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| Integral form | Differential form |
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| \frac{d}{dt} \iiint_\Omega \rho \, dV = \dot m = \frac{dm_\Omega}{dt} |
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| \frac{\partial \rho}{\partial t} + \nabla (\rho \, {\bf u}) = \frac{dm (t, {\bf r})}{dt} |
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| time |
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| body | --uriencoded--%7B\bf r %7D |
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| position vector |
| space volume (could be finite or infinite) |
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| body | --uriencoded--\rho(t, %7B\bf r%7D) |
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| continuum body spatial density distribution |
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| body | --uriencoded--%7B\bf u%7D(t, %7B\bf r) |
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| continuum body spatial velocity distribution |
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| body | --uriencoded--\displaystyle \frac%7Bdm_\Omega%7D%7Bdt%7D |
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| mass generation rate with the space volume |
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| body | --uriencoded--\dot m = \frac%7Bdm%7D%7Bdt%7D\displaystyle \frac%7Bdm(t, %7B\bf r%7D)%7D%7Bdt%7D |
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| at a given point in space | LaTeX Math Inline |
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| body | --uriencoded--%7B\bf r %7D |
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For the specific case of stationary process when density is not explicitly dependent on timeFor the stationary fluid flow:
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\frac{\partial \rho}{\partial t} = 0 \rightarrow \nabla (\rho \, {\bf u}) = 0 |
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