On MHD transient flow of a Maxwell fluid in a porous medium and rotating frame

Analytic solution for unsteady magnetohydrodynamic (MHD) flow is constructed in a rotating non-Newtonian fluid through a porous medium. Constitutive equations for a Maxwell fluid have been taken into consideration. The hydromagnetic flow in the uniformly rotating fluid is generated by a suddenly moved infinite plate in its own plane. Analytic solution of the governing flow problem is obtained by means of the Fourier sine transform. It is shown that the obtained solution satisfies both the associate partial differential equation and the initial and boundary conditions. The solution for a Navier-Stokes fluid is recovered if λ→0. The steady state solution is also obtained for t→∞.     

Generation and propagation of pulse trains with ultrashort pulse separation

We investigate the nonlinear Schrödinger equation with variable coefficients by employing perturbation method. The analysis solution of the harmonic form is presented. The solution is one of forms to describe pulse trains with ultrashort pulse separation, which is about two orders of magnitude shorter than one of sech-type solitons considered before. And we could systematically adjust the perturbation parameter to obtain different pulse separation. As an example, we consider a nonlinear dispersive system with spatial parameter variations, and the results show that, the pulse train with ultrashort pulse separation presented by analysis solution may keep its shape even if the velocity is changed. The stability of the solution is discussed numerically, and the results reveal that the finite initial perturbations, such as white noise could not influence the main character of the solution. In addition, the stability of the solution is also discussed under more general conditions.     

A New Solution to Einstein's Field Equations

We construct a new exact solution to the vacuum Einstein field equations. This solution possesses a naked physical singularity. The norm of the Riemann curvature tensor of the solution takes infinity at some points and the solution does not have any event horizon around the singularity. A detailed analysis of this new singularity is also presented.     

A new class of plane symmetric solution

A new class of static plane symmetric solution of Einstein field equation generated by a perfect fluid source is put forward. A special family of this new solution is investigated in detail. The constraints on the parameters by different energy conditions are studied. The classical stability of this solution is discussed. The junction conditions matching to Minkowski metric and Taub metric are analyzed respectively.     

On the theory of reflectivity and transmissivity of a lossless nonlinear dielectric slab

For the transverse electric polarization case (TE) we present a treatment of the optical reflectivity and transmissivity of a slab whose dielectric coefficient is a real valued function of the light intensity. If this function is numerically integrable with respect to the light intensity, our treatment can serve as an algorithm for a numerical solution of the nonlinear wave equation. If the dielectric function is proportional to the intensity, an analytical solution of the cubic wave equation is given for the electric field strength and for the phase of the field in terms of Weierstrass' elliptic functions and first elliptic theta functions, respectively. Evaluating this solution by means of a computer algebra system yields the reflectivity, transmissivity and phase dependency on the incident field intensity and on parameters characteristic for the problem. Certain combinations of the parameters lead to bistable and multivalued behavior. The solution found is used to determine the relative extrema of the reflectivity and the critical values of the thickness and of the incident intensity. The results are a generalization of linear optics results. Application of the analysis to the cubic-quintic wave equation yields the general analytic solution which is used to detemine the reflectivity of a semi-infinite nonlinear medium.     

Electric Field-Induced Fluid Velocity Field Distribution in DNA Solution

We present an analytical solution for fluid velocity field distribution of polyelectrolyte DNA. Both the electric field force and the viscous force in the DNA solution are considered under a suitable boundary condition. The solution of electric potential is analytically obtained by using the linearized Poisson-Boltzmann equation. The fluid velocity along the electric field is dependent on the cylindrical radius and concentration. It is shown that the electric field-induced fluid velocity will be increased with the increasing cylindrical radius, whose distribution also varies with the concentration     

Brownian Dynamics Simulation of Microstructures and Elongational Viscosities of Micellar Surfactant Solution

Brownian dynamics simulation is conducted for a dilute surfactant solution under a steady uniaxial elongational flow. A new inter-cluster potential is used for the interaction among surfactant micelles to determine the micellar network structures in the surfactant solution. The micellar network is successfully simulated. It is formed at low elongation rates and destroyed by high elongation rates. The computed elongational viscosities show elongation- thinning characteristics. The relationship between the elongational viscosities and the mierostructure of the surfactant solution is revealed.     

Obtaining the bidirectional transmittance distribution function of isotropically scattering materials using an integrating sphere

This paper demonstrates a method to determine the bidirectional transmittance distribution function (BTDF) using an integrating sphere. Information about the sample’s angle-dependent scattering is obtained by making transmittance measurements with the sample at different distances from the integrating sphere. Knowledge about the illuminated area of the sample and the geometry of the sphere port in combination with the measured data combines to a system of equations that includes the angle-dependent transmittance.The resulting system of equations is an ill-posed problem which rarely gives a physical solution. A solvable system is obtained by using Tikhonov regularization on the ill-posed problem. The solution to this system can then be used to obtain the BTDF.Four bulk-scattering samples were characterised using two goniophotometers and the described method to verify the validity of the new method. The agreement shown is excellent for the more diffuse samples. The solution to the low-scattering samples contains unphysical oscillations, but still gives the correct shape of the solution. The origin of the oscillations and why they are more prominent in low-scattering samples are discussed.     

A note on Burgers' equation with time delay: Instability via finite-time blow-up

Burgers' equation with time delay is considered. Using the Cole-Hopf transformation, the exact solution of this nonlinear partial differential equation (PDE) is determined in the context of a (seemingly) well-posed initial-boundary value problem (IBVP) involving homogeneous Dirichlet data. The solution obtained, however, is shown to exhibit a delay-induced instability, suffering blow-up in finite-time.     

Supersymmetric KdV-Sawada-Kotera-Ramani equation and its quasi-periodic wave solutions

In this Letter, we propose a supersymmetric KdV-Sawada-Kotera-Ramani equation. Based on a super-Riemann theta function, we devise a lucid and straightforward way for explicitly constructing a quasi-periodic wave solution of the supersymmetric KdV-Sawada-Kotera-Ramani equation. In addition, a one-soliton solution is obtained as a limiting case of the periodic wave solution under small amplitude. Indeed different from the purely bosonic case, the quasi-periodic wave observed shows that there is an “influencing band” among the waves under the presence of the Grassmann variable. The waves are symmetric about the band but collapse along with the band. Furthermore, the amplitudes of the waves increase as the waves move away from the band.     

Initial value problem of the Whitham equations for the Camassa-Holm equation

We study the Whitham equations for the Camassa-Holm equation. The equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the initial value problem of the Whitham equations. When the initial values are given by a step function, the Whitham solution is self-similar. When the initial values are given by a smooth function, the Whitham solution exists within a cusp in the x-t plane. On the boundary of the cusp, the Whitham solution matches the Burgers solution, which exists outside the cusp.     

Au-assisted electroless etching of silicon in aqueous HF/H2O2 solution

The Au-assisted electroless etching of p-type silicon substrate in HF/H₂O₂ solution at 50 °C was investigated. The dependence of the crystallographic orientation, the concentration of etching solution and the silicon resistivity on morphology of etched layer was studied. The layers formed on silicon were investigated by scanning electron microscopy (SEM). It was demonstrated that although the deposited Au on silicon is a continuous film, it can produce a layer of silicon nanowires or macropores depending on the used solution concentration.     

Shocks and finite-time singularities in Hele-Shaw flow

Hele-Shaw flow at vanishing surface tension is ill-defined. In finite time, the flow develops cusp-like singularities. We show that this ill-defined problem admits a weak dispersive solution when singularities give rise to a graph of shock waves propagating into the viscous fluid. The graph of shocks grows and branches. Velocity and pressure have finite discontinuities across the shock. We formulate a few simple physical principles which single out the dispersive solution and interpret shocks as lines of decompressed fluid. We also formulate the dispersive weak solution in algebro-geometrical terms as an evolution of the Krichever-Boutroux complex curve. We study in detail the most generic (2, 3)-cusp singularity, which gives rise to an elementary branching event. This solution is self-similar and expressed in terms of elliptic functions.     

Instability of Solution of Phase Retrieval in Direct Diffraction Phase-Contrast Imaging with Partially Coherent X-Ray Source

The theoretical model of direct diffraction phase-contrast imaging with partially coherent x-ray source is expressed by an operator of multiple integral. It is presented that the integral operator is linear. The problem of its phase retrieval is described by solving an operator equation of multiple integral. It is demonstrated that the solution of the phase retrieval is unstable. The numerical simulation is performed and the result validates that the solution of the phase retrieval is unstable.     

ESCA-studies of the structure and composition of the passive film formed on stainless steels by various immersion times in 0.1 M NaCl solution

This report deals with the structure of the passive film formed on stainless steels during immersion in a 0.1 M NaCl solution for various immersion times. The film was examined using ion etching and Electron Spectroscopy for Chemical Analysis (ESCA).Through the chemical-composition profiles a three-factor model was developed to describe the structure of the passive films: a hydrated layer in contact with the solution, an oxide layer consisting of iron and chromium oxides having maxima at depths of 3 and 10 Å, respectively, and a metallic layer enriched in nickel. There is a smooth transition between the layers, with the thickness of the outer two layers being about 15 A. The maximum concentration of iron in the oxidized state decreases with increasing immersion time concomitant with an increase in both the maximum concentration of chromium in the oxidized state and the maximum concentration of nickel in the metallic state.It is found that air-formed films have similar structures to films formed by exposure to the solution. Longer air exposure thickens the air-formed film, with the maximum of both oxidized iron and chromium nearly equal in magnitude in comparison with the solution formed films. Films formed by short-time exposure only to the solution are somewhat thinner indicating that the total history is important in determining the structure of these passive films, whereas passive films formed by exposures of 15 h to the solution are no longer strongly dependent on prior air exposure.     

Quantization of motion with friction quadratic in velocity

The motion of a point object through a viscous field is considered. The friction is assumed to depend quadratically on velocity of the particle. The inverse problem of the variational calculus is solved and the Weyl quantization procedure is employed to write a Schrödinger equation. The solution of this equation shows that the quantum mechanical wave function is oscillatory for small values of the friction. Contrarily, for large values of the friction, the wave function resembles the solution of von Neumann shock problem.     

Coupled modes of the torsion pendulum

The coupling of the swing modes of a torsion pendulum to the torsional mode has been solved analytically. Our solution provides a clear explanation of why a magnetic damper is effective in suppressing unwanted modes in most gravitational experiments. The analytic solution also shows that the amplitude of the mode coupling is greatest at the lowest frequencies. This explains why mode coupling at the lowest frequency is all that is observed in the experiments reported here.     

N-Soliton Solution in Wronskian Form for a Generalized Variable-Coefficient Korteweg--de Vries Equation

We concentrate on finding exact solutions for a generalized variable-coefficient Korteweg-de Vries equation of physically significance. The analytic N-soliton solution in Wronskian form for such a model is postulated and verified by direct substituting the solution into the bilinear form by virtue of the Wronskian technique. Additionally, the bilinear auto-Backlund transformation between the （ N - 1）- and N-soliton solutions is verified.     

IR multiple-photon dissociation by a focused uniform beam

An improved analytical method is presented in which the reaction yield in the IR multiple-photon dissociation by a focused uniform beam is expressed in a generalized form as a function of fluence. The analytical solution is derived for a real focusing geometry by assuming the cumulative log-normal distribution (CLND) for the functional form of the dissociation probability vs. fluence. Also presented is a shortcut analytical method with simple and convenient algebraic expressions which approximate the exact analytical solution, thereby speeding up the analysis of experimental data.     

Rational Chebyshev pseudospectral approach for solving Thomas-Fermi equation

In this Letter we propose a pseudospectral method for solving Thomas-Fermi equation which is a nonlinear ordinary differential equation on semi-infinite interval. This approach is based on rational Chebyshev pseudospectral method. This method reduces the solution of this problem to the solution of a system of algebraic equations. Comparison with some numerical solutions shows that the present solution is highly accurate.