Page 124 - Bird R.B. Transport phenomena

Problems 109

I Fluid in (d) Write out all the nonzero components of т for this flow.
(e) Repeat the problem for concentric spheres.
Radial flow outward
between disks
3B.12 Pressure distribution in incompressible fluids.
Penelope is staring at a beaker filled with a liquid, which
for all practical purposes can be considered as incompress-
ible; let its density be p . She tells you she is trying to un-
0
derstand how the pressure in the liquid varies with depth.
She has taken the origin of coordinates at the liquid-air in-
terface, with the positive z-axis pointing away from the liq-
Fig. 3B.10. Outward radial flow in the space between two uid. She says to you:
parallel, circular disks.
"If I simplify the equation of motion for an incom-
pressible liquid at rest, I get 0 = -dp/dz - pog. I can solve
(b) Show how the equation of continuity enables one to this and get p = p - pogz. That seems reasonable—the
alm
simplify the equation of motion to give pressure increases with increasing depth.
"But, on the other hand, the equation of state for any
fluid is p = p(p, T), and if the system is at constant temper-
ature, this just simplifies top = p(p). And, since the fluid is
in which ф = rv r is a function of z only. Why is ф indepen- incompressible, p = p(p ), and p must be a constant
0
dent of r? throughout the fluid! How can that be?"
(c) It can be shown that no solution exists for Eq. 3B.10-1 Clearly Penelope needs help. Provide a useful expla-
unless the nonlinear term containing ф is omitted. Omis- nation.
sion of this term corresponds to the "creeping flow as- 3B.13 Flow of a fluid through a sudden contraction.
sumption." Show that for creeping flow, Eq. 3B.10-1 can be (a) An incompressible liquid flows through a sudden con-
integrated with respect to r to give
traction from a pipe of diameter D l into a pipe of smaller
diameter D . What does the Bernoulli equation predict for
2
о = (зв.10-2) 9>i - 2P, the difference between the modified pressures
2
upstream and downstream of the contraction? Does this
(d) Show that further integration with respect to z gives result agree with experimental observations?
(b) Repeat the derivation for the isothermal horizontal
v ix, z) = (3B.10-3) flow of an ideal gas through a sudden contraction.
2/лг In (r /r )
r
2 }
3B.14 Torricelli's equation for efflux from a tank (Fig.
(e) Show that the mass flow rate is
3B.14). A large uncovered tank is filled with a liquid to a
height h. Near the bottom of the tank, there is a hole that
(3B.10-4) allows the fluid to exit to the atmosphere. Apply
In (r /r,)
2
Bernoulli's equation to a streamline that extends from the
(f) Sketch the curves 0>(r) and v (r, z). surface of the liquid at the top to a point in the exit
r
3B.11 Radial flow between two coaxial cylinders. Con-
sider an incompressible fluid, at constant temperature,
flowing radially between two porous cylindrical shells
with inner and outer radii KR and R. Liquid surface
(a) Show that the equation of continuity leads to v = C/r, at which
r = 0 and p =
where С is a constant. р л1т
(b) Simplify the components of the equation of motion to Typical streamline
obtain the following expressions for the modified-pressure
distribution:

=
dr ~^Tr 1в ° dz = 0 (3B.11-1) Fluid exit at which
^2 = Afflux a n d
(c) Integrate the expression for dty/dr above to get V = Patm
H^-O Fig. 3B.14. Fluid draining from a tank. Points 1 " and "2"
"
9>(r) - Ш) = Ь (3B.11-2) are on the same streamline.