Slip–Brinkman Flow Through Corrugated Microannulus with Stationary Random Roughness

This paper presents a boundary perturbation method of the Brinkman-extended Darcy model to investigate the flow in corrugated microannuli cylindrical tubes with slip surfaces. The stationary random model is used to mimic the surface roughness of the cylindrical walls. The tube is filled with a porous medium. We shall consider the two cases where corrugations are either perpendicular or parallel to the flow, and particular attention is given to the effect of the phase shift. The effects of the corrugations on the flow rate and pressure gradient are investigated as functions of wavelength, the permeability of the medium, the radius ratio and the slip parameter. Particular surface roughnesses are examined as special cases of stationary random surface. It is found that the effect of the partial slip is significant on the corrugation functions. The limiting cases of Stokes and Darcy’s flows and no-slip case are discussed.     

A Variational Model for Plastic Slip and Its Regularization via Γ-Convergence

A variational model is presented able to interpret the onset of plastic deformations, here modeled as displacement jumps occurring along slip surfaces at constant yielding stress. The corresponding strain energy functional, leading to a free-discontinuity problem set in the space of SBV functions, is then approximated by a sequence of regularized elliptic functionals following the seminal work by Ambrosio and Tortorelli (Commun. Pure Appl. Math. 43, 999–1036, 1990) within the framework of Γ-convergence. Comparisons between the results obtainable with the free-discontinuity model and its regularized approximation, in terms of stability of the pure elastic phase, irreversibility of plastic slip and response under unloading, are presented, in general, for the 2-D case of antiplane shear and exemplified, in particular, for the 1-D case.     

Existence and Uniqueness of the Flow of Second‐Grade Fluids with Slip Boundary Conditions

. This paper treats the solvability of the equations of motion for an incompressible fluid of grade two subject to nonlinear partial slip boundary conditions in a bounded simply‐connected domain. The existence of a unique classical solution, global in time, is proved under suitable regularity and growth restrictions on the initial data, the slip law and the body and surface forces. The method is based on a fixed-point formulation of the problem. (Accepted October 5, 1998)     

An example of stick–slip and stick–slip–separation waves

The dynamical problem of a brake-like mechanical system composed of an elastic cylindrical tube with Coulomb's friction in contact with a rigid and rotating cylinder is considered. This model problem enables us to give an example of non-trivial periodic solutions in the form of stick–slip or stick–slip–separation waves propagating on the contact surface. A semi-analytical analysis of stick–slip waves is obtained when the system of governing equations is reduced by condensation to a simpler system involving only the contact displacements. This reduced system, of only one space variable in addition to time, can be solved almost analytically and gives some interesting informations on the existence and the characteristics of stick–slip waves such as the wave numbers on the circumference, stick and slip proportions, wave celerities, tangential and normal forces. It is shown in particular that the stick–slip–separation solutions would occur for small normal pressures or high rotational speeds. Since the analytical discussion becomes cumbersome in this case, a second approach based on numerical analysis by the finite element method is performed. The existence and the characteristics of stick–slip and stick–slip–separation waves are discussed numerically.     

On the Camassa–Holm and K–dV Hierarchies

It is known that a transform of Liouville type allows one to pass from an equation of the Korteweg–de Vries (K–dV) hierarchy to a corresponding equation of the Camassa–Holm (CH) hierarchy (Beals et al., Adv Math 154:229–257, 2000; McKean, Commun Pure Appl Math 56(7):998–1015, 2003). We give a systematic development of the correspondence between these hierarchies by using the coefficients of asymptotic expansions of certain Green’s functions. We illustrate our procedure with some examples.     

Yielding and intrinsic plasticity of Ti–Zr–Ni–Cu–Be bulk metallic glass

Bulk metallic glass with composition Ti₄₀Zr₂₅Ni₈Cu₉Be₁₈ exhibits considerably high compressive yield stress, significant plasticity (with a concomitant vein-like fracture morphology) and relatively low density. Yielding and intrinsic plasticity of this alloy are discussed in terms of its thermal and elastic properties. An influence of normal stresses acting on the shear plane is evidenced by: (i) the fracture angle (<45°) and (ii) finite-element simulations of nanoindentation curves, which require the use of a specific yield criterion, sensitive to local normal stresses acting on the shear plane, to properly match the experimental data. The ratio between hardness and compressive yield strength (constraint factor) is analyzed in terms of several models and is best adjusted using a modified expanding cavity model incorporating a pressure-sensitivity index defined by the Drucker–Prager yield criterion. Furthermore, comparative results from compression tests and nanoindentation reveal that deformation also causes strain softening, a phenomenon which is accompanied with the occurrence of serrated plastic flow and results in a so-called indentation size effect (ISE). A new approach to model the ISE of this metallic glass using the free volume concept is presented.     

Synchronization between fractional-order Ravinovich–Fabrikant and Lotka–Volterra systems

This article examines the synchronization performance between two fractional-order systems, viz., the Ravinovich?CFabrikant chaotic system as drive system and the Lotka?CVolterra system as response system. The chaotic attractors of the systems are found for fractional-order time derivatives described in Caputo sense. Numerical simulation results which are carried out using Adams?CBoshforth?CMoulton method show that the method is reliable and effective for synchronization of nonlinear dynamical evolutionary systems. Effects on synchronization time due to the presence of fractional-order derivative are the key features of the present article.     

Global and Trajectory Attractors for a Nonlocal Cahn–Hilliard–Navier–Stokes System

The Cahn–Hilliard–Navier–Stokes system is based on a well-known diffuse interface model and describes the evolution of an incompressible isothermal mixture of binary fluids. A nonlocal variant consists of the Navier–Stokes equations suitably coupled with a nonlocal Cahn–Hilliard equation. The authors, jointly with P. Colli, have already proven the existence of a global weak solution to a nonlocal Cahn–Hilliard–Navier–Stokes system subject to no-slip and no-flux boundary conditions. Uniqueness is still an open issue even in dimension two. However, in this case, the energy identity holds. This property is exploited here to define, following J.M. Ball’s approach, a generalized semiflow which has a global attractor. Through a similar argument, we can also show the existence of a (connected) global attractor for the convective nonlocal Cahn–Hilliard equation with a given velocity field, even in dimension three. Finally, we demonstrate that any weak solution fulfilling the energy inequality also satisfies a dissipative estimate. This allows us to establish the existence of the trajectory attractor also in dimension three with a time dependent external force.     

Lumps and rogue waves of generalized Nizhnik–Novikov–Veselov equation

In this paper, via generalized bilinear forms, we consider the (\(2+1\))-dimensional bilinear p-Sawada–Kotera (SK) equation. We derive analytical rational solutions in terms of positive quadratic functions. Through applying the dependent transformation, we present a class of lump solutions of the (\(2+1\))-dimensional SK equation. Those rationally decaying solutions in all space directions exhibit two kinds of characters, i.e., bright lump wave (one peak and two valleys) and bright–dark lump wave (one peak and one valley). In addition, we also obtain three families of bright–dark lump wave solutions to the nonlinear p-SK equation for \(p=3\).     

A Reaction–Diffusion–Advection Equation with Mixed and Free Boundary Conditions

We investigate a reaction–diffusion–advection equation of the form \(u_t-u_{xx}+\beta u_x=f(u)\) \((t>0,\,0<x<h(t))\) with mixed boundary condition at \(x=0\) and Stefan free boundary condition at \(x=h(t)\). Such a model may be applied to describe the dynamical process of a new or invasive species adopting a combination of random movement and advection upward or downward along the resource gradient, with the free boundary representing the expanding front. The goal of this paper is to understand the effect of advection environment and no flux across the left boundary on the dynamics of this species. For the case \(|\beta |<c_0\), we first derive the spreading–vanishing dichotomy and sharp threshold for spreading and vanishing, and then provide a much sharper estimate for the spreading speed of h(t) and the uniform convergence of u(t, x) when spreading happens. For the case \(|\beta |\ge c_0\), some results concerning virtual spreading, vanishing and virtual vanishing are obtained. Here \(c_0\) is the minimal speed of traveling waves of the differential equation.     

The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations

In recent years two nonlinear dispersive partial differential equations have attracted much attention due to their integrable structure. We prove that both equations arise in the modeling of the propagation of shallow water waves over a flat bed. The equations capture stronger nonlinear effects than the classical nonlinear dispersive Benjamin–Bona–Mahoney and Korteweg–de Vries equations. In particular, they accommodate wave breaking phenomena.     

Mid–Span Deflection and End–Shortening of a Rod after Buckling

The load—displacement relations governing the postbuckling behavior are expressed in terms of elementary functions. An approximate solution of the elastica problem with modified expressions for the curvature is given. Equations of the elastic curve are obtained with the use of an approximate determination of elliptic integrals.     

Numerical and Experimental Study of Rayleigh–Bénard–Kelvin Convection

We performed experimental and numerical studies of combined effects of thermal buoyancy and magnetization force applied on a cubical enclosure of a paramagnetic fluid heated from below and cooled from top. The temperature difference between the hot and cold wall was kept constant. After considering neutral situation (i.e. a pure natural convection case), magnetic fields of different intensity were imposed. The magnetization force produced significant changes in flow (transition from laminar to turbulent regimes), wall-heat transfer (enhancement) and turbulence (turbulence structures reorganization). The strong magnetic field and its gradients were generated by a superconducting magnet which can generate magnetic field up to 10 T and where gradients of the magnetic induction can reach up to 900 T²/m. A good agreement between experiments and numerical simulations was obtained in predicting the integral wall heat transfer over entire range of considered working parameters. Numerical simulations provided a detailed insights into changes of the local wall-heat transfer and long-term time averaged first and second moments for different strengths of the imposed magnetic induction.     

A unified model for polystyrene–nanorod and polystyrene–nanoplatelet melt composites

We present a unified constitutive model capable of predicting the steady shear rheology of polystyrene (PS)–nanoparticle melt composites, where particles can be rods, platelets, or any geometry in between, as validated against experimental measurements. The composite model incorporates the rheological properties of the polymer matrix, the aspect ratio and characteristic length scale of the nanoparticles, the orientation of the nanoparticles, hydrodynamic particle–particle interactions, the interaction between the nanoparticles and the polymer, and flow conditions of melt processing. We demonstrate that our constitutive model predicts both the steady rheology of PS–carbon nanofiber composites and the steady rheology of PS–nanoclay composites. Along with presenting the model and validating it against experimental measurements, we evaluate three different closure approximations, an important constitutive assumption in a kinetic theory model, for both polymer–nanoparticle systems. Both composite systems are most accurately modeled with a quadratic closure approximation.     

Non-Linear Binding and the Diffusion–Migration Test

In this paper we investigate the non-linear binding effects upon the diffusion–migration test. For the diffusion test we derive the conditions required for the non-linear binding isotherm to produce an actual penetration front. When more than two ion species are present we show that the diffusion coefficient associated with a particular ion cannot be extracted from the diffusion test on account of multi-species electrical effects. In the migration test where an external electrical field is applied to the sample, we give the conditions required for the propagation of a stable travelling wave. In addition new explicit expressions of the time-lag are obtained for both tests, allowing the determination of the properties of the unknown binding isotherm whatever its physical nature. Throughout the paper the results and the method are illustrated by the diffusion of the Cl^– ion within cement-based materials, using experimental data extracted from literature. The theoretical predictions compare well to these experimental data.     

Chapman–Jouguet hypothesis 1899–1999: One century between myth and reality

The 20th century saw the rapid development of quantum mechanics and micro-scales physics. However, classical mechanics did not lose any interest, and did not cease setting severe enigmas. Among them lies detonation, observed and measured since Berthelot (1881), but whose modeling required nearly hundred years of effort. Following the fashion of celebrations, we could say that the publication by Chapman in 1899 is a reason for rewriting, in modern terms, the main facts of past century: enhancing the few brilliant steps and also mentioning their sluggish diffusion, which arises from linguistic and national fractures within the scientific world and also reflects scientists' great reluctancy to recognize and overcome the intrinsic uncertainties of modeling. Received 19 April 1999 / Accepted 27 May 1999