Project ForgeA 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 V = (2x)î + (y+3t)ĵ, and the temperature field in the fluid is T = 2x²+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 _______.
- A4
- B0
- 18
- D14
Explanation
The probe moves with the fluid, so it records the material (substantial) derivative DT/Dt = ∂T/∂t + u∂T/∂x + v∂T/∂y. At (1,1,1): u=2x=2, v=y+3t=4, ∂T/∂t=4, ∂T/∂x=4x+y=5, ∂T/∂y=x=1. DT/Dt = 4 + 2(5) + 4(1) = 4+10+4 = 18.
Source: GATE ME 2022 Session 1 (ME-1) (2022)
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