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Problem 4.87
SAE 30W oil at 20°C flows through the 9-cm-diameter pipe in Fig. P4.87 at an average velocity
of 4.3 m/s. (a) Verify that the flow is laminar. (b) Determine the volume flow rate in m3/h.
(c) Calculate the expected reading h of the mercury manometer, in cm.
Solution 4.87
(a) Check the Reynolds number. For SAE 30W oil, from Appendix A.3,
= 891 kg/m3 and
=
0.29 kg/(ms). Then
Problem 4.88
The viscous oil in Fig. P4.88 is set into steady motion by a concentric inner cylinder moving
axially at velocity U inside a fixed outer cylinder. Assuming constant pressure and density and a
purely axial fluid motion, solve Eqs. (4.38) for the fluid velocity distribution vz(r). What
are the proper boundary conditions?
Solution 4.88
If vz = fcn(r) only, the z-momentum equation (Appendix E) reduces to:
Problem 4.89
Oil flows steadily between two fixed plates that are 2 inches apart. When the pressure
gradient is 3200 pascals per meter, the average velocity is 0.8 m/s. (a) What is the flow
rate per meter of width? (b) What oil in Table A.4 fits this data? (c) Can we be sure
that the flow is laminar?
Solution 4.89
This problem fits the conditions of Example 4.10. The half-width h = 1 inch = 0.0254 m.
Problem 4.90
It is desired to pump ethanol at 20C through 25 meters of straight smooth tubing under laminar-
flow conditions, Red =
Vd/
< 2300. The available pressure drop is 10 kPa. (a) What is the
maximum possible mass flow, in kg/h? (b) What is the appropriate diameter?
Solution 4.90
For ethanol at 20C,
= 789 kg/m3 and
= 0.0012 kg/m-s. From Eq. (4.138),
Problem 4.91*
Analyze fully developed laminar pipe flow for a power-law fluid, τ = C(dvz/dr)n, for n ≠ 1, as in
Prob. P1.46. (a) Derive an expression for vz(r). (b) For extra credit, plot the velocity profile
shapes for n = 0.5, 1, and 2. [Hint: In Eq. (4.136), replace μ(dvz/dr) by τ.]
Solution 4.91*
(a) For the power-law fluid, Eq. (4.136) becomes
11
( ) [ ( ) ]
n
z
dv
d d dp
r r C
r dr r dr dr dz
==
Problem 4.92
A tank of area Ao is draining in laminar flow through a pipe of diameter D and length L, as
shown in Fig. P4.92. Neglecting the exit jet kinetic energy and assuming the pipe flow is driven
by the hydrostatic pressure at its entrance, derive a formula for the tank level h(t) if its initial
level is ho.
Solution 4.92
For laminar flow, the flow rate out is given by Eq. (4.147). A control volume mass balance
shows that this flow out is balanced by a tank level decrease:
Problem 4.93
A number of straight 25-cm-long microtubes, of diameter d, are bundled together into a
“honeycomb” whose total cross-sectional area is 0.0006 m2. The pressure drop from entrance to
exit is 1.5 kPa. It is desired that the total volume flow rate be 5 m3/h of water at 20°C. (a) What
is the appropriate microtube diameter? (b) How many microtubes are in the bundle? (c) What is
the Reynolds number of each microtube?
Solution 4.93
For water at 20°C,
= 998 kg/m3 and
= 0.001 kg/ms. Each microtube of diameter D sees the
Problem 4.94
A long solid cylinder rotates steadily in a very viscous fluid, as in Fig. P4.94. Assuming laminar
flow, solve the Navier-Stokes equation in polar coordinates to determine the resulting velocity
distribution. The fluid is at rest far from the cylinder. [HINT: the cylinder does not induce any
radial motion.]
Solution 4.94
We already have the useful hint that vr = 0. Continuity then tells us that
Problem 4.95*
Two immiscible liquids of equal thickness h are being sheared between a fixed and a moving
plate, as in Fig. P4.95. Gravity is neglected, and there is no variation with x. Find an expression
for (a) the velocity at the interface; and (b) the shear stress in each fluid. Assume steady laminar
flow.
Solution 4.95*
Treat this as a Ch. 4 problem (not Ch. 1), use continuity and Navier-Stokes:
Problem 4.96
Use the data of Prob. P1.40, with the inner cylinder rotating and outer cylinder fixed, and calculate
(a) the inner shear stress. (b) Determine whether this flow pattern is stable. [HINT: The shear stress
in (r,
) coordinates is not like plane flow.]
Solution 4.96
The exact laminar-flow velocity is Eq. (4.140), and the shear stress is Eq. (D.9):
Problem 4.97
For Couette flow between a moving and a fixed plate, Fig. 4.12 a , solve continuity and Navier-
Stokes to find the velocity distribution when there is slip at both walls.
( / ) ( / )
[]
( / ) ( / )
oo
ii
o i i o
r r r r
vr
r r r r
−
= −
Solution 4.97
We assume flow in the x direction only, with v = w = 0. Then continuity becomes
Problem 4.98
For the pressure-gradient flow between two parallel plates of Fig. 4.12( b ), reanalyze for the case of
slip flow at both walls. Use the simple slip condition uwall = l ( du / dy )wall , where l is the mean free
path of the fluid. ( a ) Sketch the expected velocity profile. ( b ) Find an expression for the shear
stress at each wall. ( c ) Find the volume flow between the plates.
Solution 4.98
(a) The velocity profile has equal slip
u all around, as shown:
Problem 4.99
For the pressure-gradient flow in a circular tube in Sec. 4.10, reanalyze for the case of slip flow at the
wall. Use the simple slip condition υz,wall = l(dvz/dr ) wall, where / is the mean free path of the fluid.
(a) Sketch the expected velocity profile. ( b ) Find an expression for the shear stress at the wall.
(c) Find the volume flow through the tube.
Solution 4.99
No solution is provided
