Changes between Version 20 and Version 21 of u/johannjc/scratchpad5

Timestamp:

10/06/15 17:58:37 (10 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad5

@@ -32 +32 @@
 or 
 
-$\rho c_v  \partial_t T = \partial_i \left [ b_i \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( b_j \partial_j T^{\lambda_\parallel+1} \right )  - \frac{ \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( b_j \partial_j  T^{\lambda_\perp + 1} \right )  \right ) + \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \delta_{ij} \partial_j T^{\lambda_\perp+1} \right ]$
-
-or 
-
-$\rho c_v \partial_t T = \partial_i b_i \frac{\kappa_\parallel}{\lambda_\parallel+1} b_j \partial_j T^{\lambda_\parallel+1} +   \partial_i n^2 \left ( \delta_{ij} - b_i b_j \right ) \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp + 1\right )} \partial_j T^{\lambda_\perp+1} $
+$\rho c_v  \partial_t T = \partial_i \left [ b_i \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( b_j \partial_j T^{\lambda_\parallel+1} \right )  - \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( b_j \partial_j  T^{\lambda_\perp + 1} \right )  \right ) + \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \delta_{ij} \partial_j T^{\lambda_\perp+1} \right ]$
+
+or 
+
+$\rho c_v \partial_t T = \partial_i b_i \frac{\kappa_\parallel}{\lambda_\parallel+1} b_j \partial_j T^{\lambda_\parallel+1} +   \partial_i \left ( \delta_{ij} - b_i b_j \right ) \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp + 1\right )} \partial_j T^{\lambda_\perp+1} $
 
 or 
@@ -52 +52 @@
 $A_\parallel = \frac{\kappa_\parallel}{\rho c_v  \left ( \lambda_\parallel + 1\right ) }$ and $A_\perp = \frac{\kappa_\perp}{\rho c_v  \left (\lambda_\perp + 1 \right )}$
 
-and $B_{\parallel, ij} = b_i b_j$ and $B_{\perp, ij} = n^2 B^-2 b_i b_j \left ( 1 - \delta_{ij} \right )$
+and $B_{\parallel, ij} = b_i b_j$ and $B_{\perp, ij} = n^2 B^{-2} b_i b_j \left ( 1 - \delta_{ij} \right )$
 
 Now to solve this implicitly, we need to replace $T$ with $T_*$ where 
@@ -129 +129 @@
 ||  ||   $\parallel$   ||   $\perp$   ||
 || $A$ || $\frac{\kappa_\parallel}{\rho c_v  \left ( \lambda_\parallel + 1 \right ) }$ || $\frac{\kappa_\perp}{\rho c_v  \left (\lambda_\perp + 1\right )}$ ||
-|| $B_{ij} $ || $b_i b_j$ || $n^2 B^-2 b_i b_j \left ( 1 - \delta_{ij} \right )$ ||
+|| $B_{ij} $ || $b_i b_j$ || $n^2 B^{-2} b_i b_j \left ( 1 - \delta_{ij} \right )$ ||
 || $C$ ||  $\left ( 1 - \phi \left ( \lambda_\parallel + 1 \right ) \right )$ ||  $\left ( 1 - \phi \left ( \lambda_\perp + 1 \right ) \right )$ ||
 || $D$ ||  $\phi \left ( \lambda_\parallel + 1 \right )$ ||  $\phi \left ( \lambda_\perp + 1 \right ) $||