Material Derivative

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The material derivative is defined along integral curves of the flow. This has some subtle implications. For instance, suppose f represents the temperature of the fluid in a swimming pool, and suppose the pool's temperature decreases as depth increases, and that this is a steady state. Then \frac{\partial f}{\partial t} = 0 everywhere in the pool, but a particle travelling from the deep end to the shallow end will have a nonzero material derivative, because \vec{u} \cdot \nabla f \neq 0.