...
to arrive at:
| LaTeX Math Block |
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| anchor | S8TNBpre_filnal |
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| alignment | left |
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| \rho \, \phi \, c_t \cdot \frac{\partial p}{\partial t} + \nabla \, ( \rho \, {\bf u}) = \rho \, \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
| | LaTeX Math Block |
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| \int_{\Sigma_k} \, {\bf u} \, d {\bf A} = q_k(t) |
|
...
Then use the following equality:
| LaTeX Math Block |
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| anchor | 1nabla_rho |
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| alignment | left |
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\nabla \, ( \rho \, {\bf u}) = \rho \, \nabla \, {\bf u} + (\nabla \rho, \, {\bf u}) = \rho \, \nabla \, {\bf u} + \frac{d\rho}{dp} \cdot (\nabla p, \, {\bf u}) = \rho \, \nabla \, {\bf u} + \rho \, c \cdot (\nabla p, \, {\bf u}) |
where
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle c(p) = \frac%7B1%7D%7B\rho%7D \frac%7Bd\rho%7D%7Bdp%7D |
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|
is
fluid compressibility.
Substituting
| LaTeX Math Block Reference |
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into | LaTeX Math Block Reference |
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|
and reducing the density one arrives to:| LaTeX Math Block |
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| anchor | pre_filnal |
|---|
| alignment | left |
|---|
| \rho \, \phi \, c_t \cdot \frac{\partial p}{\partial t} + \nabla \, ( \rho \, {\bf u}) = \rho \, \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
| | LaTeX Math Block |
|---|
| \int_{\Sigma_k} \, {\bf u} \, d {\bf A} = q_k(t) |
|
See also
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Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model / Single-phase pressure diffusion @model
...
