Interdisciplinary Applied Mathematics

Скачать в pdf «Interdisciplinary Applied Mathematics»



bH2(H + 26(1 + к)) H + 6(2 + к) )


This velocity is constant, and thus we can solve the above differential equation in terms of pressure by integrating twice and assuming that p = 0 at r ^    <x>,    and dp/dr = 0    at    r = 0    due    to    symmetry.    The    equation    for    the


pressure is then


p(r) =P ,    (10.9)


consisting of two factors, namely, the Reynolds part and the correction factor p* given by


p=



2AH



+



2H2(B — A



ln(1 + B/H) —



C-A



ln(1 + C/H) .



BC C — B    B2    C2


(10.10)


The constants A, B, C in this expression characterize the two surfaces; they are given by


A = b(2 + k),


В = 26(2 + к + Vl + к + к2), С = 26(2 + к — Vl + к + к2).


The resistance forces acting on the spheres are equal in magnitude and are primarily due to the pressure, so the force can be computed exactly from Fz = /0° p2nrdr, to obtain


6npR2ev



h



f *



(10.11)


s

FIGURE 10.26. Correction factor f * as a function of the gap to slip length ratio for the three asymptotic cases discussed in the text.


consisting also of two factors, namely, the Reynolds part and the correction factor f * given by


,* =Ah


1 BC


(™-Л>,„(1+С/,,)).    (Ю.12)


For the aforementioned three limiting cases, the above expression reduces to

Скачать в pdf «Interdisciplinary Applied Mathematics»