Jeffery equation

Concentrations of free Zn in soil solution are strongly dependent upon soil pH. Zinc solubility expressed as free Zn increases with the concentration of LL (Norvel and Lindsay, 1969 Dang et al., 1994 Tiller, 1983 Jeffery and Uren, 1983). Norvel and Lindsay (1969) proposed that the solubility of soil Zn could be described by the equation ... 

There is a class of flow situations, first identified by Jeffery [201] and Hamel [163], for which the flow has self-similar behavior. To realize the similar behavior leading to ordinary-differential-equation boundary-value problems, the analysis is restricted to steady-state, incompressible, constant property flows. After first discussing the classic analysis,... 

Consider the steady-state, fully developed, incompressible flow between parallel disks, such as illustrated in Fig. 5.9. In concert with the Jeffery-Hamel assumptions that were made in the previous configurations, one can assume that only the radial velocity is nonzero. As a consequence the continuity and momentum equations reduce to the following ... 

In concert with Jeffery-Hamel assumptions, we presume that only the u velocity is nonzero. Moreover assume that there are no circumferential variations. As a result the continuity equation reduces to... 

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... 

The distribution function is predicted to oscillate indefinitely in time with a period of T/2. The reason the period is half the Jeffery period is due to the fore-aft symmetry of the particles. The distribution starts out as a random distribution at t = 0 and returns to that state whenever t = rn f /2, where m is an integer. At intermediate times, the equation predicts a net orientation of the particles along the flow direction (the y axis in this example). 

The components of co, co and due to the velocity gradient, have been evaluated for ellipsoids of revolution by Jeffery (55) from the fundamental equations of hydrodynamics ... 

Ding and Gidaspow [16], for example, derived a two-phase flow model starting with the Boltzmann equation for the distribution function of particles and incorporated fluid-particle interactions into the macroscopic equations. The governing equations were derived using the classical concepts of kinetic theory. However, to determine the constitutive equations they used the ad hoc distribution functions proposed by Savage and Jeffery [65]. The resulting macroscopic equations contain both kinetic - and collisional pressures but only the collisional deviatoric stresses. The model is thus primarily intended for dense particle flows. 

The book has been organized into three parts to address the major issues in cosmochemistry. Part I of the book deals with stellar structure, nucleosynthesis and evolution of low and intermediate-mass stars. The lectures by Simon Jeffery outline stellar evolution with discussion on the basic equations, elementary solutions and numerical methods. Amanda Karakas s lectures discuss nucleosynthesis of low and intermediate-mass stars covering nucleosynthesis prior to the Asymptotic Giant Branch (AGB) phase, evolution during the AGB, nucleosynthesis during the AGB phase, evolution after the AGB and massive AGB stars. The slow neutron-capture process and yields from AGB stars are also discussed in detail by Karakas. The lectures by S Giridhar provide some necessary background on stellar classification. 

Figure 7. Comparison of equations predicting the reduced conductivity of random dispersions of monosized spheres with data. Maxwell Eq. (6), Jeffery Eq. (7), Prager Eq. (8), Chiew and Glandt Eq. (9).

The starting point for using an orientation tensor to predict fiber orientation is the evolution equation of Jeffery for the motion of an isolated fiber in a Newtonian fluid. Jeffery s equation is valid for dilute fiber suspensions where there are no fiber-fiber interactions. 

For an individual fiber, with a unit vector, p, directed along its length, Jeffery s equation for the time evolution of the fiber is as follows ... 

Jeffery s equation was extended to concentrated solutions by Folgar and Tucker who added a diffusion term to account for the fiber-fiber interaction. In terms of the orientation tensor, the Tucker-Folgar equation has the form... 

Ihe motion of a single fiber in the flow can be successfully computed by Jeffery s equation (Jeffery, 1922). However, most composite systems can be considered as a concentrated suspension where the interaction between fibers is not negligible. Folgar and Thcker proposed a model introducing... 

Combining Jeffery s equation with the definition of orientation tensors, the equation for the orientation tensor in concentrated suspensions can be obtained considering fiber-fiber interaction. 

Jeffery CA, Austin PH A new analytic equation of state for hquid water, J Ghem Phys 110(l) 484-496, 1999. 

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... 

Following Einstein, Jeffery (1922) investigated the motion of non-spherical particles (rigid ellipsoidal particles) in a shear field of Newtonian liquid, on the basis of the creeping flow equation, and obtained the following expression for the bulk viscosity ... 

In semidilute suspensions the oscillations present in the dilute suspensions are less common. This is a direct result of the contributing factors that can dampen the oscillations being more prevalent as the fiber concentration is increased. The common approach to predict this behavior relies on Jeffery s equation for the fiber orientation with the assumption that the fiber is infinitely long or A. = 1, in which case. 

Jeffery made the first calculations on ellipsoidal particles neglecting BROWNian motion. He solved the hydrodynamic problem, determined how the orientation of an ellipsoid changes with time and from that calculated the surplus dissipation of energy. The viscosity calculated in this way is, however, not constant and depends rather sensitively upon the initial orientations. Jeffery therefore did not give an equation for the viscosity of a suspension of ellipsoids. 

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