Changes between Version 12 and Version 13 of u/johannjc/scratchpad5
- Timestamp:
- 10/02/15 13:24:11 (9 years ago)
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u/johannjc/scratchpad5
v12 v13 2 2 The anisotropic conduction equation looks like 3 3 4 $\ frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \left ( \chi_\parallel - \chi_\perp \right ) \left ( \hat{b} \cdot \nabla T \right ) + n \chi_\perp \nabla T \right ]$4 $\rho c_v\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \left ( \chi_\parallel - \chi_\perp \right ) \left ( \hat{b} \cdot \nabla T \right ) + n \chi_\perp \nabla T \right ]$ 5 5 6 6 however, the conduction coefficients have a Temperature dependence. … … 14 14 So we can rewrite the equations as 15 15 16 $\ 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 ]$16 $\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \left ( \kappa_\parallel T^{\lambda_\parallel} - \frac{n \kappa_\perp}{B^2} T^{\lambda_\perp} \right ) \left ( \hat{b} \cdot \nabla T \right ) + \frac{n \kappa_\perp}{B^2} T^{\lambda_\perp} \nabla T \right ]$ 17 17 18 18 or 19 19 20 $\ 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 $\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( \hat{b} \cdot \nabla T^{\lambda_\parallel+1} \right ) - \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( \hat{b} \cdot \nabla T^{\lambda_\perp + 1} \right ) \right ) + \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \nabla T^{\lambda_\perp+1} \right ]$ 21 21 22 22 == Einstein simplification == … … 24 24 or in Einstein notation 25 25 26 $\ 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 ]$26 $\rho c_v \partial_t T = \partial_i \left [ n b_i \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( b_j \partial_j T^{\lambda_\parallel+1} \right ) - \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( b_j \partial_j T^{\lambda_\perp + 1} \right ) \right ) + \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \partial_i T^{\lambda_\perp+1} \right ]$ 27 27 28 28 or 29 29 30 $\ 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 ]$30 $\rho c_v \partial_t T = \partial_i \left [ n b_i \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( b_j \partial_j T^{\lambda_\parallel+1} \right ) - \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( b_j \partial_j T^{\lambda_\perp + 1} \right ) \right ) + \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \delta_{ij} \partial_j T^{\lambda_\perp+1} \right ]$ 31 31 32 32 or 33 33 34 $\ 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} $34 $\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} $ 35 35 36 36 or 37 37 38 $\ 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} $38 $\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} $ 39 39 40 40 == Implicitization == … … 46 46 where the $\parallel$ or $\perp$ subscript on $A$, $B_{ij}$, and $lambda$ is implied. 47 47 48 $A_\parallel = \frac{\kappa_\parallel}{\ lambda_\parallel + 1}$ and $A_\perp = \frac{\kappa_\perp}{\lambda_\perp + 1}$48 $A_\parallel = \frac{\kappa_\parallel}{\rho c_v \left ( \lambda_\parallel + 1\right ) }$ and $A_\perp = \frac{\kappa_\perp}{\rho c_v \left (\lambda_\perp + 1 \right )}$ 49 49 50 50 and $B_{\parallel, ij} = n b_i b_j$ and $B_{\perp, ij} = n^2 B^-2 b_i b_j \left ( 1 - \delta_{ij} \right )$ … … 137 137 ||$\alpha_{\pm i, \pm j}$ || $ \pm \pm \frac{F_{ij}\Delta t}{4 \Delta x^2}$ || 138 138 139 140 141 === Isotropic Case === 142 143 For the Isotropic Case, the diffusion equation looks like
