Spatial behavior in the electromagnetic theory of microstretch elasticity

This paper is concerned with the electromagnetic theory of microstretch elasticity. First, the initial boundary value problem is formulated in the framework of the linear dynamic theory of microstretch magnetoelectroelastic solids. Then, the spatial behavior of solutions is studied in both bounded and unbounded regions. The obtained result gives an exact idea of the domain of influence, in the sense that for each fixed time in a given interval, the entire activity vanishes at distanced from the support of the given data greater than a time-dependent threshold value. The study of spatial behavior is completed by an exponential decay estimate inside the domain of influence. As a by product a uniqueness result holding for both bounded and unbounded bodies is derived. Finally, the effect of a concentrated microstretch body force is studied.     

Thermoelastostatic equilibrium of a spatial quadrant: Green’s function and solution in integrals

In this study, a new Green??s function and a new Green-type integral formula for a 3D boundary value problem (BVP) in thermoelastostatics for a quarter-space are derived in closed form. On the boundary half-planes, twice mixed homogeneous mechanical boundary conditions are given. One boundary half-plane is free of loadings and the normal displacements and the tangential stresses are zero on the other one. The thermoelastic displacements are subjected by a heat source applied in the inner points of the quarter-space and by mixed non-homogeneous boundary heat conditions. On one of the boundary half-plane, the temperature is prescribed and the heat flux is given on the other one. When the thermoelastic Green??s function is derived, the thermoelastic displacements are generated by an inner unit point heat source, described by ??-Dirac??s function. All results are obtained in elementary functions that are formulated in a special theorem. As a particular case, when one of the boundary half-plane of the quarter-space is placed at infinity, we obtain the respective results for half-space. Exact solutions in elementary functions for two particular BVPs for a thermoelastic quarter-space and their graphical presentations are included. They demonstrate how to apply the obtained Green-type integral formula as well as the derived influence functions of an inner unit point body force on volume dilatation to solve particular BVPs of thermoelasticity. In addition, advantages of the obtained results and possibilities of the proposed method to derive new Green??s functions and new Green-type integral formulae not for quarter-space only, but also for any canonical Cartesian domain are also discussed.     

Existence of Vortex Sheets with¶Reflection Symmetry in Two Space Dimensions

The main purpose of this work is to establish the existence of a weak solution to the incompressible 2D Euler equations with initial vorticity consisting of a Radon measure with distinguished sign in H ^{? 1}, compactly supported in the closed right half-plane, superimposed on its odd reflection in the left half-plane. We make use of a new a priori estimate to control the interaction between positive and negative vorticity at the symmetry axis. We prove that a weak limit of a sequence of approximations obtained by either regularizing the initial data or by using the vanishing viscosity method is a weak solution of the incompressible 2D Euler equations. We also establish the equivalence at the level of weak solutions between mirror symmetric flows in the full plane and flows in the half-plane. Finally, we extend our existence result to odd L ¹ perturbations, without distinguished sign, of our original initial vorticity.     

Evolution Equations on Non-Flat Waveguides

We investigate the dispersive properties of evolution equations on waveguides with a non-flat shape. More precisely, we consider an operator $$H=-\Delta_{x}-\Delta_{y}+V(x,y)$$ with Dirichlet boundary conditions on an unbounded domain ??, and we introduce the notion of a repulsive waveguide along the direction of the first group of variables, x. If ?? is a repulsive waveguide, we prove a sharp estimate for the Helmholtz equation Hu???u?=?f. As consequences, we prove smoothing estimates for the Schr?dinger and wave equations associated to H, and Strichartz estimates for the Schr?dinger equation. Additionally, we deduce that the operator H does not admit eigenvalues.     

Prediction of Non-Darcy Coefficients for Inertial Flows Through the Castlegate Sandstone Using Image-Based Modeling

Near wellbore flow in high rate gas wells shows the deviation from Darcy??s law that is typical for high Reynolds number flows, and prediction requires an accurate estimate of the non-Darcy coefficient (?? factor). This numerical investigation addresses the issues of predicting non-Darcy coefficients for a realistic porous media. A CT-image of real porous medium (Castlegate Sandstone) was obtained at a resolution of 7.57???m. The segmented image provides a voxel map of pore-grain space that is used as the computational domain for the lattice Boltzmann method (LBM) based flow simulations. Results are obtained for pressure-driven flow in the above-mentioned porous media in all directions at increasing Reynolds number to capture the transition from the Darcy regime as well as quantitatively predict the macroscopic parameters such as absolute permeability and ?? factor (Forchheimer coefficient). Comparison of numerical results against experimental data and other existing correlations is also presented. It is inferred that for a well-resolved realistic porous media images, LBM can be a useful computational tool for predicting macroscopic porous media properties such as permeability and ?? factor.     

Dynamic fiber inclusions with elliptical and arbitrary cross-sections and related retarded potentials in a quasi-plane piezoelectric medium

A piezoelectric medium of transversely isotropic symmetry with continuous fiber inclusion parallel to the axis of symmetry is considered. The problem is equivalent to a two-dimensional ‘quasi-plane’ piezoelectric medium containing a 2D inclusion. The inclusion is assumed to undergo a spatially uniform δ(t)-type time domain transformation. The continuous fiber has elliptical, circular and arbitrary cross-sections. The solutions of the inclusion problem is expressed by scalar potentials. In the time domain two of these functions correspond to the retarded potential integrals of the inclusion. Their frequency domain representation which we shall call the ‘dynamic potentials of the inclusion’ are also considered. Integral formulae are derived for continuous fiber inclusions with elliptical cross-sections. Known closed-form solutions are reproduced for circular fibers. For fibers with arbitrary cross-sections a numerical method based on Gauss quadrature is applied. High accuracy and efficiency of the numerical method is confirmed. Characteristic superposition and runtime effects for the inclusions are found.     

Dynamic analysis of a simply supported beam resting on a nonlinear elastic foundation under compressive axial load using nonlinear normal modes techniques under three-to-one internal resonance condition

In this paper, the Nonlinear Normal Modes (NNMs) analysis for the case of three-to-one (3:1) internal resonance of a slender simply supported beam in presence of compressive axial load resting on a nonlinear elastic foundation is studied. Using the Euler?CBernoulli beam model, the governing nonlinear PDE of the beam??s transverse vibration and also its associated boundary conditions are extracted. These nonlinear motion equation and boundary condition relations are solved simultaneously using four different approximate-analytical solution techniques, namely the method of Multiple Time Scales, the method of Normal Forms, the method of Shaw and Pierre, and the method of King and Vakakis. The obtained results at this stage using four different methods which are all in time?Cspace domain are compared and it is concluded that all the methods result in a similar answer for the amplitude part of the transverse vibration. At the next step, the nonlinear normal modes are obtained. Furthermore, the effect of axial compressive force in the dynamic analysis of such a beam is studied. Finally, under three-to-one-internal resonance condition the NNMs of the beam and the steady-state stability analysis are performed. Then the effect of changing the values of different parameters on the beam??s dynamic response is also considered. Moreover, 3-D plots of stability analysis in the steady-state condition and the beam??s amplitude frequency response curves are presented.     

Mapping similarity between parallel and serial architecture kinematics

First the principles of mapping spatial points to surfaces is introduced in the context of the inverse kinematics of a general six revolute serial wrist partitioned robot. Then the advantage of choosing ideal frames is illustrated by showing that in the case of some architectures an image space formulation, though possible, may be an impediment to clear geometric insight and a satisfactory and much simpler solution. After showing how the general point mapping transformation is reduced to classical Blaschke-Grünwald planar mapping a novel three legged planar robot??s direct kinematics is solved in image space and then using conventional ??distance?? constraints. The purpose is to show why the latter approach yields spurious solutions and how the displacement pole rotation performed with kinematic mapping reliably avoids this problem. In conclusion certain other new and/or interesting reduced mobility parallel robots are discussed briefly to point out some advantages and insights gained with an image space approach. Particular effort is made to expose in detail how mapping simplifies and extends the solution of direct kinematics pertaining to Calvel??s ??Delta?? 3D translational robot.     

The Zero Surface Tension Limit of Two-Dimensional Interfacial Darcy Flow

We perform energy estimates for a sharp-interface model of two-dimensional, two-phase Darcy flow with surface tension. A proof of well-posedness of the initial value problem follows from these estimates. In general, the time of existence of these solutions will go to zero as the surface tension parameter vanishes. We then make two additional estimates, in the case that a stability condition is satisfied by the initial data: we make an additional energy estimate which is uniform in the surface tension parameter, and we make an estimate for the difference of two solutions with different values of the surface tension parameter. These additional estimates allow the zero surface tension limit to be taken, showing that solutions of the initial value problem in the absence of surface tension are the limit of solutions of the initial value problem with surface tension as surface tension vanishes.     

Relative Entropy and the Stability of Shocks and Contact Discontinuities for Systems of Conservation Laws with non-BV Perturbations

We develop a theory based on relative entropy to show the uniqueness and L ² stability (up to a translation) of extremal entropic Rankine?CHugoniot discontinuities for systems of conservation laws (typically 1-shocks, n-shocks, 1-contact discontinuities and n-contact discontinuities of large amplitude) among bounded entropic weak solutions having an additional trace property. The existence of a convex entropy is needed. No BV estimate is needed on the weak solutions considered. The theory holds without smallness conditions. The assumptions are quite general. For instance, strict hyperbolicity is not needed globally. For fluid mechanics, the theory handles solutions with vacuums.     

Fundamental solutions in perturbed shear flow

In this paper infinite plane Couette flow in a viscous incompressible fluid is considered subject to general three-dimensional perturbations and the equations of motion are linearized. Furthermore, initial-value problems are posed and a set of closed-form solutions are obtained for a variety of conditions, such as the system under the influence of: (i) a mass source; (ii) an external force; or (iii) initial vorticity. The result is a knowledge of both the early transient dynamics and the near spatial field behavior, as well as the state after a long time and the far field behavior. It is shown that the solutions can be considered as fundamental (in the sense that source-sink solutions are regarded fundamental for irrotational motion) and therefore are useful in analyzing other boundary-value, initial-value problems where the basic flow can be synthesized from piece-wise linear (constant shear) variations. To this end a generalized Green's function for the system is determined.     

Exact solutions and Painlevé analysis of a new (2+1)-dimensional generalized KdV equation

The new (2+1)-dimensional generalized KdV equation which exists the bilinear form is mainly discussed. We prove that the equation does not admit the Painlevé property even by taking the arbitrary constant a=0. However, this result is different from Radha and Lakshmanan??s work. In addition, based on Hirota bilinear method, periodic wave solutions in terms of Riemann theta function and rational solutions are derived, respectively. The asymptotic properties of the periodic wave solutions are analyzed in detail.     

Optimal convergence rates for three-dimensional turbulent flow equations

In this paper,the convergence rates of solutions to the three-dimensional turbulent flow equations are considered.By combining the Lp-Lq estimate for the linearized equations and an elaborate energy method,the convergence rates are obtained in various norms for the solution to the equilibrium state in the whole space when the initial perturbation of the equilibrium state is small in the H3-framework.More precisely,the optimal convergence rates of the solutions and their first-order derivatives in the L2-norm are obtained when the Lp-norm of the perturbation is bounded for some p ∈[1,6/5).     

Global Existence for the Generalized Two-Component Hunter–Saxton System

We study the global existence of solutions to a two-component generalized Hunter–Saxton system in the periodic setting. We first prove a persistence result for the solutions. Then for some particular choices of the parameters (α, κ), we show the precise blow-up scenarios and the existence of global solutions to the generalized Hunter–Saxton system under proper assumptions on the initial data. This significantly improves recent results.     

The Acoustic Limit for the Boltzmann Equation

The acoustic equations are the linearization of the compressible Euler equations about a spatially homogeneous fluid state. We first derive them directly from the Boltzmann equation as the formal limit of moment equations for an appropriately scaled family of Boltzmann solutions. We then establish this limit for the Boltzmann equation considered over a periodic spatial domain for bounded collision kernels. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that converge entropically (and hence strongly in L ¹) to a unique limit governed by a solution of the acoustic equations for all time, provided that its initial fluctuations converge entropically to an appropriate limit associated to any given L ² initial data of the acoustic equations. The associated local conservation laws are recovered in the limit. Accepted: October 22, 1999     

Large Scale Accumulation Patterns of Inertial Particles in Wall-Bounded Turbulent Flow

Turbulent internal flow in channel and pipe geometry with a diluted second phase of inertial particles is studied numerically. Direct numerical simulations (DNS) are performed at moderate Reynolds number (Re _?? ????200) in pipe and two channels??a smaller one similar in size to previous studies and a 3?×?3-times larger one??and Eulerian statistics pertaining to the particle concentration are evaluated. This simulation box constitutes the largest domain used for particle-laden flows so far. The resulting two-point correlations of the particle concentration show that in the smaller channel the particles organize in thin, streamwise elongated patterns which are very regular and long. The spanwise spacing of these structures is 120 and 160 plus units for the channel and pipe, respectively. Only in the larger box, the streamwise extent is long enough for the particle streaks to decorrelate, thus allowing the particles to move more freely. The influence of the box size on the characteristics of the turbophoresis is clearly shown; a 10% increase of the near-wall correlation is observed for particles with Stokes number St ^?+??=?50. It is thus shown that the box dimensions are an important factor in correctly assessing the motion of inertial particles, and their relation to the underlying velocity field. In addition the binning size effects on the correlation statistics of particle concentration are exploited. In particular the spanwise correlation peak values appear very sensitive to the adopted binning size, although the position of these peaks is found almost independent. Hence to allow a significant comparison between data of different configurations it is necessary to adopt the same binning spacing in inner variable.     

Radon Measure-Valued Solutions for a Class of Quasilinear Parabolic Equations

Initial value problems for quasilinear parabolic equations having Radon measures as initial data have been widely investigated, looking for solutions which for positive times take values in some function space. In contrast, it is the purpose of this paper to define and investigate solutions that for positive times take values in the space of the Radon measures of the initial data. We call such solutions measure-valued, in contrast to function-valued solutionspreviously considered in the literature. We first show that there is a natural notion of measure-valued solution of problem (P) below, in spite of its nonlinear character. A major consequence of our definition is that, if the space dimension is greater than one, the concentrated part of the solution with respect to the Newtonian capacity is constant in time. Subsequently, we prove that there exists exactly one solution of the problem, such that the diffuse part with respect to the Newtonian capacity of the singular part of the solution (with respect to the Lebesgue measure) is concentrated for almost every positive time on the set where “the regular part (with respect to the Lebesgue measure) is large”. Moreover, using a family of entropy inequalities we demonstrate that the singular part of the solution is nonincreasing in time. Finally, the regularity problem is addressed, as we give conditions (depending on the space dimension, the initial data and the rate of convergence at infinity of the nonlinearity ψ) to ensure that the measure-valued solution of problem (P) is, in fact, function-valued.     

On the Navier–Stokes Problem in Exterior Domains with Non Decaying Initial Data

We study the existence and uniqueness of regular solutions to the Navier–Stokes initial-boundary value problem with non-decaying bounded initial data, in a smooth exterior domain of ${{\mathbb R}^n, n\ge3}$ . The pressure field, p, associated to these solutions may grow, for large |x|, as O(|x| ^γ ), for some ${\gamma\in (0,1)}$ . Our class of existence is sharp for well posedeness, in that we show that uniqueness fails if p has a linear growth at infinity. We also provide a sufficient condition on the spatial growth of ${\nabla p}$ for the boundedness of v, at all times. Also this latter result is shown to be sharp.     

A class of fast diffusion p-Laplace equation with arbitrarily high initial energy

In this paper, the authors investigate a class of fast-diffusion p-Laplace equation, which was considered by Li, Han and Li (2016) [1], where, among other things, blow-up in finite time of solutions was proved for positive but suitably small initial energy. Their results will be complemented in this paper in the sense that the existence of finite time blow-up solutions for arbitrarily high initial energy will be proved. Moreover, an abstract criterion for the existence of global solutions that vanish at infinity will also be provided for high initial energy.     

Navier–Stokes Equations with Navier Boundary Conditions for an Oceanic Model

We consider the Navier–Stokes equations in a thin domain of which the top and bottom surfaces are not flat. The velocity fields are subject to the Navier conditions on those boundaries and the periodicity condition on the other sides of the domain. This toy model arises from studies of climate and oceanic flows. We show that the strong solutions exist for all time provided the initial data belong to a “large” set in the Sobolev space H ¹. Furthermore we show, for both the autonomous and the nonautonomous problems, the existence of a global attractor for the class of all strong solutions. This attractor is proved to be also the global attractor for the Leray–Hopf weak solutions of the Navier–Stokes equations. One issue that arises here is a nontrivial contribution due to the boundary terms. We show how the boundary conditions imposed on the velocity fields affect the estimates of the Stokes operator and the (nonlinear) inertial term in the Navier–Stokes equations. This results in a new estimate of the trilinear term, which in turn permits a short and simple proof of the existence of strong solutions for all time.