Drag due to the motion of a Newtonian fluid through a sparse random array of small fixed rigid objects

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Abstract
The averaged equations of slow flow in random arrays of fixed spheres are developed as a hierarchy of integro-differential equations, and an iteration procedure is described for obtaining the mean drag in the case of small volume concentration c. The leading approximation is that given by Brinkman's model of flow past a single fixed sphere, in which the effects of all other spheres are treated as a Darcy resistance. The higher approximations take account of the modification to the mean flow, particularly in the near field, due to the localized nature of the actual resistance. Thus the second approximation finds the change due to a second sphere, and averages over all its possible positions. The result for the mean drag confirms Childress’ terms in clogc and c (apart from an arithmetical correction to the latter), but indicates that for practical values of c numerical evaluation of integrals is needed, rather than expansion in powers of c and log c. The last section of the paper develops the corresponding results for flow through random arrays of fixed parallel circular cylinders.