9. Consider the channel flow shown in the figure 4. The inlet velocity is uniform $U_{in}$. The
channel height is h. The growing laminar boundary layer is shown as a dotted line. Assume
that the velocity profile in the boundary layer is parabolic (which is only an approximation),
described by $\frac{u}{U_{\infty}} = \frac{y}{\delta} - (\frac{y}{\delta})^2$, where $\delta$ is the boundary layer thickness.
a. Find the centerline velocity as a function of x.
b. If the flow is considered fully-developed when the growing boundary layers meet,
how long does the channel need to be in order for the flow to become fully-
developed? Give your answer in multiples of the channel height.
c. What is the largest magnitude of the vorticity at a given x-position? Put your answer
in terms of x, $U_{in}$, and h; but not Re, $U_{\infty}$, or $\delta$.
d. What is the largest magnitude of the vorticity anywhere in the flow?
e. What does this tell you about the parabolic velocity profile assumption?
$U_{in}$
h
Figure 4
$U_{\infty}(x)$
