Changes between Version 11 and Version 12 of ThermalConduction

Timestamp:

08/26/16 10:01:16 (8 years ago)

Author:

Jonathan

Comment:

—

ThermalConduction

@@ -52 +52 @@
 
 $\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} $
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 == Implicitization ==
@@ -102 +107 @@
 $ \partial_i B_{ij} \partial_j T^{\lambda}T' =  \left [ \frac{B^{ij}_{\hat{i}} T^{\lambda}_{\hat{i}+\hat{j}}T'_{\hat{i}+\hat{j}} - B^{ij}_{\hat{i}}T^{\lambda}_{\hat{i}-\hat{j}}T'_{\hat{i}-\hat{j}}-B^{ij}_{-\hat{i}} T^{\lambda}_{-\hat{i}+\hat{j}}T'_{-\hat{i}+\hat{j}} + B^{ij}_{-\hat{i}}T^{\lambda}_{-\hat{i}-\hat{j}}T'_{-\hat{i}-\hat{j}}}{4 \Delta x^2} \right ]  \left ( 1-\delta_{ij} \right )  $
 $+ \left [ \frac{B^{ij}_{\hat{i/2}} T^{\lambda}_{\hat{i}}T'_{\hat{i}} - B^{ij}_{\hat{i/2}}T^{\lambda}_{0}T'_{0}-B^{ij}_{-\hat{i/2}} T^{\lambda}_{0}T'_{0} + B^{ij}_{-\hat{i/2}}T^{\lambda}_{-\hat{i}}T'_{-\hat{i}}}{\Delta x^2} \right ] \delta_{ij}$
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+$ \partial_i \epsilon_{ijk} b_j \partial_k T= \frac{\epsilon_{ijk} \left ( B^j_{\hat{i}} \left (T_{\hat{i}+\hat{k}} - T_{\hat{i}-\hat{k}} \right ) - B^j_{-\hat{i}} \left (T_{-\hat{i}+\hat{k}} - T_{-\hat{i}-\hat{k}} \right ) \right )}{\Delta x}$
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 Using the above definitions, we can write the discretized equation as