The interstitial Eulerian flow

The definition (7.32) masks the local and non-local contributions from bodies to the flow. A more systematic approach to characterising the Eulerian mean velocity is to decompose the flow into (i) a far field flow contribution - far from each body but still within the group of bodies - and (ii) a near field flow contribution - local to each body. This concept, originally described qualitatively by Kowe et al. [353], is strictly valid for dilute arrays since it formally requires the bodies to be widely separated, so that there is a separation of lengthscales between the near and far field, scaling approximately as 0(a) and 0(LS) respectively. The decomposition is defined formally here for potential flows. The far field flow, u, is defined mathematically as the sum of the dipolar and source contributions from the bodies, by assuming the bodies shrink to zero, so that (from (7.31)) 

The velocity field induced by the dipole and source distribution is 

Thus the interstitial velocity, as described by Kowe et al. [353], is effectively the average velocity field experienced by a test body introduced into the flow, which is not located close to any other bodies. Notice that t Uf because u/,- includes contributions from the near field flow from each body. The dipole moment associated with each body moving parallel to the stream is (1 + Cm)(v - up)Vn/2n, where Vb is the volume of a body and Cm its added-mass coefficient. 

Boundaries and global mass conservation impose important constraints on the interstitial velocity. To relate these concepts to experimental measurements described later we focus on bounded channel flows generated by a stream of speed U through a cloud of bubbles injected into a channel and moving vertically with a speed v. When the average separation between the bodies is small relative to the separation of the channel walls, the dipole field and average flow is equivalent to a distributed dipole moment, and averaging (7.34) over the whole volume yields, 

In Section 7.5, this relationship is applied to interpret experimental results. 

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