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The paper presents analytical approach to electrostatic field for dielectric spheroids problem. For both oblate and prolate spheroids fields distributions are derived. The variable separation method is applied. Electrostatic force is evaluated with the help of the Maxwell stress tensor generalized method, material force density, coenergy and equivalent dipole. Levitation forces for both oblate and prolate dielectric spheroids versus axis permittivities and spheroids height are presented.

The present paper deals with the photogravitational restricted four-body problem, when the third primary placed at the triangular libration point of the restricted three-body problem is an oblate/prolate body. The third primary m 3 is not influencing the motion of the dominating primaries m 1 and  m 2 . We have studied the motion of  m 4 , moving under the influence of the three primaries m i , i = 1 , 2 , 3 , but the motion of the primaries is not being influenced by infinitesimal mass  m 4 . The aim of this study is to find the locations of the libration points and their stability. We obtain three collinear and five non-collinear libration points when the third body is an oblate spheroid and source of radiation. The collinear libration points are unstable for all the mass parameter. The non-collinear libration points are stable for different mass parameters and oblateness factors. Further there exist 12 libration points, depending on the mass ratio of dominating primaries, the prolateness, and the radiation pressure of the third primary. We have drawn the zero velocity surface to determine the possible allowed boundary region. We observed that for increasing values of the oblateness coefficient A , the corresponding possible boundary region increases where the particle can freely move from one side to another side. Further, for different values of the Jacobi constant C , we can find the boundary region where the particle can move in the possible allowed partitions. The stability region of the libration points expanded due to the presence of the oblateness coefficient and various values of  C . We further apply these findings to the Sun–Jupiter–asteroid–spacecraft system.

A data library is developed containing the scattering, absorption, and polarization properties of ice particles in the spectral range from 0.2 to 100 μm. The properties are computed based on a combination of the Amsterdam discrete dipole approximation (ADDA), the T-matrix method, and the improved geometric optics method (IGOM). The electromagnetic edge effect is incorporated into the extinction and absorption efficiencies computed from the IGOM. A full set of single-scattering properties is provided by considering three-dimensional random orientations for 11 ice crystal habits: droxtals, prolate spheroids, oblate spheroids, solid and hollow columns, compact aggregates composed of eight solid columns, hexagonal plates, small spatial aggregates composed of 5 plates, large spatial aggregates composed of 10 plates, and solid and hollow bullet rosettes. The maximum dimension of each habit ranges from 2 to 10 000 μm in 189 discrete sizes. For each ice crystal habit, three surface roughness conditions (i.e., smooth, moderately roughened, and severely roughened) are considered to account for the surface texture of large particles in the IGOM applicable domain. The data library contains the extinction efficiency, single-scattering albedo, asymmetry parameter, six independent nonzero elements of the phase matrix (P11, P12, P22, P33, P43, and P44), particle projected area, and particle volume to provide the basic single-scattering properties for remote sensing applications and radiative transfer simulations involving ice clouds. Furthermore, a comparison of satellite observations and theoretical simulations for the polarization characteristics of ice clouds demonstrates that ice cloud optical models assuming severely roughened ice crystals significantly outperform their counterparts assuming smooth ice crystals.

The elasto-inertial focusing and rotating characteristics of spheroids in a square channel flow of Oldroyd-B viscoelastic fluids are studied by the direct forcing/fictitious domain method. The rotational behaviours, changes in the equilibrium positions and travel distances are explored to analyse the mechanisms of spheroid migration in viscoelastic fluids. Within the present simulated parameters (1 ≤ Re ≤ 100, 0 ≤ Wi ≤ 2, 0.4 ≤ α ≤3), the results show that there are four kinds of equilibrium positions and six (five) kinds of rotational behaviours for the elasto-inertial migration of prolate (oblate) spheroids. We are the first to identify a new rotational mode for the migration of prolate spheroids. Only when the particles are initially located at a corner and wall bisector, some special initial orientations of the spheroids have an impact on the final equilibrium position and rotational mode. In other general initial positions, the initial orientation of the spheroid has a negligible effect. A higher Weissenberg number means the faster the particles migrate to the equilibrium position. The spheroid gradually changes from the corner (CO), channel centreline (CC), diagonal line (DL) and cross-section midline (CSM) equilibrium positions as the elastic number decreases, depending on the aspect ratio, initial orientation and rotational behaviour of the particles and the elastic number of the fluid. When the elastic number is less than the critical value, the types of rotational modes of the spheroids are reduced. By controlling the elastic number near the critical value, spheroids with different aspect ratios can be efficiently separated.

The flow around different prolate (needle-like) and oblate (disc-like) spheroids is studied using a multi-relaxation-time lattice Boltzmann method. We compute the mean drag coefficient $C_{D,\\unicode[STIX]{x1D719}}$ at different incident angles $\\unicode[STIX]{x1D719}$ for a wide range of Reynolds numbers ( $\\mathit{Re}$ ). We show that the sine-squared drag law $C_{D,\\unicode[STIX]{x1D719}}=C_{D,\\unicode[STIX]{x1D719}=0^{\\circ }}+(C_{D,\\unicode[STIX]{x1D719}=90^{\\circ }}-C_{D,\\unicode[STIX]{x1D719}=0^{\\circ }})\\sin ^{2}\\unicode[STIX]{x1D719}$ holds up to large Reynolds numbers, $\\mathit{Re}=2000$ . Further, we explore the physical origin behind the sine-squared law, and reveal that, surprisingly, this does not occur due to linearity of flow fields. Instead, it occurs due to an interesting pattern of pressure distribution contributing to the drag at higher $\\mathit{Re}$ for different incident angles. The present results demonstrate that it is possible to perform just two simulations at $\\unicode[STIX]{x1D719}=0^{\\circ }$ and $\\unicode[STIX]{x1D719}=90^{\\circ }$ for a given $\\mathit{Re}$ and obtain particle-shape-specific $C_{D}$ at arbitrary incident angles. However, the model has limited applicability to flatter oblate spheroids, which do not exhibit the sine-squared interpolation, even for $\\mathit{Re}=100$ , due to stronger wake-induced drag. Regarding lift coefficients, we find that the equivalent theoretical equation can provide a reasonable approximation, even at high $\\mathit{Re}$ , for prolate spheroids.

This article presents a computational approach for determining the optimal slip velocities on any given shape of an axisymmetric micro-swimmer suspended in a viscous fluid. The objective is to minimize the power loss to maintain a target swimming speed, or equivalently to maximize the efficiency of the micro-swimmer. Owing to the linearity of the Stokes equations governing the fluid motion, we show that this PDE-constrained optimization problem reduces to a simpler quadratic optimization problem, whose solution is found using a high-order accurate boundary integral method. We consider various families of shapes parameterized by the reduced volume and compute their swimming efficiency. Among those, prolate spheroids were found to be the most efficient micro-swimmer shapes for a given reduced volume. We propose a simple shape-based scalar metric that can determine whether the optimal slip on a given shape makes it a pusher, a puller or a neutral swimmer.

The dynamics of sedimenting particles under gravity are surprisingly complex due to the presence of effective long-ranged forces. When the particles are polar with a well-defined symmetry axis and non-uniform density, recent theoretical predictions suggest that prolate objects will repel and oblate ones will weakly attract. We tested these predictions using mass polar prolate spheroids, which are composed of 2 mm spheres glued together. We probed different aspect ratios ($\\kappa$) and centre of mass variations ($\\chi$) by combining spheres of different densities. Experiments were done in both quasi-two-dimensional (2-D) and three-dimensional (3-D) chambers. By optically tracking the motion of single particles, we found that the dynamics were well described by a reduced mobility matrix model that could be solved analytically. Pairs of particles exhibited an effective repulsion, and their separation roughly scaled as $(\\kappa - 1)/\\chi ^{0.39}$, i.e. particles that were more prolate or had smaller mass asymmetry had stronger repulsion effects. In three dimensions, particles with $\\chi >0$ were distributed more uniformly than $\\chi =0$ particles, and the degree of uniformity increased with $\\kappa$, indicating that the effective 2-body repulsion manifests for a large number of particles.

In this study we investigate the sedimentation of prolate spheroids in a quiescent fluid by means of the particle-resolved direct numerical simulation. With the increase of the particle volume fraction $\\phi$ from $0.1\\,\\%$ to $10\\,\\%$, we observe a non-monotonic variation of the mean settling velocity of particles, $\\langle V_s \\rangle$. By virtue of the Voronoi analysis, we find that the degree of particle clustering is highest when $\\langle V_s \\rangle$ reaches the local maximum at $\\phi =1\\,\\%$. Under the swarm effect, clustered particles are found to preferentially sample downward fluid flows in the wake regions, leading to the enhancement of the settling speed. As for lower or higher volume fractions, the tendency of particle clustering and the preferential sampling of downward flows are attenuated. The hindrance effect becomes predominant when the volume fraction exceeds 5 % and reduces $\\langle V_s \\rangle$ to less than the isolated settling velocity. Particle orientation plays a minor role in the mean settling velocity, although individual prolate particles still tend to settle faster in suspensions when they deviate more from the broad-side-on alignment. Moreover, we also demonstrate that particles are prone to form column-like microstructures in dilute suspensions under the effect of wake-induced hydrodynamic attractions. The radial distribution function is higher at a lower volume fraction. As a result, the collision rate scaled by the particle number density decreases with the increasing volume fraction. By contrast, as another contribution to the particle collision rate, the relative radial velocity for nearby particles shows a minor degree of variation due to the lubrication effect.

An experimental study has been conducted on the near wake of a 6 : 1 spheroid, in both uniform and stratified backgrounds. The pitch angle, $\\theta$, was varied from $0^\\circ \\text { to }20^\\circ$. When $\\theta = 0^\\circ$, stratification decreases the characteristic wake element spacing so a characteristic Strouhal number ($St$) increases from 0.32 to 0.4. However, a similar measure scaled on wake momentum thickness shows the wake spacing to converge on those measured for other bluff and streamlined bodies. There is an apparent effect of Reynolds number, which changes the location of separation lines and hence the initial wake thickness. When $\\theta > 0^\\circ$, the wake is a combination of the usual drag wake together with a collection of streamwise vortices that have separated from the body, and this wake geometry can evolve in ways that are measurably different from the zero incidence case. These differences may be limited to the near wake, as the later evolution appears to converge with previous bluff- and streamlined bodies, with normalised wake height, $L_V = 0.5$ and centreline velocity, $\\bar {u}_0 = 0.3$ at $Nt = 10$, as the early wake enters the non-equilibrium regime with similar values to previously studied stratified wakes. In the presence of density stratification, the inclined wake itself generates large-scale internal wave undulations with time scale $2{\\rm \\pi} /N$, even when the background stratification is not strong and a body-based Froude number is $O(10)$. The geometry and strengths of the primary streamwise vortices are not symmetric, mirroring previous results from experiments and computations in the literature.

We derive analytically the angular velocity of a spheroid, of an arbitrary aspect ratio $\\kappa$, sedimenting in a linearly stratified fluid. The analysis demarcates regions in parameter space corresponding to broadside-on and edgewise settling in the limit $Re, Ri_v \\ll 1$, where $Re = \\rho _0UL/\\mu$ and $Ri_v =\\gamma L^3\\,g/\\mu U$, the Reynolds and viscous Richardson numbers, respectively, are dimensionless measures of the importance of inertial and buoyancy forces relative to viscous ones. Here, $L$ is the spheroid semi-major axis, $U$ an appropriate settling velocity scale, $\\mu$ the fluid viscosity and $\\gamma \\ (>0)$ the (constant) density gradient characterizing the stably stratified ambient, with the fluid density $\\rho_0$ taken to be a constant within the Boussinesq framework. A reciprocal theorem formulation identifies three contributions to the angular velocity: (1) an $O(Re)$ inertial contribution that already exists in a homogeneous ambient, and orients the spheroid broadside-on; (2) an $O(Ri_v)$ hydrostatic contribution due to the ambient stratification that also orients the spheroid broadside-on; and (3) a hydrodynamic contribution arising from the perturbation of the ambient stratification whose nature depends on $Pe$; $Pe = UL/D$ being the Péclet number with $D$ the diffusivity of the stratifying agent. For $Pe \\ll 1$, this contribution is $O(Ri_v)$ and orients prolate spheroids edgewise for all $\\kappa \\ (>1)$. For oblate spheroids, it changes sign across a critical aspect ratio $\\kappa _c \\approx 0.41$, orienting oblate spheroids with $\\kappa _c < \\kappa < 1$ edgewise and those with $\\kappa < \\kappa _c$ broadside-on. For $Pe \\ll 1$, the hydrodynamic component is always smaller in magnitude than the hydrostatic one, so a sedimenting spheroid in this limit always orients broadside-on. For $Pe \\gg 1$, the hydrodynamic contribution is dominant, being $O(Ri_v^{{2}/{3}}$) in the Stokes stratification regime characterized by $Re \\ll Ri_v^{{1}/{3}}$, and orients the spheroid edgewise regardless of $\\kappa$. Consideration of the inertial and large-$Pe$ stratification-induced angular velocities leads to two critical curves which separate the broadside-on and edgewise settling regimes in the $Ri_v/Re^{{3}/{2}}$–$\\kappa$ plane, with the region between the curves corresponding to stable intermediate equilibrium orientations. The predictions for large $Pe$ are broadly consistent with observations.