323
Chapter 4 x Differential Relations for a Fluid Particle
4.35 From the Navier-Stokes equations
for incompressible flow in polar coordinates
(App. E for cylindrical coordinates), find
the most general case of purely circulating
motion XT(r), Xr Xz 0, for flow with no
slip between two fixed concentric cylinders,
as in Fig. P4.35.
υθ (r)
r
r=a
No slip
r=b
Fig. P4.35
Solution: The preliminary work for this
problem is identical to Prob. 4.32 on an earlier page. That is, there are two possible
solutions for purely circulating motion XT(r), hence
C1r vT
C2
, subject to vT (a) 0
r
This requires C1
C2 0, or vT
Cy(2h – y) v
w
C1b C2 /b
0 (no steady motion possible between fixed walls) Ans.
4.36 A constant-thickness film of viscous
liquid flows in laminar motion down a
plate inclined at angle T, as in Fig. P4.36.
The velocity profile is
u
C1a C2 /a and vT (b) 0
y
g
h
0
(y)
Find the constant C in terms of the specific
weight and viscosity and the angle T. Find
the volume flux Q per unit width in terms
of these parameters.
θ
x
Fig. P4.36
Solution: There is atmospheric pressure all along the surface at y h, hence wp/wx
The x-momentum equation can easily be evaluated from the known velocity profile:
§ wu
wu·
v ¸
© wx
wy¹
U¨u
wp
Ug x P’ 2 u, or: 0 0 Ug sinT + P (2C)
wx
U g sinT
Solve for C
Ans. (a)
2P
The flow rate per unit width is found by integrating the velocity profile and using C:
h
Q
³
0
h
u dy
³
0
Cy(2h y) dy
2 3
Ch
3
U gh 3 sinT
per unit width Ans. (b)
3P
0.