Rayleigh Regime Research Articles

Experimental study on splash phenomena of liquid jet impinging on a vertical wall

In this study, the splash phenomena of a plain orifice jet impinging on a vertical wall were examined. Experiments were carried out by changing the jet speed (i.e., inlet pressure) and the distance of a vertical wall with rounded and sharp-edged nozzles of different diameters, and the splashed fraction was measured by weighing the mass drained on the vertical wall. In addition, the breakup of the liquid jet and the flow patterns resulting from the jet impinging on the vertical wall were visualized.By increasing the velocity of a liquid jet emerging from a rounded nozzle, the Rayleigh, first wind-induced, and second wind-induced regimes could be observed in order. For the cavitating jet, the jet velocity became significantly larger than the volume flow rate divided by the nozzle area. Nevertheless, the trends of jet breakup as well as the splashed fraction results between the rounded and sharp-edged nozzles were similar for the same jet velocity. The splashed fraction reached its maximum value when small droplets resulted from the secondary breakup of large fragments started impinging on the vertical wall. In addition, the jet velocity at which the maximum splashed fraction attained decreased with increasing distance of the vertical wall from the nozzle exit. These experimental results were explained qualitatively in terms of the conditions of the liquid jet, breakup regime, drop impact, and splashing phenomena.

Gradient, scattering and other kinds of longitudinal optical forces exerted by off-axis Bessel beams in the Rayleigh regime in the framework of generalized Lorenz-Mie theory

After Arthur Ashkin’s pioneering work in optical levitation and manipulation, the study of optical forces exerted by laser beams on particles has become an active field of research. The present paper is a contribution to this issue. The interest of Bessel beams is that their intensity does not have any longitudinal gradient along the direction of propagation leading to a trivial separation between gradient and scattering forces. Beside the classical gradient and scattering forces, we shall however exhibit a new kind of optical forces associated with the existence of non zero axicon angles.

Polarized emission by aligned grains in the Mie regime: Application to protoplanetary disks observed by ALMA

Context. The azimuthal polarization patterns observed in some protoplanetary disks by the Atacama Large Millimetre Array (ALMA) at millimeter wavelengths have raised doubts about whether they are truly produced by dust grains that are aligned with the magnetic field lines. These conclusions were based on the calculations of dust polarized emission in the Rayleigh regime, that is, for grain sizes that are much smaller than the wavelength. However, the grain size in such disks is typically estimated to be in the range of 0.1−1 mm from independent observations. Aims. We study the dust polarization properties of aligned grains in emission in the Mie regime, that is, when the mean grain size approaches the wavelength. Methods. By using the T-MATRIX and DustEM codes, we computed the spectral dependence of the polarization fraction in emission for grains in perfect spinning alignment for various grain size distributions. We restricted our study to weakly-elongated oblate and prolate grains of astrosilicate composition that have a mean size ranging from 10 μm to 1 mm. Results. In the submillimeter and millimeter wavelength range, the polarization by B-field aligned grains becomes negative for grains larger than ∼250 μm, meaning that the polarization vector becomes parallel to the B-field. The transition from the positive to the negative polarization occurs at a wavelength of λ ∼ 1 mm. The regime of negative polarization does not exist for grains that are smaller than ∼100 μm. Conclusions. When using realistic grain size distributions for disks with grains up to the submillimeter sizes, the polarization direction of thermal emission by aligned grains is shown to be parallel to the direction of the magnetic field over a significant fraction of the wavelengths typically used to observe young protoplanetary disks. This property may explain the peculiar azimuthal orientation of the polarization vectors in some of the disks observed with ALMA and attest to the conserved ability of dust polarized emission to trace the magnetic field in disks.

Analysis of the Transverse Force in the Rayleigh and Mie Approximations for a Capture Beam TEM00 and TEM*01

Optical tweezers use a highly-focused laser beam to capture and manipulate micro- and nanometric objects. These have been demonstrated to be a promising devices for state-of-the-art research in several fields, such as microbiology and biophysics. The prediction of the optical forces that are present in this phenomenon is a current problem in continuous evolution. Additionally, the recent use of vortex beams with exotic properties as the orbital angular momentum, with advantages as the rotational manipulation of the captured microparticles and reduction of the optical damage in biological samples (Opticution), makes the problem even more complex. Mathematical expressions in the Rayleigh and Mie regimes for the radiation force on a dielectric sphere captured by TEM00 and TEM*01 mode beams are presented. Theoretical results are then compared with experimental measurements obtained with a direct force measurement device based on light-momentum detection.

Validity of cylindrical approximation for spherical birefringent microparticles in rotational optical tweezers

Rotational manipulation of microscopic birefringent particles has conventionally been done by manoeuvring the polarization of the trapping light in optical tweezers. The torque on the particle is a sum of contributions from the linear polarization and the circular polarization, while assuming that the difference in optical path lengths between the extraordinary and the ordinary components of polarization depends upon the wavelength of light, the thickness of the particle and the birefringence. Generally, the thickness of spherical microparticles is assumed to be the diameter which renders the particle appear cylindrical. We test this hypothesis for sizes relevant towards optical tweezers manipulation. We find that for a range of particles from the Rayleigh regime to the early Mie regime, the approximation holds good.

Nodularity Control of Spheroid Ductile Cast Iron by Rayleigh Backscattering

Spheroid Ductile Cast Iron finds increasingly industrial use as a high strength material of high fatigue resistance. The material properties can be designed by density, shape and size of the graphite spheres—the nodularity—and the matrix microstructure. Until today, the near surface nodularity cannot be controlled nondestructively to take full advantage of the material quality especially at high loaded surface areas. A viable nondestructive approach we propose is based on ultrasonic backscattering preferably in the Rayleigh regime. First experiments with model samples proved significant contrast to distinguish materials with different graphite morphology and matrices. The claim of our tests is a fast and easy industrial control of nodularity in reference to a validated sample. The results encourage us for the development of a prototype equipment with optimized probe systems and inspection parameters. Quantified acceptance criteria still require systematic tests for optimized parameter settings and probe configurations on more representative samples provided by industry.

Effects of surface tension and viscosity on liquid jet breakup

The breakup process of liquid jet was explored through visualization using a high-speed camera to develop a prediction model for the impact frequency (the number of droplet passing per unit time). The experiments were conducted for the three jet regimes of Rayleigh, first wind-induced and second wind-induced. The five liquids (water, two ethanol aqueous solutions, and two glycerin aqueous solutions) were used as the test liquids to explore the effects of surface tension and viscosity. Since the droplets were produced due to jet breakup, the impact frequency was zero just downstream of the nozzle and increased asymptotically with an increase in the distance from the nozzle. Thus, auxiliary correlations were developed for the minimum breakup length where the jet breakup is initiated, the maximum breakup length where the jet breakup is completed, and the mean droplet size and the impact frequency in the equilibrium region downstream of the maximum breakup length. Since the correlations were dependent on the flow regime, the boundaries between the three jet regimes were also determined using the present observation results. It was demonstrated that the impact frequencies calculated by the proposed model agree with the experimental data accumulated in this work fairly well in the Rayleigh and second wind-induced regimes, while agreement was deteriorated to some extent in the transition (first wind-induced) regime. For the application to the prediction of splashing rate during coolant leakage in sodium-cooled fast reactors, a simple method to ensure conservative estimation of the impact frequency was also proposed.

Radiation forces study of a Laguerre Gaussian beam type TEM∗01 on a dielectric sphere in the Rayleigh scattering regime

From the invention of the Optical Tweezer (OT) in 1986, these devices have been considered as high-level tools for research in the areas such as biology and microbiology. A theoretical study obtaining equations for gradient and scattering forces that exert an OT when the illumination beam is a doughnut-shaped mode TEM∗01 linearly polarized is realized. This work focuses on the behavior of radiation forces on a dielectric sphere in the Rayleigh regime. In order to facilitate the phenomenological analysis of the behavior of the radiation forces a graphical user interface is created.

Radiation force and torque on perfect electrically–conducting (PEC) corrugated circular and elliptical cylinders in TE or TM polarized plane progressive waves with arbitrary incidence

In this work, theoretical modeling and numerical computations for the electromagnetic (EM) radiation force and torque (per-length) on perfect electrically conducting (PEC) cylinders of corrugated circular and elliptical geometrical cross-sections are developed and presented. A TE-polarized or TM-polarized plane progressive wave illumination with arbitrary incidence (in the polar plane) is assumed at normal incidence with respect to the axis of the cylindrical particle. The multipole expansion method in cylindrical coordinates is used and the scattering coefficients of the object of arbitrary shape (in 2D) are determined by imposing appropriate boundary conditions and solving numerically a linear system of equations by matrix inversion. Numerical computations are performed for the non-dimensional longitudinal and transverse radiation force functions as well as the axial radiation torque function. Suitable convergence plots confirm the validity of the multipole expansion approach to evaluate the radiation force and torque with no limitation to a particular frequency range (i.e. Rayleigh, Mie or geometrical optics regimes can be considered using the presented formalism). Particular emphases are given on the shape of the particle (i.e., circular or elliptical), its non-dimensional size, the corrugation/waviness characteristic of its surface, the polarization of the incident plane wave field, and the angle of incidence in the polar plane. The numerical predictions show that the longitudinal and transverse components of the radiation force vector are positive regardless of particle shape, size, corrugation properties, polarization and incidence angle [0 ≤ α ≤ 90°], while the axial torque component reverses sign at particular values of these parameters. The results are predominantly relevant in understanding the fundamentals of the optical/EM radiation force and torque theories and possible applications dealing with the interactions of EM waves with elongated tubular particles with circular or noncircular ribbed surfaces in particle manipulation and other areas. The acoustical analogue is also noted, which shows the universal characteristic of the radiation force and torque phenomena.

Thermal forcing and ‘classical’ and ‘ultimate’ regimes of Rayleigh–Bénard convection

The fundamental challenge to characterize and quantify thermal transport in the strongly nonlinear regime of Rayleigh–Bénard convection – the buoyancy-driven flow of a horizontal layer of fluid heated from below – has perplexed the fluid dynamics community for decades. Rayleigh proposed controlling the temperature of thermally conducting boundaries in order to study the onset of convection, in which case vertical heat transport gauges the system response. Conflicting experimental results for ostensibly similar set-ups have confounded efforts to discriminate between two competing theories for how boundary layers and interior flows interact to determine transport through the convecting layer asymptotically far beyond onset. In a conceptually new approach, Bouillaut, Lepot, Aumaître and Gallet (J. Fluid Mech., vol. 861, 2019, R5) devised a procedure to radiatively heat a portion of the fluid domain bypassing rigid conductive boundaries and allowing for dissociation of thermal and viscous boundary layers. Their experiments reveal a new level of complexity in the problem suggesting that heat transport scaling predictions of both theories may be realized depending on details of the thermal forcing.

Effect of dust size and structure on scattered-light images of protoplanetary discs

We study scattered light properties of protoplanetary discs at near-infrared wavelengths for various dust size and structure by performing radiative transfer simulations. We show that different dust structures might be probed by measuring disk polarisation fraction as long as the dust radius is larger than the wavelength. When the radius is larger than the wavelength, disc scattered light will be highly polarised for highly porous dust aggregates, whereas more compact dust structure tends to show low polarisation fraction. Next, roles of monomer radius and fractal dimension for scattered light colours are studied. We find that, outside the Rayleigh regime, as fractal dimension or monomer radius increases, colours of the effective albedo at near-infrared wavelengths vary from blue to red. Our results imply that discs showing grey or slightly blue colours and high polarisation fraction in near-infrared wavelengths might be explained by the presence of large porous aggregates containing sub-microns sized monomers.

Simultaneous trapping and pulling by acoustical Bessel beams as stable tractor beams

A stable tractor beam for long range pulling of particles by acoustic Bessel beams requires a simultaneous trapping in the lateral direction. Trapping force acting on a sphere centered on the axis of Bessel beams is examined to guide the selection of material and beam parameters. It is found that a heavy and rigid sphere in the Rayleigh regime can be trapped at central pressure maximum of zero-order Bessel beam, which is contrary to the repelling behavior at the pressure anti-nodes of one- or two-dimensional standing waves. The trapping results from the three-dimensional features of the velocity fields and the momentum projection. The projection leads the trapping of an elastic sphere or a droplet at the axis of ordinary and vortex Bessel beams in the Rayleigh regime to be reversed when reducing the conical angle of the beam at the situation when there is a strong contrast of mass density between the particle and the surrounding fluid. Ranges of conical angle and material parameters for simultaneous trapping and pulling a spherical object are identified.A stable tractor beam for long range pulling of particles by acoustic Bessel beams requires a simultaneous trapping in the lateral direction. Trapping force acting on a sphere centered on the axis of Bessel beams is examined to guide the selection of material and beam parameters. It is found that a heavy and rigid sphere in the Rayleigh regime can be trapped at central pressure maximum of zero-order Bessel beam, which is contrary to the repelling behavior at the pressure anti-nodes of one- or two-dimensional standing waves. The trapping results from the three-dimensional features of the velocity fields and the momentum projection. The projection leads the trapping of an elastic sphere or a droplet at the axis of ordinary and vortex Bessel beams in the Rayleigh regime to be reversed when reducing the conical angle of the beam at the situation when there is a strong contrast of mass density between the particle and the surrounding fluid. Ranges of conical angle and material parameters for simultaneous trapp...

Scattering of electromagnetic waves by cylinder inside uniaxial hyperbolic medium.

We study electromagnetic wave scattering inside a hyperbolic medium by a circular cylinder. The hyperbolic medium's unique properties result in scattering behaviors that differ greatly from scattering by the same cylinder inside a positive isotropic medium. Incident wave polarization is preserved for all angles of incidence and the scattered waves have the same polarization in the far-field region. TM-polarized plane wave scattering is highly anisotropic. At any given frequency, the dielectric cylinder's scattering properties can vary from the Rayleigh regime, to the resonant regime, to the evanescent regime, by simply changing the angle of incidence. At a given angle, as a function of frequency, the scattering efficiency exhibits narrow resonant features associated with Fano interference. We also show that a magnetic cylinder's scattering can be suppressed. All of these effects stem from the hyperbolic medium's properties, as well as point to the interesting opportunities of tailoring scattering properties by controlling the surrounding medium.

Trapping Force of Acoustical Bessel Beams on a Sphere and Stable Tractor Beams

In noncontact particle manipulation by acoustic or optical beams, which is important for applications in the biological sciences and materials engineering, a $t\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}r$ $b\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}m$ can pull particles toward the beam's source, but long-range pulling is impossible without simultaneous trapping in the transverse direction. This study uses theory to gain insight into the parameters and conditions for simultaneous trapping and pulling of a spherical particle by an acoustic Bessel beam. Trapping at the pressure maximum of an axisymmetric Bessel beam in the Rayleigh regime is more flexible than the trapping by focused beams currently used for acoustic tweezers.

Optical radiation force (per–length) on an electrically conducting elliptical cylinder having a smooth or ribbed surface

The aim of this work is to develop a formal semi-analytical model using the modal expansion method in cylindrical coordinates to calculate the optical/electromagnetic (EM) radiation force-per-length experienced by an infinitely long electrically-conducting elliptical cylinder having a smooth or wavy/corrugated surface in EM plane progressive waves with different polarizations. One of the semi-axes of the elliptical cylinder coincides with the direction of the incident field. The modal matching method is used to determine the scattering coefficients by matrix inversion. Standard cylindrical (Bessel and Hankel) wave functions are used. Simplified expressions leading to exact series expansions for the optical/EM radiation forces assuming either electric (TM) of magnetic (TE) plane wave incidences are provided without any approximations, in addition to integral equations demonstrating the direct relationship of the radiation force with the square of the scattered field magnitude. Numerical computations for the non-dimensional radiation force function are performed for electrically conducting elliptic and circular cylinders having a smooth or ribbed/corrugated surface. Adequate convergence plots confirm the validity and correctness of the method to evaluate the radiation force with no limitation to a particular frequency range (i.e. Rayleigh, Mie or geometrical optics regimes). Emphases are given on the aspect ratio, the non-dimensional size of the cylinder, the corrugation characteristic of its surface, and the polarization of the incident field. The results are particularly relevant in optical tweezers and other related applications in fluid dynamics, where the shape and stability of a cylindrical drop stressed by a uniform external electric/magnetic field are altered. Furthermore, a direct analogy with the acoustical counterpart is discussed.

Spherical particle absorption over a broad range of imaginary refractive index

This paper presents studies of the optical absorption cross section for homogeneous spheres of radius R interacting with light of wavelength λ as calculated with the Mie equations. Four regimes of behavior are disclosed: the Rayleigh Regime, the Geometric Regime, the Reflection Regime, and a Crossover Regime. Two parameters govern these regimes, the imaginary part of the refractive index, κ, and the product of κ with the sphere size parameter kR where k = 2π/λ, i.e. κkR. Simple, approximate functionalities on κ, k, sphere volume and projected geometric cross section are derived for these regimes. Interesting aspects of our observations include: Rayleigh absorption can apply to all particle sizes, Fresnel reflection can occur for sub-wavelength spheres, and while κ is the agent of absorption, large κ can increase scattering to the detriment of absorption.

The Foldy--Lax Approximation for the Full Electromagnetic Scattering by Small Conductive Bodies of Arbitrary Shapes

We deal with the electromagnetic wave propagation in the harmonic regime. We derive the Foldy--Lax approximation of the scattered fields generated by a cluster of small conductive inhomogeneities of arbitrary shapes and arbitrarily distributed in a bounded domain $\Omega$ of $\mathbb{R}^3$. The dominating term in the approximation of the electric fields (respectively, the magnetic fields) is a combination of both electric and magnetic dipoles through scattering coefficients which are solutions of an algebraic system, i.e., the Foldy system. This algebraic system describes the multiple electromagnetic interactions between the small conductivities. This approximation is valid under a sufficient but general condition on the number of such inhomogeneities $m$, their maximum radii $\epsilon$, and the minimum distances between them, $\delta$, of the form $(\ln m)^{\frac{1}{3}}\frac{\epsilon}{\delta} \leq C, $ where $C$ is a constant depending only on the Lipschitz characters of the scaled inhomogeneities. In addition, we provide explicit error estimates of this approximation in terms of the aforementioned parameters, $m, \epsilon, \delta$ but also the used frequencies $k$ under the Rayleigh regime. Both the far fields and the near fields (stated at a distance $\delta$ to the cluster) are estimated. In particular, for a moderate number of small inhomogeneities $m$, the derived expansions are valid in the mesoscale regime where $\delta \sim \epsilon$.

Measurement and density normalisation of acoustic attenuation and backscattering constants of arbitrary suspensions within the Rayleigh scattering regime

The scattering and attenuation of megahertz frequency acoustic backscatter in liquid suspensions, is examined for a range of fine organic and inorganic particles in the Rayleigh regime, 10−4 < ka < 100 (where k is the wavenumber and a the particle radius) which are widely industrially relevant, but with limited existing data. In particular, colloidal latex, mineral titania and barytes sediments, as well as larger glass powders were investigated. A manipulation of the backscatter voltage equation was used to directly measure the sediment attenuation constants, ξ. Decoupling of the combined backscattering-transducer constant, allowing explicit measurement of the backscattering constant, ks, was achieved through calibration of the transducer constant, kt. Additionally, the methodology was streamlined via averaging between a number of intermediate concentrations to reduce data variability. This approach enabled the form function, f, and the corresponding total normalized scattering cross-sections, χ, to be determined for all species. While f and χ are available in the literature for large glass and sand, this methodology allowed extension for the colloidal organic and inorganic particles. Specific gravity normalisation of f collapsed all data onto a single distribution, with the exception of titania, due to scattering complexities associated with agglomeration. There was some additional variation in χ, with measured values of the fine particles up to of magnitude greater than the density-normalised prediction at low ka. Mechanisms accounting for these variations from theory are however analysed, and include viscous attenuation effects, the polydispersity of the particle type and increasing influence of the solvent attenuation. Additionally, thermoacoustic losses appeared to dominate the attenuation behaviour of the organic latex particles. This study demonstrates that particles close to the colloidal regime can be measured successfully with acoustic backscatter, and highlights the great potential of this technique to be applied for in situ or online monitoring purposes in such systems.

Electromagnetic binding and radiation force reversal on a pair of electrically conducting cylinders of arbitrary geometrical cross-section with smooth and corrugated surfaces

The electromagnetic (EM) radiation force-per-length exerted on a pair of electrically-conducting cylindrical particles of circular and non-circular cross-sections is examined using a formal semi-analytical method based on boundary matching in cylindrical coordinates. Initially, the scattering coefficients of the particle pair are determined by imposing suitable boundary conditions leading linear systems of equations computed via matrix inversion and numerical integration procedures. Standard cylindrical (Bessel and Hankel) wave functions are used and closed-form expressions for the dimensionless longitudinal and transverse radiation force functions are evaluated assuming either magnetic (TE) or electric (TM) plane wave incidences. Particle pairs with smooth and corrugated surfaces are considered and numerical computations are performed with emphasis on the distance separating their centers of mass, the angle of incidence of the incident illuminating field and the surface roughness. Adequate convergence plots confirm the validity of the method to evaluate the radiation force functions, and the model is adaptable to any frequency range (i.e. Rayleigh, Mie or geometrical optics regimes). The results can find potential applications in optical tweezers and other related applications in fluid dynamics. In addition, the acoustical analogue is discussed.

Dipolar and quadrupolar mode dissipation of spherical probes spinning in vortex beam acoustical tweezers

In this work, the possibility of simultaneously trapping and rotating single polystyrene beads, or clusters, against gravity with an ultrasonic vortex beam is demonstrated. The induced rotation of a single particle is compared to a torque balance model accounting for the acoustic response of the particle. Two dominating dissipation mechanisms of the acoustic orbital angular momentum responsible for the observed rotation are examined. The first takes place in the bulk of the absorbing particle, while the second arises as the angular momentum flux is dissipated in the viscous boundary layer surrounding the particle. Importantly, even in the long-wavelength (Rayleigh) regime, it is crucial to fully model the dissipation of the dipolar and quadrupolar vibrational modes of the particle in a slightly viscous fluid such as water. Further results suggest that, while the induced outer (Eckart) rotational flow can be neglected in water, it can play an important role in other viscous and complex fluids.