Rosenzweig instability in a thin layer of a magnetic fluid

A simple mathematical model of the initial stage of nonlinear evolution of the Rosenzweig instability in a thin layer of a nonlinearly magnetized viscous ferrofluid coating a horizontal nonmagnetizable plate is constructed on the basis of the system of equations and boundary conditions of ferrofluid dynamics. A dispersion relation is derived and analyzed using the linearized equations of this model. The critical magnetization of the initial layer with a flat free surface, the threshold wavenumber, and the characteristic time of evolution of the most rapidly growing mode are determined. The equation for the neutral stability curve, which is applicable for any physically admissible law of magnetization of a ferrofluid, is derived analytically.     

Kuramoto—Sivashinsky equation for modulation of surface relief of molten layer and formation of surface periodic microstructures under pulsed laser irradiation of solids

A closed nonlinear two dimensional equation of Kuramoto-Sivashinsky type for the thickness of the laser pulse induced viscous molten layer is derived in the long wavelength and weak nonlinearity approximation. Linear stability analysis shows that under the condition that the temperature gradient at the surface is directed from the surface to the bulk, the surface instability sets in that leads to formation of surface relief structures with wavelength proportional to the thickness of the liquid layer. Computer simulations predict the subsequent formation of lamellar and disordered, quasihexagonal structures of surface relief when the time of irradiation is increased. Obtained results are used for the interpretation of experimental data on formation of lamellar and quasihexagonal surface relief microstructures upon multiple nanosecond pulse laser irradiation of silicon in water confinement.     

Large-scale Marangoni convection in a liquid layer with insoluble surfactant of low concentration

We derive a system of amplitude equations describing the evolution of a large-scale Marangoni patterns in a liquid layer with poorly conducting boundaries in the presence of a small amount of an insoluble surfactant on the free flat interface. The presence of quadratic nonlinear terms in the amplitude equation leads to the selection of hexagonal patterns. The type of hexagons bifurcating into the subcritical region, depends on the parameters of the system.     

Nonlinear periodic waves on the charged surface of a finite-thickness ideal liquid layer

In the fourth order of smallness in the amplitude of a periodic capillary-gravitational wave travelling over the uniformly charged free surface of an ideal incompressible conducting liquid of a finite depth, analytical expressions for the evolution of the nonlinear wave, velocity field potential of the liquid, electrostatic field potential above the liquid, and nonlinear frequency correction that is quadratic in a small parameter are derived. It is found that the dependence of the amplitude of the nonlinear correction to the frequency on the charge density on the free liquid surface and on the thickness of the liquid layer changes qualitatively when the layer gets thinner. In thin liquid layers, the resonant wavenumber depends on the surface charge density, while in thick layers, this dependence is absent.     

平顶光束经表面有缺陷的厚非线性介质后的光强分布

基于非线性薛定谔方程和分步傅里叶算法, 研究了高功率平顶高斯光束经过厚非线性介质后在自由空间传输的光强演变过程, 详细分析了介质表面缺陷对光强演化规律的影响.研究结果表明, 平顶阶数越大, 光束的自聚焦效果越差, 最大光强点越靠近介质后表面.入射光束的初始场强越强, 介质的厚度和非线性系数越大, 光束的自聚焦越强, 且极值点越靠近介质后表面.介质表面的缺陷使得光束在靠近介质后表面有较大的光强. 并且相位调制型缺陷比振幅调制型缺陷对光强的影响更大. 关键词： 平顶光束 自聚焦 缺陷 光强分布     

Surface waves in a vertically excited circular cylindrical container

The nonlinear free surface amplitude equation, which has been derived from the inviscid fluid by solving the potential equation of water waves with a singular perturbation theory in a vertically oscillating rigid circular cylinder, is investigated successively in the fourth-order Runge－Kutta approach with an equivalent time-step. Computational results include the evolution of the amplitude with time, the characteristics of phase plane determined by the real and imaginary parts of the amplitude, the single-mode selection rules of the surface waves in different forced frequencies, contours of free surface displacement and corresponding three-dimensional evolution of surface waves, etc. In addition, the comparison of the surface wave modes is made between theoretical calculations and experimental measurements, and the results are reasonable although there are some differences in the forced frequency.     

Stability of liquid film falling down a vertical non-uniformly heated wall

The main object in this paper is to study the stability of a viscous film flowing down a vertical non-uniformly heated wall under gravity. The wall temperature is assumed linearly distributed along the wall and the free surface is taken to be adiabatic. A long wave perturbation method is used to derive the nonlinear evolution equation for the falling film. Using the method of multiple scale, the nonlinear stability analysis is studied for travelling wave solution of the evolution equation. The complex Ginzburg-Landau equation is determined to discuss the bifurcation analysis of the evolution equation. The results indicate that the supercritical unstable region increases and the subcritical stable region decreases with the increase of Peclet number. It has been also shown that the spatial uniform solution corresponding to the sideband disturbance may be stable in the unstable region.     

Collisionless Boltzmann equations and integrable moment equations

A collisionless Boltzmann equation, describing long waves in a dense gas of particles interacting via short-range forces, is shown to be equivalent to the Benney equations, which describe long waves in a perfect two-dimensional fluid with a free surface. These equations also describe, in a random phase approximation, the evolution, on long space and time scales, of multiply periodic solutions of the nonlinear Schrödinger equation. The derivative nonlinear Schrödinger equation is likewise shown to be related to an integrable system of moment equations.     

Effect of surface tension on the mode selection of vertically excited surface waves in a circular cylindrical vessel

Singular perturbation theory of two-time-scale expansions was developed in inviscid fluids to investigate patternforming, structure of the single surface standing wave, and its evolution with time in a circular cylindrical vessel subject to a vertical oscillation. A nonlinear slowly varying complex amplitude equation, which involves a cubic nonlinear term,an external excitation and the influence of surface tension, was derived from the potential flow equation. Surface tensionwas introduced by the boundary condition of the free surface in an ideal and incompressible fluid. The results show that when forced frequency is low, the effect of surface tension on the mode selection of surface waves is not important.However, when the forced frequency is high, the surface tension cannot be neglected. This manifests that the function of surface tension is to cause the free surface to return to its equilibrium configuration. In addition, the effect of surface tension seems to make the theoretical results much closer to experimental results.     

Numerical simulation of the transient multiphase field during plasma deposition manufacturing composite materials

A solid/liquid/gas unified model has been developed to investigate the gradient composition formation during the plasma deposition manufacturing (PDM) composite materials process. In this model, an enthalpy porosity model was applied to deal with the melting and solidification of the deposited layer, and a level-set approach was introduced to track the evolution of the free surface of the molten pool and the deposited layer. Moreover, complicated physical phenomena occurring at the liquid/gas interface, including forced convection heat loss, heat emission and plasma heat source, have been incorporated into the governing equations by source terms. In this study, the numerical experiment of nickel base alloy powder deposited on the medium steel substrate by PDM technique was implemented based on the staggered grid and SIMPLEC algorithm. Concentration gradient distribution of the solute material at the composite material interface, fluid flow and temperature distribution in the molten pool and the deposited layer have been investigated in detail. Supported by the National Natural Science Foundation of China (Grant No. 50474053) and the High Technology Research and Development Program of China (Grant No. 2007AA04Z142)     

The effect of the nonparabolicity of the free carriers dispersion law on the propagation of surface polaritons in a semiconductor-metal structure

The results of a study on the influence of the nonparabolicity of the free carriers dispersion law on the propagation of surface polaritons (SPs) located near the interface between an n-type semiconductor and a metal arc reported. The semiconductor plasma is assumed to be warm and nonisothermal. The nonparabolicity of the electron dispersion law has two effects. The first one is associated with nonlinear self-interaction of the SPs. The nonlinear dispersion equation and the nonlinear Schrodinger equation for the amplitude of the SP envelope are obtained. The nonlinear evolution of the SP is studied on the base of the above mentioned equations. The second effect results in third harmonics generation. Analysis shows that these third harmonics may appear as a pure surface polariton, a pseudosurface polariton, or a superposition of a volume wave and a SP depending on the wave frequency, electron density and lattice dielectric constant.     

The Kuramoto-Sivashinsky equation for the defect-deformation instability of a surface-stressed nanolayer

It is demonstrated that the evolution of the surface relief of the stressed elastic nanolayer that interacts with mobile point defects can be described using a nonlinear equation similar to the Kuramoto-Sivashinsky equation. The solution to the equation describes the threshold (with respect to the defect concentration or the mechanical stress) transition to the periodic spatially bent state of the layer with the simultaneous generation of spatially periodic defect clusters at the extrema of the spontaneously emerging surface relief. In this case, the layer deformation corresponds to the displacements in the static bending Lamb wave. The self-organizing periodic cellular deformation-defect surface structure can serve as a universal seed for the subsequent growth of nanoparticles in the processes of laser, ion, or electrochemical etching and in molecular beam epitaxy of nanostructures. Original Text ? Astro, Ltd., 2009.     

A nonlinear evolution equation for Bénard-Marangoni convection with deformable boundary

For the case of small Biot number, i.e. small heat transfer, a nonlinear evolution equation is derived for the deformable upper surface in a liquid layer heated from below and open to the ambient air (Bénard-Marangoni convection). As a byproduct of our analysis, threshold values and other relevant findings are obtained for the onset of convection.     

Strong nonlinear rupture theory of thin free liquid films

A simplified governing equation with high-order effects is formulated after a procedure of evaluating the order of magnitude. Furthermore, the nonlinear evolution equations are derived by the Kármán-Polhausen integral method with a specified velocity profile. Particularly, the effects of surface tension, van der Waals potential, inertia and high-order viscous dissipation are taken into consideration in these equation. The numerical results reveal that the rupture time of free film is much shorter than that of a film on a flat plate. It is shown that because of a more complete high-order viscous dissipation effect discussed in the present study, the rupture process of present model is slower than is predicted by the high-order long wave theory.     

Generalized and Improved （G＇/G）-Expansion Method Combined with Jacobi Elliptic Equation

In this article, we propose an alternative approach of the generalized and improved （G＇/G）-expansion method and build some new exact traveling wave solutions of three nonlinear evolution equations, namely the Boiti- Leon-Pempinelle equation, the Pochhammer-Chree equations and the Painleve integrable Burgers equation with free parameters. When the free parameters receive particular values, solitary wave solutions are constructed from the traveling waves. We use the Jacob/elliptic equation as an auxiliary equation in place of the second order linear equation. It is established that the proposed algorithm offers a further influential mathematical tool for constructing exact solutions of nonlinear evolution equations.     

Dispersion-Modified Burgers Description for Nonlinear Surface Wave Excitations in Bénard-Marangoni Convection

The evolution of long nonlinear surface wave excitations in a Bénard-Marangoni shallow liquid layer is studied when the system is far from its instability threshold. For (M -M_c)/M_c = 0(1), thus for surface wave excitations with possible large amplitude, where M is the Marangoni number of the system and M_c is its critical value bifurcating to oscillatory convecting state from a motionless one, the surface wave displacement is found to obey a dispersion-modified Burgers equation whose kink-like solutions are also studied.     

Observation of solitary elastic surface pulses

The formation of solitary elastic surface pulses from laser-generated pulselike initial conditions is reported. The nonlinearity of the medium is compensated by both normal dispersion and anomalous dispersion, which were realized by coating isotropic fused silica by a metal and titanium nitride film, respectively. As an anisotropic material, silicon covered with an oxide layer was studied. The experimental results agree with numerical simulations carried out with a nonlocal evolution equation, which describes nonlinear propagation of surface acoustic waves in a dispersive medium.     

Instability and breakup of a thick layer of a viscous magnetic fluid in a tilted magnetic field

A linear partial differential equation describing the evolution of an initial disturbance of a flat free surface of a thin layer of a viscous magnetic fluid covering a horizontal plate in the presence of a uniform magnetic field, is derived within a system of ferrohydrodynamic and magnetostatic equations. The effect of magnetizing the plate on the stability of the flat free surface is investigated. An estimate is obtained for the minimum value of the tangential component of the magnetization vector of the fluid sufficient to radically alter the pattern of the final breakup of the continuous layer. Zh. Tekh. Fiz. 69, 14–22 (October 1999)     

Nanoscale pore formation dynamics during aluminum anodization

A theoretical analysis of nanoscale pore formation during anodization reveals its fundamental instability mechanism to be a field focusing phenomenon when perturbations on the minima of the two oxide interfaces are in phase. Lateral leakage of the layer potential at high wave number introduces a layer tension effect that balances the previous destabilizing effect to produce a long-wave instability and a selected pore separation that scales linearly with respect to voltage. At pH higher than 1.77, pores do not form due to a very thick barrier layer. A weakly nonlinear theory based on long-wave expansion of double free surface problem yields two coupled interface evolution equations that can be reduced to one without altering the dispersion relationship by assuming an equal and in-phase amplitude for the two interfaces. This interface evolution equation faithfully reproduces the initial pore ordering and their dynamics. A hodograph transformation technique is then used to determine the interior dimension of the well-developed pores in two dimensions. The ratio of pore diameter to pore separation is found to be a factor independent of voltage but varies with the pH of the electrolyte. Both the predicted pH range where pores are formed and the predicted pore dimensions are favorably compared to experimental data. (c) 2002 American Institute of Physics.     

一类非线性发展方程的复合型双孤子新解

通过下列步骤,构造了一类非线性发展方程的无穷序列复合型双孤子新解: 步骤一, 给出两种函数变换,把一类非线性发展方程化为二阶非线性常微分方程; 步骤二, 再通过函数变换, 二阶非线性常微分方程转化为一阶非线性常微分方程组,并获得了该方程组的首次积分; 步骤三, 利用首次积分与两种椭圆方程的新解与Bäcklund 变换, 构造了一类非线性发展方程的无穷序列复合型双孤子新解.