Changes between Version 22 and Version 23 of u/johannjc/scratchpad

Timestamp:

04/21/14 13:24:36 (11 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad

@@ -1 @@
-
-[[CollapsibleBegin(Single Temperature Solution)]]
 If we plug the expressions for the radiation 4-momentum back into the gas equations and keep terms necessary to maintain accuracy we get:
 
@@ -8 +6 @@
 
   [[latex($\frac{\partial E}{\partial t}  \color{red}{ - \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E}=\color{red}{\kappa_{0P} (4 \pi B-cE)} \color{green}{-\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1\right)\mathbf{v}\cdot \nabla E} \color{green}{-\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E\right )}\color{blue}{+\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E}  $)]]  
-[[CollasibleEnd()]]
+
+
+Now if
+
+[[latex($E=a_RT^4$)]]
+
+and
+
+[[latex($e=\rho c_v T$)]]
+
+we can combine the gas and radiation diffusion equations to arrive at:
+
+  [[latex($\frac{\partial \left (e+E\right) }{\partial t}+\nabla\cdot\left[\left(e+P\right)\mathbf{v}\right]=\color{red}{ \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E} \color{green}{-\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E\right )}$)]]