Question

The velocity profile for a fully developed laminar flow of an incompressible Newtonian fluid, at steady state, through a horizontal infinitely wide and long slit of gap 0.01 m is given by the following equation.
\[ u(y) = 7.5 - 3\times10^{5}y^2 \]
where \(y\) is the vertical distance measured from the center of the slit. The average velocity of the fluid (in m s^-1) is ______ (rounded off to the nearest integer).

Hint:

For a parabolic velocity profile in a slit, the average velocity equals two-thirds of the centerline (maximum) velocity.

Answer 1

Correct Answer: 5

Centerline velocity $u_{max}=u(0)=7.5$. Checking $u(h)=7.5-3\times10^5(0.005)^2=0$, confirming the standard parabolic form $u(y)=u_{max}[1-(y/h)^2]$, for which $\bar u = (2/3)u_{max} = (2/3)(7.5)=5.0$ m/s.

\[ \boxed{\bar u \approx 5\ m/s} \]