Excitation control for three-dimensional Peregrine solution and combined breather of a partially nonlocal variable-coefficient nonlinear Schrödinger equation

Nonlinear Dynamics - A (2 + 1)-dimensional variable-coefficient partially nonlocal nonlinear Schrödinger equation is considered, and analytical Peregrine solution (PS) and combined Akhmediev...     

Localized traveling wave solution for a logarithmic nonlinear Schrödinger equation

We investigate localized traveling wave solutions for a Schrödinger equation with two logarithmic nonlinear terms under no external potential. It is shown that it can have solitary wave type solutions whose envelope profile depends on the two types of nonlinearity. Remarkably, the profile has cutoffs in the coordinate of propagation. We argue also some fundamental properties that discriminate it from power law type nonlinear Schrödinger equations.     

Exact solution of the capillary diffusion equation for a three‐component mixture

The paper gives an exact solution of the steady system of equations for stable threecomponent diffusion in the entire range of concentrations for a long capillary under a controlled capillary pressure differential. The solution allows one to calculate the distributions of component concentrations and mixture density along the capillary. It is shown that if the diffusion coefficients are markedly different, an extremum of mixture density can arise inside the capillary. In particular, if the density of the mixture in the upper flask is higher than that in the lower flask and the stratification of the system is generally stable, a region with a reverse density gradient that is unstable against gravity convection can appear inside the capillary. A comparison with experimental results shows that the resistance to gravity convection is disturbed when an extremum of mixture density arises in the channel during steady diffusion.     

Application of solution structure theorems to Cattaneo–Vernotte heat conduction equation with non-homogeneous boundary conditions

In this study, a non-Fourier heat conduction problem formulated using the Cattaneo–Vernotte (C–V) model with non-homogeneous boundary conditions is solved with the superposition principle in conjunction with solution structure theorems. It is well known that the aforementioned analytical method is not suitable for such a class of thermal problems. However, by performing a functional transformation, the original non-homogeneous partial differential equation governing the physical problem can be cast into a new form such that it consists of a homogeneous part and an additional auxiliary function. As a result, the modified homogeneous governing equation can then be solved with solution structure theorems for temperatures inside a finite planar medium. The methodology provides a convenient, accurate, and efficient solution to the C–V heat conduction equation with non-homogeneous boundary conditions.     

Analytical solution for functionally graded anisotropic cantilever beam subjected to linearly distributed load

The bending problem of a functionally graded anisotropic cantilever beam subjected to a linearly distributed load is investigated.The analysis is based on the exact elasticity equations for the plane stress problem.The stress function is introduced and assumed in the form of a polynomial of the longitudinal coordinate.The expressions for stress components are then educed from the stress function by simple differentiation. The stress function is determined from the compatibility equation as well as the bound- ary conditions by a skilful deduction.The analytical solution is compared with FEM calculation,indicating a good agreement.     

Modelling the three-dimensional flow of a semi-dilute polymer solution in microfluidics—on the effect of aspect ratio

The flow of polymer solutions in microfluidic devices is inherently three-dimensional, especially in the non-linear flow regime, and often results in flow phenomena that might not even be encountered in macro-devices. Using a multi-mode Phan-Thien–Tanner model, three-dimensional (3-D) simulations of a semi-dilute polyethylene oxide (PEO) solution through 8:1 planar contraction micro-channels with various depths have been carried out to systematically study the effect of the aspect ratio on the flow fields. Vortex dynamics in the upstream flow section and excess pressure drop are quantified in detail. A transition from a salient-corner vortex mechanism to a lip vortex mechanism is observed as the aspect ratio is varied from 1 to 1/4, which corresponds to the elasticity numbers El = 36.3 to 48.3. The numerical results show that varying the aspect ratio of microfluidic channels has similar effects to varying other parameters, such as fluid properties, which influence the elasticity number. Thus, our results support the view that vortex growth mechanism is determined by the elasticity number, which is fixed for a given fluid and geometry. The principle is of significance to the design of new microfluidic chips for a wide range of applications.     

Analytical solution for a strained reinforcement layer bonded to lip-shaped crack under remote mode Ⅲ uniform load and concentrated load

In this paper,the analytical solution of stress field for a strained reinforcement layer bonded to a lip-shaped crack under a remote mode Ⅲ uniform load and a concentrated load is obtained explicitly in the series form by using the technical of conformal mapping and the method of analytic continuation.The effects of material combinations,bond of interface and geometric configurations on interfacial stresses generated by eigenstrain,remote load and concentrated load are studied.The results show that the stress concentration and interfacial stresses can be reduced by rational material combinations and geometric configurations designs for different load forms.     

Analytical solution for buckling of asymmetrically delaminated Reissner’s elastic columns including transverse shear

The exact analytical solution of buckling in delaminated columns is presented. In order to investigate analytically the influence of axial and shear strains on buckling loads the geometrically exact beam theory is employed with no simplification of the governing equations. The critical forces are then obtained by the linearized stability theory. In the paper, we limit the studies to linear elastic columns with a single delamination, but with arbitrary longitudinal and vertical asymmetry of delamination and arbitrary boundary conditions. The studies of quantitative and qualitative influence of transverse shear are shown in detail and extensive results for buckling loads with respect to delamination length, thickness and longitudinal position are presented.     

Planar flow of a three-layer plastic material through a converging wedge-shaped channel. Part 1 – Analytical solution

Flow of a three-layer strip consisting of two different materials through an infinite converging channel is studied. Each material is assumed to be incompressible, rigid perfectly plastic. It is shown that the structure of the solution strongly depends on the ratio of the yield stresses and the friction factors, namely the problem may have the unique solution, multiple solutions, and may have no solution. This makes a significant difference between the new solution and other known solutions to the problem of flow through infinite channels.     

Analytical solution for squeeze film damping of MEMS perforated circular plates using Green’s function

Squeezed air film between two closely spaced vibrating microstructures is the important source of energy dissipation and has profound effects on the dynamics of microelectromechanical systems (MEMS). Perforations in the design are one of the methods to model these damping effects. The literature reveals that the analytical modeling of squeeze film damping of perforated circular microplates is less explored; however, these microplates are also an imperative part of the numerous MEMS devices. Here, we derive an analytical model of transverse and rocking motions of a perforated circular microplate. A modified Reynolds equation that incorporates compressibility and rarefaction effects is utilized in the analysis. Pressure distribution under the vibrating microplate is derived by using Green’s function and also derived by finite element method (FEM) to visualize the pressure distribution under perforated and non-perforated areas of the microplate. The analytical damping results are validated with previous renowned analytical models and also with the FEM results. The outcomes confirm the potential of the present analytical model to accurately predict the squeeze film damping parameters.     

Analytical solution for the general mechanochemical corrosion of an ideal elastic–plastic thick-walled tube under pressure

The paper presents analytical solutions for the equal-rate mechanochemical wear of an ideal elastic–plastic thick-walled cylindrical tube subjected to any combination of internal and external pressure. The rates of corrosion at the inner and outer surfaces are supposed to be proportional to the equivalent tensile stress at the surface involved when it exceeds a given threshold. Furthermore, the corrosion rate can decay exponentially with time. The obtained solutions allow to assess the time of the initial yielding at the bore of the tube and the time of fully plastic yielding. Calculations showed that the time of plastic-zone propagation throughout the tube wall can be much greater than the length of the pure elastic stage. The proposed analytical solutions are to be used for design purposes and as benchmark solutions for numerical analysis.     

Pattern formation in a nonequilibrium phase transition for a generalized Burgers–Fisher equation

A nonequilibrium phase transition of a generalized Burgers–Fisher equation describing biological pattern formation with a periodic boundary condition is examined. In the presence of a weak external force, some approximate bifurcation solutions near a critical point and new spatially periodic patterns are obtained by using the perturbation method in an infinite-dimensional space. The result shows that the external force delays the bifurcation.     

Explicit solutions for depressed masonry arches loaded until collapse—Part II: a solution method for statically indeterminate systems

Meccanica - In this second part of the paper we use the results obtained in the first part in order to build the solution to two statically indeterminate cases of a depressed masonry arch. As in...     

Two-breather solutions for the class I infinitely extended nonlinear Schrödinger equation and their special cases

Nonlinear Dynamics - We derive the two-breather solution of the class I infinitely extended nonlinear Schrödinger equation. We present a general form of this multi-parameter solution that...     

Analysis of mesoscale effects in high-shear granulation through a computational fluid dynamics–population balance coupled compartment model

There is a need for mesoscale resolution and coupling between flow-field information and the evolution of particle properties in high-shear granulation. We have developed a modelling framework that compartmentalizes the high-shear granulation process based on relevant process parameters in time and space. The model comprises a coupled-flow-field and population-balance solver and is used to resolve and analyze the effects of mesoscales on the evolution of particle properties. A Diosna high-shear mixer was modelled with microcrystalline cellulose powder as the granulation material. An analysis of the flow-field solution and compartmentalization allows for a resolution of the stress and collision peak at the impeller blades. Different compartmentalizations showed the importance of resolving the impeller region, for aggregating systems and systems with breakage. An independent study investigated the time evolution of the flow field by changing the particle properties in three discrete steps that represent powder mixing, the initial granulation stage mixing and the late stage granular mixing. The results of the temporal resolution study show clear changes in collision behavior, especially from powder to granular mixing, which indicates the importance of resolving mesoscale phenomena in time and space.     

Multi-layer analytic solution for k-ω model equations via a symmetry approach

Despite being one of the oldest and most widely-used turbulence models in engineering computational fluid dynamics (CFD), the k-ω model has not been fully understood theoretically because of its high nonlinearity and complex model parameter setting. Here, a multi-layer analytic expression is postulated for two lengths (stress and kinetic energy lengths), yielding an analytic solution for the k-ω model equations in pipe flow. Approximate local balance equations are analyzed to determine the key parameters in the solution, which are shown to be rather close to the empirically-measured values from the numerical solution of the Wilcox k-ω model, and hence the analytic construction is fully validated. The results provide clear evidence that the k-ω model sets in it a multilayer structure, which is similar to but different, in some insignificant details, from the Navier-Stokes (N-S) turbulence. This finding explains why the k-ω model is so popular, especially in computing the near-wall flow. Finally, the analysis is extended to a newly-refined k-ω model called the structural ensemble dynamics (SED) k-ω model, showing that the SED k-ω model has improved the multi-layer structure in the outer flow but preserved the setting of the k-ω model in the inner region.     

Painlevé integrability and N-soliton solution for the variable-coefficient Zakharov–Kuznetsov equation from plasmas

With the help of symbolic computation, this paper investigates the variable-coefficient Zakharov–Kuznetsov equation which governs the two-dimensional ion-acoustic waves obliquely propagating in an inhomogeneous magnetized two-ion-temperature dusty plasma. The integrability of this model is examined through the Painlevé analysis. Via the Hirota method, the bilinear form of such model is derived. Based on the obtained bilinear form, the N-soliton solution is constructed. Propagation characteristics and interaction behaviors of the solitons are discussed through the graphical analysis.     

Rogue-wave solutions for a discrete Ablowitz–Ladik equation with variable coefficients for an electrical lattice

Nonlinear Dynamics - We investigate a discrete Ablowitz–Ladik equation with variable coefficients, which models the modulated waves in an electrical lattice. Employing the similarity...     

Exact solution of the Muskat–Leibenzon problem for a growing elliptic bubble

The exact solution of the two-phase time-dependent Hele–Shaw problem (in other words, the plane Muskat–Leibenzon problem) in which a fluid occupied an unbounded channel is displaced by another fluid incoming through a slitted cut in the channel. In this case the interface between the phases, namely, fluids of different viscosity, evolves as an ellipse whose area and eccentricity vary continuously.