Creeping flow particle orientation

Since the net drag on an arbitrary particle is generally not parallel to the direction of motion, a particle falls vertically without rotation only if it possesses a certain symmetry or a specific orientation. The following guidelines for solid particles with uniform density are derived from general results for creeping flow (H3) ... 

It is common practice to define a hydraulic equivalent sphere as the sphere with the same density and terminal settling velocity as the particle in question. For a spheroid in creeping flow, the hydraulic equivalent sphere diameter is 2a- E/A and thus depends on orientation. 

For a particle which is spherically isotropic (see Chapter 2), the three principal resistances to translation are all equal. It may then be shown (H3) that the net drag is — judJ regardless of orientation. Hence a spherically isotropic particle settling through a fluid in creeping flow falls vertically with its velocity independent of orientation. 

The critical difficulty with this problem is that the solution depends on the orientations of U and f2 relative to axes fixed in the particle, as well as on the relative magnitudes of U and f2. Thus, for every possible orientation of U and/or f2, a new solution appears to be required to calculate u, p, F, or G. Fortunately, however, the possibility of constructing solutions of a problem as a sum (or superposition) of solutions to a set of simpler problems means that this is not actually necessary in the creeping-flow limit. Rather, to evaluate u, p, F, or G for any arbitrary choice of U and f2, we will show that it is sufficient to obtain detailed solutions for translation in three mutually orthogonal directions (relative to axes fixed in the particle) with unit velocity U = e, and il = 0, and for rotation about three mutually orthogonal axes with unit angular velocity il = e, and U = 0. 

In reviewing our analysis of (9-152) leading to (9-141), we may note that we have used the conditions (9-143) and (9-144) only on the velocity field. Thus, as stated earlier, the result (9-141) or (9-142) is valid in streaming flow for any heated body with a uniform surface temperature provided that these conditions are satisfied and that /V Higher-order terms in (9-142) will depend on the details of the flow, and thus on the Reynolds number Re, as well as the shape and orientation of the body relative to the free stream. However, in the creeping-flow limit, Brenner was able to extend (9-142) to one additional term for a particle of arbitrary shape,... 

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