GATE 2022 AE – Question 50
Air flow is $\vec V=ay\hat i+bx\hat j$, with a=10 s⁻¹, b=20 s⁻¹ and ρ=1 kg/m³. Pressure at (0,0) is 100 kPa. Neglecting gravity, pressure at (6,8) m (nearest integer kPa) is ________.
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Correct answer: 90
Explanation
**Step 1: acceleration** of a fluid particle in this steady flow, $\vec V=(ay,\ bx)$:
$$a_x=u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=(ay)(0)+(bx)(a)=ab\,x,$$
$$a_y=u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=(ay)(b)+(bx)(0)=ab\,y .$$
**Step 2: Euler's equation** (no body force): $\nabla p=-\rho\vec a$, so
$$\frac{\partial p}{\partial x}=-\rho ab\,x,\qquad\frac{\partial p}{\partial y}=-\rho ab\,y .$$
**Step 3: integrate** from $(0,0)$ to $(x,y)$:
$$p=p_0-\frac{\rho ab}{2}\left(x^2+y^2\right).$$
**Numbers.** $\rho ab=1\times10\times20=200$ and $x^2+y^2=36+64=100$:
$$p=100\,000-\frac{200}{2}(100)=100\,000-10\,000=90\,000\text{ Pa}=\mathbf{90\ kPa}.$$