Jeffery orbit

If u is a unit vector parallel to the axis of symmetry of a spheroidal particle, then in the absence of Brownian motion (Pe — 00) and of interparticle interactions, the time rate of change of u in a flow field is 

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  

P oo for either p — oo, which is the limit of an infinitely thin prolate ellipsoid, or p — 0, which is the limit of an infinitely flat oblate one. In these two limits, the particle rotates until a long axis is parallel to the flow direction, and then rotation slows to a halt. For large, but finite, aspect ratios, the particle rotates slowly when its long axis is nearly parallel to the flow direction, and rapidly otherwise. The Jeffery orbits of rod-like and disklike particles have been observed directly (Anczurowski and Mason 1967a, 1967b) and indirectly by optical dichroism (Frattini and Fuller 1986). 

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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... 

The period of oscillation is shortest for a sphere and increases as the particles become either oblate or prolate. In either case, the motion of the particle will be a periodic orbit where the symmetry axis tumbles about the vorticity axis of the flow and is referred to as a Jeffery orbit. ... 

The Jeffery orbits are deterministic, and the particles will precess indefinitely in the flow. The following effects can perturb a particle and deflect it from an orbit and send it into a new one ... 

It is now necessary to use the Jeffery orbit equations (7.107) to express the current angles, 9 and <)>, existing of time t, in terns of the initial angles, 0Q and < >0. After some... 

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. 

Stress optical coefficient, (7.123), Jeffery orbit constant, (7.108). 

Brownian motion and interparticle interactions can produce deviations from this Jeffery orbit. When particle rotations are disturbed by Brownian motion, the orbits become stochastic... 

In fact, the fiber contribution to the shear viscosity of a fiber suspension at steady state is modest, at most. The reason is that, without Brownian motion, the fibers quickly rotate in a shear flow until they come to the flow direction in this orientation they contribute little to the viscosity. Of course, the finite aspect ratio of a fiber causes it to occasionally flip through an angle of n in its Jeffery orbit, during which it dissipates energy and contributes more substantially to the viscosity. The contribution of these rotations to the shear viscosity is proportional to the ensemble- or time-averaged quantity (u u ), where is the component of fiber orientation in the flow direction and Uy is the component in the shear gradient direction. Figure 6-21 shows as a function of vL for rods of aspect... 

Even very weak hydrodynamic interactions among fibers in the semidilute regime are enough to disturb the Jeffery orbits of individual fibers enough that a steady-state distribution of fiber orientations is obtained after long shearing times without such interactions, no steady state would ever be reached, since the initial fiber orientation would just be revisited over and over with periodicity equal to that of the Jeffery orbit. 

Kamis and Mason 1966 Iso et al. 1996a) and analysis (Leal 1975) show that the Jeffery orbit of a single isolated fiber is modified by a spiraling drift of the fiber axis towards the vorticity direction, which is the stable orientation in this case. This drift can be accounted... 

In shear flows of non-interacting fibers, the fibers exhibit a closed periodie rotation known as a Jeffery orbit with a period Tr = 2n ar + /j, where y is the... 

Huilgol RR, You Z (2006) On the importance of the pressure dependence of viscosity in steady non-isothermal shearing flows of compressible and incompressible fluids and in the isothermal fountain flow. J Non-Newtonian Fluid Mech 136 106-117 Hulsen MA, Van Heel APG, Van den Brule BHAA (1997) Simulation of viscoelastic flows using Brownian configuration fields. J Non-Newtonian Fluid Mech 70 79-101 Ingber MS, Mondy LA (1994) A numerical study of three-dimensional Jeffery orbits in shear flow. J Rheol 38 1829-1843... 

Much of the work on particle rotation at low Rqq follows from the early work of Jeffery (J2) who considered a rigid, neutrally buoyant spheroid subject to the uniform shear field defined by Eq. (10-30). Jeffery showed that the particle center moves with the velocity which the continuous fluid would have at that point in the absence of the particle, while the axis of the spheroid undergoes rotation in one of a family of periodic orbits with angular velocities... 

Polymer processing involves shear flow as well as simple extensional flow. Shear flow is a more complex arrangement with a three-dimensional nature to it (Fig. 3.6) and a weak extensional flow component. Rigid rods in a viscous fluid were first considered theoretically by Jeffery (1922). Jeffery showed that objects exhibited a trajectory often called a Jeffery s orbit which rotated the axis of the... 

A major significance of Jeffery s studies lies in the derivation of equations of anisotropic particle motion in a Newtonian liquid. He predicted that shear flow rotates the disc/rod in shear planes. Jeffery hypothesized that in time the orbital motions would evolve to the one that corresponds to the least viscous dissipation. The motion of single ellipsoids in the shear... 

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. 

Nonetheless, the few rod-shaped clusters that have oriented to the rotational axis, indicated in Fig. 6, show a unique behavior. Their orientation hints at the pioneer works by Jeffery on spheroids in shear flow, where for the dilute limit, these spheroids tend to have periodic orbits. This period scales with the inverse of the shear rate and becomes longer with an increasing deviation from sphericity [62], Since aggregates tend to retain their orientation, we would like to propose that if the orientation of clusters is variable, a more uniform compaction can be expected. Yet these elongated shapes then only form by keeping the rotational axis constant throughout the simulation, thus giving an anisotropic compaction. 

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