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GATE 2022 ME (ME1) – Question 24

Fluid Mechanics · Bernoulli's equation and fluid acceleration · 1 mark · Multiple choice

A tiny temperature probe is fully immersed in a flowing fluid and is moving with zero relative velocity with respect to the fluid. The velocity field in the fluid is $\vec{V} = (2x)\hat{i} + (y + 3t)\hat{j}$, and the temperature field in the fluid is $T = 2x^2 + xy + 4t$, where x and y are the spatial coordinates, and t is the time. The time rate of change of temperature recorded by the probe at ($x = 1$, $y = 1$, $t = 1$) is _______.

  1. 4
  2. 0
  3. 18
  4. 14

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Correct answer: (C) 18

Explanation

The probe moves with the fluid, so it records the material derivative $\frac{DT}{Dt} = \frac{\partial T}{\partial t} + u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y}$. Here $\frac{\partial T}{\partial t} = 4$, $\frac{\partial T}{\partial x} = 4x + y = 5$, $\frac{\partial T}{\partial y} = x = 1$, $u = 2$ and $v = 1 + 3 = 4$. So $\frac{DT}{Dt} = 4 + 2 \times 5 + 4 \times 1 = 18$.