Page 158 - Bird R.B. Transport phenomena

142 Chapter 4 Velocity Distributions with More Than One Independent Variable

4A.6 Use of boundary-layer equations. Air at 1 atm and 20°C flows tangentially on both sides of a
thin, smooth flat plate of width W = 10 ft, and of length L = 3 ft in the direction of the flow.
The velocity outside the boundary layer is constant at 20 ft/s.
(a) Compute the local Reynolds number Re . = xv /v at the trailing edge.
x
A
(b) Assuming laminar flow, compute the approximate boundary-layer thickness, in inches, at
the trailing edge. Use the results of Example 4.4-1.
(c) Assuming laminar flow, compute the total drag of the plate in \b. Use the results of Exam-
f
ples 4.4-1 and 2.
4A.7 Entrance flow in conduits.
(a) Estimate the entrance length for laminar flow in a circular tube. Assume that the boundary-
layer thickness 8 is given adequately by Eq. 4.4-17, with v x of the flat-plate problem corre-
sponding to v max in the tube-flow problem. Assume further that the entrance length L can be
c
taken to be the value of x at which 8 = R. Compare your result with the expression for L cited
e
in §2.3—namely, L = 0.035D Re.
e
(b) Rewrite the transition Reynolds number xv^lv ~ 3.5 X 10 5 (for the flat plate) by inserting
8 from Eq. 4.4-17 in place of x as the characteristic length. Compare the quantity 8v /v thus
x
obtained with the corresponding minimum transition Reynolds number for the flow through
long smooth tubes.
(c) Use the method of (a) to estimate the entrance length in the flat duct shown in Fig. 4C.1.
Compare the result with that given in Problem 4C.l(d).
4B.1 Flow of a fluid with a suddenly applied constant wall stress. In the system studied in Ex-
ample 4.1-1, let the fluid be at rest before t = 0. At time t = 0 a constant force is applied to the
fluid at the wall in the positive x direction, so that the shear stress т takes on a new constant
ух
value r at у = 0 for t > 0.
0
(a) Differentiate Eq. 4.1-1 with respect to у and multiply by -/x to obtain a partial differential
equation for т (у, t).
ух
(b) Write the boundary and initial conditions for this equation.
(c) Solve using the method in Example 4.1-1 to obtain

1
(d) Use the result in (c) to obtain the velocity profile. The following relation will be helpful
J; (1 - erf u)du = -1= e~ - x{\ - erf x) (4B.1-2)
x2

4В.2 Flow near a wall suddenly set in motion (approximate solution) (Fig. 4B.2). Apply a proce-
dure like that of Example 4.4-1 to get an approximate solution for Example 4.1.1.
(a) Integrate Eq. 4.4-1 over у to get
: x
dv x dv x
-^dy = v-± (4B.2-1)
ot dy Q
Make use of the boundary conditions and the Leibniz rule for differentiating an integral
(Eq. C.3-2) to rewrite Eq. 4B.2-1 in the form
j - I pv dy = r \ (4B.2-2)
x
ux u=0
ut J о
Interpret this result physically.

1
A useful summary of error functions and their properties can be found in H. S. Carslaw and
J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, 2nd edition (1959), Appendix II.