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GATE 2021 ME (ME2) – Question 21

Fluid Mechanics · Differential equations of continuity and momentum, velocity potential · 1 mark · Multiple choice

For a two-dimensional, incompressible flow having velocity components $u$ and $v$ in the $x$ and $y$ directions, respectively, the expression $\frac{\partial(u^2)}{\partial x}+\frac{\partial(uv)}{\partial y}$ can be simplified to

  1. $uu_x+uv_y$
  2. $2uu_x+uv_y$
  3. $2uu_x+vu_y$
  4. $uu_x+vu_y$

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Correct answer: (D) $uu_x+vu_y$

Explanation

Expand by the product rule to get $2uu_x+vu_y+uv_y$. Incompressible continuity says uₓ+vᵧ=0.

Group the expression as $uu_x+vu_y+u(u_x+v_y)$; the final bracket vanishes. **Result: uuₓ+vuᵧ (D)**.