# Why is the advective term positive when the flow velocity points in the same direction of the gradient of temperature?

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?

• 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)? Commented Mar 20, 2020 at 15:19

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