Changes between Version 6 and Version 7 of u/bliu/AblativeRT/Equations

Timestamp:

05/29/15 13:47:07 (10 years ago)

Author:

Baowei Liu

Comment:

—

u/bliu/AblativeRT/Equations

@@ -1 @@
-The current version of the code has changed the temperature from computational units to Kelvin or solves the energy equation instead of temperature. This note is to check the parameter and equations changes need to do to be consistent comparing the old ways of blog:bliu01092014  and blog:bliu01142014 . So the term $\gamma_{7}$ won't show in the thermal diffusion equations.
+The current version of the code has changed the temperature from computational units to Kelvin or solves the energy equation instead of temperature. This note is to check the parameter and equations changes need to do to be consistent comparing the old ways of [https://astrobear.pas.rochester.edu/trac/wiki/u/bliu/AblativeRT/TempEquations thermal diffusion equation in Astrobear]. So lhs of the equation was replaced by the derivative of the energy.
 
-1. Equations
-$$\frac{1}{\gamma-1}\rho_{cgs}\frac{\partial T}{\partial t}=\frac{10\kappa_{0}K_{B}^{n}}{(\gamma-1)C_{v}}\frac{\partial}{\partial x_{cgs}}\left[T^{n}\frac{\partial T}{\partial x_{cgs}}\right]$$
+=== 1. Equations ===
+The heat diffusion equation solved in AstroBEAR is equivalent to 
 
-where T is in cgs unit or Kelvin.
+$$C_{v}\rho_{cgs}\frac{\partial T}{\partial t}=\frac{\partial}{\partial x_{cgs}}\left[\kappa_{1}T^{n}\frac{\partial T}{\partial x_{cgs}}\right]$$
 
+where T is in cgs unit or Kelvin and $C_{v}$ is the specific heat capacity.
+
+Here 
+ $$\kappa_{1}=\frac{10\kappa_{0}K_{B}^{n}}{(\gamma-1)}$$
 In current version of AstroBEAR, we have
 $$\frac{\partial E}{\partial T}\frac{\partial T}{\partial t}=\frac{10\kappa_{0}K_{B}^{n}}{(\gamma-1)C_{v}}\frac{\partial}{\partial x_{cgs}}\left[T^{n}\frac{\partial T}{\partial x_{cgs}}\right]$$