The Fast Exact Closure for Jeffery’s equation with diffusion

Jeffery’s equation with diffusion is widely used to predict the motion of concentrated fiber suspensions in flows with low Reynold’s numbers. Unfortunately, the evaluation of the fiber orientation distribution can require excessive computation, which is often avoided by solving the related second order moment tensor equation. This approach requires a ‘closure’ that approximates the distribution function’s fourth order moment tensor from its second order moment tensor. This paper presents the Fast Exact Closure (FEC) which uses conversion tensors to obtain a pair of related ordinary differential equations; avoiding approximations of the higher order moment tensors altogether. The FEC is exact in that when there are no fiber interactions, it exactly solves Jeffery’s equation. Numerical examples for dense fiber suspensions are provided with both a Folgar–Tucker (1984) [3] diffusion term and the recent anisotropic rotary diffusion term proposed by Phelps and Tucker (2009) [9]. Computations demonstrate that the FEC exhibits improved accuracy with computational speeds equivalent to or better than existing closure approximations.     

Weakly Nonlinear-Dissipative Approximations of Hyperbolic–Parabolic Systems with Entropy

Hyperbolic–parabolic systems have spatially homogenous stationary states. When the dissipation is weak, one can derive weakly nonlinear-dissipative approximations that govern perturbations of these constant states. These approximations are quadratically nonlinear. When the original system has an entropy, the approximation is formally dissipative in a natural Hilbert space. We show that when the approximation is strictly dissipative it has global weak solutions for all initial data in that Hilbert space. We also prove a weak-strong uniqueness theorem for it. In addition, we give a Kawashima type criterion for this approximation to be strictly dissipative. We apply the theory to the compressible Navier–Stokes system.     

Entropy analysis of integer and fractional dynamical systems

This paper investigates the adoption of entropy for analyzing the dynamics of a multiple independent particles system. Several entropy definitions and types of particle dynamics with integer and fractional behavior are studied. The results reveal the adequacy of the entropy concept in the analysis of complex dynamical systems.     

A generalization of Prandtl’s and Spencer’s solutions on axisymmetric viscous flow

New analytical solutions for axisymmetric deformation of a viscous hollow circular cylinder on a rigid fibre are given. One of the solutions generalizes the famous Prandtl’s solution for compression of a rigid perfectly plastic layer between two rough, parallel plates and the other is a modification of Spencer’s solution for compression of an axisymmetric rigid perfectly plastic layer on a rigid fibre. All equations are satisfied exactly whereas some boundary conditions are approximated in a standard manner. Special attention is devoted to frictional interface conditions since these conditions result in additional limitations of the applicability of the solution when compared to that based on a rigid perfectly plastic models. In particular, difficulties with the convergence of numerical solutions under certain conditions can be explained with the use of results obtained. Therefore, the solutions can serve as benchmark problems for verifying numerical codes. The solutions are also adopted to predict the brittle fracture of fibres by means of an approach used in previous studies and confirmed by experiment.     

Effective Young’s modulus of biopolymer composites with imperfect interface

The effective elastic behaviour of a biopolymer composite with random microstructure is studied. The effective elasticity modulus is computed as function of the protein fraction with the hypothesis of an imperfect interface. This interface is handled considering normal and tangential stiffness parameters. The methodology assumes the direct import of real microstructures of starch–protein (zein) composite, obtained using Confocal Laser Scanning Microscopy (CLSM), into a finite element model. A static linear analysis is conducted to determine the influence of interface rigidities, zein fraction and interface quantity on the effective properties. Predictions show that the effective modulus is nonlinearly correlated to the product of stiffness parameters for which the normal stiffness is the most influential parameter. The effective property dependence with respect to interface related variables and zein fraction shows a logarithmic influence of the stiffness parameters when increasing zein fraction.     

Hölder Continuity and Injectivity of Optimal Maps

Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x, y). If the source density f ⁺(x) is bounded away from zero and infinity in an open region ${U' \subset \mathbf{R}^n}$ , and the target density f ^?(y) is bounded away from zero and infinity on its support ${\overline{V} \subset \mathbf{R}^n}$ , which is strongly c-convex with respect to U′, and the transportation cost c satisfies the ${\mathbf{A3}_{\rm w}}$ condition of Trudinger and Wang (Ann Sc Norm Super Pisa Cl Sci 5, 8(1):143–174, 2009), we deduce the local Hölder continuity and injectivity of the optimal map inside U′ (so that the associated potential u belongs to ${C^{1,\alpha}_{loc}(U')}$ ). Here the exponent α > 0 depends only on the dimension and the bounds on the densities, but not on c. Our result provides a crucial step in the low/interior regularity setting: in a sequel (Figalli et al., J Eur Math Soc (JEMS), 1131–1166, 2013), we use it to establish regularity of optimal maps with respect to the Riemannian distance squared on arbitrary products of spheres. Three key tools are introduced in the present paper. Namely, we first find a transformation that under ${\mathbf{A3}_{\rm w}}$ makes c-convex functions level-set convex (as was also obtained independently from us by Liu (Calc Var Partial Diff Eq 34:435–451, 2009)). We then derive new Alexandrov type estimates for the level-set convex c-convex functions, and a topological lemma showing that optimal maps do not mix the interior with the boundary. This topological lemma, which does not require ${\mathbf{A3}_{\rm w}}$ , is needed by Figalli and Loeper (Calc Var Partial Diff Eq 35:537–550, 2009) to conclude the continuity of optimal maps in two dimensions. In higher dimensions, if the densities f ^± are Hölder continuous, our result permits continuous differentiability of the map inside U′ (in fact, ${C^{2,\alpha}_{loc}}$ regularity of the associated potential) to be deduced from the work of Liu et al. (Comm Partial Diff Eq 35(1):165–184, 2010).     

Dynamical behavior of a stochastic Nicholson’s blowflies model with distributed delay and degenerate diffusion

Nonlinear Dynamics - The mathematical model with time delay is often more practical because it is subject to current and past state. What remains unclear are the details, such as how time delay and...     

A study of shallow water waves with Gardner’s equation

In this paper the dynamics of solitary waves governed by Gardner’s equation for shallow water waves is studied. The mapping method is employed to carry out the integration of the equation. Subsequently, the perturbed Gardner equation is studied, and the fixed point of the soliton width is obtained. This fixed point is then classified. The integration of the perturbed Gardner equation is also carried out with the aid of He’s semi-inverse variational principle. Finally, Gardner’s equation with full nonlinearity is solved with the aid of the solitary wave ansatz method.     

Investigation of generalized Fick’s and Fourier’s laws in the second-grade fluid flow

The second-grade fluid flow due to a rotating porous stretchable disk is modeled and analyzed. A porous medium is characterized by the Darcy relation. The heat and mass transport are characterized through Cattaneo-Christov double diffusions. The thermal and solutal stratifications at the surface are also accounted. The relevant nonlinear ordinary differential systems after using appropriate transformations are solved for the solutions with the homotopy analysis method (HAM). The effects of various involved variables on the temperature, velocity, concentration, skin friction, mass transfer rate, and heat transfer rate are discussed through graphs. From the obtained results, decreasing tendencies for the radial, axial, and tangential velocities are observed. Temperature is a decreasing function of the Reynolds number, thermal relaxation parameter, and Prandtl number. Moreover, the mass diffusivity decreases with the Schmidt number.     

The integrable Boussinesq equation and it’s breather,lump and soliton solutions

The fourth-order nonlinear Boussinesq water wave equation, which explains the propagation of long waves in shallow water, is explored in this article. We used the Lie symmetry approach to analyze the Lie symmetries and vector fields. Then, by using similarity variables, we obtained the symmetry reductions and soliton wave solutions. In addition, the Kudryashov method and its modification are used to explore the bright and singular solitons while the Hirota bilinear method is effectively used to obtain a form of breather and lump wave solutions. The physical explanation of the extracted solutions was shown with the free choice of different parameters by depicting some 2-D, 3-D, and their corresponding contour plots.

     

Floquet’s Theorem and Stability of Periodic Solitary Waves

This paper is concerned with the spectrum the Hill operator L(y) = −y′′ + Q(x) y in L²_per[0, p]{L^{2}_{\rm per}[0, \pi]} . We show that the eigenvalues of L can be characterized by knowing one of its eigenfunctions. Applications are given to nonlinear stability of a class of periodic problems.     

Dynamic homogenisation of Maxwell’s equations with applications to photonic crystals and localised waveforms on gratings

A two-scale asymptotic theory is developed to generate continuum equations that model the macroscopic behaviour of electromagnetic waves in periodic photonic structures when the wavelength is not necessarily long relative to the periodic cell dimensions; potentially highly-oscillatory short-scale detail is encapsulated through integrated quantities. The resulting equations include tensors that represent effective refractive indices near band edge frequencies along all principal axes directions, and these govern scalar functions providing long-scale modulation of short-scale Bloch eigenstates, which can be used to predict the propagation of waves at frequencies outside of the long wavelength regime; these results are outside of the remit of typical homogenisation schemes.The theory we develop is applied to two topical examples, the first being the case of aligned dielectric cylinders, which has great importance in modelling photonic crystal fibres. Results of the asymptotic theory are verified against numerical simulations by comparing photonic band diagrams and evanescent decay rates for guided modes. The second example is the propagation of electromagnetic waves localised within a planar array of dielectric spheres; at certain frequencies strongly directional propagation is observed, commonly described as dynamic anisotropy. Computationally this is a challenging three-dimensional calculation, which we perform, and then demonstrate that the asymptotic theory captures the effect, giving highly accurate qualitative and quantitative comparisons as well as providing interpretation for the underlying change from elliptic to hyperbolic behaviour.     

Maxwell’s Equations with Vector Hysteresis

Electromagnetic processes in magnetic materials are described by Maxwells equations. In ferrimagnetic insulators, assuming that D = E, we have the equationIn ferromagnetic metals, neglecting displacement currents and assuming Ohms law, we instead getAlternatively, under quasi-stationary conditions, for either material we can also deal with the magnetostatic equations:(Here f_ext and J_ext are prescribed time-dependent fields.) In any of these settings, the dependence of M on H is represented by a constitutive law accounting for hysteresis: M= (H), being a vector extension of the relay model. This is characterized by a rectangular hysteresis loop in a prescribed x-dependent direction, and accounts for high anisotropy and nonhomogeneity. The discontinuity in this constitutive relation corresponds to the possible occurrence of free boundaries.Weak formulations are provided for Cauchy problems associated with the above equations; existence of a solution is proved via approximation by time-discretization, derivation of energy-type estimates, and passage to the limit. An analogous representation is given for hysteresis in the dependence of P on E in ferroelectric materials. A model accounting for coupled ferrimagnetic and ferroelectric hysteresis is considered, too.Acknowledgement This research was partly supported by the project Free boundary problems in applied sciences of Italian M.I.U.R.. I gratefully acknowledge the useful suggestions from the reviewers.