Page 157 - Bird R.B. Transport phenomena

Problems 141

3. What happens in Example 4.1-2 if one tries to solve Eq. 4.1-21 by the method of sepa-
ration of variables without first recognizing that the solution can be written as the
sum of a steady-state solution and a transient solution?
4. What happens if the separation constant after Eq. 4.1-27 is taken to be с or c instead of
2
2
-c ?
5. Try solving the problem in Example 4.1-3 using trigonometric quantities in lieu of
complex quantities.
6. How is the vorticity equation obtained and how may it be used?
7. How is the stream function defined, and why is it useful?
8. In what sense are the potential flow solutions and the boundary-layer flow solutions
complementary?
9. List all approximate forms of the equations of change encountered thus far, and indi-
cate their range of applicability.

PROBLEMS 4A.1 Time for attainment of steady flow in tube flow.
4
2
(a) A heavy oil, with a kinematic viscosity of 3.45 X 10~ m /s, is at rest in a long vertical tube
with a radius of 0.7 cm. The fluid is suddenly allowed to flow from the bottom of the tube by
virtue of gravity. After what time will the velocity at the tube center be within 10% of its final
value?
(b) What is the result if water at 68°F is used?
Note: The result shown in Fig. 4D.2 should be used.
2
Answers: (a) 6.4 X 10~ s; (b) 22 s
4A.2 Velocity near a moving sphere. A sphere of radius R is falling in creeping flow with a termi-
nal velocity v x through a quiescent fluid of viscosity fx. At what horizontal distance from the
sphere does the velocity of the fluid fall to 1 % of the terminal velocity of the sphere?
Answer: About 37 diameters
4A.3 Construction of streamlines for the potential flow around a cylinder. Plot the streamlines for
the flow around a cylinder using the information in Example 4.3-1 by the following procedure:
(a) Select a value of ¥ = С (that is, select a streamline).
2
(b) Plot У = С + К (straight lines parallel to the X-axis) and У = K(X 2 + У) (circles with ra-
dius 1/2K, tangent to the X-axis at the origin).
(c) Plot the intersections of the lines and circles that have the same value of K.
(d) Join these points to get the streamline for ¥ = C.
Then select other values of С and repeat the process until the pattern of streamlines is clear.
4A.4 Comparison of exact and approximate profiles for flow along a flat plate. Compare the val-
ues of v /v x obtained from Eq. 4.4-18 with those from Fig. 4.4-3, at the following values of
x
y^/v /vx: (a) 1.5, (b) 3.0, (с) 4.О. Express the results as the ratio of the approximate to the exact
x
values.
Answers: (a) 0.96; (b) 0.99; (c) 1.01
4A.5 Numerical demonstration of the von Karman momentum balance.
(a) Evaluate the integrals in Eq. 4.4-13 numerically for the Blasius velocity profile given in
Fig. 4.4-3.
(b) Use the results of (a) to determine the magnitude of the wall shear stress r | .
l/v 1/=0
(c) Calculate the total drag force, F , for a plate of width W and length L, wetted on both
x
sides. Compare your result with that obtained in Eq. 4.4-30.
Answers: (a) I pv (v - v )dy = 0.664Vp/xidx
•Jo x e x
p(v c - v )dy = 1.73Vp/xv x
x
x