By Z. Zapryanov

1. aim and Scope Bubbles, drops and inflexible debris happen all over the place in existence, from beneficial business operations like gas-liquid contracting, fluidized beds and extraction to such important average techniques as fermentation, evaporation, and sedimentation. As we develop into more and more conscious of their primary function in commercial and organic platforms, we're pushed to understand extra approximately those attention-grabbing debris. it truly is no shock, for that reason, that their useful and theoretical implications have aroused nice curiosity one of the medical neighborhood and feature encouraged an increasing number of reviews and courses. during the last ten years advances within the box of small Reynolds numbers flows and their technological and organic functions have given upward push to numerous definitive monographs and textbooks within the quarter. furthermore, the earlier 3 many years have witnessed huge, immense growth in describing quantitatively the behaviour of those debris. besides the fact that, to the simplest of our wisdom, there are nonetheless no on hand books that mirror such achievements within the parts of bubble and drop deformation, hydrodynamic interactions of deformable fluid debris at low and average Reynolds numbers and hydrodynamic interactions of debris in oscillatory flows. certainly, just one extra ebook is devoted solely to the behaviour of bubbles, drops and inflexible debris ["Bubbles, Drops and debris" via Clift et al. (1978)] and the authors kingdom its barriers truly within the preface: "We deal with in basic terms phenomena within which particle-particle interactions are of negligible significance. as a result, direct software of the booklet is proscribed to single-particle structures of dilute suspensions.

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**Example text**

R dk . :. F

Ii)] da. n))da, sP s~ where n = -n. Application 2. As a second application we shall derive Brenner's result (Brenner, 1964b) for the drag on a rigid particle of arbitrary shape immersed in an ambient fluid of velocity v""(r). Let us consider two flow fields (v·, T•) and (v 0 , T 0 ), the first of which is induced by a rigid particle of arbitrary shape translating with a steady velocity v • = U = const. and the second is induced by the same particle placed in an arbitrary ambient flow field, v""(r).

Hill and Power Drag Statements. Theorem 1. If for a given fluid motion the deformation velocity tensor e,J is equal to zero, then the fluid is moving as a rigid body. Therefore, we have to prove that, if eiJ = 0, (i, j = 1, 2, 3) the fluid velocity v consists of translation U and rotation eii = 0 it follows m [see (Kim and Karrila, 1991)]. ovi ovj oxj- oxi In fact, from (i,j=1, 2, 3). After differentiating this equality with respect to xk, we obtain: o 2 v. 1) oxioxk ' This equality is true for every j.