Page 162 - Bird R.B. Transport phenomena

146 Chapter 4 Velocity Distributions with More Than One Independent Variable

= 2B Fig- 4C.1. Entrance flow into a slit.

(a) Use the above two equations to get the mass flow rate w through an arbitrary cross sec-
tion in the region 0 < x < L . Then evaluate zv from the inlet conditions and obtain
e
v c (x) _ В (4C.1-3)
<v x ) В - 1
(b) Next use Eqs. 4.4-13,14, and 15 with °° replaced by В (why?) to obtain a differential equa-
tion for the quantity A = 8/B:
6Д + 7Л 2 dA = 1Q ( v (4C.1-4)
(3 - Д) 2 dx \(v )B 2
x
(c) Integrate this equation with a suitable initial condition to obtain the following relation be-
tween the boundary-layer thickness and the distance down the duct:

48 In (1 - U (4C.1-5)
(v )B 2
x
(d) Compute the entrance length L from Eq. 4C.1-5, where L is that value of x for which
e
c
8(x) = B.
(e) Using potential flow theory, evaluate 9* - <3> in the entrance region, where 2P is the value
0
0
of the modified pressure at x = 0. г
2
2
Answers: (d) L = O.W4(v )B /i>; (e) <3> - % = \p(v ) \ 1 - L
c x x
2 | \3 - ,
4C.2 Torsional oscillatory viscometer (Fig. 4C.2). In the torsional oscillatory viscometer, the fluid
is placed between a "cup" and "bob" as shown in the figure. The cup is made to undergo
small sinusoidal oscillations in the tangential direction. This motion causes the bob, sus-
pended by a torsion wire, to oscillate with the same frequency, but with a different amplitude
Torsion wire

-"Bob"
-"Cup"

Forced oscillation of
outer cylinder F i . 4 C > 2 . Sketch of a torsional oscillatory viscometer.
g