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GATE 2022 ME (ME2) – Question 29

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

The velocity field in a fluid is given to be $\mathbf V=(4xy)\hat i+2(x^2-y^2)\hat j$. Which of the following statement(s) is/are correct?

  1. The velocity field is one-dimensional.
  2. The flow is incompressible.
  3. The flow is irrotational.
  4. The acceleration experienced by a fluid particle is zero at $(x=0,y=0)$.

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Correct answer: (B) The flow is incompressible.; (C) The flow is irrotational.; (D) The acceleration experienced by a fluid particle is zero at $(x=0,y=0)$.

Explanation

Take $u=4xy$ and $v=2(x^2-y^2)$.

**A, one-dimensional?** The field has two non-zero components that depend on both $x$ and $y$, so the flow is two-dimensional. False.

**B, incompressible?**
$$\nabla\cdot\mathbf V=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=4y-4y=0.$$
True.

**C, irrotational?** The vorticity is
$$\omega_z=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}=4x-4x=0.$$
True.

**D, acceleration at the origin?** The flow is steady, so the acceleration is the convective part $(\mathbf V\cdot\nabla)\mathbf V$. At the origin $u=v=0$, so every term is a product with a zero velocity and the acceleration is zero. True.

Correct statements: **B, C and D**.