Dynamics of Particulates

The dynamics of rigid, isolated spheroids was first analyzed for the case of shear flow by Jeffery[95]. When subject to a general linear flow with velocity gradient tensor G, the time rate of change of the unit vector defining the orientation of the symmetry axis of such a particle will have the following general form, 

and C depend only on the aspect ratio of the particles, p =, where a is the length 

In the limit of an infinite aspect ratio (p - °°), this result tends to the dynamics of the rigid dumbbell shown in equation (7.61). 

It is instructive to present the solution to equation (7.106) for the case of simple shear flow. For a spheroid oriented with its symmetry axis defined by the polar angle 9 relative to the z axis, and azimuthal axis ( measured in the (x, y) plane relative to the x axis, equation (7.106) produces the following two equations for a simple shear flow of the form, v = G (0, x, 0)  

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.  

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TABLE 20-37 Methods of Characterizing Wetting Dynamics of Particulate Systems ... 

Barnea E, J Mizrahi. A generalized approach to the fluid dynamics of particulate systems, Part I. Chem Eng J 5 171-189, 1973. 

Method of Lines. The method of lines is used to solve partial differential equations (12) and was already used by Cooper (I3.) and Tsuruoka (l4) in the derivation of state space models for the dynamics of particulate processes. In the method, the size-axis is discretized and the partial differential a[G(L,t)n(L,t)]/3L is approximated by a finite difference. Several choices are possible for the accuracy of the finite difference. The method will be demonstrated for a fourth-order central difference and an equidistant grid. For non-equidistant grids, the Lagrange interpolation formulaes as described in (15 ) are to be used. 

Lock, M. A. 1994. Dynamics of particulate and dissolved organic matter over the substratum... 

Bianchi, T.S., and Argyrou, M.E. (1997) Temporal and spatial dynamics of particulate organic carbon in the Lake Pontchartrain estuary, southeast Louisiana, USA. Estuar. Coastal Shelf Sci. 45, 557-569. 

Archer SD, Smith GC, Nightingale PD, Widdicombe CE, Tarran GA, Rees AP, Burkill PH (2002) Dynamics of particulate dimethylsulphoniopropionate during a Lagrangian experiment in the northern North Sea. Deep-Sea Res Part II 49 2979-2999... 

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