For fully developed laminar flow through a parallel-plate channel, the x-momentum
equation has the form
?\(\frac{d^2u}{dy^2}\) = \frac{dp}{dx} = constant
The purpose of this problem is to develop expressions for the velocity distribution and
pressure gradient analogous to those for the circular tube in Section 8.1.
(a) Show that the velocity profile, u(y), is parabolic and of the form
u(y) = \frac{3}{2}u_m[1 - \frac{y^2}{(a/2)^2}]
where $u_m$ is the mean velocity
u_m = \frac{-a^2}{12?} \(\frac{dp}{dx}\)
and -dp/dx = ?p/L, where ?p is the pressure drop across the channel of length L.
(d) Airflow in a parallel-plate channel with a separation of 5 mm and a length of 200
mm experiences a pressure drop of ?p = 3.75 N/m². Calculate the mean velocity and the
Reynolds number for air at atmospheric pressure and 300 K. Is the assumption of fully
developed flow reasonable for this application? If not, what is the effect on the estimate
for $u_m$?
