Changes between Version 19 and Version 20 of u/johannjc/scratchpad5


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Timestamp:
10/06/15 17:55:34 (9 years ago)
Author:
Jonathan
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  • u/johannjc/scratchpad5

    v19 v20  
    1010however, the conduction coefficients have a Temperature dependence.
    1111
    12 $\chi_\parallel = \kappa_\parallel T^{5/2}$
     12$\chi_\parallel = \kappa_\parallel T^{5/2} n^{-1}$
    1313
    1414$\chi_\perp = \kappa_\perp \frac{n}{B^2 T^{1/2}}$
    … …  
    1818So we can rewrite the equations as
    1919
    20 $\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \left ( \kappa_\parallel T^{\lambda_\parallel} - \frac{n \kappa_\perp}{B^2} T^{\lambda_\perp} \right ) \left ( \hat{b} \cdot \nabla T \right ) + \frac{n \kappa_\perp}{B^2} T^{\lambda_\perp} \nabla T \right ]$
    21 
    22 or
    23 
    24 $\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( \hat{b} \cdot \nabla T^{\lambda_\parallel+1} \right )  - \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( \hat{b} \cdot \nabla T^{\lambda_\perp + 1} \right )  \right ) + \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \nabla T^{\lambda_\perp+1} \right ]$
     20$\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \left [ \hat{b} \left ( \kappa_\parallel T^{\lambda_\parallel} - \frac{n^2 \kappa_\perp}{B^2} T^{\lambda_\perp} \right ) \left ( \hat{b} \cdot \nabla T \right ) + \frac{n^2 \kappa_\perp}{B^2} T^{\lambda_\perp} \nabla T \right ]$
     21
     22or
     23
     24$\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \left [ \hat{b} \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( \hat{b} \cdot \nabla T^{\lambda_\parallel+1} \right )  - \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( \hat{b} \cdot \nabla T^{\lambda_\perp + 1} \right )  \right ) + \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \nabla T^{\lambda_\perp+1} \right ]$
    2525
    2626== Einstein simplification ==
    … …  
    2828or in Einstein notation
    2929
    30 $\rho c_v \partial_t T = \partial_i \left [ n b_i \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( b_j \partial_j T^{\lambda_\parallel+1} \right )  - \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( b_j \partial_j  T^{\lambda_\perp + 1} \right )  \right ) + \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \partial_i T^{\lambda_\perp+1} \right ]$
    31 
    32 or
    33 
    34 $\rho c_v  \partial_t T = \partial_i \left [ n b_i \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( b_j \partial_j T^{\lambda_\parallel+1} \right )  - \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( b_j \partial_j  T^{\lambda_\perp + 1} \right )  \right ) + \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \delta_{ij} \partial_j T^{\lambda_\perp+1} \right ]$
    35 
    36 or
    37 
    38 $\rho c_v \partial_t T = \partial_i n b_i \frac{\kappa_\parallel}{\lambda_\parallel+1} b_j \partial_j T^{\lambda_\parallel+1} +   \partial_i n \left ( \delta_{ij} - b_i b_j \right ) \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp + 1\right )} \partial_j T^{\lambda_\perp+1} $
    39 
    40 or
    41 
    42 $\rho c_v  \partial_t T = \frac{\kappa_\parallel}{\lambda_\parallel+1}  \partial_i n b_i  b_j \partial_j T^{\lambda_\parallel+1} +   \frac{\kappa_\perp}{\left ( \lambda_\perp + 1\right )} \partial_i n^2 B^{-2} \left ( \delta_{ij} - b_i b_j \right )  \partial_j T^{\lambda_\perp +1} $
     30$\rho c_v \partial_t T = \partial_i \left [ b_i \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( b_j \partial_j T^{\lambda_\parallel+1} \right )  - \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( b_j \partial_j  T^{\lambda_\perp + 1} \right )  \right ) + \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \partial_i T^{\lambda_\perp+1} \right ]$
     31
     32or
     33
     34$\rho c_v  \partial_t T = \partial_i \left [ b_i \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( b_j \partial_j T^{\lambda_\parallel+1} \right )  - \frac{ \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( b_j \partial_j  T^{\lambda_\perp + 1} \right )  \right ) + \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \delta_{ij} \partial_j T^{\lambda_\perp+1} \right ]$
     35
     36or
     37
     38$\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} $
     39
     40or
     41
     42$\rho c_v  \partial_t T = \frac{\kappa_\parallel}{\lambda_\parallel+1}  \partial_i b_i  b_j \partial_j T^{\lambda_\parallel+1} +   \frac{\kappa_\perp}{\left ( \lambda_\perp + 1\right )} \partial_i n^2 B^{-2} \left ( \delta_{ij} - b_i b_j \right )  \partial_j T^{\lambda_\perp +1} $
    4343
    4444== Implicitization ==
    … …  
    5252$A_\parallel = \frac{\kappa_\parallel}{\rho c_v  \left ( \lambda_\parallel + 1\right ) }$ and $A_\perp = \frac{\kappa_\perp}{\rho c_v  \left (\lambda_\perp + 1 \right )}$
    5353
    54 and $B_{\parallel, ij} = n b_i b_j$ and $B_{\perp, ij} = n^2 B^-2 b_i b_j \left ( 1 - \delta_{ij} \right )$
     54and $B_{\parallel, ij} = b_i b_j$ and $B_{\perp, ij} = n^2 B^-2 b_i b_j \left ( 1 - \delta_{ij} \right )$
    5555
    5656Now to solve this implicitly, we need to replace $T$ with $T_*$ where
    … …  
    129129||  ||   $\parallel$   ||   $\perp$   ||
    130130|| $A$ || $\frac{\kappa_\parallel}{\rho c_v  \left ( \lambda_\parallel + 1 \right ) }$ || $\frac{\kappa_\perp}{\rho c_v  \left (\lambda_\perp + 1\right )}$ ||
    131 || $B_{ij} $ || $n b_i b_j$ || $n^2 B^-2 b_i b_j \left ( 1 - \delta_{ij} \right )$ ||
     131|| $B_{ij} $ || $b_i b_j$ || $n^2 B^-2 b_i b_j \left ( 1 - \delta_{ij} \right )$ ||
    132132|| $C$ ||  $\left ( 1 - \phi \left ( \lambda_\parallel + 1 \right ) \right )$ ||  $\left ( 1 - \phi \left ( \lambda_\perp + 1 \right ) \right )$ ||
    133133|| $D$ ||  $\phi \left ( \lambda_\parallel + 1 \right )$ ||  $\phi \left ( \lambda_\perp + 1 \right ) $||