Interdisciplinary Applied Mathematics

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1    3h


/* = -(1 + — К1 + /?./(46))ln(l + 46//?.) — 1] for к = -1,


h


f* = — [(1 + /?/(66)) ln(l + 66//?) — 1] for к = 0, h


/* = — [(1 + /?/(36)) ln(l + 36//?) — 1] for к —s- oo.


These three cases are plotted in Figure 10.26 and show, as expected, that the correction factor is always less than one; the no-slip case corresponds to f *    =    p*    = 1.    Also,    both    correction    factors depend    on    the    length    scale


ratios, namely h/Ь and h/[(k+1)6]. In the limit of very small gap, the case of


f * ^ 0 (corresponding to к > —1) represents a configuration of two bubbles approaching each    other,    while    the    case    of    f *    ^    1/4    (corresponding to


к = —1) represents the flow resistance for a hydrophilic sphere interacting with a bubble.


In the standard Reynolds theory (f * = 1) the hydrodynamic resistance is inversely proportional to the gap and diverges for h ^ 0. However, the new physical result in the solutions of Vinogradova is for two hydrophobic surfaces, i.e., к > —1 and h ^ 0, where the friction coefficient


f*    2


L- =    g [(1 — A/В) In B/h — (1 — A/C) In C/h


depends logarithmically on h and is inversely proportional to the slip length b. The above result is valid for h ^ C < B. This dependence is more clearly seen for the case in which A and C are approximately of the same order of magnitude. In this case, (Vinogradova, 1995) has derived that

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