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If the total derivative of a fluid property (say, temperature) is given by $$ \frac{dT}{dt} = \frac{\partial T}{\partial t} + (\vec{V}\cdot \nabla) T $$ then when $ \nabla T $ and $\vec{V}$ point in the same direction, we have that the flow is coming from a region where the temperature is lower. At the same time, we have that $(\vec{V} \cdot \nabla) T$ is positive in this case, giving a positive contribution to the total derivative $ \frac{dT}{dt} $, which doesn't make sense to me. If the flow is coming from a region where temperatures are lower, shouldn't the contribution of advection to the total derivative be negative?

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  • $\begingroup$ I think you're ignoring the $\partial_{t} T$ term in your evaluation. The effect of a temperature gradient is to get rid of itself, right? Can you use this type of derivative on the temperature without including other effects (e.g., heat flux)? $\endgroup$ Commented Mar 20, 2020 at 15:19

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You've written the definition of the temperature total derivative, but that isn't the conservation law so it doesn't tell you anything about how temperature is changing. It's just the definition.

If we assume just fluid motion and no diffusion, the equation is going to be:

$$ \frac{dT}{dt} = 0 $$ or $$ \frac{\partial T}{\partial t} = -(\vec{V}\cdot\nabla)T$$

And now you can see that if velocity transports things along the temperature gradient, the temperature at a given point will decrease with time, just as you would expect.

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