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GATE 2024 CH – Question 16

Fluid Mechanics and Mechanical Operations · Equations of continuity, motion and mechanical energy, Euler and Bernoulli equations · 1 mark · Multiple choice

The velocity field in an incompressible flow is $\mathbf{v} = \alpha xy\hat{i} + v_y\hat{j} + \beta\hat{k}$, where $\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit-vectors in the $(x, y, z)$ Cartesian coordinate system. Given that $\alpha$ and $\beta$ are constants, and $v_y = 0$ at $y = 0$, the correct expression for $v_y$ is

  1. $-\frac{\alpha xy}{2}$
  2. $-\frac{\alpha y^2}{2}$
  3. $\frac{\alpha y^2}{2}$
  4. $\frac{\alpha xy}{2}$

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Correct answer: (B) $-\frac{\alpha y^2}{2}$

Explanation

Incompressibility gives $\nabla \cdot \mathbf{v} = \frac{\partial(\alpha xy)}{\partial x} + \frac{\partial v_y}{\partial y} = \alpha y + \frac{\partial v_y}{\partial y} = 0$. So $\frac{\partial v_y}{\partial y} = -\alpha y$ and $v_y = -\frac{\alpha y^2}{2}$ (using $v_y = 0$ at $y = 0$).