I am having some trouble understanding some things about the Eulerian description of a continuum.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose we have a fluid that is continuously deforming and moving in time. If x are the spatial coordinates that the fluid is passing through, t is time, and p is a scalar function p(x,t), for example density, then for a path [itex]\gamma(t)[/itex] through space the chain rule gives,

[tex]\frac{d}{dt}\rho = \frac{\partial \rho}{\partial t} + \dot{\gamma} \cdot \nabla \rho[/tex]

for the derivative of p in along the path [itex]\gamma[/itex].

So far so good. But then, multiple sources I have read make the jump to the following statement without justification: In general (not on a path),

[tex]\frac{d}{dt}\rho = \frac{\partial \rho}{\partial t} + v \cdot \nabla \rho[/tex]

where v is the velocity of the fluid at that point x and time t. This seems nonsensical though. If p(x,t) really describes the exact density at a spatial point x and time t, then shouldn't we simply have [itex]\frac{d\rho}{dt} = \frac{\partial \rho}{\partial t}[/itex]? In the Eulerian description, isn't x just a static point in space? Plus, what is the fluid velocity v doing in there?

I've been stuck on this for several hours, and I'm pretty sure there's an important but subtle point I'm missing.

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# Eulerian description

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