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Mathematical form of Mass Conservation for continuum body:
Integral form | Differential form |
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LaTeX Math Block |
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anchor | ContinuityIntegral |
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alignment | left |
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| \frac{d}{dt} \iiint_\Omega \rho \, dV = \frac{dm_\Omega}{dt} |
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...
ContinuityDifferential | alignment | left |
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| \frac{\partial \rho}{\partial t} + \nabla (\rho \, {\bf u}) = \frac{d\rho (t, {\bf r})}{dt} |
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where
| time | LaTeX Math Inline |
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body | --uriencoded--\rho(t, %7B\bf r%7D) |
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| continuum body spatial density distribution |
LaTeX Math Inline |
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body | --uriencoded--%7B\bf r %7D |
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| position vector | LaTeX Math Inline |
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body | --uriencoded--%7B\bf u%7D(t, %7B\bf r) |
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| continuum body spatial velocity distribution |
| space volume (could be finite or infinite) | LaTeX Math Inline |
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body | --uriencoded--\displaystyle \frac%7Bdm_\Omega%7D%7Bdt%7D |
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| mass generation rate with the space volume |
| gradient operator | LaTeX Math Inline |
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body | --uriencoded--\displaystyle \frac%7Bd\rho(t, %7B\bf r%7D)%7D%7Bdt%7D |
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| volume-specific mass generation rate 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 time:
LaTeX Math Block |
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\frac{\partial \rho}{\partial t} = 0 \rightarrow \nabla (\rho \, {\bf u}) = 0 |
For the specific case of finite number of mass generation locations the differential equation
LaTeX Math Block Reference |
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anchor | ContinuityDifferential |
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takes form: LaTeX Math Block |
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anchor | ContinuityDifferential |
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alignment | left |
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| \frac{\partial \rho}{\partial t} + \nabla (\rho \, {\bf u}) = \sum_k \dot m_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
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where
LaTeX Math Inline |
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body | --uriencoded--%7B\bf r %7D_k |
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| position vector of the -th source/sink |
| mass generation rate at -th source/sink: LaTeX Math Inline |
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body | --uriencoded--\displaystyle \dot m_k(t) = \frac%7Bdm_k%7D%7Bdt%7D |
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| Dirac delta function |
Alternatively it can be written as:
LaTeX Math Block |
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anchor | ContinuityDifferential |
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alignment | left |
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| \frac{\partial \rho}{\partial t} + \nabla (\rho \, {\bf u}) = \sum_k \rho(t, {\bf r}) \cdot \dot q_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
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wehre
| volumetric value of body mass generation rate at -th source/sink: LaTeX Math Inline |
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body | --uriencoded--\displaystyle q_k(t) = \frac%7BdV_k%7D%7Bdt%7D |
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See also
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Natural Science / Physics / Mechanics / Continuum mechanics
[ Mass Conservation ]