fluid_mechanics

6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Fluids 313
atmospheric 1or zero gage pressure2. The equation of motion in
the y direction 1Eq. 6.127b2 thus reduces to
or
Integration of Eq. 1 yields
On the film surface 1x h2 we assume the shearing stress is
zero—that is, the drag of the air on the film is negligible. The
shearing stress at the free surface 1or any interior parallel surface2
is designated as , where from Eq. 6.125d
Thus, if t xy 0 at x h, it follows from Eq. 2 that
A second integration of Eq. 2 gives the velocity distribution in
the film as
At the belt 1x 02 the fluid velocity must match the belt velocity,
V 0 , so that
and the velocity distribution is therefore
With the velocity distribution known we can determine the
flowrate per unit width, q, from the relationship
h
h
q v dx
0
t xy
0 rg m d 2 v
dx 2
d 2 v
dx 2 g m
dv
dx g m x c 1
t xy m a dv
dx b
c 1 gh
m
v g
2m x2 gh
m x c 2
v g
2m x2 gh
m x V 0
0
c 2 V 0
a g
2m x2 gh
m x V 0b dx
(1)
(2)
(3)
and thus
The average film velocity, V 1where q Vh2, is therefore
COMMENT
form as
q V 0 h gh3
3m
V V 0 gh2
3m
(Ans)
Equation (3) can be written in dimensionless
v
c a x 2
V 0 h b 2c a x h b 1
where c h 2 /2V 0 . This velocity profile is shown in Fig. E6.9b.
Note that even though the belt is moving upward, for c 1 (e.g.,
for fluids with small enough viscosity or with a small enough belt
speed) there are portions of the fluid that flow downward (as indicated
by v/V 0 0).
It is interesting to note from this result that there will be a net
upward flow of liquid (positive V) only if V 0 h 2 /3. It takes a
relatively large belt speed to lift a small viscosity fluid.
v/V 0
1
0.8
0.6
0.4
0.2
c = 1.0
0
0 0.2 0.4 0.6 0.8 1
–0.2
–0.4
c = 1.5
–0.6
–0.8
–1
F I G U R E E6.9b
x/h
c = 0
c = 0.5
c = 2.0
An exact solution
can be obtained for
steady, incompressible,
laminar flow in
circular tubes.
6.9.3 Steady, Laminar Flow in Circular Tubes
Probably the best known exact solution to the Navier–Stokes equations is for steady, incompressible,
laminar flow through a straight circular tube of constant cross section. This type of flow is commonly
called Hagen–Poiseuille flow, or simply Poiseuille flow. It is named in honor of J. L. Poiseuille 11799–
18692, a French physician, and G. H. L. Hagen 11797–18842, a German hydraulic engineer. Poiseuille
was interested in blood flow through capillaries and deduced experimentally the resistance laws
for laminar flow through circular tubes. Hagen’s investigation of flow in tubes was also
experimental. It was actually after the work of Hagen and Poiseuille that the theoretical results
presented in this section were determined, but their names are commonly associated with the
solution of this problem.
Consider the flow through a horizontal circular tube of radius R as is shown in Fig. 6.34a.
Because of the cylindrical geometry it is convenient to use cylindrical coordinates. We assume that
the flow is parallel to the walls so that v r 0 and v u 0, and from the continuity equation 16.342
0v z0z 0. Also, for steady, axisymmetric flow, is not a function of t or u so the velocity, v z ,
v z