new low-Reynolds source term is introduced and a model parameter is written in terms of dimensionless rate-of-strain and vorticity. The model is checked on different realizability constraints. It is shown that practically all constraints are fulfilled. The model is numerically tested on a fully developed channel and pipe flow, both stationary and rotating. The plane jet–round jet anomaly is addressed. Finally, the model is applied to the flow over a backward-facing step. Results are compared with a linear low-Reynolds k–ε model and the shear stress transport model. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical calculation of nonlinear normal modes in structural systems

This paper presents two methods for numerical calculation of nonlinear normal modes (NNMs) in multi-degree-of-freedom, conservative, nonlinear structural dynamics models. The approaches used are briefly described as follows. Method 1: Starting with small amplitude initial conditions determined by a selected mode of the associated linear system, a small amount of negative damping is added in order to “artificially destabilize” the system; numerical integration of the system equations of motion then produces a simulated response in which orbits spiral outward essentially in the nonlinear modal manifold of interest, approximately generating this manifold for moderate to strong nonlinearity. Method 2: Starting with moderate to large amplitude initial conditions proportional to a selected linear mode shape, perform numerical integration with the coefficient ε of the nonlinearity contrived to vary slowly from an initial value of zero; this simulation methodology gradually transforms the initially flat eigenspace for ε = 0 into the manifold existing quasi-statically for instantaneous values of ε. The two methods are efficient and reasonably accurate and are intended for use in finding NNMs, as well as interesting behavior associated with them, for moderately and strongly nonlinear systems with relatively many degrees of freedom (DOFs).     

The extended Graetz problem at low Peclet numbers

The temperature profile in a circular tube of infinite extent through which a fluid is moving under conditions of small Péclet numbersε is determined by means of an asymptotic analysis inε. The walls of the tube are heated forx>0 and are insulated whenx<0. It is shown that the heated region extends anO(ε ⁻¹) distance — relative to the radius of the tube — upstream of the pointx=0, and that convective effects remain important even whenε→0. These results apply to a wider class of problems in which the Péclet number is small.     

Numerical prediction of turbulent flow in a conical diffuser usingk-ε model

Fully developed incompressible turbulent flow in a conical diffuser having a total divergence angle of 8° and an area ratio of 4∶1 has been simulated by ak-ε turbulence model with high Reynolds number and adverse pressure gradient. The research has been done for pipe entry Reynolds numbers of 1.16×10⁵ and 2.93×10⁵. The mean flow velocity and turbulence energy are predicted successfully and the advantage of Boundary Fit Coordinates approach is discussed. Furthermore, thek-ε turbulence model is applied to a flow in a conical diffuser having a total divergence angle of 30° with a perforated screen. A simplified mathematical model, where only the pressure drop is considered, has been used for describing the effect of the perforated screen. The optimum combination of the resistance coefficient and the location of the perforated screen is predicted for high diffuser efficiency or the uniform velocity distribution.     

The method of composite expansions applied to boundary layer problems in symmetric bending of the spherical shells

In this paper,the method of composite expansions which was proposed by W. Z. Chien (1948)[5]is extended to investigate two-parameter boundary layer problems.For the problems of symmetric deformations of the spherical shells under the action of uniformly distribution load q, its nonlinear equilibrium equations can be written as follows: where ε and δ are undetermined parameters.If δ=1 and ε is a small parameter, the above-mentioned problem is called first boundary layer problem; if ε is a small parameter, and δ is a small parameter, too, the above-mentioned problem is called second boundary layer problem.For the above-mentioned problems, however, we assume that the constants ε, δ and p satisfy the following equation: εp=1-ε In defining this condition by using the extended method of composite expansions, we find the asymptotic solution of the above-mentioned problems with the clamped boundary conditions.     

Spinodal Decomposition for the¶Cahn-Hilliard Equation in Higher Dimensions:¶Nonlinear Dynamics

This paper addresses the phenomenon of spinodal decomposition for the Cahn-Hilliard equation

Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity

For elliptic equations ε²Δu − V(x) u + f(u) = 0, x ∈ R ^N , N ≧ 3, we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of V, as ε → 0, under conditions on f which we believe to be almost optimal. An erratum to this article can be found at     

Thin-walled Beams: A Derivation of Vlassov Theory via Γ-Convergence

This paper deals with the asymptotic analysis of the three-dimensional problem for a linearly elastic cantilever having an open cross-section which is the union of rectangles with sides of order ε and ε ², as ε goes to zero. Under suitable assumptions on the given loads and for homogeneous and isotropic material, we show that the three-dimensional problem Γ-converges to the classical one-dimensional Vlassov model for thin-walled beams.      

Simulation of the thermal method for reducing turbulent friction

The results of calculating a supersonic turbulent boundary layer on a heated surface on the basis of the algebraic two-parameter (k-ε) and four-parameter (k-ε-θ ²-ε ₆) models of turbulence are compared with experimental data. Emphasis is placed on the ability of the models to predict the behavior of the friction and heat-transfer coefficients on a heated surface. The optimal model of turbulence is chosen. The possibility of improving the efficiency of viscous drag reduction by localizing the regions of heat addition to the boundary layer is demonstrated on the basis of numerical calculations. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–68, January–February, 1998. This research was carried out with financial support from the International Scientific and Technological Center (project No. 199).     

Effect of couple stresses on the flow in a constricted annulus

The flow of an incompressible couple stress fluid in an annulus with local constriction at the outer wall is considered. This configuration is intended as a simple model for studying blood flow in a stenosed artery when a catheter is inserted into it. The effects couple stress fluid parameters α and σ, height of the constriction (ε), and ratio of radii (k) on the impedance and wall shear stresses are studied graphically. Graphical results show that the resistance to the flow as well as the wall shear stress increases as the ratio of the radii increases and decreases as the couple stress fluid parameters increases.     

Rheo-Dielectric behavior of low molecular weight liquid crystals.

Dielectric relaxation behavior was examined for 4-4′-n-pentyl-cyanobiphenyl (5CB) and 4-4′-n-heptyl-cyanobiphenyl (7CB) under flow. In quiescent states at all temperatures examined, both 5CB and 7CB exhibited dispersions in their complex dielectric constant ε*(ω) at characteristic frequencies ω _c above 10⁶ rad s^–1. This dispersion reflected orientational fluctuation of individual 5CB and 7CB molecules having large dipoles parallel to their principal axis (in the direction of C≡N bond). In the isotropic state at high temperatures, these molecules exhibited no detectable changes of ε*(ω) under flow at shear rates . In contrast, in the nematic state at lower temperatures the terminal relaxation intensity of ε*(ω) as well as the static dielectric constant ε′(0) decreased under flow at . This rheo-dielectric change was discussed in relation to the flow effects on the nematic texture (director distribution) and anisotropy in motion of individual molecules with respect to the director. Received: 14 April 1998 Accepted: 29 July 1998     

Testing of Model Equations for the Mean Dissipation using Kolmogorov Flows

Direct Numerical Simulations (DNS) of Kolmogorov flows are performed at three different Reynolds numbers Re _λ between 110 and 190 by imposing a mean velocity profile in y-direction of the form U(y) = F sin(y) in a periodic box of volume (2π)³. After a few integral times the turbulent flow turns out to be statistically steady. Profiles of mean quantities are then obtained by averaging over planes at constant y. Based on these profiles two different model equations for the mean dissipation ε in the context of two-equation RANS (Reynolds Averaged Navier–Stokes) modelling of turbulence are compared to each other. The high Reynolds number version of the k-ε-model (Jones and Launder, Int J Heat Mass Transfer 15:301–314, 1972), to be called the standard model and a new model by Menter et al. (2006), to be called the Menter–Egorov model, are tested against the DNS results. Both models are solved numerically and it is found that the standard model does not provide a steady solution for the present case, while the Menter–Egorov model does. In addition a fairly good quantitative agreement of the model solution and the DNS data is found for the averaged profiles of the kinetic energy k and the dissipation ε. Furthermore, an analysis based on flow-inherent geometries, called dissipation elements (Wang and Peters, J Fluid Mech 608:113–138, 2008), is used to examine the Menter–Egorov ε model equation. An expression for the evolution of ε is derived by taking appropriate moments of the equation for the evolution of the probability density function (pdf) of the length of dissipation elements. A term-by-term comparison with the model equation allows a prediction of the constants, which with increasing Reynolds number approach the empirical values.