Pathlines around freely rotating spheroids in simple shear flow

The pathlines around oblate and prolate spheroids freely rotating in shear flow according to Jeffery's equations have been calculated numerically. When the spheroid is aligned with the vorticity axis, open and closed pathlines exist separated by a surface of limiting pathlines. This is very similar to pathlines around spheres and similarly aligned (infinite) cylinders. For spheroids with an arbitrary orientation, four kinds of pathline exist: (i) closed pathlines: (ii) open (single pass) pathlines; (iii) transient orbits; and (iv) permanent non-closed orbits. In general the permanent (closed and non-closed) orbits are separated from the open pathlines by a region occupied by transient orbits.The relevance of pathlines around spheroids to problems of heat and mass transfer and particle deposition in flowing sols is discussed.     

Particle shape effect on hydrodynamics and heat transfer in spouted bed: A CFD–DEM study

Spouted bed has drawn much attention due to its good heat and mass transfer efficiency in many chemical units. Investigating the flow patterns and heat and mass transfer inside a spouted bed can help optimize the spouting process. Therefore, in this study, the effects of particle shape on the hydrodynamics and heat transfer in a spouted bed are investigated. This is done by using a validated computational fluid dynamics–discrete element method (CFD–DEM) model, considering volume–equivalent spheres and oblate and prolate spheroids. The results are analysed in detail in terms of the flow pattern, microstructure, and heat transfer characteristics. The numerical results show that the prolate spheroids (Ar = 2.4) form the largest bubble from the beginning of the spouting process and rise the highest because the fluid drag forces can overcome the interlocking and particle–particle frictional forces. Compared with spherical particles, ellipsoidal spheroids have better mobility because of the stronger rotational kinetic energy resulting from the rough surfaces and nonuniform torques. In addition, the oblate spheroid system exhibits better heat transfer performance benefiting from the larger surface area, while prolate spheroids have poor heat transfer efficiency because of their orientation distribution. These findings can serve as a reference for optimizing the design and operation of complex spouted beds.     

Numerical investigation on motion of an ellipsoidal particle inside confined microcavity flow

The behaviors of a neutrally buoyant ellipsoidal particle in vortical flow confined by a microcavity are numerically studied using the Lattice-Boltzmann method. For specific initial position, an isolated ellipsoid may develop a stable limit cycle orbit inside microcavity due to the interaction between particle and the carrier flow. It is observed that ellipsoidal particles of different shapes exhibit two different stable rotational modes depending on the initial orientation and lateral position. A prolate spheroid tends to enter a tumbling mode whereas an oblate spheroid is apt to achieve a rolling mode. The evolution of rotational velocities along the stable orbit is also analyzed for particles of different shapes.     

Physically Nonlinear Ellipsoidal Inclusion in a Linearly Elastic Medium

This paper considers a physically nonlinear ellipsoidal inclusion in an elastic space loaded at infinity by uniform external forces. Relations are obtained that link the stresses and strains at infinite points of the medium and in the inclusion (in the latter, a homogeneous stress–strain state occurs). Some examples, in particular, inclusions in the shape of oblate and prolate spheroids exhibiting nonlinear creep properties, are discussed.     

The three-dimensional fundamental solution to Stokes flow in the oblate spheroidal coordinates with applications to multiples spheroid problems

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Stokes flow between two confocal rotating spheroids with slip

The steady axisymmetric flow problem of a viscous fluid confined between two confocal spheroids that are rotating about their axis of revolution with different angular velocities is considered. A linear slip, of Basset type, boundary condition on both surfaces of the spheroidal particle and the container is used. Under the Stokesian assumption, a general solution is constructed from the superposition of basic solutions in prolate and oblate spheroidal coordinates. The boundary conditions on the particle’s surface and spheroidal container are satisfied by a collocation technique. The torque exerted on the spheroidal particle by the fluid is evaluated with good convergence for various values of the slip parameters, the relative angular velocity and aspect ratios of the spheroids. The limiting case of no-slip is in good agreement with the available values in the literature.     

Translational Steady Fall of Symmetric Bodies in a Navier-Stokes Liquid, with Application to Particle Sedimentation

Let B \cal B ; be a homogeneous body of revolution around an axis a, with fore-and-aft symmetry. Typical examples are bodies of constant density having the shape of cylinders of circular cross-section, of prolate and oblate spheroids, etc. In this paper we prove that, provided a certain geometric condition is satisfied, the only possible orientations that B \cal B ; can eventually achieve when dropped in a Navier-Stokes fluid under the action of the acceleration of gravity g and at a small and nonzero Reynolds number, is with a either parallel or perpendicular to g. This result is obtained by a rigorous calculation of the torque exerted by the fluid on the body. We also show that the above geometric condition is certainly satisfied if B \cal B ; is a prolate spheroid. Moreover, in this case, we prove, by a "quasi-steady" argument, that, at first order in 5, the configuration with a perpendicular to g is stable to small disorientation, while the other is unstable, in accordance with experiments.     

Determination of stresses in ellipsoidal rigid inclusions

A problem of determining stresses in isolated ellipsoidal rigid inclusions contained in an isotropic elastic space exposed to the impact of external forces uniformly distributed at infinity is considered. Examples of inclusions in the form of oblate and prolate spheroids are studied when the problem has a unique solution.     

Void-shape effects on strength properties of nanoporous materials

In this paper, strength properties of nanoporous materials with spheroidal nanocavities are investigated via a Molecular Dynamics approach applied to a nanovoided aluminium single crystal, in the case of a fixed porosity level, and for prolate, oblate and spherical void shapes. Estimates of the effective strength domain are provided, by considering several mechanical loadings including axisymmetric and shear strain-rate states. Void-shape effects are quantified for different values of the void aspect ratio, mainly resulting in an overall weakening of the sample as the spheroidal nanovoid assumes either an oblate or a prolate shape, in comparison to the case of a spherical void. Finally, it is observed that the computed strength profiles exhibit the following specific features: (i) a strong dependence on the hydrostatic, second-order and third-order deviatoric stress invariants, (ii) more significant void-shape effects for triaxial-expansion stress states with a small hydrostatic component, and (iii) a more pronounced influence of the spheroid shape, as the aspect ratio is varied, in the presence of an oblate nanovoid rather than of a prolate one.     

Dynamics of disk-like particles in turbulent vertical channel flow

The dynamical behavior of inertial disk-like particles in turbulent vertical channel flow is investigated by an Eulerian–Lagrangian point-particle approach. Gravity effects on distribution, translation, rotation and orientation statistics of non-spherical particles modeled as oblate spheroids have been studied both in an upward and a downward flow and compared with results obtained in the absence of gravity. Altogether 12 different particle classes have been studied, with inertia and shape parameterized by means of Stokes number St and aspect ratio λ ≤ 1. The St = 5 disk-like particles distribute more evenly across the channel in upward than in downward flow. The gravity effect on the particle concentration diminishes with large inertia and the spheroid shape has only a modest influence. Although the gravity significantly affects the streamwise and wall-normal mean slip velocities with increasing inertia, the particle shape rarely has any impact on the translational motion, except for the mean wall-normal velocity. The fluctuations of the velocity of disk-like particles are mainly ascribed to inertia, whereas the gravity and shape only have marginal effects. The presence of gravity is moreover found to have a negligible effect on the particles’ orientation and rotation, in spite of the striking effect of λ on the orientation and rotation seen in the near-wall region. The tendency of the disks to align their symmetry axis orthogonal to the fluid vorticity in the channel center is stronger for particles with modest inertia. In the near-wall region, however, oblate spheroids preferentially align with the fluid vorticity for St >> 1. The observed behavior is believed to be caused by the influence of the gravity force on the turbophoresis; i.e. that inertial particles move towards low-turbulence regions.     

Scattering of acoustic waves by elastic and viscoelastic obstacles immersed in a fluid

A scattering or T-matrix approach is presented for studying the scattering of acoustic waves by elastic and viscoelastic obstacles immersed in a fluid. A Kelvin-Voigt model is used to obtain the complex elastic moduli of the viscoelastic solid. The T-matris formulation is somewhat complicated because the wave equations and fields are quite different in the solid and fluid regions and are coupled by continuity conditions at the interface. We have obtained fairly extensive numerical results for prolate and oblate spheroids for a variety of scattering geometries. The backscattering, bistatic, absorption and extinction cross-section are presented as a function of the frequency of the incident wave.     

Creeping flow past and within a permeable spheroid

Flow past and within an isolated permeable spheroid directed along its axis of symmetry is studied. The flow velocity field is solved using the Stokes creeping flow equations governing the fluid motion outside the spheroid, and the Darcy equation within the spheroid. Expressions for the hydrodynamic resistance experienced by oblate and prolate spheroids are derived and analyzed. The limiting cases of permeable circular disks and elongated rods are examined. It is shown that the spheroid’s resistance varies significantly with its aspect ratio and permeability, expressed via the Brinkman parameter.     

Dynamics of heteroclinic tangles with an unbroken saddle connection

This paper studies a second-order differential equation with two heteroclinic solutions to two saddle fixed points. When an equation is periodically perturbed, one heteroclinic solution generates tangle while the other remains unbroken. We illustrate chaotic dynamics in the sense of Smale horseshoes and Hénon-like attractors with SRB measures. More explicitly, we obtain three different dynamical phenomena, namely the transient heteroclinic tangles containing no physical measures, heteroclinic tangles dominated by sinks representing stable dynamical behavior, and heteroclinic tangles with Hénon-like attractors admitting SRB measures representing chaos. We also demonstrate that three types of phenomena repeat periodically as the forcing magnitude goes to zero.     

Rapid distortion of turbulent structures

A simple two scale rapid distortion model of turbulence is used to investigate the generation of coherent structures and to explain some dynamical effects (vorticity alignment with the intermediate eigenvector of the rate of strain, and vorticity production) which have been observed in recent Direct Numerical Simulations (Vincent & Meneguzzi 1991, Ashurstet al. 1987).A three dimensional homogeneous, isotropic, turbulent velocity field with the Von Karman energy spectrum is generated from random Fourier modes (Kinematic Simulation), and dynamics are added by subjecting this flow to a variety of plane large scale distortions calculated using Rapid Distortion Theory (RDT). Five non-overlapping zones (eddy, donor, convergence, shear and streaming regions) are defined. Eddy, convergence and donor regions increase with the proportion of rotational straining by the large scales, while stream regions increase with irrotational distortion. Shear regions show the largest overall change in volume.After large scale irrotational straining the small scale vorticity aligns with themiddle eigenvector of the small scale rate of strain, but with the largest eigenvector of the large scale rate of strain. There is no net production (destruction) of vorticity except under large scale irrotational strain in regions with two positive (negative) eigenvalues of the small scale rate of strain. These vorticity alignment and production results may be deduced analytically from Rapid Distortion by assuming that the initial turbulence is homogeneous and isotropic.     

Stokes flow in a cylindrical tube containing a line of spheroidal particles

Viscous flow in a circular cylindrical tube containing an infinite line of rigid spheroidal particles equally spaced along the axis of the tube is considered for (a) uniform axial translation of the spheroids (b) flow past a line of stationary spheriods and (c) flow of the suspending fluid and spheroids under an imposed pressure gradient. The fluid is assumed to be incompressible and Newtonian. The Reynolds number is assumed to be small and the equations of creeping flow are used. Two types of solutions are developed: (i) an exact solution in the form of an infinite series which is valid for ratios of the spheroid diameter to the tube diameter up to 0.80, (ii) an approximate solution using lubrication theory which is valid for spheroids which nearly fill the tube. The drag on each spheroid and the pressure drop are computed for all cases. Both prolate and oblate spheroids are considered. The results show that the drag and pressure drop depend on the spheroidal diameter perpendicular to the axis of tube primarily and the effects of the spheroidal thickness and spacing are secondary. The results are of interest in connection with mechanics of capillary blood flow, sedimentation, fluidized beds, and fluid-solid transport.     

Indirect boundary element method for unsteady linearized flow over prolate and oblate spheroids and hemispheroidal protuberances

The indirect boundary element method was used to study the hydrodynamics of oscillatory viscous flow over prolate and oblate spheroids, and over hemispheroidal bodies hinged to a plate. Analytic techniques, such as spheroidal coordinates, method of images, and series representations, were used to make the numerical methods more efficient. A novel method for computing the hydrodynamic torque was used, since for oscillatory flow the torque cannot be computed directly from the weightings. Instead, a Green's function for torque was derived to compute the torque indirectly from the weightings. For full spheroids, the method was checked by comparing the results to exact solutions at low and high frequencies, and to results computed using the singularity method. For hemispheroids hinged to a plate, the method for low frequencies was checked by comparing the results to previous results, and to exact solutions at high frequencies. Copyright © 2004 John Wiley & Sons, Ltd.     

The motion of a slightly deformed sphere in a viscoelastic fluid

Summary To study the motion of an arbitrary near sphere immersed in a homogeneous shear flow of an incompressible viscoelastic fluid we impose the restriction that the flow is dynamically and rheologically slow. This allows us to derive an expression for the hydrodynamic force and couple, respectively, which are exerted upon the particle. With the exception of elongational flows marked differences to the behavior in a newtonian fluid show up: sedimentation in a quiescent fluid is accompanied by a rotation until a stable terminal orientation is attained. For prolate spheroids the symmetry axis thus ends up parallel to the direction of the external force and perpendicular to it if the spheroid is oblate. In simple shear the difference between prolate and oblate spheroids manifests itself in the direction in which the rotating symmetry axis drifts (orbit-drift): prolate spheroids drift towards the orbitC = 0 while for oblate ones the drift is towardsC = 0. For deviations from the spheroidal shape (but still fore-aft symmetry) another orbitC ^* comes into play. Some particles drift towardsC ^* while others towardsC = 0 if initiallyC < C ^* and towardsC = if initiallyC > C ^*. If no longer matters whether the particle is slender or not. Although this is perhaps the most interesting result obtained one should also mention the behavior of an ovoid in simple shear. If the symmetry axis is parallel to the vorticity vector the resulting translational slip velocity causes the particle to migrate out of the flow-shear plane in the direction of its pointed end.

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Zusammenfassung Um die Bewegung eines nahezu kugelförmigen Teilchens, das sich in einer homogenen Scherströmung einer viskoelastischen Flüssigkeit befindet, theoretisch erfassen zu können, beschränken wir uns auf dynamisch und rheologisch langsame Strömungen. Das ermöglicht uns, Ausdrücke für die hydrodynamische Kraft und das Drehmoment zu berechnen. Mit Ausnahme von Dehnströmungen ergeben sich daraus fundamentale Unterschiede gegenüber dem Verhalten in einer newtonschen Flüssigkeit. So wird die Sedimentation in einer ruhenden Flüssigkeit von einer Rotation begleitet, die das Teilchen in eine stabile Endlage bringt. Bei länglichen Rotationsellipsoiden bedeutet dies, daß sich die Symmetrieachse parallel zur Richtung der äußeren Kraft einstellt, bei abgeplatteten dagegen senkrecht dazu. Ähnliche Unterschiede treten in der einfachen Scherströmung auf. So driftet die Symmetrieachse länglicher Rotationsellipsoide in den OrbitC = 0, währendC = der Endorbit für abgeplattete Rotationsellipsoide ist. Sind dagegen Abweichungen von dieser Körpergestalt vorhanden, wobei aber immer noch Spiegelsymmetrie vorliegt, so tritt ein anderer ausgezeichneter OrbitC ^* auf. Die Symmetrieachsen einiger Teilchen driften in diesen Orbit, während andere in den OrbitC = 0 wandern, wenn anfänglichC < C ^* war, und in den OrbitC = , wenn anfänglichC > C ^* war. Dabei spielt es keine Rolle, ob das Teilchen länglich oder abgeplattet ist. Dies ist wahrscheinlich das interessanteste Resultat, obwohl auch das Verhalten eines Ovoids in der gleichen Strömung Beachtung verdient; Steht die Symmetrieachse senkrecht zur Strömungsscherebene, so zieht die translatorische Gleitungsgeschwindigkeit eine Bewegung aus dieser Ebene in Richtung des spitzen Endes nach sich.

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With 4 figures and 1 table     

Slow flow of a non-Newtonian fluid past a micro-capsule

The streaming motion past a spherical microcapsule is studied. The particle consists of a thin elastic membrane enclosing an incompressible fluid. Since the problem is highly nonlinear, a perturbation solution is sought in the limiting case where the deviation from sphericity is small. Obviously, the capsule remains nearly spherical when λ, the ratio of viscous forces to elastic (shape-restoring) membrane forces is small. In this limit, the rheology of the inside fluid is immaterial and the problem is essentially characterized by three parameters: λ, the Reynolds number Re (interia effect), and the Weissenberg number We (non-newtonian effect). The deformation is obtained explicitly under the restriction We<1, Re<1. It is shown that to leading order, the capsule deforms exactly into a spheroid which can be either oblate or prolate, depending mainly upon the elasticity number We/Re: for We/Re<0.57 the spheroid is oblate, while for We/Re>0.81 a prolate spheroid results. For 0.57<We/Re<0.81 additional details of the rheology of the membrane and of the suspending fluid are needed. The degree of the deformation is governed by the parameters λ Re. All parameters of the problem enter into the expression of the drag force. On a qualitative basis, these results are similar to those for droplets although major differences exist quantitatively.     

Numerical study of the sedimentation of spheroidal particles

The gravity-driven motion of rigid particles in a viscous fluid is relevant in many natural and industrial processes, yet this has mainly been investigated for spherical particles. We therefore consider the sedimentation of non-spherical (spheroidal) isolated and particle pairs in a viscous fluid via numerical simulations using the Immersed Boundary Method. The simulations performed here show that the critical Galileo number for the onset of secondary motions decreases as the spheroid aspect ratio departs from 1. Above this critical threshold, oblate particles perform a zigzagging motion whereas prolate particles rotate around the vertical axis while having their broad side facing the falling direction. Instabilities of the vortices in the wake follow when farther increasing the Galileo number. We also study the drafting-kissing-tumbling associated with the settling of particle pairs. We find that the interaction time increases significantly for non-spherical particles and, more interestingly, spheroidal particles are attracted from larger lateral displacements. This has important implications for the estimation of collision kernels and can result in increasing clustering in suspensions of sedimenting spheroids.     

Generation of amplitude death and rhythmogenesis in coupled hidden attractor system with experimental demonstration

Nonlinear Dynamics - Third-order nonlinear dynamical systems with attractors (one with no fixed point and the other with a stable fixed point) are conjugately coupled. It is observed that the...