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I am doing an introductory course on fluid mechanics, and I'd like some intuition (that's why I'm posting on the engineering forums and not on the math forums, even if I'm studying for a degree in math) about the concept of the total derivative and, particularly, its convective component.

The total derivative of a vector field is defined as:

[tex]\frac {D \bf u}{Dt}= \frac {\partial u} {\partial t} + ({\bf u} \cdot \nabla){\bf u}[/tex]where the first term is the 'local' acceleration and the second the 'convective' acceleration.

I have no problem (I think) with the purely symbolic manipulation on the second term,

[tex]({\bf u} \cdot \nabla){\bf u} = u_1 \frac {\partial u_1}{\partial x} + u_2 \frac {\partial u_2}{\partial y} + u_3 \frac {\partial u_3}{\partial z}[/tex]where [itex]{\bf u} = u_1 {\bf i} + u_2 {\bf j} + u_3 {\bf k}[/itex] is the velocity vector. I just need to wrap my head around about the physical interpretation of it. The explanation of it being the 'acceleration following the motion' is a bit too vague, and I was wishing for some intuition about it.

A similar operation appears also applied to scalar fields, as in the continuity equation for (possibly compressible) flows,

[tex]\frac {\partial \rho} {\partial t} + ({\bf \rho} \cdot \nabla){\bf u} = 0[/tex]where [itex]\rho[/itex] is the density field, and which reduces to [itex]\mbox{div } {\bf u} = 0[/itex] if the flow is incompressible (constant [itex]\rho[/itex]). I believe I have a half-decent understanding of what 'divergence' means (in this context, if no sources or sinks are in the region, a compression*** on x has to be compensated by a decompression*** on y, in a two-dimensional example, just as a consequence of conservation of mass), and I hoped for a similar intuition for the 'convective' term, in both the scalar and vector field contexts.

Thanks a million if you have read so far! :)

*** P.S.: sorry, 'compression' and 'decompression' when talking about divergence is a poor choice of words; I meant 'change in speed' rather than 'change in density', since I was referring at that point to the continuity equation for an incompressible flow.

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# Fluid mechanics: intuition about the 'convective' term

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