diff --git a/docs/source/FluidEquations.rst b/docs/source/FluidEquations.rst index eda92b7f1cb03523650feb1efc76d1c4934f4a6c..d60e9d9aa5ae83c239098967a3b0d804f321b12a 100644 --- a/docs/source/FluidEquations.rst +++ b/docs/source/FluidEquations.rst @@ -1,25 +1,22 @@ -Fluid Equations +Fluid Variables =============== -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: