3.5.6 Convective derivative In equation (33), the derivative d/dt is the total time derivative seen by an observer travelling with

the fluid velocity v:

d dt

=

∂ ∂t

+

v · ∇.

(34)

For instance, when entering a warm house on a cold winter day, one feels the temperature changing with time: at the time when you are outside, the temperature you feel is low, an instant later, when you are inside, the temperature is high. By dividing by the time it took you to go inside, you get a derivative dT/dt. This is completely due to your motion, v · ∇T , the change of temperature at any fixed point, ∂T/∂t, being essentially zero.

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