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GATE 2024 ME – Question 16

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

The velocity field of a two-dimensional, incompressible flow is given by

$$\vec{V} = 2\sinh x\,\hat{i} + v(x, y)\,\hat{j}$$

where $\hat{i}$ and $\hat{j}$ denote the unit vectors in $x$ and $y$ directions, respectively. If $v(x, 0) = \cosh x$, then $v(0, -1)$ is

  1. 1
  2. 2
  3. 3
  4. 4

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Correct answer: (C) 3

Explanation

Incompressible flow needs $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$. Here $\frac{\partial u}{\partial x} = 2\cosh x$, so $\frac{\partial v}{\partial y} = -2\cosh x$ and $v = -2y\cosh x + g(x)$. The condition $v(x, 0) = \cosh x$ gives $g(x) = \cosh x$. Then $v(0, -1) = -2(-1)(1) + 1 = 3$.