Particle axisymmetric

The terminal velocity of axisymmetric particles in axial motion... 

Two particularly useful equations can be derived by applying the thin concentration boundary layer approximation to steady-state transfer from an axisymmetric particle (L2). The particle and the appropriate boundary layer coordinates are sketched in Fig. 1.1. The x coordinate is parallel to the surface x == 0 at the front stagnation point), while the y coordinate is normal to the surface. The distance from the axis of symmetry to the surface is R. Equation (1-38), subject to the thin boundary layer approximation, then becomes... 

A body has a plane of symmetry if the shape is unchanged by reflection in the plane. Orthotropic particles have three mutually perpendicular planes of symmetry. An axisymmetric particle is symmetric with respect to all planes containing its axis, so that it is orthotropic if it has a plane of symmetry normal to the axis, i.e., if it has fore-and-aft symmetry. 

Unlike the sphericity, can be determined from microscopic or photographic observation. Use of is only justified on empirical grounds, but it has the potential advantage of allowing correlation of the dependence of flow behavior on particle orientation. For an axisymmetric particle projected parallel to its axis, is unity. 

Fig. 4.1 Arbitrary axisymmetric particle in steady translation, perpendicular to the axis of symmetry ...

For prolate spheroids, Eq. (4-37) with k — 0.5 again agrees with the limiting exact result for -> oo. The validity of these equations for cylinders is demonstrated in Figs. 4.7 and 4.8. Comparison of Eqs. (4-36) and (4-37) shows that the ratio of C2 to cq tends to 2 as -> 00. This result holds for any axisymmetric particle, while cq < 2c for finite aspect ratios (W2). Consequently a needlelike particle falls twice as fast when oriented vertically at low Re than when its axis is horizontal. 

Fig. 4.9 Bowen and Masliyah correlation for axial resistance of axisymmetric particles — Eq. (4-42) — Spheroids (exact). 

Fig. 4.10 Resistance of axisymmetric particles to translation normal to the axis — Eq. (4-43) --- Spheroids (exact).

For an orthotropic particle in steady translation through an unbounded viscous fluid, the total drag is given by Eq. (4-5). In principle, it is possible to follow a development similar to that given in Section IT.B.l for axisymmetric particles, to deduce the general behavior of orthotropic bodies in free fall. This is of limited interest, since no analytic results are available for the principal resistances of orthotropic particles which are not bodies of revolution. General conclusions from the analysis were given in TLA. 

The conductance of arbitrary axisymmetric particles may be approximated using the correlation given in Fig. 4.13. By analogy with the drag ratio, a conductance factor is defined as... 

Fig. 4.13 Correlation for conductance factor of axisymmetric particles in stagnant media (based on perimeter-equivalent sphere).

Consider a single, freely suspended axisymmetric particle in a homogeneous shear flow held of an incompressible Newtonian liquid. The free suspension condition implies that the net instantaneous force and torque on the particle vanish. There is, however, a finite net force along the axis that one half of the particle exerts on the other, as shown schematically in Fig. 7.25. 

Fig. 7.25 An axisymmetric particle freely rotating in a simple shear field. The force F exerted hy one half of the particle on the other is zero when the main axis is perpendicular to the flow direction it reaches a maximum tensile strength at 45° and it drops to zero at 90°. Then at 135° it will reach maximum compression and return to zero at 180°. If the ellipsoid is at a certain angle to the direction of shear, the same phenomenon takes place, except that the tensile and compressive forces will he smaller and the particle will rotate and wobble. If the agglomerate is spherical it will smoothly rotate and a maximum tensile strength will be generated along an axis at 45° to the direction of shear.

Note that G is time-periodic a non-Brownian axisymmetric particle rotates indefinitely in a shearing flow. This rotation is called a Jeffery orbit (Jeffery 1922). The period P required for a rotation of tt in a Jeffery orbit is ... 

The problem for the interactions upon central collisions of two axisymmetric particles (bubbles, droplets, or solid spheres) at small surface-to-surface distances was first solved by Reynolds" and Taylor for solid surfaces and by Ivanov et for films of uneven thickness. Equation 5.255... 

In the case of two axisymmetric particles moving along the z axis toward each other with velocity = -dhidt Equation 5.255 can be integrated, and from Equation 5.256 the resistance force can be calculated. The latter turns out to be proportional to the velocity and bulk viscosity and depends on the shape in a complex way. For particles with tangentially immobile smfaces and without surface electric charge (u, = U2 = 0, O = 0), Charles and Mason have derived... 

Problem 7-9. Motion of a Force- and Torque-Free Axisymmetric Particle in a General Linear Flow. We consider a force- and torque-free axisymmetric particle whose geometry can be characterized by a single vector d immersed in a general linear flow, which takes the form far from the particle y°°(r) = U00 + r A fl00 + r E00, where U°°, il, and Ex are constants. Note that E00 is the symmetric rate-of-strain tensor and il is the vorticity vector, both defined in terms of the undisturbed flow. The Reynolds number for the particle motion is small so that the creeping-motion approximation can be applied. 

Figure 2.7. Relative drag coefficient for axisymmetric particles moving along the axis. Solid line, the approximate formula (2.6.28) dashed line, the exact solution for an ellipsoid of revolution...

If the velocity of a spherical particle in Stokes settling is always codirected with the gravity force, even for homogeneous axisymmetric particles the velocity is directed vertically if and only if the vertical coincides with one of the principal axes of the translational tensor K. If the angle between the symmetry axis and the vertical is [Pg.85]

If the settling direction is not vertical, this means that a falling particle is subject to the action of a transverse force, which leads to its horizontal displacement. An additional complication is that the center of hydrodynamic reaction (including the buoyancy force) does not coincide with the particle center of mass. In this case, in addition to the translational motion, the particle is subject to rotation under the action of the arising moment of forces (e.g., the somersault of a bullet with displaced center of mass). For axisymmetric particles, this rotation stops when the system the mass center + the reaction center becomes stable, that is, the mass center is ahead of the reaction center. In this case, the settling trajectory becomes stable and rectilinear. 

This idea is an example of what is referred to as effective field theory [270], and it has been used for a host of vastly diverse problems, ranging from black holes in general relativity [271, 272] to finite-size radiation corrections in electrodynamics [273]. The first application in the context of fluid soft surfaces was given by Yolcu et al. [274, 275]. For two axisymmetric particles on a membrane, Yolcu and Desemo showed that Eq. (19) extends as follows [269] ... 

The book by Clift et al. (1978) contains an extensive review on this subject. The treatment is, however, mostly for the axisymmetric particles such as spheroids and cylinders and orthotropic particles such as rectangular parallelepipeds. For particles of arbitrary shape. 

The problem for the interactions upon central collisions of two axisymmetric particles (bubbles, droplets, or solid spheres) at small surface-to-surface distances was first solved by Reynolds [646] and Taylor [653,654] for solid surfaces and by Ivanov et al. [655,656] for films of uneven thickness. Equation 4.266 is referred to as the general equation for films with deformable surfaces [655,656] (see also the more recent reviews [240,657,658]). The asymptotic analysis [659-661] of the dependence of the drag and torque coefficient of a sphere, which is translating and rotating in the neighborhood of a solid plate, is also based on Equation 4.266 applied to the special case of stationary conditions. 

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