Optohydrodynamics of soft fluid interfaces: Optical and viscous nonlinear effects

Recent experimental developments showed that the use of the radiation pressure, induced by a continuous laser wave, to control fluid-fluid interface deformations at the microscale, represents a very promising alternative to electric or magnetic actuation. In this article, we solve numerically the dynamics and steady state of the fluid interface under the effects of buoyancy, capillarity, optical radiation pressure and viscous stress. A precise quantitative validation is shown by comparison with experimental data. New results due to the nonlinear dependence of the optical pressure on the angle of incidence are presented, showing different morphologies of the deformed interface going from needle-like to finger-like shapes, depending on the refractive index contrast. In the transient regime, we show that the viscosity ratio influences the time taken for the deformation to reach steady state.     

Nano-indentation Test and Numerical Evaluation of Cu–Cr Interface Structure in Micro/Nano Manufacturing

Interface design is an important topic in micro/nano electronic manufacturing. Interfaces of dissimilar materials in micro/nano electronic manufacturing are prone to crack initiations, leading to delaminations. The objective of this paper is to provide a systematic investigation to design or evaluate a bilayer film structure between Cu and Cr in micro/nano electronic manufacturing. In this paper, the Cu/Cr bilayer film prototype was deposited on the quartz glass by using RF magnetron sputtering. The elastic modulus and the hardness of the bilayer film prototype can be tested by using a nano-indenter. The test results show that the elastic modulus and the hardness of the bilayer film prototype are different at the difference maximum depth, h _max. The elastic modulus and the hardness of the Cu/Cr interface are influenced by the nanometer indentation size effect and each single film. The elastic modulus of the bilayer film shows nonlinear characteristics which include increase at first and decrease on second stage. The change trend of the hardness also shows nonlinear characteristics which include a fast steady decline at first and a slow nonlinear decline on second stage. These results show that there are scale domino effects in micro/nano electronic manufacturing. Based on the test results, the mechanical properties of the interface are not the simple average of each composition film. There is very great difference in the interface. In the meantime, the finite element method is used to simulate the plastic property of the interface. The comparison between the simulation and the test shows that the modeling method is a valid investigating method to analyze mechanical properties for nano-interface structure. It builds a basis for a progressive study of the mechanical properties of a Cu/Cr interface structure.     

On Methods of Finding Bäcklund Transformations in Systems with More than Two Independent Variables

Abstract

Bäcklund transformations, which are relations among solutions of partial differential equations–usually nonlinear–have been found and applied mainly for systems with two independent variables. A few are known for equations like the Kadomtsev-Petviashvili equation [1], which has three independent variables, but they are rare. Wahlquist and Estabrook [2] discovered a systematic method for searching for Bäcklund transformations, using an auxiliary linear system called a prolongation structure. The integrability conditions for the prolongation structure are to be the original differential equation system, most of which systems have just two independent variables. This paper discusses how the Wahlquist-Estabrook method might be applied to systems with larger numbers of variables, with the Kadomtsev-Petviashvili equation as an example. The Zakharov-Shabat method is also discussed. Applications to other equations, such as the Davey-Stewartson and Einstein equation systems, are presented.     

An iterative method to solve a regularized model for strongly nonlinear long internal waves

We present a simple iterative scheme to solve numerically a regularized internal wave model describing the large amplitude motion of the interface between two layers of different densities. Compared with the original strongly nonlinear internal wave model of Miyata [10] and Choi and Camassa [2], the regularized model adopted here suppresses shear instability associated with a velocity jump across the interface, but the coupling between the upper and lower layers is more complicated so that an additional system of coupled linear equations must be solved at every time step after a set of nonlinear evolution equations are integrated in time. Therefore, an efficient numerical scheme is desirable. In our iterative scheme, the linear system is decoupled and simple linear operators with constant coefficients are required to be inverted. Through linear analysis, it is shown that the scheme converges fast with an optimum choice of iteration parameters. After demonstrating its effectiveness for a model problem, the iterative scheme is applied to solve the regularized internal wave model using a pseudo-spectral method for the propagation of a single internal solitary wave and the head-on collision between two solitary waves of different wave amplitudes.     

An algebraic approach to soliton solutions

Within the scope of a new algebraic method for the construction of soliton solutions of nonlinear equations [10–13], concrete examples of the system are considered which are connected with the algebra sl(2, R).     

Nd0.7Sr0.3MnO3陶瓷EPIR效应的起源

用两线法和四线法对球磨合成后热处理的Nd_0.7Sr_0.3MnO₃陶瓷样品的电脉冲诱导电阻转变(EPIR)效应进行了测试.结果表明:在Nd_0.7Sr_0.3MnO₃块体中用四线法测得的I-V曲线为非线性,说明Nd_0.7Sr_0.3MnO₃样品在晶(相)界处存在空间电荷层和界面电阻.但用四线法测量EPIR效应时,没有EPIR效应发生,表明晶(相)界处的空间电荷层和肖特基势垒不能产生EPIR效应;对同一样品采用直流两线法测量,其I-V曲线也为非线性,但却发生明显的EPIR效应.说明在Nd_0.7Sr_0.3MnO₃陶瓷中发现的EPIR效应起源于样品与电极之间的接触界面,块体内的晶(相)界处虽能产生相似I-V特性,但却不能发生EPIR效应. 关键词： 电脉冲诱导电阻转变(EPIR) 亚锰酸盐 空间电荷层 I-V非线性')" href="#">I-V非线性     

A fully second order implicit/explicit time integration technique for hydrodynamics plus nonlinear heat conduction problems

We present a fully second order implicit/explicit time integration technique for solving hydrodynamics coupled with nonlinear heat conduction problems. The idea is to hybridize an implicit and an explicit discretization in such a way to achieve second order time convergent calculations. In this scope, the hydrodynamics equations are discretized explicitly making use of the capability of well-understood explicit schemes. On the other hand, the nonlinear heat conduction is solved implicitly. Such methods are often referred to as IMEX methods [2], [1], [3]. The Jacobian-Free Newton Krylov (JFNK) method (e.g. [10], [9]) is applied to the problem in such a way as to render a nonlinearly iterated IMEX method. We solve three test problems in order to validate the numerical order of the scheme. For each test, we established second order time convergence. We support these numerical results with a modified equation analysis (MEA) [21], [20]. The set of equations studied here constitute a base model for radiation hydrodynamics.     

Nanoscale deformation of a liquid surface

We study the interaction between a solid particle and a liquid interface. A semianalytical solution of the nonlinear equation that describes the interface deformation points out the existence of a bifurcation behavior for the apex deformation as a function of the distance. We show that the apex curvature obeys a simple power-law dependency on the deformation. Relationships between physical parameters disclose the threshold distance at which the particle can approach the liquid before capillarity provokes a "jump to contact." A prediction of the interface original position before deformation takes place, as well as the attraction force measured by an approaching probe, are produced. The results of our analysis agree with the force curves obtained from atomic force microscopy experiments over a liquid puddle.     

Series solutions of stagnation slip flow and heat transfer by the homotopy analysis method

An analytical approximation for the similarity solutions of the two- and three-dimensional stagnation slip flow and heat transfer is obtained by using the homotopy analysis method. This method is a series expansion method, but it is different from the perturbation technique, because it is independent of small physical parameters at all. Instead, it is based on a continuous mapping in topology so that it is applicable for not only weakly but also strongly nonlinear flow phenomena. Convergent [m,m] homotopy Padé approximants are obtained and compared with the numerical results and the asymptotic approximations. It is found that the homotopy Padé approximants agree well with the numerical results. The effects of the slip length ℓ and the thermal slip constant β on the heat transfer characteristics are investigated and discussed. Supported by the National Natural Science Foundation of China (Grant No. 10872129)     

A gradient stable scheme for a phase field model for the moving contact line problem

In this paper, an efficient numerical scheme is designed for a phase field model for the moving contact line problem, which consists of a coupled system of the Cahn–Hilliard and Navier–Stokes equations with the generalized Navier boundary condition [1], [2], [4]. The nonlinear version of the scheme is semi-implicit in time and is based on a convex splitting of the Cahn–Hilliard free energy (including the boundary energy) together with a projection method for the Navier–Stokes equations. We show, under certain conditions, the scheme has the total energy decaying property and is unconditionally stable. The linearized scheme is easy to implement and introduces only mild CFL time constraint. Numerical tests are carried out to verify the accuracy and stability of the scheme. The behavior of the solution near the contact line is examined. It is verified that, when the interface intersects with the boundary, the consistent splitting scheme [21], [22] for the Navier Stokes equations has the better accuracy for pressure.     

Potential-based reduced Newton algorithm for nonlinear multiphase flow in porous media

We present a phase-based potential ordering that is an extension of the Cascade ordering introduced by Appleyard and Cheshire [John R. Appleyard, Ian M. Cheshire, The cascade method for accelerated convergence in implicit simulators, in: European Petroleum Conference, 1982, pp. 113–122]. The proposed ordering is valid for both two-phase and three-phase flow, and it can handle countercurrent flow due to gravity and/or capillarity. We show how this ordering can be used to reduce the nonlinear algebraic system that arises from the fully-implicit method (FIM) into one with only pressure dependence. The potential-based reduced Newton algorithm is then obtained by applying Newton’s method to this reduced-order system. Numerical evidence shows that our potential-based reduced Newton solver is able to converge for time steps that are much larger than what the standard Newton’s method can handle. In addition, whenever standard Newton converges, so does the reduced Newton algorithm, and the number of global nonlinear iterations required for convergence is significantly reduced compared with the standard Newton’s method.     

Computing Ground State Solution of Bose-Einstein Condensates Trapped in One-Dimensional Harmonic Potential

For Bose-Einstein condensates trapped in a harmonic potential well, we present numerical results from solving the tlme-dependent nonlinear Schrolinger equation based on the Crank-Nicolson method. With this method we are able to find the ground state wave function and energy by evolving the trial initial wave function in real and imaginary time spaces, respectively. In real time space, the results are in agreement with [Phys. Rev. A 51 （1995） 4704], but the trial wave function is restricted in a very small range. On the contrary, in imaginary time space, the trial wave function can be chosen widely, moreover, the results are stable.     

Some features of one-soliton solutions of the Korteweg - de Vries equation

Using the method of inverse scattering problem [1, 2], we study solutions of the Korteweg - de Vries equation under initial conditions in the form of two nonsoliton pulses with not very large amplitudes. It is shown that if the distance between these pulses is not large, then they evolve to one soliton and an oscillating nonlinear tail for t → ∞. As the distance between the pulses or the pulse amplitudes increase, two solitons and an oscillating nonlinear tail are formed. Similar behavior is observed for solutions of the nonlinear Schrödinger equation. The only difference is that three, but not two, solitons are formed if the distance between two initial inphase pulses increases. The results of analytical consideration are illustrated by the numerical solution of the Korteweg - de Vries equation.     

Converse KAM: Theory and practice

We unify, extend, reinterpret and apply criteria of Birkhoff [1], Herman [9], Mather [2, 3], Aubry et al. [4, 5], and Newman and Percival [6] for the nonexistence of invariant circles for area preserving twist maps. The criteria enable one to establish regions of phase space through which no rotational invariant circles pass. For families of maps the same can be done for regions of the combined space of phase points and parameters. The criteria can be implemented rigorously on a computer, and give a practical method of proving quite strong results. As an example, we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value |k|63/64.     

A Method for q-Calculus

Abstract

We present a notation for q-calculus, which leads to a new method for computations and classifications of q-special functions. With this notation many formulas of q-calculus become very natural, and the q-analogues of many orthogonal polynomials and functions assume a very pleasant form reminding directly of their classical counterparts.

The first main topic of the method is the tilde operator, which is an involution operating on the parameters in a q-hypergeometric series. The second topic is the q-addition, which consists of the Ward–AlSalam q-addition invented by Ward 1936 [102, p. 256] and Al-Salam 1959 [5, p. 240], and the Hahn q-addition.

In contrast to the the Ward–AlSalam q-addition, the Hahn q-addition, compare [57, p. 362] is neither commutative nor associative, but on the other hand, it can be written as a finite product.

We will use the generating function technique by Rainville [76] to prove recurrences for q-Laguerre polynomials, which are q-analogues of results in [76]. We will also find q-analogues of Carlitz’ [26] operator expression for Laguerre polynomials. The notation for Cigler’s [37] operational calculus will be used when needed. As an application, q-analogues of bilinear generating formulas for Laguerre polynomials of Chatterjea [33, p. 57], [32, p. 88] will be found.     

Nonlinear optimized expansions and the Gross-Neuveu model

An alternative approach to the investigations of the phase structure of the Gross-Neveu model by the optimized expansion method is proposed. Contrary to [6], in order to conserve the Lorentz invariance, the auxiliary scalar fields technique is used. As a result, the interpolation Lagrangian becomes nonlinear in the artificial parameter . The effective potential is constructed by the canonical optimization procedure. The predictions for the phase structure of the model are as in the 1/N expansion method.     

Accurate numerical solutions to stationary free surface problems from capillarity

The paper considers two static problems from capillarity. The first one consists in the determination of the surface of a liquid in a capillary tube and the second in the computation of the shape of a sessile or pendent drop of liquid of a given volume, both configurations being considered in the gravitational field. From a mathematical point of view, both problems are nonlinear two-point boundary value problems which include in their formulations additional geometrical unknowns, i.e., the so-called free boundary value problems. Computational, they are laborious problems owing to the fact that they involve nonnormal Jacobian matrices. The resulting numerical difficulties are considerably reduced by treating these problems by a collocation method in conjunction with the continuation with respect to the parameters. The method is implemented by the MATLAB code bvp4c. The numerical evaluations of the free surfaces are in reasonable accordance with asymptotic estimations. The text was submitted by the author in English.     

Organic chemistry at interfaces studied by optical second-harmonic and IR-vis sum-frequency generation

Three selected systems are presented which demonstrate the versatility of second-order nonlinear techniques for interface analysis and thin film monitoring. With second-harmonic generation processes at buried interfaces can be monitored in real time as is shown by thiol adsorption on polycrystalline gold from solution. The time dependent second-harmonic signal shows a strong wavelength dependence due to resonances. In situ monitoring of thin-film formation by second-harmonic generation can differentiate between the reaction at the interface and the subsequent film growth. The chemical vapor deposition of polyamic acid is presented as example. Hydrogen adsorption on diamond C(111) is analysed by sum-frequency generation. It is demonstrated that sum-frequency generation has the sensitivity and specificity to trace chemical reactions at surfaces and to infer on the orientation of chemical bonds.     

界面动力学对过冷熔体中球晶生长界面形态的影响

考虑了在非平衡凝固条件下球晶生长过程中界面动力学系数随界面温度的变化,利用渐近分析方法求出了在过冷熔体中球晶生长温度场和界面的近似解析解,研究了非线性界面动力学过冷对于过冷熔体中球晶界面形态和生长速度的影响.研究表明,界面动力学系数越大,球晶的生长速度越快; 反之,表明界面动力学系数越小,球晶的生长速度越慢.与忽略界面动力学的情形比较,在球晶生长过程中依赖于界面温度变化的界面动力学显著地减缓了晶体生长的速度. 关键词： 球晶 界面形态 渐近分析