Changes between Version 9 and Version 10 of ThermalConduction

Timestamp:

08/17/16 12:59:49 (8 years ago)

Author:

ehansen

Comment:

—

ThermalConduction

@@ -15 +15 @@
 $\chi_\perp = \kappa_\perp \frac{n}{B^2 T^{1/2}}$
 
+Just FYI, some references will define variables a bit differently, and write these equations like this:
+
+$\rho c_v\frac{\partial T}{\partial t} = \nabla \cdot \left [\hat{b} \left ( \kappa_\parallel - \kappa_\perp \right ) \left ( \hat{b} \cdot \nabla T \right ) + \kappa_\perp \nabla T \right ]$
+
+$\kappa_\parallel = \c_\parallel T^{5/2}$ 
+
+$\kappa_\perp = \c_\perp \frac{n^2}{B^2 T^{1/2}}$
+
+where the c's are coefficients usually given as some number in references.
+
 == Collecting power's of T ==
 
@@ -24 +34 @@
 
 $\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \left [ \hat{b} \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( \hat{b} \cdot \nabla T^{\lambda_\parallel+1} \right )  - \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( \hat{b} \cdot \nabla T^{\lambda_\perp + 1} \right )  \right ) + \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \nabla T^{\lambda_\perp+1} \right ]$
-
-Just FYI, some references put T, n, and B dependencies into kappa, and write equation like this
-
-$\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \left [ \hat{b} \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( \hat{b} \cdot \nabla T^{\lambda_\parallel+1} \right )  - \frac{\kappa_\perp}{\left ( \lambda_\perp+1 \right )} \left ( \hat{b} \cdot \nabla T^{\lambda_\perp + 1} \right )  \right ) + \frac{\kappa_\perp}{\left (\lambda_\perp +1 \right )} \nabla T^{\lambda_\perp+1} \right ]$
 
 == Einstein simplification ==