Page 159 - Bird R.B. Transport phenomena

Problems 143
Fig. 4B.2. Comparison
of true and approximate
8(t 3 ) velocity profiles near a
wall suddenly set in mo-
tion with velocity %
8(t 2 )
Increasing t

v 0 —•
(a) True solution (b) Boundary-layer approximation

(b) We know roughly what the velocity profiles look like. We can make the following reason-
able postulate for the profiles:

(4B.2-3)
2 5(0 2\8(t)
= 1 for у > 5(0 (4B.2-4)
Here 5(0 is a time-dependent boundary-layer thickness. Insert this approximate expression
into Eq. 4B.2-2 to obtain

(4B.2-5)

(c) Integrate Eq. 4B.2-5 with a suitable initial value of 5(0, and insert the result into Eq. 4B.2-3
to get the approximate velocity profiles.
(d) Compare the values of v /v x obtained from (c) with those from Eq. 4.1-15 at y/y/ivi =
x
0.2, 0.5, and 1.0. Express the results as the ratio of the approximate value to the exact value.
Answer (d) 1.015,1.026, 0.738
4B.3 Creeping flow around a spherical bubble. When a liquid flows around a gas bubble, circula-
tion takes place within the bubble. This circulation lowers the interfacial shear stress, and, to a
first approximation, we may assume that it is entirely eliminated. Repeat the development of
Ex. 4.2-1 for such a gas bubble, assuming it is spherical.
(a) Show that B.C. 2 of Ex. 4.2-1 is replaced by

B.C. 2: 2 (4B.3-1)
dr \ r dr
and that the problem set-up is otherwise the same.
(b) Obtain the following velocity components:
.-[f
= V, cos0 (4B.3-2)
V r

(4B.3-3)
(c) Next obtain the pressure distribution by using the equation of motion:

c
p = p 0 - pgh - I — II - I os^ (4B.3-4)