Motion of ellipsoids

Fiber motion — Jeffery orbits. The motion of ellipsoids in uniform, viscous shear flow of a Newtonian fluid was analyzed by Jeffery [32, 33] in 1922. For a prolate spheroid of aspect ratio a (defined as the ratio between the major axis and the minor axis) in simple shear flow, u°° = (zj), the angular motion of the spheroid is described... 

Jefferey, The motion of ellipsoidal particles immersed in a viscous fluid, Proc. R. Soc. London, A, 102, 161 (1922). 

Jeffery, G. B. The Motion of Ellipsoidal Particles Immersed in a Viscous... 

Figure 3.5 Rotational diffusional motion of ellipsoids in liquids. The ratio of the friction coefficients calculated using slip boundary conditions to that calculated using stick boundary conditions, for prolate ellipsoids and for oblate ellipsoids, plotted against a/b, the ratio of the shorter axis to the longer axis. From Ref. [1 l,b].

Kim, S., The motion of ellipsoids in a second order fluid, J. Non-Newtonian Fluid Mech. 27 255-269 (1986). 

As noted earlier, in the 1920s Taylor [38] experimentally verified Jeffery s [37] analysis for the motion of ellipsoids. There were subsequent studies by Taylor in the 1930s [39,40]. In the 1950s Mason and his coworkers [41 to 44] made extensive efforts to visualize anisotropic particle motions in dilute suspensions during flow of rigid rod- and disk-shaped particles. They observed a distribution of orbits. 

The viscosity of a suspension of ellipsoids depends on the orientation of the particle with respect to the flow streamlines. The ellipsoidal particle causes more disruption of the flow when it is perpendicular to the streamlines than when it is aligned with them the viscosity in the former case is greater than in the latter. For small particles the randomizing effect of Brownian motion is assumed to override any tendency to assume a preferred orientation in the flow. 

A modified version of the TAB model, called dynamic drop breakup (DDB) model, has been used by Ibrahim et aU556l to study droplet distortion and breakup. The DDB model is based on the dynamics of the motion of the center of a half-drop mass. In the DDB model, a liquid droplet is assumed to be deformed by extensional flow from an initial spherical shape to an oblate spheroid of an ellipsoidal cross section. Mass conservation constraints are enforced as the droplet distorts. The model predictions agree well with the experimental results of Krzeczkowski. 311 ... 

Surface-active contaminants play an important role in damping out internal circulation in deformed bubbles and drops, as in spherical fluid particles (see Chapters 3 and 5). No systematic visualization of internal motion in ellipsoidal bubbles and drops has been reported. However, there are indications that deformations tend to decrease internal circulation velocities significantly (MI2), while shape oscillations tend to disrupt the internal circulation pattern of droplets and promote rapid mixing (R3). No secondary vortex of opposite sense to the prime internal vortex has been observed, even when the external boundary layer was found to separate (Sll). 

We noted above that either solvation or ellipticity could cause the intrinsic viscosity to exceed the Einstein value. Simha and others have derived extensions of the Einstein equation for the case of ellipsoids of revolution. As we saw in Section 1.5a, such particles are characterized by their axial ratio. If the particles are too large, they will adopt a preferred orientation in the flowing liquid. However, if they are small enough to be swept through all orientations by Brownian motion, then they will increase [17] more than a spherical particle of the same mass would. Again, this is very reminiscent of the situation shown in Figure 2.4. 

ORTEP is an acronym For Oak Ridge Thermal Ellipsoid Program, a ccropuier program frequently used in structural analysis. The acronym is often used as a short label to indicate a drawing in which ellipsoids indicate the extern of Ihermal motions Of the atoms... 

Figure 3. Analcime. Apparent thermal-motion probability ellipsoids of the T and 0 atoms in the A structure and the displacements from the symmetrized position obtained by DLS. Ellipsoids are based on thermal parameters reported by Knowles, Rinaldi, and Smith (7) and are scaled to enclose 50% probability. The diagrams were generated with the aid of computer program ortep by C.K. Johnson...

Eq. (3.21) discussed in Section 3.3.2 is only valid if the motion of the molecules under study has no preferential orientation, i.e. is not anisotropic. Strictly speaking, this applies only for approximately spherical bodies such as adamantane. Even an ellipsoidal molecule like trans-decalin performs anisotropic motion in solution it will preferentially undergo rotation and translation such that it displaces as few as possible of the other molecules present. This anisotropic rotation during translation is described by the three diagonal components Rlt R2, and R3 of the rotational diffusion tensor. If the principal axes of this tensor coincide with those of the moment of inertia - as can frequently be assumed in practice - then Rl, R2, and R3 indicate the speed at which the molecule rotates about its three principal axes. 

Applications of optical methods to study dilute colloidal dispersions subject to flow were pioneered by Mason and coworkers. These authors used simple turbidity measurements to follow the orientation dynamics of ellipsoidal particles during transient shear flow experiments [175,176], In addition, the superposition of shear and electric fields were studied. The goal of this work was to verify the predictions of theories predicting the orientation distributions of prolate and oblate particles, such as that discussed in section 7.2.I.2. This simple technique clearly demonstrated the phenomena of particle rotations within Jeffery orbits, as well as the effects of Brownian motion and particle size distributions. The method employed a parallel plate flow cell with the light sent down the velocity gradient axis. 

However, for nonspherical particles, rotational Brownian motion effects already arise at 0(0). In the case of ellipsoidal particles, such calculations have a long history, dating back to early polymer-solution rheologists such as Simha and Kirkwood. Some of the history of early incorrect attempts to include such rotary Brownian effects is documented by Haber and Brenner (1984) in a paper addressed to calculating the 0(0) coefficient and normal stress coefficients for general triaxiai ellipsoidal particles in the case where the rotary Brownian motion is dominant over the shear (small rotary Peclet numbers)—a problem first resolved by Rallison (1978). 

Fig. 3.2. Thermal ellipsoids (at 99% probability) for 1,2,4-triazole by neutron diffraction at 15 K illustrating the relative thermal motion of hydrogen and nonhydrogen atoms. That of the hydrogen bonded H(l) is only slightly less than that of H(3) and H(5), and the corrections of the X-H bond lengths are +0.005 A for N-H versus +0.006 for the C-H bonds at 15 K [199]...

Fio. 1. A stereodiagram showing the structure and conformation of reserpine (6), one of the first crystal-structure determinations without the presence of a heavy atom. The ellipsoids represent the thermal motions of the atoms, drawn at the 50% probability level. 

---