GATE 2023 ME – Question 62
The velocity field of a certain two-dimensional flow is given by $\mathbf{V}(x, y) = k(x\hat{i} - y\hat{j})$ where $k = 2$ s$^{-1}$. The coordinates $x$ and $y$ are in meters. Assume gravitational effects to be negligible.
If the density of the fluid is 1000 kg/m$^3$ and the pressure at the origin is 100 kPa, the pressure at the location (2 m, 2 m) is _____________ kPa.
(Answer in integer)
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Correct answer: 84
Explanation
The flow is steady, incompressible ($\nabla \cdot \mathbf{V} = k - k = 0$) and irrotational (the curl is zero), and gravity is negligible, so Bernoulli's equation holds everywhere: $p + \frac{1}{2}\rho V^2 = $ constant. At the origin $V = 0$. At $(2, 2)$, $V^2 = k^2(x^2 + y^2) = 4 \times 8 = 32$ m$^2$/s$^2$. So $p = 100000 - \frac{1}{2} \times 1000 \times 32 = 84000$ Pa $= 84$ kPa.