Pair Of Spheres Research Articles

The Gromov–Wasserstein Distance Between Spheres

AbstractThe Gromov–Wasserstein distance—a generalization of the usual Wasserstein distance—permits comparing probability measures defined on possibly different metric spaces. Recently, this notion of distance has found several applications in Data Science and in Machine Learning. With the goal of aiding both the interpretability of dissimilarity measures computed through the Gromov–Wasserstein distance and the assessment of the approximation quality of computational techniques designed to estimate the Gromov–Wasserstein distance, we determine the precise value of a certain variant of the Gromov–Wasserstein distance between unit spheres of different dimensions. Indeed, we consider a two-parameter family $$\{d_{{{\text {GW}}}p,q}\}_{p,q=1}^{\infty }$$ { d GW p , q } p , q = 1 ∞ of Gromov–Wasserstein distances between metric measure spaces. By exploiting a suitable interaction between specific values of the parameters p and q and the metric of the underlying spaces, we are able to determine the exact value of the distance $$d_{{{\text {GW}}}4,2}$$ d GW 4 , 2 between all pairs of unit spheres of different dimensions endowed with their Euclidean distance and their uniform measure.

An efficient direct‐forcing immersed boundary method for flow around a pair of spheres

AbstractThe numerical study of flow around a pair of spheres and a square array of spheres is investigated by using a direct‐forcing immersed boundary method. Using high resolution three‐dimensional computations, we analyzed the flow around several configurations: a sphere, a pair of spheres in a tandem arrangement with center‐to‐center streamwise ratio L/D ranging from 1 to 6, and a square array with 9 spheres in a uniform arrangement. In the latter case, we explore the ratio of array diameter (DG) to sphere diameter (D) at 4, 5, 6 and 7. The center‐to‐center streamwise and transverse pitch is the same, varied from L/D = 1.5, 2, 2.5 to 3, and they were arranged in a square periodic array to allow uniform distribution within the array. Based on the effective direct‐forcing immersed boundary projection method, the fractional time marching methodology is applied for solving four field variables involving three velocities and one pressure component. The pressure Poisson equation is advanced in space by using the fast Fourier transform (FFT) and a tridiagonal matrix algorithm (TDMA), effectively solving for the diagonally dominant tridiagonal matrix equations. A direct‐forcing immersed boundary method is involved to treat the interfacial terms by adding the appropriate sources as force function at the boundary, separating the phases. Geometries featuring the stationary solid obstacles in the flow are embedded in the Cartesian grid with special discretizations near the embedded boundary using a discrete Dirac delta function to ensure the accuracy of the solution in the cut cells. An important characteristic of flow over the multiple spheres is devised by comparing with the drag and lift coefficients, as well as vortex shedding.

Free convection from a single and a pair of spheres in power-law fluids at very small Grashof numbers

Free convection from a single and a pair of spheres in power-law fluids at very small Grashof numbers

Collision efficiency of like-charged spheres settling in a quiescent environment

We study the gravity-induced collisions of charged spheres of dielectric materials dispersed in a gaseous medium. When the gap thickness between the surfaces of two spheres is shorter than the mean free path of the surrounding fluid medium, continuum assumptions for the hydrodynamics interactions are no longer valid, and the non-continuum lubrication interactions result in surface-to-surface contact in finite time. Two like-charged dielectric spheres attract each other at close separations for a wide range of size and charge ratio values. We use trajectory analysis to calculate the collision rate and, thus, explore the role of electrostatic interactions in the collision dynamics of a pair of like-charged dielectric spheres. We present the modifications of pair trajectories due to electrostatic forces and show how collision efficiencies vary with the non-dimensional parameter capturing the relative strength of the electrostatic force to gravity as well as the charge ratio and size ratio.

Hydrodynamic spin-orbit coupling in asynchronous optically driven micro-rotors

Vortical flows of rotating particles describe interactions ranging from molecular machines to atmospheric dynamics. Yet to date, direct observation of the hydrodynamic coupling between artificial micro-rotors has been restricted by the details of the chosen drive, either through synchronization (using external magnetic fields) or confinement (using optical tweezers). Here we present a new active system that illuminates the interplay of rotation and translation in free rotors. We develop a non-tweezing circularly polarized beam that simultaneously rotates hundreds of silica-coated birefringent colloids. The particles rotate asynchronously in the optical torque field while freely diffusing in the plane. We observe that neighboring particles orbit each other with an angular velocity that depends on their spins. We derive an analytical model in the Stokes limit for pairs of spheres that quantitatively explains the observed dynamics. We then find that the geometrical nature of the low Reynolds fluid flow results in a universal hydrodynamic spin-orbit coupling. Our findings are of significance for the understanding and development of far-from-equilibrium materials.

Interaction force and dynamics for a pair of Rayleigh spherical particles located in nodal or anti-nodal planes for Gauss standing wave

The acoustic radiation force acting on a pair of spherical particles in a Gauss standing wave field is investigated, with the particle size considered to be much smaller than the wave length. Within the Rayleigh scattering limit, analytical expressions are presented for both the primary and secondary radiation forces using their corresponding potential functions. Force and dynamic analyses are performed for the particles located in nodal or anti-nodal planes where the axial primary radiation force vanishes. Two equal-sized rigid spheres in air and two highly compressible gas bubbles in water are selected as computational examples, whose corresponding equations of motion are solved using numerical methods to facilitate the analysis of their dynamic behaviors. The acoustic interaction force dominates only in the short-range limit for a pair of rigid spheres but without the range limit for a pair of gas bubbles. This study provides a theoretical guide for precise manipulation in standing acoustic waves, especially when there are several particles in the acoustic field.

On the absence of collective motion in a bulk suspension of spontaneously rotating dielectric particles.

A suspension of dielectric particles rotating spontaneously when subjected to a DC electric field in two dimensions next to a no-slip electrode has proven to be an ideal model for active matter [Bricard et al., Nature, 2013, 503, 95-98]. In this system, an electrohydrodynamic (EHD) instability called Quincke rotation was exploited to create self-propelling particles which aligned with each other due to EHD interactions, giving rise to collective motion on large length scales. It is natural to question whether a suspension of such particles in three dimensions will also display collective motion and spontaneously flow like bacterial suspensions do. Using molecular dynamics type simulations, we show that dielectrophoretic forces responsible for chaining in the direction of the applied electric field in conventional electrorheological fluids and the counter-rotation of neighboring particles in these chains prevent collective motion in suspensions undergoing spontaneous particle rotations. Our simulations discover that the fundamental microstructural unit of a suspension under Quincke rotation is a pair of counter-rotating spheres aligned in the direction of the electric field. We perform a linear stability analysis that explains this observation.

Using a Pair of Different-Sized Spheres to Calibrate Radar Data in the Microwave Anechoic Chamber

Traditional radar calibration methods often use one standard sphere as the standard calibration object, but researchers seldom discuss the size and the material content of the calibration sphere. We propose using a pair of different-sized spheres to verify the standard method. In a microwave anechoic chamber, the simulated and measured results show that the radar cross-section (RCS) values of two spheres with radii of 0.15 m and 0.05 m in the Mie region differ from their physical cross-sections in the optic region, and only in the optic region, their difference of RCS value (i.e., 9.5424 dB) can be approximately equal to the theoretical one (i.e., 9.6803 dB). Thus, radar calibration should be conducted in the same scattering region for both the calibration object and the calibrated target. The use of two different-sized spheres can aid in three applications: (1) verifying the scattering regions, (2) searching the pure area in the microwave anechoic chamber, and (3) locating the positions of the targets in the range profiles.

A parallel algorithm for computing Voronoi diagram of a set of spheres using restricted lower envelope approach and topology matching

We present a parallel algorithm for computing the Voronoi diagram of a set of spheres, S in R3. The spheres have varying radii and do not intersect. We compute each Voronoi cell independently using a two-stage iterative procedure, assuming the input spheres are in general position. In the first stage, an initial Voronoi cell for a sphere si is computed using an iterative lower envelope approach restricted to a subset of spheres Li⊂S. This helps to avoid defining the bisectors between all pairs of input spheres and develop a distributed memory parallel algorithm. We use the Delaunay graph of sample points from the input spheres to select the subset Li for computing each Voronoi cell. In the second stage, Voronoi cells obtained from the first stage are matched for updating the subsets. If additional spheres are added to a subset Li, the correctness of the computed vertices is verified with the bisectors of spheres newly added to Li. Results and performance of the algorithm show robustness and speed of the algorithm in handling a large set of spheres.

Numerical study of a pair of spheres in an oscillating box filled with viscous fluid

When two identical spherical particles are submerged in an oscillating fluid, they align themselves perpendicularly to the direction of the flow, leaving a small gap between them. The formation of this structure is attributed to a non-zero residual flow known as steady streaming. We performed direct numerical simulations of a fully resolved, oscillating flow in which the pair of particles is modeled using an immersed boundary method. In equilibrium, the particles oscillate both parallel and perpendicularly to the oscillating flow. Two scaling regimes for the particle dynamics, which are related to changes in the (average) flow field, are observed.

Porpoises adjust echolocation clicks to distinguish objects from clutter

Hollywood representations of Cold War sonar depict torpedoes blinking unhindered across green screens. Yet, the reality of navigating by echolocation is far more demanding for the animals that depend on it. Porpoises and bats must differentiate between the echoes generated by background objects and the juicy morsels they are intent upon catching. And the difficulties of interpreting the environment are even more challenging for aquatic creatures; sound travels in water at 4.5 times the speed in air, leaving porpoises and dolphins with even less time to distinguish between echoes returning from objects that are close together. Chloe Malinka from the University of Aarhus, Denmark, explains that porpoises were thought to be unable to differentiate sounds that appear within 264 μs of each other, placing a natural limit – around 20 cm – on the separation between objects that they should be able to distinguish by echolocation, making her wonder how echolocating porpoises differentiate objects that are close to each other.Malinka travelled to the Fjord & Bælt Centre, Denmark, home to harbour porpoises Freja and Sif who have been collaborators of Peter T. Madsen's for almost 20 years. ‘It's always a pleasure to work with Freja and Sif – two of the world's most studied echolocators’, says Malinka, who set the animals the task of distinguishing between a 5 cm diameter aluminium sphere, which they had been trained to recognise, and a 5 cm wide steel sphere placed 13.5, 27, 54 or 108 cm away. ‘We gently attached opaque cups over the eyes of the porpoises to act as a blindfold, so that they could only use echolocation to solve the task’, says Malinka. She also attached an underwater microphone behind the animals’ blow holes to record the clicks that they produced while approaching the objects and the echoes they heard. ‘The porpoises were given fish as a reward for correctly selecting the aluminium target’, says Malinka, who also filmed the animals as they approached the spheres.Impressively, both porpoises successfully selected the aluminium sphere more than 93% of the time, even when the spheres were only 13.5 cm apart – closer than the scientists thought the animals could distinguish. ‘Peter, Laia [Rojan-Doñate] and I were surprised to find that Freja and Sif were able to do this, even with small differences in the timings of the returning echoes from each of the targets’, says Malinka. But how were the porpoises pulling off the feat?The secret was in the rapid cascade of clicks – known as buzzes – produced by the animals as they closed in on their target. ‘We observed that when the targets were closely spaced and the discrimination task was more difficult the porpoises buzzed for longer and started buzzing from farther away’, says Malinka. In addition, the porpoises clicked faster as they approached the closest pairs of spheres and more softly in the final moments of the approach. As the porpoises closed in, they also moved their heads from side to side more, directing their acoustic focus toward one sphere and then then other, before making their decision.‘We think that the porpoises were benefiting from their highly directional biosonar beam’, says Malinka, explaining that objects directly in line with the powerful beam of sound will return stronger reflections than objects off to the side, allowing the porpoise to home in precisely on the aluminium target. So, porpoises are not hampered by being unable to distinguish echoes that arrive within 264 μs of each other. They adapt their echolocation clicks while closing in to gather more information during their approach, allowing them to directly pursue a target despite being bombarded by distracting echoes from surrounding clutter.

Flow Past a Pair of Separated Porous Spheres using Brinkman model

In this work, we obtained an analytical solution to the problem of viscous fluid flow past a pair of porous separated spheres. Stokesian approximation of the Navier-Stokes equation for the viscous fluid governs the flow in the region outside the two spheres, whereas Brinkman's model describes the flow in the porous region (within the spheres). Since the bipolar coordinate system is the most convenient system to represent separated spheres' geometry, we formulated this problem in the bipolar coordinate system. We then eliminated the pressure term from the equations governing the flow in the region outside the spheres, and they got reduced to a separable equation in terms of the stream function. Further, the flow governing equations inside the porous spheres gave rise to the Helmholtz equation. As the Helmholtz equation is not separable in the bipolar system, we used the spherical coordinate system to describe the fluid flow within each separated spheres and solved the resulting problem in the spherical coordinate system. Because of this, the flow variables on either side of the interface (spheres) are in different coordinate systems. To match the values of the field variables at the boundary, we used the transformation equations between the bipolar and spherical systems and transformed all the variables into the bipolar coordinate system. We then solved the governing equations with appropriate boundary conditions for the arbitrary constants and derived expressions for the stream and pressure functions. We plotted the respective functions for various values of the mathematical model's parameters to understand the flow pattern and pressure distribution in the flow domain and noted our observations. This study revealed an intuiting insight that the pressure distribution inside the porous spheres is independent of the non-dimensional parameter related to the medium's permeability and the fluid's viscosity.

Collision rate of bidisperse, hydrodynamically interacting spheres settling in a turbulent flow

Abstract

Construction of soil–water characteristic curve of granular materials with toroidal model and artificially generated packings

The soil–water characteristic curve (SWCC) of granular materials is crucial for many emerging engineering applications, such as permeable pavement and methane hydrate extraction. Laboratory determination of the SWCC of granular materials suffers from inaccurate volume readings by the diffused air bubbles in the hanging column and sudden desaturation at small matric suction intervals. Theoretical determination of the SWCC of granular materials also suffers from semi-empirical nature in the prediction from grain size distribution, or from the limitation of assumed cubic packing or face-centred cubic packing with a toroid meniscus water model. In this study, real three-dimensional particle packing was first rendered with the discrete element method using approximation of spheres. Then, the Young–Laplace equation was applied to calculate the volume of toroidal meniscus water between each pair of spheres, which adds to the water content in the pendular regime of the SWCC. Additionally, a digitized image algorithm was used to identify the pore throats and calculate the air entry value and residual matric suction, the connection of which yields a straight line approximating the funicular regime. The SWCC was thus constructed. Comparison with laboratory-measured SWCCs suggested that although reasonable agreement was reached in general for glass beads, residual water content was underestimated, especially for non-spherical granular materials. Several possible reasons were discussed including the existence of patchy water accounting for the major portion of water in the beginning of pendular regime of granular materials, which was also observed in microscopic photographs through a special desaturation experiment.

Triazine and Porphyrin-Based Cross-Linked Conjugated Polymers: Protonation-Assisted Dissolution and Thermoelectric Properties

As two pivotal functional segments, triazine and porphyrin can be coupled to form a highly cross-linked conjugated polymer. Although the obtained conjugated polymers are almost insoluble in most so...

Fiddler crabs ignore near misses when threatened from all sides

Everyone suffers with attention overload from time to time. Sometimes there are simply too many meetings and deadlines for one brain to keep track of, so you have to prioritise. Yet, compared with the life and death attention grabbers that stalk many creatures’ lives, we've got it easy. ‘Prey animals are often exposed to multiple simultaneous threats, which significantly complicates the decision-making process’, says Zahra Bagheri from the University of Western Australia. So how do animals that are facing several threats simultaneously prioritise which one, or ones, to take seriously?To find out how fiddler crabs respond when confronted with two looming threats, Bagheri, Callum Donohue and Jan Hemmi, also from the University of Western Australia, rigged up a pair of black spheres suspended on fishing line travelling threateningly toward the crustaceans: one on a direct collision course and the second coming in at an angle to produce an alarming near miss. Intriguingly, the crabs that were on a near-miss collision course began scuttling for safety when the alarming ball was as much as 300 cm away. However, the crabs that were in direct line with the incoming sphere stood their ground until it was within 100 cm; only then did they beat a retreat. In contrast, when confronted with both inbound balls simultaneously, instead of turning tail when the near-miss ball was within 300 cm, the crabs held on until the ball heading straight for them came within a threatening 100 cm. The crabs appeared to be completely disregarding the less dangerous threat, focusing exclusively on the inbound ball that could genuinely bowl them over.‘Crabs that face multiple predators simultaneously behave as if they only face the single, directly approaching predator’, says Bagheri, adding, ‘this suggests that the crabs do not perceive, or do not respond to, the increased risk of predation posed by two simultaneous predators’. The team suspects that the crabs weigh up the relative dangers of each impending threat and focus on the one that poses the greatest risk, ignoring others that would simply result in a narrow escape. After all, there's only so much attention to go around, so the key to survival is to use it wisely.

Realization of a push-me-pull-you swimmer at low Reynolds numbers

Locomotion at low Reynolds numbers encounters stringent physical constraints due to the dominance of viscous over inertial forces. A variety of swimming microorganisms have demonstrated diverse strategies to generate self-propulsion in the absence of inertia. In particular, ameboid and euglenoid movements exploit shape deformations of the cell body for locomotion. Inspired by these biological organisms, the ‘push-me-pull-you’ (PMPY) swimmer (Avron J E et al 2005 New J. Phys. 7 234) represents an elegant artificial swimmer that can escape from the constraints of the scallop theorem and generate self-propulsion in highly viscous fluid environments. In this work, we present the first experimental realization of the PMPY swimmer, which consists of a pair of expandable spheres connected by an extensible link. We designed and constructed robotic PMPY swimmers and characterized their propulsion performance in highly viscous silicone oil in dynamically similar, macroscopic experiments. The proof-of-concept demonstrates the feasibility and robustness of the PMPY mechanism as a viable locomotion strategy at low Reynolds numbers.

A theory for like-charge attraction of polarizable ions

A theory that explains how like-charge attraction of polarizable ions could occur is presented. Based on an effective-dipole description, the theory predicts that, for a pair of spherical polarizable ions in vacuum, like-charge attraction is possible but opposite-charge repulsion could not occur, where any occurrence of like-charge attraction is necessarily conditional upon dissimilarities in net charge or polarizable volume between the particles. It also predicts a correlation between the nature of interaction and the alignments of effective-dipole moments. The findings, which echo recent theoretical predictions of like-charge attraction for a pair of dielectric spheres in vacuum, suggest that this counterintuitive phenomenon could be realized in systems of different length scales through similar mechanisms of particle dissimilarities.

The role of heating and cooling in viscous sintering of pairs of spheres and pairs of cylinders

PurposeThis study aims to develop mathematical models for the determination of the effects of heating or cooling on neck growth in Selective Laser Sintering (SLS) and Fused Filament Fabrication (FFF). Two particle shapes are studied: spherical and cylindrical.Design/methodology/approachThe time required for the coalescence (sintering) process is determined by balancing the work of surface tension forces and viscous dissipation. Heating and cooling effects are studied by incorporating temperature dependence of viscosity in an exponential form. Heating by a laser, convective and/or radiative heat transfer is assumed. It is also assumed that there are no temperature gradients within the coalescing molten polymers (lumped parameter heat transfer analysis).FindingsThe models predict faster sintering with heating and slower with cooling, as expected because of the effect of temperature on viscosity. For the isothermal case of pairs of cylinders, the present model predicts significantly longer time for completion of sintering than a previously developed and frequently cited model by Hopper.Originality/valueAn isothermal sintering model for two spheres was reworked for two long cylinders, and for the first time it has been compared to other models available in the literature. The mathematical models are capable of predicting neck growth under non-isothermal conditions for both spheres and cylinders. They are useful in assessment of bonding in selective laser sintering and fused deposition fabrication.

Creeping flow of a viscous fluid past a pair of porous separated spheres

Creeping flow of a viscous fluid past a pair of porous separated spheres