The theory of nonstationary thermophoresis of a solid spherical particle

The theory of nonstationary thermophoresis of a solid spherical particle in a viscous gaseous medium is presented. The theory is constructed on the solutions of fluid-dynamics and thermal problems, each of which is split into stationary and strictly nonstationary parts. The solution of the stationary parts of the problems gives the final formula for determining the stationary component of the thermophoretic velocity of this particle. To determine the nonstationary component of the thermophoretic velocity of the particle, the corresponding formula in the space of Laplace transforms is derived. The limiting value theorems from operational calculus are used for obtaining the dependence of the nonstationary component of the thermophoretic velocity of the spherical particle on the strictly nonstationary temperature gradient for large and small values of time. The factors determining the thermophoretic velocity of the particle under investigation are determined.     

Radial motion of a solid spherical body in a magnetic field

The object of the present paper is to investigate the radial motion of a solid spherical body, assumed to be homogeneous, isotropic and elastic, in presence of a magnetic field in the azimuthal direction. The body is assumed to be in a state of initial stress which is hydrostatic in nature. This theory of radial motion of a solid spherical body in a magnetic field has been utilised to find the small radial motion of a solid Earth assumed to be homogeneous isotropic elastic sphere in presence of a magnetic field in the azimuthal direction. Considering the effect of gravity and the initial stress produced by slow process of creep due to extra masses over the surface of the Earth, the fundamental equations of motion are derived which are non-linear in character and are solved. The times of a desired radial displacement are calculated in presence of a magnetic field only and in presence of the same magnetic field, initial stress and gravitational field, which are compared and exhibited numerically.     

Collective atomic motion in an optical lattice formed inside a high finesse cavity

We report on collective nonlinear dynamics in an optical lattice formed inside a high finesse ring cavity in a so far unexplored regime, where the light shift per photon times the number of trapped atoms exceeds the cavity resonance linewidth. We observe bistability and self-induced squeezing oscillations resulting from the retroaction of the atoms upon the optical potential wells. We can well understand most of our observations within a simplified model assuming adiabaticity of the atomic motion. Nonadiabatic aspects of the atomic motion are reproduced by solving the complete system of coupled nonlinear equations of motion.     

Intensity of optical field inside a weakly absorbing spherical particle irradiated by a femtosecond laser pulse

Using the Fourier technique in combination with the Mie theory, we study numerically the spatiotemporal evolution of the intensity of the internal optical field inside micron-sized weakly absorbing spherical particles upon diffraction by these particles of a femtosecond laser field. A number of specific features of the dynamics of the spatial intensity distribution of the femtosecond pulses inside the particles are found to depend on the pulse width, the shape of the laser beam, the size of the particles, and the geometry of their irradiation. It is shown that, under conditions of nonstationary diffraction, the internal optical field is usually excited in a resonance way, with the eigenfrequencies of one or several high-Q resonance modes of the particle falling into the central part of the original pulse spectrum. This causes a time delay of the light in the particle and a reduction of the absolute maximum in the time dependence of the internal field intensity as compared with a stationary regime. The greatest reduction of the peak occurs at exact resonance. In this case, the decrease in the peak intensity may reach several orders of magnitude. Irradiation of a particle by a narrow Gaussian beam of femtosecond duration directed toward the particle center enhances the internal field intensity as compared with the case of near-edge incidence.     

On the gravitational motion of a nonuniformly heated solid particle in a gaseous medium

The steady motion of a nonuniformly heated spherical aerosol particle through a viscous gaseous medium is theoretically studied in the Stokes approximation. It is assumed that the mean temperature of the particle surface may differ appreciably from the ambient temperature. The solution of gasdynamic equations yields an analytical expression for the drag of the medium and the gravitational fall velocity of the nonuniformly heated spherical solid particle with allowance for the temperature dependence of the density of the medium and molecular transfer coefficients (viscosity and thermal conductivity). Numerical estimates show that heating of the particle surface considerably influences the drag force and gravitational fall velocity.     

Resonant motion of a spherical pendulum

The weakly nonlinear, resonant response of a damped, spherical pendulum (length l, damping ratio δ, natural frequency ω₀) to the planar displacement εl cos ωt (ε ? 1) of its point of suspension is examined in a four-dimensional phase space in which the coordinates are slowly varying amplitudes of a sinusoidal motion. The loci of equilibrium points and the corresponding bifurcation points in this space are determined. The control parameters are $\text{α= 2δ/ε}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$ and . If α < 0.441 there is a finite interval of v within which no stable equilibrium points exist. As v decreases through the upper bound (a Hopf-bifurcation point) of this interval the motion in the phase space becomes periodic and then, following a period-doubling cascade, chaotic. There may be alternating sub-intervals of chaotic and periodic motion. The chaotic trajectories in the phase space appear to lie on fractal attractors.     

Motion of fluid and a solid core in a spherical cavity rotating in an external force field

The motion of a fluid in a rotating spherical cavity with a free light spherical body under the perturbing effect of an external force field perpendicular to the rotation axis is investigated experimentally. It is shown that the external field excites the lagging differential rotation of the core occupying the central position in the cavity under the action of a centrifugal force. The regularities of the averaged rotation of the body and the motion of fluid shaped as the Taylor-Proudman column are investigated. The sequential threshold manifestation of various instability types of the Taylor-Proudman column with an increase in the velocity of the differential body rotation is found. Initially a new type of instability manifests itself, and a two-dimensional system of vortexes elongated along the rotation axis is formed inside the column. Then the development of azimuthal two-dimensional waves at the column boundary is observed. It is shown that the Reynolds number calculated through the velocity of the differential body rotation determines the threshold transitions. A map of motion modes of a fluid in a spherical layer on a plane of dimensionless parameters is plotted.     

Parametric scattering of high-frequency elastic waves by a small spherical cavity oscillating under the action of a Rayleigh wave

Scattering of high-frequency transverse and longitudinal plane waves incident on a spherical cavity located at a small depth under the surface of a half-space is considered. The cavity oscillates as a whole in the field of a low-frequency Rayleigh surface wave, the oscillation vectors of the longitudinal, transverse, and surface waves being coplanar. The cavity radius is assumed to be small compared to the wavelengths of the sounding wave and the pumping surface wave. The scattered compression and shear waves at the combination frequencies ω±Ω are calculated in the dipole approximation. Expressions obtained describe the qualitative behavior of the combination-frequency signal levels produced at the outputs of horizontally and vertically oriented geophones moving over the free surface of the elastic half-space.     

Interaction of a spherical acoustic wave with an elastic spherical shell

The interaction of a spherical acoustic wave with an elastic spherical shell is treated analytically. The solution includes the coupling between the acoustic sound field and vibration of the shell with any degree of fluid loading. The formulation for the far-field acoustic pressure is derived in terms of natural spherical wave functions, the properties of the acoustic medium, and the material constants of the shell. The far acoustic field is computed for a thin aluminum shell and several sound source locations over a large range of ka, where k is the wavenumber, and a is the shell radius. It is shown that the acoustic pressure depends significantly on whether the shell is in air or is submerged in water, particularly when the sound source is very near the surface. In air, the sound field of the shell is nearly identical to that of a rigid sphere but, in water, the shell is more compliant, which results in a damped radiation field that is characterized by vibrational resonances throughout the range of frequencies considered. As the sound sources is moved further away from the surface, however, this resonance response decreases very rapidly, and the sound field corresponds more closely to that of the shell in air.     

Effect of internal heat evolution on the motion of a solid particle in a viscous fluid

The problem of the effect of internal heat evolution on the motion of a heated solid spherical particle in a viscous fluid is analytically solved in the Stokes approximation at small Reynolds and Peclet numbers. The temperature drop between the surface of the particle and the area away from it is assumed to be arbitrary. In solving hydrodynamic equations, the thermal conductivity of the particle is set to be a power function of temperature and the viscosity of the fluid, an exponential-power function of temperature. The observability of this effect is discussed.     

Electromagnetic vacuum fields in a spherical cavity

The vacuum fluctuations of the electromagnetic fields are calculated within a spherical cavity with perfectly conducting boundary.     

Soret motion of a charged spherical colloid

The thermophoretic motion of a charged spherical colloidal particle and its accompanying cloud of counterions and coions in a temperature gradient is studied theoretically. Using the Debye-Hückel approximation, the Soret drift velocity of a weakly charged colloid is calculated analytically. For highly charged colloids, the nonlinear system of electrokinetic equations is solved numerically, and the effects of high surface potential, dielectrophoresis, and convection are examined. Our results are in good agreement with some of the recent experiments on highly charged colloids without using adjustable parameters.     

Multipole polarizability of a graded spherical particle

We have studied the multipole polarizability of a graded spherical particle in a nonuniform electric field, in which the conductivity can vary radially inside the particle. The main objective of this work is to access the effects of multipole interactions at small interparticle separations, which can be important in non-dilute suspensions of functionally graded materials. The nonuniform electric field arises either from that applied on the particle or from the local field of all other particles. We developed a differential effective multipole moment approximation (DEMMA) to compute the multipole moment of a graded spherical particle in a nonuniform external field. Moreover, we compare the DEMMA results with the exact results of the power-law graded profile and the agreement is excellent. The extension to anisotropic DEMMA will be studied in an Appendix.     

振动不敏感球形光学参考腔研究

本文主要研究用于超稳激光研制的实验室内水平放置的球参考腔的振动敏感度问题.研究了支撑点高度和支撑面积对腔长变化的影响,并讨论了加速度大小的变化对腔长变化的影响.在支撑点完全固定的情况下,振动敏感度被降低到3.0×10-10/g以内.通过大量的数值模拟获得了球参考腔的最佳支撑位置.根据优化的结果提出了这种切割球参考腔放置平台的方案.考虑了加工误差、近水平放置和非对称支撑对参考腔振动敏感度的影响,定量地给出了这三种因素引起的腔长改变量,而且讨论了二阶效应对腔长变化的贡献.     

Broadband resonant cavity inside a two-dimensional sonic crystal

A rectangular cavity inside a two-dimensional sonic crystal was theoretically and experimentally characterized by examining its response to a cylindrical source emitting narrow-band filtered noise bursts with central frequencies ranging from 2 to 12 kHz. A broadband intensity resonance was observed for frequencies within the full band-gap region of the sonic crystal (5.5–6.5 kHz). Unlike ordinary resonances, this broadband resonance depends on the reflection properties of the sonic crystal forming the surrounding walls rather than on the geometry of the cavity.     

A spherical scatterer inside a circular hollow dielectric waveguide

A theoretical investigation is presented of scattering characteristics of a perfectly conducting spherical object placed inside a quasi-optical transmission line of the millimeter and submillimeter waves in the form of a circular hollow dielectric waveguide (HDW). From the analytical expressions obtained, backscattering and extinction cross sections of this object are derived via the excitation coefficient of the dominantHE ₁₁ mode of the circular HDW and the HDW geometrical parameters. The agreement of the results with the corresponding data for the scattering of a plane homogeneous wave is shown.     

Scattering of a light pulse by a spherical particle

In this work the scattering of a brief light pulse by a homogeneous spherical particle is studied within the framework of linear electrodynamics. Convenient formulas are obtained for the scattered fields of the pulse. As a concrete example the Rayleigh scattering of a light pulse is considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 80–84, August, 1972.