Hill’s vortex

The FRC in the pilot reactor above is assumed to correspond roughly to a spherical Hill s vortex equilibrium in a uniform external field. Weak mirror and quadrupole fields are assumed to stabilize the ring. The ring current is assumed to... 

Fig. 2 Comparison of classical spheromak configuration (solid line) with Hill s vortex configuration (dotted line).

The internal motion given by Eq. (3-8) is that of Hill s spherical vortex (H6). Streamlines are plotted in Figs. 3.1 and 3.2 for k = 0 and k = 2, and show the fore-and-aft symmetry required by the creeping flow equation. It may also be noted in Fig. 3.2 that the streamlines are not closed for any value of k, the solution predicts that outer fluid is entrained along with the moving sphere. This entrainment, sometimes known as drift, is infinite in creeping flow. This problem is discussed further in Chapter 4. 

Treatment of liquid drops is considerably more complex than bubbles, since the internal motion must be considered and internal boundary layers are difficult to handle. Early attempts to deal with boundary layers on liquid drops were made by Conkie and Savic (C8), McDonald (M9), and Chao (C4, W7). More useful results have been obtained by Harper and Moore (HIO) and Parlange (PI). The unperturbed internal flow field is given by Hill s spherical vortex (HI3) which, coupled with irrotational flow of the external fluid, leads to a first estimate of drag for a spherical droplet for Re 1 and Rep 1. The internal flow field is then modified to account for convection of vorticity by the internal fluid to the front of the drop from the rear. The drag coefficient. 

Bhaga (B3) determined the fluid motion in wakes using hydrogen bubble tracers. Closed wakes were shown to contain a toroidal vortex with its core in the horizontal plane where the wake has its widest cross section. The core diameter is about 70% of the maximum wake diameter, similar to a Hill s spherical vortex. When the base of the fluid particle is indented, the toroidal motion extends into the indentation. Liquid within the closed wake moves considerably more slowly relative to the drop or bubble than the terminal velocity Uj, If a skirt forms, the basic toroidal motion in the wake is still present (see Fig. 8.5), but the strength of the vortex is reduced. Momentum considerations require that there be a velocity defect behind closed wakes and this accounts for the tail observed by some workers (S5). Crabtree and Bridgwater (C8) and Bhaga (B3) measured the velocity decay and drift in the far wake region. 

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