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).
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).
\[ u(y) = 7.5 - 3\times10^{5}y^2 \]
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).
Show Hint
For a parabolic velocity profile in a slit, the average velocity equals two-thirds of the centerline (maximum) velocity.
Updated On: Jul 17, 2026
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Correct Answer: 5
Solution and Explanation
Step 1: Half-gap h = 0.005 m.
\[ \bar{u} = \frac{1}{2h}\int_{-h}^{h}u(y)\,dy \]
\[ \bar{u} = \frac{1}{2h}\int_{-h}^{h}u(y)\,dy \]
Step 2: Evaluate.
\[ \int_{-h}^{h}7.5\,dy = 15h = 0.075 \]
\[ \int_{-h}^{h}3\times10^5y^2dy = 2\times10^5h^3 = 0.025 \]
\[ \int u\,dy = 0.075-0.025=0.05 \]
\[ \int_{-h}^{h}7.5\,dy = 15h = 0.075 \]
\[ \int_{-h}^{h}3\times10^5y^2dy = 2\times10^5h^3 = 0.025 \]
\[ \int u\,dy = 0.075-0.025=0.05 \]
Step 3: Average.
\[ \bar{u}=0.05/0.01=5.0\ m/s \]
\[ \bar{u}=0.05/0.01=5.0\ m/s \]
\[ \boxed{\bar{u} \approx 5\ m/s} \]
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