Aeronautical Engineering Chapter 4 Homework Oil flows steadily between two fixed plates that

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/(ms). 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 20C 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 20C,



= 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/ms. 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