# 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»