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William D. Fullmer
docs
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db9ac5b5
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db9ac5b5
authored
6 years ago
by
Ann Almgren
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Put variable definitions in a table
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docs/source/FluidEquations.rst
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Fluid
Equation
s
Fluid
Variable
s
===============
We define the following fluid variables:
where :math:`\rho_g =` fluid density (assumed to be constant in the absence of reactions)
where :math:`\varepsilon_g =` volume fraction of fluid (accounts only for displacement of fluid by particle, does not account for the EB walls)
where :math:`U_g =` fluid velocity
where :math:`p_g =` fluid pressure
where :math:`tau =` viscous stress tensor
+-----------------------+--------------------------------------------------+
| Variable | Definition |
+=======================+==================================================+
| :math:`\rho_g` | Fluid density |
+------------+-------------------------------------+-----------------------+
| :math:`\varepsilon_g` | Volume fraction of fluid (= 1 if no particles) |
+------------+-------------------------------------+-----------------------+
| :math:`U_g` | Fluid velocity |
+------------+-------------------------------------+-----------------------+
| :math:`\tau` | Viscous stress tensor |
+------------+-------------------------------------+-----------------------+
| :math:`g` | Gravitational acceleration |
+------------+-------------------------------------+-----------------------+
where :math:`g =` gravitational acceleration
where :math:`\beta_p =` drag coefficient associated with a particle
where :math:`V_p =` velocity associated with a particle
Below are the governing equations for the fluid:
Fluid Equations
===============
Conservation of fluid mass:
...
...
@@ -30,7 +27,8 @@ 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
+ \sum_{part} \beta_p (V_p - U_g) + \rho_g g
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_p \beta_p (V_p - U_g)` is the drag term in which :math:`V_p` represents the particle velocity and
:math:`\beta_p` is the drag coefficient associated with that particle
Conservation of fluid volume:
...
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