Test Academy | 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 V → = ( 2 x ) i ^ + ( y + 3 t ) 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 ______

Question

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 $\vec{V} = (2 x) \hat{i} + (y + 3 t) \hat{j},$ 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 _______.

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Answer :

Solution :

$\vec{V} = (2 x) \hat{i} + (y + 3 t) \hat{j},$ T = 2x² + xy + 4t

at, x = 1, y = 1, t = 1 to find : $\frac{D T}{D t}$

$\frac{D T}{D t} = \frac{\partial T}{\partial x} . \frac{d x}{d t} + \frac{\partial T}{\partial y} . \frac{d y}{d t} + \frac{\partial T}{\partial t} . \frac{d t}{d t}$

$\frac{d x}{d t}$ = u = 2x, $\frac{d y}{d t}$ = v = y + 3t

$\frac{\partial T}{\partial x}$ = 4x + y, $\frac{d y}{d t}$ = x, $\frac{\partial T}{\partial t}$ = 4

$\frac{D T}{D t}$ = (4x + y).2x + x.(y + 3t) + 4

put x = 1 , y = 1, t = 1

$\frac{D T}{D t}$ = (4 + 1).2 + 1.(1 + 3) + 4

$\frac{D T}{D t}$ = 10 + 4 + 4 = 18

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