On nonlinear corrections to the frequencies of oscillations of a charged drop in an external incompressible medium

Analytical expressions are derived for the shape generatrix of an ideally conducting drop immersed in an incompressible dielectric medium as well as for nonlinear corrections to the frequencies of the oscillations of the drop. The solutions are obtained in an approximation of the third order of smallness with respect to the amplitude of the initial deformation of the equilibrium spherical shape of the drop. It is shown that the presence of the ambient liquid results in a reduction of the absolute magnitudes of corrections both to the oscillation frequencies and the self-charge critical for the development of instability of the drop.     

On nonlinear vibrations of a charged drop in the third-order approximation in the amplitude of initial single-mode excitation

Nonlinear axisymmetric motions of the free surface of a charged drop of an ideal liquid under the single-mode initial deformation of its equilibrium shape is investigated in the third-order approximation in the initial perturbation amplitude. An analytical expression for the drop shape generatrix is derived. Nonlinear corrections to the vibration frequencies for the initial perturbation of an arbitrary mode are found for the first time. The effect of vibration nonlinearity on the instability of the drop against its self-charge is studied.     

Nonlinear vibrations of a charged drop in an electrostatic suspension

The problem of nonlinear vibrations of a charged drop of an ideal incompressible conducting fluid in an electrostatic suspension is analytically solved in an approximation quadratic in two small parameters: vibration amplitude and equilibrium deformation of the shape of the drop in an electrostatic field. To solve the problem analytically, the desired quantities are expanded in semiinteger powers of the small parameters. It is shown that the charge of the drop and the gravitational field influence the shape of the drop, nonlinear corrections to the vibration frequencies, and critical conditions for instability of the drop against the surface charge. At near-critical values of the charge, the shape of the nonlinearly vibrating drop falls far short of being a sphere or a spheroid, which should be taken into account in treating experimental data.     

Nonlinear vibrations of a charged drop in a third-order approximation in the amplitude of multimode initial deformation

A solution to the problem of nonlinear surface vibration of a charged ideal liquid drop is found in a third-order approximation in initial multimode deformation of the equilibrium spherical shape by the method of many scales. It is shown that the spectrum of modes that are responsible for the shape of the drop at an arbitrary time instant depends considerably on the spectrum of modes governing the initial deformation of the drop. The latter spectrum also has an effect on nonlinear corrections to the vibration frequencies and, consequently, on a nonlinear correction to the critical Rayleigh parameter, which specifies the stability of the drop against self-charge.     

High order quadratures for the evaluation of interfacial velocities in axi-symmetric Stokes flows

We propose new high order accurate methods to compute the evolution of axi-symmetric interfacial Stokes flow. The velocity at a point on the interface is given by an integral over the surface. Quadrature rules to evaluate these integrals are developed using asymptotic expansions of the integrands, both locally about the point of evaluation, and about the poles, where the interface crosses the axis of symmetry. The local expansions yield methods that converge to the chosen order pointwise, for fixed evaluation point. The pole expansions yield corrections that remove maximal errors of low order, introduced by singular behaviour of the integrands as the evaluation point approaches the poles. An interesting example of roundoff error amplification due to cancellation is also addressed. The result is a uniformly accurate fifth order method. Second order, pointwise fifth order, and uniform fifth order methods are applied to compute three sample flows, each of which presents a different computational difficulty: an initially bar-belled drop that pinches in finite time, a drop in a strain flow that approaches a steady state, and a continuously extending drop. In each case, the fifth order methods significantly improve the ability to resolve the flow. The examples furthermore give insight into the effect of the corrections needed for uniformity. We determine conditions under which the pointwise method is sufficient to obtain resolved results, and others under which the corrections significantly improve the results.     

Deformation and nonlinear corrections to critical conditions of charged drop instability in an electrostatic suspension

Nonlinear asymptotic analysis of a charged drop placed in electrostatic and gravitational fields reveals a correction to the oscillation frequency and, accordingly, to the critical Rayleigh parameter. The analysis uses approximations quadratic in oscillation amplitude and linear in dimensionless equilibrium deformation of the drop. The correction is found to be proportional to the product of the oscillation amplitude and deformation. It is natural to name this correction deformational. In computations of the third order of smallness in oscillation dimensionless amplitude, a correction to the frequency and Rayleigh parameter appears, which is due to a nonlinear interaction between oscillation modes. This correction is larger than the deformational one in magnitude. Deformational corrections can be eliminated by experimenting under no-gravity conditions, but corrections due to the nonlinearity of hydrodynamic equations cannot be eliminated in this way. It is these corrections that are responsible for a critical Rayleigh parameter measurement inaccuracy.     

The Semiclassical Study of Nuclear High Spin States

Nuclear high spin states are studied in the extended Thomas-Fermi approximation.Explicit formula for rotational energy and various corrections are given.Concrete calculations are performed for three typical nuclei:¹¹⁶Sn,¹⁵⁸Yb and ²⁴⁰Pu.Qualitatively,the results are similar to that of liquid drop model.Quantum corrections to rotational energy (besides shell corrections) cannot be neglected in a quantitative study.     

Effect of the initial deformation of a charged drop on nonlinear corrections to critical conditions for instability

A nonlinear (proportional to the vibration amplitude squared) decrease in the critical (in terms of instability) charge of a vibrating drop is found to be limited, as follows from third-order asymptotic calculations. This effect occurs when the spectrum of modes specifying the initial deformation of the drop contains, along with the fundamental mode, higher modes. The influence of the environment density on nonlinear corrections to the critical conditions for instability is analyzed.     

Modelling of cascaded Mach-Zehnder add/drop multiplexer with film mode matching method

A semi-analytical method for modelling cascaded Mach-Zehnder add/drop multiplexers with a mode solver based on the film mode matching method is presented. Examples for an inverted rib waveguide structure are given. They focus on the polarization dependence, etching tolerances and on the coupling which takes place at the circular access waveguides of each directional coupler. This revised version was published online in November 2006 with corrections to the Cover Date.     

Semi-empirical interpretation of nuclear fission based on deformed-shell effects

The main features of the fragment mass and charge distribution in the fission of nuclides from Po to Fm are accounted for using a correlation of minimum potential energy with fragment yield. The potential energy is calculated from a liquid drop surface with deformed-fragment shell and pairing corrections at a particular deformation for the nascent fission fragments.     

基于光滑粒子动力学方法的液滴冲击固壁面问题数值模拟

采用改进的光滑粒子动力学(SPH)方法对液滴冲击固壁面问题进行了数值模拟. 为了提高传统SPH方法的计算精度和数值稳定性, 在传统的SPH方法的基础上对粒子方法中的密度和核梯度进行了修正, 采用了考虑黎曼解法的SPH流体控制方程, 构造了一种新型的粒子间相互作用力(IIF)模型来模拟表面张力的影响. 应用改进的SPH方法对液滴冲击固壁面问题进行了数值模拟. 计算结果表明：新型的IIF 模型能够较好地模拟表面张力的影响, 改进的SPH方法能够精细地描述液滴与固壁面相互作用过程中液滴的内部压力场演变和自由面形态变化, 液滴的铺展因子随初始韦伯数的增大而增大, 数值模拟结果与实验得到的结果基本一致. 关键词： 液滴 固壁面 光滑粒子动力学 表面张力     

Macroscopic-microscopic theory of semi-spheroidal atomic cluster

The macroscopic-microscopic method is adapted to atomic clusters deposited on a surface. Analytical relationships for the deformation-dependent liquid drop model (LDM) energies of oblate and prolate semi-spheroidal atomic clusters have been obtained. A superdeformed prolate semi-spheroid is the most stable semi-spheroidal shape within LDM. It is also the shape with maximum degeneracy of quantum states of the semi-spheroidal harmonic oscillator used to compute the shell and pairing corrections. The microscopic corrections as well as total deformation energy show parabolic valley and ridges of the potential energy surfaces in the plane (deformation, number of atoms). The ground state and isomeric state deformation of clusters of various sizes depends on the interplay between the minima of LDM and shell correction energies.     

Finite size effects for the dilute coupling Derrida model

There is a certain correspondence between the problems of information theory and spin-glass physics. The Derrida model occupies an important place here, as it is optimal in a certain sense. We have considered finite size corrections in the paramagnetic, spin-glass and ferromagnetic phases. At T = 0 the finite size corrections in the ferromagnetic phase have an important information theoretic meaning. We found singular corrections in the paramagnetic region and finite size corrections in the spin-glass region, which are responsible for the zero-modes. The connection between information theory and spin-glass physics and the physical meaning of finite size corrections are discussed in detail.     

NATURAL-CONVECTION AND DROP MORPHOLOGY IN LIQUID-LIQUID DIRECT CONTACT

An optical Fourier transform system is applied to visualize natural-convectire flow patterns inside an evaporating liquid drop floating on an immiscible liquid. Strong cellular convective patterns prevail inside the drop at room temperature, which play an important role in the evaporation of the drop. A grating-shearing interferometry with a 4t system is developed for a real-time determination of a change in the drop morphology during the process of evaporation. Various types of drop geometry, which are derived from the moire fringes, are identified. An unstable, interfacial evaporation mode is disclosed and the mechanism for the augmentation of drop evaporation is determined. The study has potential applications in the process industry and in environmental control.     

On the difficulty of computing higher-twist corrections

We discuss the evaluation of power corrections to hard scattering and decay processes for which an operator product expansion is applicable. The Wilson coefficient of the leading-twist operator is the difference of two perturbative series, each of which has a renirmalon ambiguity of the same order as the power corrections themselves, but which cancel in the difference. We stress the necessity of calculating this coefficient function to sufficiently high orders in perturbation theory so as to make the uncertainty of the same order of or smaller than the relevant power corrections. We investigate in some simple examples whether this can be achieved. Our conclusion is that in most of the theoretical calculations which include power corrections, the uncertainties are at least comparable to the power corrections themselves, and that it will be a very difficult task to improve the situation.     

QED correction to asymmetry for polarized <Emphasis Type="Italic">ep</Emphasis> scattering from the method of electron structure functions

The electron structure function method is applied to calculate model-independent QED radiative corrections to the asymmetry of electron-proton scattering. Representations for both spin-independent and spin-dependent parts of the cross section are derived. Master formulas include the leading corrections in all orders and the main contribution of the second order and provide accuracy of the QED corrections at the level of one per mill. Numerical calculations illustrate our analytic results for both elastic and deep inelastic events.     

Long-range Coulomb forces in DIS: missed radiative corrections?

The Born approximation, one photon exchange, used for DIS (deep inelastic scattering) is subject to virtual radiative corrections which are related to the long-range Coulomb forces. They may be sizeable for heavy nuclei since Zα is not a small parameter. So far, these corrections are known only for two processes, elastic scattering and bremsstrahlung on the Coulomb field of a point-like target. While the former amplitude acquires only a phase, in the latter case also the cross-section is modified. Although the problem of Coulomb corrections for DIS on nuclei is extremely difficult, it should be challenged rather than “swept under the carpet”. The importance of these radiative corrections is questioned in the present paper. We show that, in the simplest case of a constant hadronic current, the Coulomb corrections provide a phase to the Born amplitude, therefore the cross-section remains the same. Inclusion of more realistic hadronic dynamics changes this conclusion. The example of coherent production of vector mesons off nuclei reveals large effects. So far a little progress has been made deriving lepton wave functions in the Coulomb field of an extended target. Employing available results based on the first-order approximation in Zα, we conclude that the Coulomb corrections are still important for heavy nuclei. We also consider an alternative approach for extended nuclear targets, the eikonal approximation, which we demonstrate to reproduce the known exact results for Coulomb corrections. Calculating electroproduction of vector mesons, we again arrive at a large deviation from the Born approximation. We conclude that one should accept with caution the experimental results for nuclear effects in DIS based on analyses done in the Born approximation. Received: 16 May 2001 / Accepted: 4 July 2001     

Strong QCD corrections of the wave function in S-wave quarkonium decays

The first non-leading QCD corrections to the coulombic wave function of an S-state in quarkonium have been computed. The result is presented for general principal quantum number n of the decaying state. The corrections increase with n; in most cases, they dominate by far the decay corrections in the simplest example, the leptonic decay of quarkonium.     

Nonlinear analysis of the equilibrium shape of a charged drop in the tornado funnel wall

From the pressure balance condition on the free surface of a conducting liquid charged drop, an expression is derived for the equilibrium shape of the drop placed in the field of centrifugal forces acting in the tornado wall. The analysis is carried out in an approximation quadratic in small parameter (the ratio of the deformation amplitude to the radius of the initially spherical drop). In the linear approximation, the drop is a spheroid oblate in the direction normal to the tornado axis. The eccentricity of the spheroid squared is proportional to the angular velocity squared and the radius of the drop cubed. In the quadratic approximation, the equilibrium shape of the drop is other than spheroidal.