GATE 2026 CH – Question 65

Fluid Mechanics and Mechanical Operations · Equations of continuity, motion and mechanical energy, Euler and Bernoulli equations · 2 marks · Numerical answer

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.

$u(y) = 7.5 - 3 \times 10^5 y^2$

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).

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Correct answer: 5

Explanation

The walls are at $y = \pm 0.005$ m and $u$ is zero there: $7.5 - 3 \times 10^5 \times 2.5 \times 10^{-5} = 0$. The average velocity is $\bar{u} = \frac{1}{b}\int_0^b u\,dy = 7.5 - 3 \times 10^5\frac{b^2}{3} = 7.5 - 10^5 \times 2.5 \times 10^{-5} = 7.5 - 2.5 = 5$ m/s (two thirds of the maximum 7.5 m/s).