Axisymmetric particles rotation

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 ... 

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.

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. 

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. 

The use of distributed vector spherical wave functions is most effective for axisymmetric particles because, in this case, the T matrix is diagonal with respect to the azimuthal indices. For elongated particles, the sources are distributed on the axis of rotation, while for flattened particles, the sources are distributed in the complex plane (which is the dual of the symmetry plane). 

The use of distributed vector spherical wave functions improves the numerical stability of the null-field method for highly elongated and flattened layered particles. Although the above formalism is valid for nonaxisymmetric particles, the method is most effective for axisymmetric particles, in which case the 2 -axis of the particle coordinate system is the axis of rotation. Applications of the null-field method with distributed sources to axisymmetric layered spheroids with large aspect ratios have been given by Doicu and Wriedt [50]. 

The above system of vector functions is also known as the system of localized vector spherical wave functions. Another system of vector functions which is suitable for analyzing axisymmetric particles with extreme geometries is the system of distributed vector spherical wave functions [49]. For an axisymmetric particle with the axis of rotation along the z-axis, the distributed vector spherical wave functions are defined as... 

A. T. Chwang and T. Y. Wu, Hydromechanics of low-Reynolds number flow, Part 1, Rotation of axisymmetric prolate bodies, J. Fluid Mech. 63, 607-22 (1974) A. T. Chwang and T. Y. Wu, Hydromechanics of low-Reynolds number flow, Part 2, Singularity method for Stokes flows, J. Fluid Mech. 67, 787-815 (1975) A. T. Chwang, Hydromechanics of low-Reynolds number flow, Part 3, Motion of a spheroidal particle in quadratic flows, J. Fluid Mech. 72, 17-34 (1975) A. T. Chwang and T. Y. Wu, Hydromechanics of low-Reynolds number flow, Part 4, Translation of spheroids, J. Fluid Mech. 75, 677-89 (1975). 

A symmetric tensor ft is called a rotational tensor. It depends both on the shape and size of the particle and on the choice of the origin. The rotational tensor characterizes the drag under rotation of the body and has the diagonal form with entries fii, Q2, 3 in the principal axes (the positions of the principal axes of the rotational and translational tensors in space are different). For axisymmetric bodies, one of the major axes (for instance, the first) is parallel to the symmetry axis, and in this case = O3. For a spherical particle, we have fii = fl2 = flj. 

The first particle moves toward the second immobile particle and rotates around the line of centers (see Figure 4.44a). This is an axisymmetric rotation problem (a 2D hydrodynamic problem) which was solved by Jeffery [674] and Stimson and Jeffery [675] for two identical spheres moving with equal velocities along their line of centers. Cooley and O Neill [676,677] calculated the forces for two nonidentical spheres moving with the same speed in the same direction, or alternatively. 

The centrifugal flows considered in this chapter are those dominated by rotatioa The azimuthal component of the velocity is preponderant, that is, Ur ue and Uz Ue in the cylindrical coordinate system. In such a configuration, the flow tends to become two-dimensional in a plane perpendicular to the Oz axis. The Ur and Ue components of the velocity are quasi-independent from coordinate z. The proof of this property goes beyond the scope of this chapter. Our goal is to describe the centrifugation of solid particles in a rotating flow. We simply choose to consider steady-state axisymmetric fluid flows, whose velocity and pressure fields possess the following kinematic characteristics ... 

Taneda [124] studied the flow past a sphere at particle Reynolds numbers between lO and 10, and found that the wake is not axisymmetric and that it rotates slowly and randomly about the sfleam-wise axis. The sphere is thus subject to a random side force. 

Electromagnetic scattering by axisymmetric, composite particles can be computed with the TCOMP routine. An axisymmetric, composite particle consists of several nonenclosing, rotationally symmetric regions with a common axis of symmetry. In contrast to the TMULT routine, TCOMP is based on a formalism which avoid the use of any local origin translation. The scattering... 

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