Spheroid Aspect Ratio Research Articles

Modeling drag coefficients of spheroidal particles in rarefied flow conditions

Transport of particles in flows is often modeled in a combined Eulerian–Lagrangian framework. The flow is evaluated on an Eulerian grid, while particles are modeled as Lagrangian points whose positions and velocities are evolved in time, resulting in particle trajectories embedded in the time-dependent flow field. The method essentially resolves the flow field in complex geometries in detail but uses a closure model for the particle dynamics aimed at including the essential particle–fluid interactions at the cost of detailed small-scale physics. Rarefaction effects are usually included through the phenomenological Cunningham correction on the drag force experienced by the particles. In this Lagrangian point-particle approach, any explicit reference to the finite size and the shape of the particles, and their local orientation in the flow field, is typically ignored. In this work we aim to address this gap by deriving, from fully-resolved Direct Simulation Monte Carlo (DSMC) studies, heuristic or closure models for the drag force acting on prolate and oblate spheroidal particles with different aspect ratios, and a fixed orientation, in uniform ambient rarefied flows covering the transition regime between the continuum and free-molecular limits. These closure models predict the drag in the transition regime for all considered parameter settings (validated with DSMC data). The continuum limit is enforced a priori and we retrieve the free-molecular limit with reasonable accuracy (based on comparisons with literature data). We also include in the models the capability to predict effects related to basic gas-surface interactions via the tangential momentum accommodation coefficient. We furthermore assess the validity of the proposed closure model for particle dynamics in proximity to solid walls. This investigation extends our previous work, which focused on small aspect ratio spheroids with exclusively diffusive gas-surface interactions [see Livi et al. (2022)]. The derived models are obtained for isothermal, subsonic flows relevant for particle contamination control in semiconductor manufacturing.

Inertial migration of a neutrally buoyant spheroid in plane Poiseuille flow

We study the cross-stream inertial migration of a torque-free neutrally buoyant spheroid, of an arbitrary aspect ratio $\kappa$ , in wall-bounded plane Poiseuille flow for small particle Reynolds numbers ( $Re_p\ll 1$ ) and confinement ratios ( $\lambda \ll 1$ ), with the channel Reynolds number, $Re_c = Re_p/\lambda ^2$ , assumed to be arbitrary; here $\lambda =L/H$ , where $L$ is the semi-major axis of the spheroid and $H$ denotes the separation between the channel walls. In the Stokes limit ( $Re_p =0)$ , and for $\lambda \ll 1$ , a spheroid rotates along any of an infinite number of Jeffery orbits parameterized by an orbit constant $C$ , while translating with a time-dependent speed along a given ambient streamline. Weak inertial effects stabilize either the spinning ( $C=0$ ) or tumbling orbit ( $C=\infty$ ), or both, depending on $\kappa$ . The asymptotic separation of the Jeffery rotation and orbital drift time scales, from that associated with cross-stream migration, implies that migration occurs due to a Jeffery-averaged lift velocity. Although the magnitude of this averaged lift velocity depends on $\kappa$ and $C$ , the shape of the lift profiles are identical to those for a sphere, regardless of $Re_c$ . In particular, the equilibrium positions for a spheroid remain identical to the classical Segre–Silberberg ones for a sphere, starting off at a distance of about $0.6(H/2)$ from the channel centreline for small $Re_c$ , and migrating wallward with increasing $Re_c$ . For spheroids with $\kappa \sim O(1)$ , the Jeffery-averaged analysis is valid for $Re_p\ll 1$ ; for extreme aspect ratio spheroids, the regime of validity becomes more restrictive being given by $Re_p \kappa /\ln \kappa \ll 1$ and $Re_p/\kappa \ll 1$ for $\kappa \rightarrow \infty$ (slender fibres) and $\kappa \rightarrow 0$ (flat disks), respectively.

On the clustering of low-aspect-ratio oblate spheroids settling in ambient fluid

We have performed particle-resolved direct numerical simulations of many heavy non-spherical particles settling under gravity in the dilute regime. The particles are oblate spheroids of aspect ratio $1.5$ and density ratio $1.5$ . Two Galileo numbers are considered, namely $111$ and $152$ , for which a single oblate spheroid follows a steady vertical and a steady oblique path, respectively. In both cases, a strongly inhomogeneous spatial distribution of the disperse phase in the form of columnar clusters is observed, with a significantly enhanced average settling velocity as a consequence. Thus, in contrast to previous results for spheres, the qualitative difference in the single-particle regime does not result in a qualitatively different behaviour of the many-particle cases. In addition, we have carried out an analysis of pairwise interactions of particles in the well-known drafting–kissing–tumbling set-up, for oblate spheroids of aspect ratio $1.5$ and for spheres. We have varied systematically the relative initial position between the particle pair and we have considered free-to-rotate particles and rotationally locked ones. We have found that the region of attraction for both particle shapes, with and without rotation, is very similar. However, significant differences occur during the drafting and tumbling phases. In particular, free-to-rotate spheres present longer drafting phases and separate quickly after the collision. Spheroids remain close to each other for longer times after the collision, and free-to-rotate ones experience two or more collision events. Therefore, we have observed a shape-induced increase in the interaction time which might explain the increased tendency to cluster of the many-particle cases.

Size effects on spheroidal voids by Finite Fracture Mechanics and application to corrosion pits

AbstractThe present work aims at investigating the failure size effect of a spheroidal void in an infinite linear elastic solid under remote tension by means of the coupled Finite Fracture Mechanics (FFM) approach. The opening stress field and the stress intensity factor (SIF) of an annular crack surrounding the cavity —necessary for the FFM implementation— are obtained numerically through parametric axisymmetric finite element analyses (FEAs): The spheroid aspect ratio is varied between 0.1 and 10 and Poisson's ratio between 0.1 and 0.5. Accordingly, semi‐analytical functions approximating the stress concentration factor and the SIF are put forward. Finally, the failure size effect on spheroidal voids is reported, and FFM predictions are compared with experimental results on the fatigue limit arising from corrosion pits, showing a fairly good agreement.

Effect of neutrally buoyant oblate spheroid's aspect ratio on its equilibrium position in a square duct

The inertial migration of a neutrally buoyant oblate spheroid in a square duct is numerically simulated via the multiple-relaxation-time lattice Boltzmann method. This study aims to investigate the effect of a particle's aspect ratio $(AR)$ on its equilibrium position at different initial positions, orientations, and Reynolds numbers $(Re)$. The spheroid's equatorial radius remains unchanged here, and its $AR$ is adjusted by changing its polar radius, which is distinct from previous studies. Therefore, the particle's volume and mass increase with the $AR$. At a low $Re$ $(Re~=~80)$, the particle's initial position and orientation affect its final motion. There are three possible motion modes, tumbling, log-rolling and inclined log-rolling. Although the oblate spheroids with different $ARs$ may experience different motion modes at their final equilibrium states, their rotational diameters are constant. For the tumbling mode, the $AR$ affects the equilibrium position, and the particle gradually approaches the duct center as the $AR$ increases. For the inclined log-rolling mode, the $AR$ partially affects the equilibrium position, where the corresponding equilibrium positions remain unchanged for the spheroids with $AR~=~0.5$ and $0.75$, approaching closer to the duct center for the spherical particle with $AR~=~1$. Finally, for the log-rolling mode, $AR$ has a negligible effect on the equilibrium positions. Furthermore, in the same motion mode, the particle's equilibrium position approaches the wall as the $Re$ increases from 40 to 160.

Accurate hydrodynamic force and torque correlations for prolate spheroids from Stokes regime to high Reynolds numbers

Detailed simulations of flow around various prolate spheroids are performed using multi-relaxation time lattice Boltzmann method (LBM). The simulations are performed in the Reynolds numbers range 0.1≤Re≤2000 at different incident angles 0∘≤ϕ≤90∘ for various prolate spheroids of aspect ratios 1≤λ≤8. The Re is based on the volume equivalent sphere diameter. From the simulations, accurate correlations for average drag, lift and torque coefficients (CD, CL, and CT respectively) are proposed. The mean deviations between the correlations and the simulation data for CD, CL, and CT are 2%, 6.5%, and 4.7% respectively. Furthermore, the correlations are fitted with the physics in mind such that they can be extrapolated beyond the regimes simulated. The fits are designed such that correlations reduce to analytical (CD and CL) solutions available for prolate spheroids in the limit Re≈0. At high Re, the correlations mimic the asymptotic flattening of CD as for spheres and therefore, we expect the correlations to be valid until the critical Re (≈105). Furthermore, extensive comparison of our correlations with other high-fidelity simulations from literature demonstrate the accuracy of our work compared to other correlations from literature. This work enables researchers to perform accurate unresolved Euler-Lagrangian simulations for a wide range of elongated particles.

Investigation of convective heat transfer from a high temperature prolate spheroid to moving fluid

Most of the currently used fluid-solid heat transfer models assume that solid particles are perfect spheres. Heat transfer between non-spherical particles and flowing fluids acquires great importance in many engineering appliances. A numerical study is performed on airflow past a stationary confined prolate spheroid under the forced convective heat transfer regime, and the influence of non-sphericity of the particle on their heating rate is examined. The prolate spheroid is assumed to be maintained at a constant wall temperature. The effect of Reynolds number (Re) and Aspect Ratio (AR) on convective heat transfer rate and Nusselt number was investigated by solving steady-state Navier-Stokes and energy equations. The influence of the flow in terms of Reynolds number and the prolate spheroid aspect ratio was investigated. The spheroid’s surface temperature (Ts) was varied between 294 K and 1500 K. The simulations show that the mean Nusselt number has a positive dependence on Ts, AR, and Re. A new correlation was developed to predict the mean Nusselt number for a wide range of temperature differences. The new correlation consolidates the influence of aspect ratio, surface temperature, and Reynolds number, in contrast to the existing correlations that assume merely isothermal conditions to calculate the mean Nusselt number for spheroid particles.

Orientation Dynamics of Sedimenting Anisotropic Particles in Turbulence.

We examine the dynamics of small anisotropic particles (spheroids) sedimenting through homogeneous isotropic turbulence using direct numerical simulations and theory. The gravity-induced inertial torque acting on sub-Kolmogorov spheroids leads to pronouncedly non-Gaussian orientation distributions localized about the broadside-on (to gravity) orientation. Orientation distributions and average settling velocities are obtained over a wide range of spheroid aspect ratios, Stokes, and Froude numbers. Orientational moments from the simulations compare well with analytical predictions in the inertialess rapid-settling limit, with both exhibiting a nonmonotonic dependence on spheroid aspect ratio. Deviations arise at Stokes numbers of order unity due to a spatially inhomogeneous particle concentration field resulting from a preferential sweeping effect; as a consequence, the time-averaged particle settling velocities exceed the orientationally averaged estimates.

Effects of void shape and orientation on the elastoplastic properties of spheroidally voided single-crystal and nanotwinned copper

ABSTRACT Void nucleation, growth and coalescence have been identified as the leading cause of ductile damage in metallic materials. To understand the underlying deformation and damage mechanisms, extensive theoretical, experimental and simulation efforts have been attempted on spherically voided metals. In this work, molecular dynamics simulations are performed to analyze the uniaxial straining deformation behaviours of both single-crystal and nanotwinned copper materials embedded with a preexisting spheroidal void. The coupling effects among twin boundary, spheroidal void aspect ratio and orientation on unidirectional elastoplastic behaviours are systematically examined. The dislocation-induced plastic deformation mechanism is also examined and compared with the one due to a perfectly spherical cavity. Simulation results show that elastic modulus increases with both spheroidal void aspect ratio and orientation. So do the yield stress, the first peak stress and the plasticity index. Another peak stress exists for most cases, except for a prolate void embedded in nanotwinned specimens. The slope between peak stresses decreases with both the spheroidal aspect ratio and orientation. The incorporation of a twin boundary results in lower elastic modulus, higher yield strength and smaller plasticity index. For an oblate void, the twin boundary gives rise to more severe strain softening behaviour. The dislocation extraction algorithm illustrates that the continuous nucleation, propagation and reaction of dislocations emanated from both the void front and twin boundary are responsible for the ductile damage of spheroidally voided crystals. The lower dislocation densities found in nanotwinned specimens indicate the desired suppression effects of twin boundary on dislocation activities.

Nonlinear viscoelasticity of a dilute suspension of Brownian spheroids in oscillatory shear flow

The nonlinear viscoelasticity of a dilute suspension of Brownian spheroids subject to an oscillatory shear deformation is calculated numerically. This is achieved by determining the suspension microstructure, parameterized via the orientation distribution function. Specifically, the long-time periodic orientation distribution function is obtained via a numerical solution to the Fokker–Planck equation by combining a finite-difference approximation in space with a Fourier series in time. From an ensemble average of the particle stresslet, weighted by the orientation distribution function, the entire stress tensor and relevant birefringence parameters, namely, the average orientation angle and linear dichroism, are calculated; this is done over a range of the Weissenberg number (Wi) and the Deborah number (De), or dimensionless strain-rate amplitude and oscillation frequency, respectively. Detailed calculations are presented for prolate spheroids of aspect ratio r=20; however, our methodology is general and can be applied to spheroids of arbitrary aspect ratio. We provide results in four viscoelastic regimes: linear viscoelastic (Wi≪1), quasilinear viscoelastic (Wi>1 and Wi/De≪1), quasisteady viscoelastic (De→0), and finally, the nonlinear viscoelastic regime (Wi ≳ 1 and Wi/De ≳ 1), which is our main emphasis. In this last regime, where the nonlinear and unsteady viscoelasticity of the material is probed, multiple overshoots are observed in the shear stress and first normal stress difference. The mechanistic origin of these overshoots can be understood from the periodic orientation dynamics (i.e., Jeffery orbits) of a particle under steady shear in the absence of Brownian rotation (Wi→∞). This is achieved by simultaneously analyzing the microstructure, shear stress, first normal stress difference, and birefringence parameters specifically at Wi=20 and De=1. For these values of Wi,De, and r, the period of a Jeffery orbit is comparable to the period of an oscillation cycle, allowing sufficient time for a single Jeffery orbit (and subsequent overshoot) to occur during an oscillation cycle. We contrast this behavior to recent work by Khair [J. Fluid. Mech. 791, R5 (2016)] on nearly spherical particles, for which many more overshoots are observed during an oscillation cycle, due to the shorter period of the Jeffery orbit. We briefly provide results for oblate spheroids of aspect ratio r=0.05 and compare to the results for r=20. Finally, we comment on the relevance of the present micro-mechanical analysis to the nonlinear viscoelasticity of other complex fluid materials.

Effect of particle shape on fluid statistics and particle dynamics in turbulent pipe flow

.Anisotropic particles are present in many natural and industrial flows. Here we perform direct numerical simulation (DNS) of turbulent pipe flows with dispersed finite-size prolate spheroids simulated by means of the lattice Boltzmann method (LBM). We consider three different particle shapes: spheroidal (aspect ratio 2 and 3) and spherical. These three simulations are complemented with a reference simulation of a single-phase flow. For the sake of comparison, all simulations, laden or unladen have the same energy input. The flow geometry used is a straight pipe with length eight times its radius where the fluid is randomly seeded with 256 finite-size particles. The volume fraction of particles in the flow has been kept fixed at 0.48% by varying the major and minor axis of each particle such that their volume remains the same. We studied the effect of different particle shapes on particle dynamics and orientation, as well as on the flow modulation. We show that the local accumulation of spheres close to the wall decreases for spheroids with increasing aspect ratio. These spheroidal particles rotate slower than spheres near to the wall and tend to stay with their major axes aligned to the flow streamwise direction. Despite the lower rotation rates, a higher intermittency in the rotational rates was observed for spheroids and this increase at increasing the aspect ratio. The drag reduction observed for particles with higher aspect ratio have also been investigated using the one-dimensional energy and dissipation spectra. These results point to the relevance of particle shapes on their dynamics and their influence on the turbulent flow.Graphical abstract

Lattice Boltzmann simulation of resolved oblate spheroids in wall turbulence

The present study focuses on the behaviour of fully resolved oblate spheroids in turbulent channel flows using the lattice Boltzmann method (LBM). Mean and maximum drag reductions of 1.3 % and 4.4 %, respectively, are observed at a solid-phase volume fraction of 10 % by using oblate spheroids of aspect ratio $\unicode[STIX]{x1D706}=1/3$ and equivalent diameter $D_{eq}/H=1/6.5$. The behaviour of oblate spheroids and spheres are found to be different in the near-wall region, where oblate spheroids, in contrast to spheres, do not augment the near-wall turbulence. Strong reduction of spanwise and wall-normal fluid velocity fluctuations in the buffer layer by using oblate spheroids is observed. It is also seen that reduction of Reynolds shear stress cannot solely guarantee the occurrence of drag reduction, as spherical particles increase the drag in spite of the reduction in Reynolds shear stress. Moreover, in drag-reduced particle-laden flows, although the spanwise spacing of streaks increases, vortices are stronger and smaller. Thus, in addition to the streak cycle of turbulence regeneration, the vortices generated by particles contribute to higher vortex strengths. By examining the role of the oblate spheroid concentration, it is found to mainly influence the streak spacing, with a minor effect on the strength of vortices. Drag reduction in oblate-spheroid-laden flows is thus attributed to the reduction of fluid velocity fluctuations in the transverse directions and enhancement of the spanwise streak spacing. Higher vortex strength has less effect on the initiation of drag reduction but can influence the level of drag reduction and explains the lower drag-reduction effect of rigid particles compared to polymers. By quadrant analysis, sweeps and ejections are seen to have less contribution to the Reynolds shear stress by using oblate spheroids. With respect to the particle data, oblate spheroids tend to orient with their symmetry axis mainly along the wall-normal direction. Very close to the wall, spanwise orientation is dominant and wall-normal alignment of the symmetry axis is reduced. Furthermore, oblate spheroids move faster than the fluid, before reaching a similar streamwise velocity in the channel centre. In contrast, spheres move slower than the fluid in the buffer region, as well as showing a local concentration peak near the wall. It is also seen that the angular velocity of oblate spheroids decays very rapidly whereas spheres have a higher rotation rate which shows a smoother reduction towards the centre.

Snowfall retrieval at X, Ka and W bands: consistency of backscattering and microphysical properties using BAECC ground-based measurements

Abstract. Radar-based snowfall intensity retrieval is investigated at centimeter and millimeter wavelengths using co-located ground-based multi-frequency radar and video-disdrometer observations. Using data from four snowfall events, recorded during the Biogenic Aerosols Effects on Clouds and Climate (BAECC) campaign in Finland, measurements of liquid-water-equivalent snowfall rate S are correlated to radar equivalent reflectivity factors Ze, measured by the Atmospheric Radiation Measurement (ARM) cloud radars operating at X, Ka and W frequency bands. From these combined observations, power-law Ze–S relationships are derived for all three frequencies considering the influence of riming. Using microwave radiometer observations of liquid water path, the measured precipitation is divided into lightly, moderately and heavily rimed snow. Interestingly lightly rimed snow events show a spectrally distinct signature of Ze–S with respect to moderately or heavily rimed snow cases. In order to understand the connection between snowflake microphysical and multi-frequency backscattering properties, numerical simulations are performed by using the particle size distribution provided by the in situ video disdrometer and retrieved ice particle masses. The latter are carried out by using both the T-matrix method (TMM) applied to soft-spheroid particle models with different aspect ratios and exploiting a pre-computed discrete dipole approximation (DDA) database for rimed aggregates. Based on the presented results, it is concluded that the soft-spheroid approximation can be adopted to explain the observed multi-frequency Ze–S relations if a proper spheroid aspect ratio is selected. The latter may depend on the degree of riming in snowfall. A further analysis of the backscattering simulations reveals that TMM cross sections are higher than the DDA ones for small ice particles, but lower for larger particles. The differences of computed cross sections for larger and smaller particles are compensating for each other. This may explain why the soft-spheroid approximation is satisfactory for radar reflectivity simulations under study.

The inertial orientation dynamics of anisotropic particles in planar linear flows

In the Stokes limit, the trajectories of neutrally buoyant torque-free non-Brownian spheroids in ambient planar linear flows are well known. These flows form a one-parameter family, with the velocity gradient tensor given by $\unicode[STIX]{x1D735}\boldsymbol{u}^{\infty \dagger }=\dot{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{x}^{\prime }\mathbf{1}_{y}^{\prime }+\unicode[STIX]{x1D706}\mathbf{1}_{y}^{\prime }\mathbf{1}_{x}^{\prime })$. The parameter $\unicode[STIX]{x1D706}$ is related to the ratio of the vorticity to the extension (given by $(1-\unicode[STIX]{x1D706})/(1+\unicode[STIX]{x1D706})$), and ranges from $-1$ to 1, with $\unicode[STIX]{x1D706}=1\,,0$ and $-1$ being planar extensional flow, simple shear flow and solid-body rotation respectively. The unit vectors $\mathbf{1}_{x}^{\prime }$ and $\mathbf{1}_{y}^{\prime }$ are unit vectors along the flow and gradient axes of the simple shear flow ($\unicode[STIX]{x1D706}=0$). The trajectories, as described by a unit vector along the spheroid symmetry axis, are closed orbits for $\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{crit}$, where $\unicode[STIX]{x1D706}_{crit}=\unicode[STIX]{x1D705}^{2}(1/\unicode[STIX]{x1D705}^{2})$ for an oblate (a prolate) spheroid of aspect ratio $\unicode[STIX]{x1D705}$. We investigate analytically the orientation dynamics of such a spheroid in the presence of weak inertial effects. The inertial corrections to the angular velocities at $O(Re)$ and $O(St)$, where $Re$ and $St$ are the Reynolds ($Re=\unicode[STIX]{x1D70C}_{f}\dot{\unicode[STIX]{x1D6FE}}L^{2}/\unicode[STIX]{x1D707}$) and Stokes numbers ($St=\unicode[STIX]{x1D70C}_{p}\dot{\unicode[STIX]{x1D6FE}}L^{2}/\unicode[STIX]{x1D707}$) respectively, are derived using a reciprocal theorem formulation. Here, $L$ is the semimajor axis of the spheroid, $\unicode[STIX]{x1D707}$ is the viscosity of the suspending fluid, $\dot{\unicode[STIX]{x1D6FE}}$ is the shear rate, and $\unicode[STIX]{x1D70C}_{p}$ and $\unicode[STIX]{x1D70C}_{f}$ are the particle and fluid densities respectively. A spheroidal harmonics formalism is then used to evaluate the reciprocal theorem integrals and obtain closed-form expressions for the inertial corrections. The detailed examination of these corrections is restricted to the aforementioned Stokesian closed-orbit regime ($\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{crit}$). Here, even weak inertia, for asymptotically long times, of $O(1/(\dot{\unicode[STIX]{x1D6FE}}Re))$ or $O(1/(\dot{\unicode[STIX]{x1D6FE}}St))$, will affect the leading-order orientation distribution on account of the indeterminate nature of the distribution across orbits in the Stokes limit. For $\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{crit}$, inertia results in a drift across the closed orbits in Stokes flow, and this orbital drift is characterized using a multiple time scale analysis. The orbits stabilized by the inertial drift, at $O(Re)$ and $O(St)$, are identified in the $\unicode[STIX]{x1D706}{-}\unicode[STIX]{x1D705}$ plane. For the majority of ($\unicode[STIX]{x1D706},\unicode[STIX]{x1D705}$) combinations, the stabilized orbit is either one confined to the plane of symmetry (the flow-gradient plane) of the ambient flow (the tumbling orbit) or one where the spheroid is aligned with the ambient vorticity vector (the spinning orbit). However, for some ($\unicode[STIX]{x1D706},\unicode[STIX]{x1D705}$) combinations, depending on the initial orientation, the orbit stabilized can be either the spinning or the tumbling orbit, since both orbits have non-trivial basins of attraction, separated by a pair of unstable (repelling) limit cycles, on the unit sphere of orientations. A stochastic orientation decorrelation mechanism in the form of rotary Brownian motion, characterized by a Péclet number, $Pe_{r}$ ($Pe_{r}=\dot{\unicode[STIX]{x1D6FE}}/D_{r}$, where $D_{r}$ is the rotary Brownian diffusivity), is included to eliminate the aforementioned dependence on the initial orientation distribution for certain ($\unicode[STIX]{x1D706}$, $\unicode[STIX]{x1D705}$) combinations. The unique steady-state orientation distribution determined by the combined effect of Brownian motion and inertia is obtained by solving a closed-orbit-averaged drift–diffusion equation. The steady-state orientation dynamics of an inertial spheroid in a planar linear flow, in the presence of weak thermal orientation fluctuations, has similarities to the thermodynamic description of a one-component system. Thus, we identify a tumbling–spinning transition in a $C{-}\unicode[STIX]{x1D705}{-}Re\,Pe_{r}$ space. Here, $C$ is the orbital coordinate that acts as a label for the closed orbits in the Stokes limit. This transition implies hysteretic orientation dynamics in certain regions in the $C$–$\unicode[STIX]{x1D705}$–$Re\,Pe_{r}$ space, although the hysteretic volume shrinks rapidly on either side of simple shear flow. In the hysteretic region, one requires exceedingly large times to achieve the unique steady-state distribution (underlying the thermodynamic interpretation), and for durations relevant to experiments, the system may instead attain an initial-condition-dependent metastable distribution.

Diffusion-influenced reaction rates for active "sphere-prolate spheroid" pairs and Janus dimers.

The purpose of this paper is twofold. First, we provide a concise introduction to the generalized method of separation of variables for solving diffusion problems in canonical domains beyond conventional arrays of spheres. Second, as an important example of its application in the theory of diffusion-influenced reactions, we present an exact solution of the axially symmetric problem on diffusive competition in an array of two active particles (including Janus dumbbells) constructed of a prolate spheroid and a sphere. In particular, we investigate how the reaction rate depends on sizes of active particles, spheroid aspect ratio, particles' surface reactivity, and distance between their centers.

Path instabilities of oblate spheroids

In the present paper, we investigate the path instabilities and the transition scenario for oblate homogeneous spheroids falling (or ascending) freely in a quiescent and unconfined Newtonian fluid under the action of gravity, buoyancy and hydrodynamic forces. The problem depends on three independent external parameters: the aspect ratio $\unicode[STIX]{x1D712}=d/a$, where $d$ is the diameter and $a$ the length of the axisymmetry axis of the spheroid; the non-dimensionalized mass $m^{\ast }=m/(\unicode[STIX]{x1D70C}d^{3})$, $m$ being the mass of the spheroid and $\unicode[STIX]{x1D70C}$ the fluid density; and the Galileo number, defined as $G=\sqrt{(m^{\ast }-V^{\ast })gd^{3}}/\unicode[STIX]{x1D708}$. In the definition of the Galileo number, $V^{\ast }=V/d^{3}$ is the non-dimensionalized volume, $g$ the gravitational acceleration and $\unicode[STIX]{x1D708}$ is the kinematic viscosity. Asymptotic solutions (regimes) are investigated in seven $\unicode[STIX]{x1D712}=\text{const}.$ parameter planes going from $\unicode[STIX]{x1D712}=10$ (very flat spheroid) to $\unicode[STIX]{x1D712}=1.1$ (an almost spherical shape), for $m^{\ast }$ going from 0 to 5 and Galileo numbers up to 300 (i.e. Reynolds numbers roughly up to 500). The obtained results provide a link between the known scenario of a homogeneous disk and that, well known, of a sphere. The scenario of the flat spheroid of aspect ratio 10 has many common features with that of an infinitely thin disk, but the finite thickness brings about significant quantitative differences. At the opposite side of the investigated aspect ratio interval, the dynamics of the spheroid of aspect ratio 1.1 is found very close to that of a perfect sphere except for small density ratios (smaller than approximately 0.5). Very light spheroids of aspect ratio 1.1 move along vertical zig-zagging trajectories. At intermediate aspect ratios, the strong subcritical effects and the characteristic zig-zagging and fluttering motion, typical for flat bodies, progressively disappear. The tumbling regime remains remarkably stable and is shown to be present down to $\unicode[STIX]{x1D712}=2$. An interesting result consists in the evidence of the first two bifurcations typical for the sphere scenario (leading to steady oblique and oblique oscillating trajectories) present both for very flat and thick spheroids but absent at intermediate aspect ratios. The results pertaining to spheroids might be more useful in practical applications than those obtained for too idealized thin disks and perfect spheres.

The effect of inertia on the time period of rotation of an anisotropic particle in simple shear flow

We calculate the leading-order correction to the time period of rotation of a neutrally buoyant spheroid of arbitrary aspect ratio, in a simple shear flow ($\boldsymbol{u}^{\infty }=\dot{\unicode[STIX]{x1D6FE}}y\mathbf{1}_{1}$; $\mathbf{1}_{1}$ is the unit vector in the flow direction, $y$ being the coordinate along the gradient direction), in its long-time orbit set up by the weak fluid inertial drift at $O(Re)$. Here, $Re$ is the microscale Reynolds number, a dimensionless measure of the fluid inertial effects on the length scale of the spheroid, and is defined as $Re=\dot{\unicode[STIX]{x1D6FE}}L^{2}\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D707}$, where $L$ is the semimajor axis of the spheroid, $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D70C}$ are respectively the viscosity and density of the fluid, and $\dot{\unicode[STIX]{x1D6FE}}$ is the shear rate. This long-time orbit is the tumbling orbit for prolate spheroids; for oblate spheroids, it is the spinning orbit for aspect ratios greater than $0.137$, and can be either the tumbling or the spinning orbit for oblate spheroids of aspect ratios less than $0.137$. We also calculate the leading-order correction to the time period of rotation of a neutrally buoyant triaxial ellipsoid in a simple shear flow, rotating with its intermediate principal axis aligned along the vorticity of the flow; the latter calculation is in light of recent evidence, by way of numerical simulations (Rosen, PhD dissertation, 2016, Stockholm), of the aforementioned rotation being stabilized by weak inertia. The correction to the time period for arbitrary $Re$ is expressed as a volume integral using a generalized reciprocal theorem formulation. For $Re\ll 1$, it is shown that the correction at $O(Re)$ is zero for spheroids (with aspect ratios of order unity) as well as triaxial ellipsoids in their long-time orbits. The first correction to the time period therefore occurs at $O(Re^{3/2})$, and has a singular origin, arising from fluid inertial effects in the outer region (distances from the spheroid or triaxial ellipsoid of the order of the inertial screening length of $O(LRe^{-1/2})$), where the leading-order Stokes approximation ceases to be valid. Since the correction comes from the effects of inertia in the far field, the rotating spheroid (triaxial ellipsoid) is approximated as a time-dependent point-force-dipole singularity, allowing for the reciprocal theorem integral to be evaluated in Fourier space. It is shown for all relevant cases that fluid inertia at $O(Re^{3/2})$ leads to an increase in the time period of rotation compared with that in the Stokes limit, consistent with the results of recent numerical simulations at finite $Re$. Finally, combination of the $O(Re^{3/2})$ correction derived here with the $O(Re)$ correction derived earlier by Dabade et al. (J. Fluid Mech., vol. 791, 2016, 631703) yields a uniformly valid description of the first effects of inertia for spheroids of all aspect ratios, including prediction of the arrest of rotation for extreme-aspect-ratio spheroids.

Fully-resolved prolate spheroids in turbulent channel flows: A lattice Boltzmann study

Particles are present in many natural and industrial multiphase flows. In most practical cases, particle shape is not spherical, leading to additional difficulties for numerical studies. In this paper, DNS of turbulent channel flows with finite-size prolate spheroids is performed. The geometry includes a straight wall-bounded channel at a frictional Reynolds number of 180 seeded with particles. Three different particle shapes are considered, either spheroidal (aspect ratio λ=2 or 4) or spherical (λ=1). Solid-phase volume fraction has been varied between 0.75% and 1.5%. Lattice Boltzmann method (LBM) is used to model the fluid flow. The influence of the particles on the flow field is simulated by immersed boundary method (IBM). In this Eulerian-Lagrangian framework, the trajectory of each particle is computed individually. All particle-particle and particle-fluid interactions are considered (four-way coupling). Results show that, in the range of examined volume fractions, mean fluid velocity is reduced by addition of particles. However, velocity reduction by spheroids is much lower than that by spheres; 2% and 1.6%, compared to 4.6%. Maximum streamwise velocity fluctuations are reduced by addition of particle. By comparing particle and fluid velocities, it is seen that spheroids move faster than the fluid before reaching the same speed in the channel center. Spheres, on the other hand, move slower than the fluid in the buffer layer. Close to the wall, all particle types move faster than the fluid. Moreover, prolate spheroids show a preferential orientation in the streamwise direction, which is stronger close to the wall. Far from the wall, the orientation of spheroidal particles tends to isotropy.

Optical Bessel tractor beam on active dielectric Rayleigh prolate and oblate spheroids

Optical Bessel tractor beams, designed to produce a negative pulling force on a particle, are gaining increased attention for applications in noncontact remote sampling, particle manipulation, and handling, to name some examples. In the long-wavelength (Rayleigh) limit, known also as the electric dipole approximation, earlier investigations demonstrated that a zeroth-order Bessel beam incident upon a passive dielectric sphere (i.e., no radiating sources in its core) always acts as a repulsor beam, which causes the particle to be pushed away from the source in the forward direction of the linear momentum. In contrast to what has already been established, this work shows that the incident wave field can act as a tractor beam (where a small spheroid is pulled backwards towards the source due to a negative attractive force) in the dipole approximation (Rayleigh) limit, provided that the particle is made of an active material, i.e., a dielectric spheroid acting as an oscillating source for which the extinction energy efficiency is negative. Numerical computations for the Cartesian components of the optical radiation force on active prolate and oblate spheroids with arbitrary orientation are performed. Emphasis is placed on the emergence of the tractor beam behavior and its dependence upon the half-cone angle, the polarization type of the incident beam, the spheroid aspect ratio, as well as its orientation in space. The analysis is extended to calculate the Cartesian components of the spin radiation torque, which causes a rotation of the spheroid around its center of mass in either the counterclockwise or the clockwise (negative) direction of spinning. Unlike the case of a sphere, the optical spin torque arises for a nonabsorptive oblate or prolate spheroid with arbitrary orientation in the field of a zeroth-order Bessel beam. Potential applications in optically engineered metamaterials, optical tractor beams, tweezers, particle manipulation, rotation, and handling would benefit from the results of this study.

Analytical treatment on the effective material properties of a composite material with spheroidal and ellipsoidal inhomogeneities in an isotropic matrix

Effective material properties of a composite with spheroidal and ellipsoidal inhomogeneities in an isotropic matrix are investigated analytically using the dilute approximation and the Mori–Tanaka approximation together with the Eshelby's equivalent inclusion method. Both uniaxially aligned and uniformly randomly oriented spheroidal and ellipsoidal inhomogeneities are treated. For a spheroid, both oblate and prolate spheroidal inhomogeneities are considered. It is analytically shown that a composite with uniaxially aligned anisotropic ellipsoidal inhomogeneities in an isotropic matrix is anisotropic in general in thermal conductivity. It is also analytically shown that a composite with uniformly randomly oriented anisotropic ellipsoidal inhomogeneities in an isotropic matrix is exactly isotropic in thermal conductivity. Various special cases are also treated for the effective thermal conductivity of a composite with ellipsoidal and spheroidal inhomogeneities. Similar results are also obtained for the effective elastic moduli. Newly obtained expressions for the effective elastic moduli of a composite with isotropic spheroidal inhomogeneities are rather involved. Conversely, an effective thermal conductivity of a composite with anisotropic ellipsoidal inhomogeneities is relatively simple. An effective thermal conductivity of a composite with isotropic spheroidal inhomogeneities reduces to a known result (Kerner, E. H. [1956] “The electrical conductivity of composite media,” Proceedings of the Physical Society London Section B 69, 802–807; Hashin, Z. and Shtrikman, S. [1962] “A variational approach to the theory of the effective magnetic permeability of multiphase materials,” Journal of Applied Physics 33, 3125–3131.) as the spheroid aspect ratio approaches 1 (i.e., a sphere). The effective thermal conductivity of a composite with uniformly randomly oriented isotropic spheroidal inhomogeneities in an isotropic matrix obtained in this paper as a special case is similar to the one obtained by Hatta and Taya (Hatta, H. and Taya, M. [1985] “Effective thermal conductivity of a misoriented short fiber composite,” Journal Applied Physics 58, 2478–2486.) in some respects, but is different. Numerical results are shown for a composite with oblate spheroidal voids in an isotropic matrix.