Multiscale Dynamics of Heterogeneous Media in?the?Peridynamic Formulation

A methodology is presented for investigating the dynamics of heterogeneous media using the nonlocal continuum model given by the peridynamic formulation. The approach presented here provides the ability to model the macroscopic dynamics while at the same time resolving the dynamics at the length scales of the microstructure. Central to the methodology is a novel two-scale evolution equation. The rescaled solution of this equation is shown to provide a strong approximation to the actual deformation inside the peridynamic material. The two scale evolution can be split into a microscopic component tracking the dynamics at the length scale of the heterogeneities and a macroscopic component tracking the volume averaged (homogenized) dynamics. The interplay between the microscopic and macroscopic dynamics is given by a coupled system of evolution equations. The equations show that the forces generated by the homogenized deformation inside the medium are related to the homogenized deformation through a history dependent constitutive relation.     

On the Dynamics of Solitons in the Nonlinear Schr?dinger Equation

We study the behavior of the soliton solutions of the equation i\frac?y?t = - \frac12m Dy+ \frac12W_e^￠(y) + V(x)y,i\frac{\partial\psi}{{\partial}t} = - \frac{1}{2m} \Delta\psi + \frac{1}{2}W_{\varepsilon}^{\prime}(\psi) + V(x){\psi},     

Back to the Future? A Re-examination of the Aerodynamics of Flettner-Thom Rotors for Maritime Propulsion

The paper examines, by way of a 3-dimensional, unsteady-RANS analysis, the high-Reynolds-number turbulent flow normal to a cylinder rotating about its axis, thus continuing the re-evaluation of this flow configuration as a potential means of providing low-cost ship propulsion. Comparisons are made between the available experimental and LES data both for the bare cylinder, as employed in Flettner’s notable Atlantic crossing, and the variant advocated by Thom in which close-packed discs are distributed along the cylinder. Our results display close agreement with available experimental and LES data other than the results of Thom (1934). We conclude that the addition of discs by the latter led, at the relatively low Reynolds numbers of his experiments, to the boundary layer on the cylinder being thinned to a point at which the boundary layers became laminar or transitional, thus leading to higher lift coefficients than pertain in turbulent flow.     

Infinitesimal Stability of the Equilibrium States of?an?Incompressible, Isotropic Elastic Tube Under?Pressure

The infinitesimal stability of the equilibrium states of an arbitrary incompressible, isotropic and homogeneous elastic cylindrical shell in a pure radial expansion under a constant inflation pressure is studied for both thick- and thin-walled shells. The classical criterion of infinitesimal stability yields a general stability theorem relating the frequency and pressure response and reveals that points at which the pressure is stationary define the domain of unstable or neutrally stable states. All results are expressed in terms of a general shear response function, and specific results are provided for the Mooney-Rivlin, Gent and Ogden models, the second having limited extensibility, the last including experimental data. Every static state of a Mooney-Rivlin tube is stable so long as the pressure is less than an asymptotic limit that increases with the thickness. Otherwise, only the Ogden model exhibits static states of instability for all long cylindrical tubes of thickness less than a transitional value above which all static states are infinitesimally stable. A long cylindrical cavity in all three unbounded models is stable for all pressures. All results are illustrated graphically.     

What is the Correct Definition of Average Pressure?

Use of a correct definition of average pressure is important in numerical modeling of oil reservoirs and aquifers, where the simulated domain can be very large. Also, the average pressure needs to be defined in the application of pore-network modeling of (two-phase) flow in porous media, as well as in the (theoretical) upscaling of flow equations. Almost always the so-called intrinsic phase-volume average operator, which weighs point pressure values with point saturation values, is employed. Here, we introduce and investigate four other potentially plausible averaging operators. Among them is the centroid-corrected phase-average pressure, which corrects the intrinsic phase-volume average pressure for the distance between the centroid of the averaging volume and the phase. We consider static equilibrium of two immiscible fluids in a homogeneous, one-dimensional, vertical porous medium domain under a series of (static) drainage conditions. An important feature of static equilibrium is that the total potential (i.e., the sum of pressure and gravity potentials) is constant for each phase over the whole domain. Therefore, its average will be equal to the same constant. It is argued that the correct average pressure must preserve the fact that fluid potentials are constant. We have found that the intrinsic phase-volume average pressure results in a gradient in the total phase potential, i.e., the above criterion is violated. In fact, only the centroid-corrected operator satisfies this criterion. However, at high saturations, use of the centroid-corrected average can give rise to negative values of the difference between the average nonwetting and wetting phase pressures. For main drainage, differences among various averaging operators are significantly less because both phases are present initially, such that the difference between the centroids of phases, and the middle of the domain are relatively small.     

The monotonicity and critical periods of periodic waves?of?the �� 6 field model

In this paper, we study the relationship between period and energy of periodic traveling wave solutions for the ?? ⁶ field model. The various topological phase portraits with periodic annulus are given by using standard phase portrait analytical technique. Some analytic behaviors (convexity, monotonicity and number of critical periods) of the period functions associated with periodic waves are investigated. We prove that the period function has exactly one critical period under certain conditions. Moreover, the numerical simulation is made. The results show that our theoretical analysis agrees with the numerical simulation.     

Can one measure the temperature of a curve?

The entropy of a plane curve is defined in terms of the number of intersection points with a random line. The Gibbs distribution which maximizes the entropy enables one to define the temperature of the curve. At 0 temperature, the curve reduces to a straight segment. At high temperature, the curve is somewhat chaotic and behaves like a perfect gas. We attempt to show that thermodynamic formalism can be used for the study of plane curves. The curves we discuss have finite length, unlike Mandelbrot's fractal curves [1], yet we feel our approach to the mathematics is not far from his.     

Estimation of the largest Lyapunov exponent from?the?perturbation vector and its derivative dot product

Nowadays, the largest Lyapunov exponent (LLE) is employed in many different areas of the scientific research. Thus, there is still need to elaborate fast and simple methods of LLE calculation. The new method of the LLE estimation is presented in this paper. The method (LLEDP) applies the perturbation vector and its derivative dot product to calculate Lyapunov exponent in the direction of disturbance. Value of this exponent is the LLE. The theoretical improvement was introduced. Results of the numerical simulations were shown and compared with the Stefański method (Stefański in Chaos Solitons Fractals 11(15):2443–2451, 2000; Stefański and Kapitaniak in Chaos Solitons Fractals 15:233–244, 2003; Stefański et al. in Chaos Solitons Fractals 23:1651–1659, 2005; Stefański in J. Theor. Appl. Mech. 46:665–678, 2008). Investigations of the Duffing oscillators with external excitation and three Duffing oscillators coupled in ring scheme without external excitation were made with use of the presented method. Fast time LLE estimation results convergence was shown.     

Bio-Marangoni convection ?ow of Casson nanoliquid througha porous medium in the presence of chemicallyreactive activation energy?

正J. K. MADHUKESH1, G. K. RAMESH2, B. C. PRASANNAKUMARA1,S. A. SHEHZAD3,, F. M. ABBASI4     

Graphical Illustrations for the Nur-Byerlee-Carroll Proof of the Formula for the Biot Effective Stress Coefficient in?Poroelasticity

This work presents a graphically illustrated version of the Nur-Byerlee-Carroll proof of the formula for the Biot effective stress coefficient in poroelasticity. The original elegant proof was provided by Nur and Byerlee (J. Geophys. Res. 76:6414, 1971) for isotropic materials and extended by Carroll (J. Geophys. Res. 84:7510?C7512, 1979) to anisotropic materials. Although the application of this result is in poroelasticity or in the analysis of composite materials, the proof is an analytical thought experiment in linear elasticity, and should be appreciated as such.     

On the Motive Power of Chemical Transformations in?Open Systems

The classical basic concepts of cyclic processes and the efficiency of heat engines are used here to conjecture about the laws of thermodynamics for open systems that can exchange matter with a surrounding environment. An ideal chemomechanical elastic bar is envisioned that changes its stiffness while undergoing a chemical transformation which is, in turn, influenced by the axial strain of the bar. Stable equilibrium states are identified as minimizers of the total energy, which is assumed to be nonconvex in type. If the bar is loaded and then alternatively placed in environments at chemical potentials either ?? _i or ?? _s >?? _i , a reversible cycle analogous to the classical Carnot cycle may be traced. In this case, the environmental ??chemical potential?? plays the role of the temperature and the ??chemical work?? the role of heat. For the system, the main form of interaction with the exterior, other than mechanical work, is the exchange of mass of a component at different environmental chemical potentials. It is then possible to obtain an elementary theory of chemical engines in which efficiency estimates (in terms of environmental chemical potentials) and related pertinent issues can be discussed. This model may serve as a basis for analyzing coupled chemo-mechanical processes occurring in materials such as ionized gels for possible applications as actuators, and to interpret complex phenomena in biological systems, such as the chemical kinetics of smooth muscles.     

Two Problems of the Semi-Conductor Physics Discussed with the Point of View of the Fluid Dynamics

In this article,we discuss two problems of the semi-conductor physics from the point of view ofthe fluid dynamics.Firstly,we discuss the problem of the p-n junction,and find that the previoustreatment and the previous conclusion of the problem are somewhat erroneous.Secondly,we discussthe coefficient C of the block resistance,and find that the mathematical method of the previoustreatment is erroneous.     

On the Time Decay of the Solutions of the Navier–Stokes System

We investigate the relationship between the time decay of the solutions u of the Navier–Stokes system on a bounded open subset of and the time decay of the right-hand sides f. In suitable function spaces, we prove that u always inherits at least part of the decay of f, up to exponential, and that the decay properties of u depend only upon the amount and type (e.g., exponential, or power-like) of decay of f. This is done by first making clear what is meant by “type” and “amount” of decay and by next elaborating upon recent abstract results pointing to the fact that, in linear and nonlinear PDEs, the decay of the solutions is often intimately related to the Fredholmness of the differential operator. This work was done while the second author was visiting the Bernoulli Center, EPFL, Switzerland, whose support is gratefully acknowledged.     

the future of PARTICUOLOGY

I congratulate the Chinese Society of Particuology(and especially my friend Professor Mooson Kwauk,the Society's distinguished President Emeritus) forits initiative in launching a new journal devoted to studiesof particle science and technology.     

How far the wake of a wind farm can persist for?

With the increased penetration of wind energy in our nation’s energy portfolio, wind farms are placed in a way close to each other. Thus, their wakes have to be fully considered in the design and operation of a wind farm. In this study, we investigate the wake of a wind farm using large-eddy simulation with wind turbine rotor modelled by the actuator disk model. The simulated results show that the wake of a wind farm can persist for a long distance in its downstream. For the considered wind farm layout, the velocity in the wake recovers 95% of that of the undisturbed inflow at 55 rotor diameters downstream from its last row, suggesting that the wake of a wind farm should be fully considered in the optimal design and operation for its downstream wind farms.     

Group foliation of the Lamé equations of the classical dynamical theory of elasticity

We perform the group foliation of the system of Lamé equations of the classical dynamical theory of elasticity for an infinite subgroup contained in a normal divisor of the main group. The resolving system of this foliation includes the following two classical systems of mathematical physics: the system of equations of vortex-free acoustics and the system of Maxwell equations, which allows one to use wider groups to obtain exact solutions of the Lamé equations. We obtain a first-order conformal-invariant system, which describes shear waves in a three-dimensional elastic medium. We also give examples of partially invariant solutions.