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William D. Fullmer
docs
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506a8ef1
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506a8ef1
authored
6 years ago
by
Ann Almgren
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Define variables
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docs/source/FluidEquations.rst
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506a8ef1
Fluid Equations
===============
We define the following fluid variables:
:math:`\rho_g = ` fluid density (assumed to be constant in the absence of reactions)
:math:`\varepsilon_g = ` volume fraction of fluid (accounts only for displacement of fluid by particle, does not account for the EB walls)
:math:`U_g = ` fluid velocity
:math:`p_g = ` fluid pressure
:math:`tau = ` viscous stress tensor
:math:`g = ` gravitational acceleration
:math:`\beta_p = ` drag coefficient associated with a particle
:math:`V_p = ` velocity associated with a particle
Below are the governing equations for the fluid:
Conservation of fluid mass:
...
...
@@ -9,12 +27,12 @@ Conservation of fluid mass:
Conservation of fluid momentum:
.. math:: \frac{ \partial (\varepsilon_g \rho_g U)}{\partial t} + \nabla \cdot (\varepsilon_g \rho_g U_g U_g) + \varepsilon_g \nabla p_g = \nabla \cdot \tau
+ {\bf g}
+ \sum_{part} \beta (V_
{part}
- U_g)
.. math:: \frac{ \partial (\varepsilon_g \rho_g U)}{\partial t} + \nabla \cdot (\varepsilon_g \rho_g U_g U_g) + \varepsilon_g \nabla p_g = \nabla \cdot \tau
+ \sum_{part} \beta
_p
(V_
p
- U_g)
+ \rho_g g
Conservation of fluid volume:
where :math:`\sum_p \beta_p (V_p - U_g)` is the drag term in which :math:`V_p` represents the particle velocity.
where :math:`\sum_{part} \beta (V_{part} - U_g)` is the drag term in which :math:`V_{part}` represents the particle velocity.
Conservation of fluid volume:
.. math:: \frac{\partial \varepsilon_g}{\partial t} + \nabla \cdot (\varepsilon_g U_g) = 0
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