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GATE 2022 AE – Question 16

Aerodynamics · Basic fluid mechanics: kinematics, conservation laws, dimensional analysis, incompressibility and Newtonian fluids · 1 mark · Multiple choice

For steady two-dimensional incompressible flat-plate flow in control volume PQRS, inlet PQ speed is uniform $u_\infty$ and outlet RS speed is $u=A_0(y/h)^n$, n positive integer. The value of $A_0$ giving zero flow through PS is

source diagram and its labels are reproduced in the attached image.
  1. $u_\infty/(n+1)$
  2. $u_\infty(n+1)$
  3. $u_\infty/(n-1)$
  4. $u_\infty(n-1)$

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Correct answer: (B) $u_\infty(n+1)$

Explanation

**Mass conservation.** The flow is steady and incompressible, and no fluid crosses the plate surface $QR$ or the top boundary $PS$ (the question asks for zero flow through $PS$), so the volume flux entering through $PQ$ must equal the flux leaving through $RS$.

**Inlet flux** (per unit width), with uniform speed $u_\infty$ over the height $h$:
$$Q_{in}=u_\infty h .$$

**Outlet flux** with $u=A_0(y/h)^n$:
$$Q_{out}=\int_0^hA_0\left(\frac yh\right)^ndy=A_0\,\frac{h}{n+1}.$$

**Equate:**
$$u_\infty h=\frac{A_0h}{n+1}\;\Rightarrow\;A_0=(n+1)\,u_\infty\quad(\text{option B}).$$