Symmetries of the hyperbolic shallow water equations and the Green–Naghdi model in Lagrangian coordinates

The observation that the hyperbolic shallow water equations and the Green–Naghdi equations in Lagrangian coordinates have the form of an Euler–Lagrange equation with a natural Lagrangian allows us to apply Noether's theorem for constructing conservation laws for these equations. In this study the complete group analysis of these equations is given: admitted Lie groups of point and contact transformations, classification of the point symmetries and all invariant solutions are studied. For the hyperbolic shallow water equations new conservation laws which have no analog in Eulerian coordinates are obtained. Using Noether's theorem a new conservation law of the Green–Naghdi equations is found. The dependence of solutions on the parameter is illustrated by self-similar solutions which are invariant solutions of both models.     

A Revised Exposition of the Green–Naghdi Theory of Heat Propagation

We offer a revised exposition of the three types of heat-propagation theories proposed by Green and Naghdi. Those theories, which make use of the notion of thermal displacement and allow for heat waves, are at variance with the standard Fourier theory; they have attracted considerable interest, and have been applied in a number of disparate physical circumstances, where heat propagation is coupled with elasticity, viscous flows, etc. (Straughan in Heat waves. Applied mathematical sciences, vol. 177. Springer, Berlin, 2011). However, their derivation is not exempt from criticisms, that we here detail, in hopes of opening the way to reconsideration of old applications and proposition of new ones.     

Classification of traveling wave solutions to the Green–Naghdi model

In this paper, the Green–Naghdi model is investigated by employing the qualitative method. We classify all traveling wave solutions to this model in specified parameter region of the parameter space. Especially, we study the limiting behavior of all smooth and non-smooth periodic solutions as the parameters tend to some special values. Based on the qualitative results, all exact traveling wave solutions as well as their profiles are also given.     

A two-step back–forth algorithm for the solution of the unsteady adjoint equations

A new algorithm for the solution of the unsteady adjoint equations is proposed in this article, aiming at overcoming the excessive computational cost and memory requirements of the conventional adjoint approach for the optimisation of unsteady problems in computational mechanics. The total cost is equal to four times the cost of the unsteady state solution, which is twice the cost of the conventional backward-in-time adjoint calculation but the memory requirements are very small, equivalent to those of a steady-state problem, while stability is acceptable. The proposed algorithm is validated in the case of the 1D unsteady Burgers equation with non-smooth source terms.     

Numerical study of the exact solution of the Navier–Stokes equations describing free-boundary fluid flow

This paper presents a numerical study of the partially invariant solution of the Navier–Stokes equations for the plane case which describes unsteady flow in a layer bounded by a straight solid wall and a free boundary parallel to it. It is found that for different initial flow velocities, a steady state can be established with a decrease or an increase in the initial layer thickness or the layer thickness can be increased infinitely due to fluid inflow from infinity.     

On the steady vibrations problem in linear theory of micropolar solid–fluid mixture

In this paper we study the steady vibrations problem in linear theory of isothermal micropolar solid–fluid mixture. With the help of fundamental solution we establish representations of Somigliana type. Then, using the potentials of single layer and double layer, we reduce the boundary value problems to singular integral equations for which the Fredholm’s theorems are valid. Existence and uniqueness results for interior and exterior problems are presented.     

The effect of a dipolar structure on the Hölder stability in Green–Naghdi thermoelasticity

This study is concerned with the mixed initial boundary value problem for a dipolar body in the context of the thermoelastic theory proposed by Green and Naghdi. For the solutions of this problem we prove a result of Hölder’s-type stability on the supply terms. We impose middle restrictions on the thermoelastic coefficients, which are common in continuum mechanics. For the same conditions we propose a continuous dependence result with regard to the initial data.     

On the long-time limit of the Garvin–Alterman–Loewenthal solution for a buried blast load

The Garvin–Alterman–Loewenthal solution refers to the problem of a line blast load suddenly applied in the interior of an elastic half-space. It is expected that the long-time asymptotic limit of this solution should be equal to the solution of a related static problem. This expectation is justified here. First, the solution of the static problem is constructed. Then, the asymptotic limit of the transient problem is found, correcting previously published results.     

On exact solutions of a class of fractional Euler–Lagrange equations

In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles. We find a fractional Lagrangian L(x(t), where _a ^c D _t ^α x(t)) and 0<α<1, such that the following is the corresponding Euler–Lagrange

On the thermodynamics of the Swift–Hohenberg theory

We present the microbalance including the microforces, the first- and second-order microstresses for the Swift–Hohenberg equation concomitantly with their constitutive equations, which are consistent with the free-energy imbalance. We provide an explicit form for the microstress structure for a free-energy functional endowed with second-order spatial derivatives. Additionally, we generalize the Swift–Hohenberg theory via a proper constitutive process. Finally, we present one highly resolved three-dimensional numerical simulation to demonstrate the particular form of the resulting microstresses and their interactions in the evolution of the Swift–Hohenberg equation.     

On a class of quasilinear Schrodinger equations

A type of quasilinear Schrodinger equations in two space dimensions which describe attractive Bose-Einstein condensates in physics is discussed. By establishing the property of the equation and applying the energy method, the blowup of solutions to the equation are proved under certain conditions. At the same time, by the variational method, a sutficient condition of global existence which is related to the ground state of a classical elliptic equation is obtained.     

Exact solution of the multi-cracked Euler–Bernoulli column

The use of distributions (generalized functions) is a powerful tool to treat singularities in structural mechanics and, besides providing a mathematical modelling, their capability of leading to closed form exact solutions is shown in this paper. In particular, the problem of stability of the uniform Euler–Bernoulli column in presence of multiple concentrated cracks, subjected to an axial compression load, under general boundary conditions is tackled. Concentrated cracks are modelled by means of Dirac’s delta distributions. An integration procedure of the fourth order differential governing equation, which is not allowed by the classical distribution theory, is proposed. The exact buckling mode solution of the column, as functions of four integration constants, and the corresponding exact buckling load equation for any number, position and intensity of the cracks are presented. As an example a parametric study of the multi-cracked simply supported and clamped–clamped Euler–Bernoulli columns is presented.     

On fractional Euler–Lagrange and Hamilton equations and the fractional generalization of total time derivative

Fractional mechanics describe both conservative and nonconservative systems. The fractional variational principles gained importance in studying the fractional mechanics and several versions are proposed. In classical mechanics, the equivalent Lagrangians play an important role because they admit the same Euler–Lagrange equations. By adding a total time derivative of a suitable function to a given classical Lagrangian or by multiplying with a constant, the Lagrangian we obtain are the same equations of motion. In this study, the fractional discrete Lagrangians which differs by a fractional derivative are analyzed within Riemann–Liouville fractional derivatives. As a consequence of applying this procedure, the classical results are reobtained as a special case. The fractional generalization of Faà di Bruno formula is used in order to obtain the concrete expression of the fractional Lagrangians which differs from a given fractional Lagrangian by adding a fractional derivative. The fractional Euler–Lagrange and Hamilton equations corresponding to the obtained fractional Lagrangians are investigated, and two examples are analyzed in detail.     

Numerical solution of the Schrödinger equations by using Delta-shaped basis functions

Numerical simulations of the nonlinear Schrödinger equations are studied using Delta-shaped basis functions, which recently proposed by Reutskiy. Propagation of a soliton, interaction of two solitons, birth of standing and mobile solitons and bound state solutions are simulated. Some conserved quantities are computed numerically for all cases. Then we extend application of the method to solve some coupled nonlinear Schrödinger equations. Obtained systems of ordinary differential equations are solved via the fourth- order Runge–Kutta method. Numerical solutions coincide with the exact solutions in desired machine precision and invariant quantities are conserved sensibly. Some comparisons with the methods applied in the literature are carried out.     

On the rheology of ethylene–octene copolymers

It has previously been shown that the plateau modulus, G_N^o, and thus the entanglement molecular weight, M_e, of flexible polymers can be correlated to the unperturbed chain dimension, <R²>_o/M, and mass density, , via the use of the packing length, p. For polyolefins, a method was recently proposed whereby knowledge of the average molecular weight per backbone bond, m_b, allows <R²>_o/M and consequently G_N^o and M_e to be estimated. This is particularly valuable for polyolefin copolymers since the melt chain dimensions are often unknown. This work corroborates these theoretical predictions by studying the rheology of a series of carefully synthesized ethylene/octene copolymers with varying octene content (19–92 wt%). Furthermore, the results reported herein also allow the advancement of rheological characterization techniques of polymer melts. For instance, based on the analysis of the linear viscoelastic properties of these copolymers, it has been found that several rheological parameters scale with the copolymer comonomer content. Analysis of the viscoelastic material functions in terms of the evolution of the phase angle, , as a function of the absolute value of the complex modulus, |G*|, (the so-called van Gurp–Palmen plots), provides a fast and reliable rheological means for determining the composition of ethylene/-olefin copolymers. The crossover parameters, G_co(=G=G) and _co(=1/_co) also scale with copolymer composition.Submitted for publication to Rheologica Acta

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An erratum to this article is available at .     

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.     

On the Local Smoothness of Solutions of the Navier–Stokes Equations

We consider the Cauchy problem for incompressible Navier–Stokes equations with initial data in , and study in some detail the smoothing effect of the equation. We prove that for T < ∞ and for any positive integers n and m we have , as long as stays finite.     

On the viscoelastic characterization of the Jeffreys–Lomnitz law of creep

In 1958, Jeffreys (Geophys J?R Astron Soc 1:92–95) proposed a power law of creep, generalizing the logarithmic law earlier introduced by Lomnitz, to broaden the geophysical applications to fluid-like materials including igneous rocks. This generalized law, however, can be applied also to solid-like viscoelastic materials. We revisit the Jeffreys–Lomnitz law of creep by allowing its power law exponent α, usually limited to the range 0?≤?α?≤?1 to all negative values. This is consistent with the linear theory of viscoelasticity because the creep function still remains a Bernstein function, that is positive with a completely monotone derivative, with a related spectrum of retardation times. The complete range α?≤?1 yields a continuous transition from a Hooke elastic solid with no creep $\left(\alpha \,\to\, -\infty\right)$ to a Maxwell fluid with linear creep $\left(\alpha \,=\,1\right)$ passing through the Lomnitz viscoelastic body with logarithmic creep $\left(\alpha\, =0\right)$ , which separates solid-like from fluid-like behaviors. Furthermore, we numerically compute the relaxation modulus and provide the analytical expression of the spectrum of retardation times corresponding to the Jeffreys–Lomnitz creep law extended to all α?≤?1.