Changes between Version 13 and Version 14 of ThermalConduction
- Timestamp:
- 12/20/21 12:34:33 (3 years ago)
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ThermalConduction
v13 v14 145 145 or in Einstein notation 146 146 147 $\rho c_v \partial_t T = \partial_i \ frac{\kappa}{\lambda+1} \partial_i T^{\lambda+1}$147 $\rho c_v \partial_t T = \partial_i \kappa \partial_i T$ 148 148 149 149 which we can rewrite as 150 150 151 $\partial_t T = A \partial_i B \partial i T ^{\lambda+1}$152 153 $\partial_t T = A \partial_i B \delta_{ij} \partial j T ^{\lambda+1}$151 $\partial_t T = A \partial_i B \partial i T$ 152 153 $\partial_t T = A \partial_i B \delta_{ij} \partial j T$ 154 154 155 155 This is the exact same form as before, but now … … 165 165 So in summary we have 166 166 167 || $A$ || $\frac{1}{\rho c_v \left ( \lambda + 1 \right )}$ ||167 || $A$ || $\frac{1}{\rho c_v }$ || 168 168 || $B_{ij} $ || $\kappa \delta{ij}$|| 169 || $C$ || $\left ( 1 - \phi \ left ( \lambda+ 1 \right ) \right )$ ||170 || $D$ || $\phi \left ( \lambda + 1 \right )$ ||169 || $C$ || $\left ( 1 - \phi \right )$ || 170 || $D$ || $\phi $ || 171 171 172 172 and … … 179 179 And our equation becomes 180 180 181 $T_0 + \Delta t \displaystyle \sum_{\parallel, \perp} AC \left [ \alpha_0 T ^{\lambda+1}_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} \right ] = $182 $T'_0 - \Delta t \displaystyle \sum_{\parallel, \perp} AD \left [ \alpha_0 T ^{\lambda}_0T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda}_{\pm \hat{j}} T'_{\pm \hat{j}} \right ]$181 $T_0 + \Delta t \displaystyle \sum_{\parallel, \perp} AC \left [ \alpha_0 T_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T_{\pm \hat{j}} \right ] = $ 182 $T'_0 - \Delta t \displaystyle \sum_{\parallel, \perp} AD \left [ \alpha_0 T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T'_{\pm \hat{j}} \right ]$ 183 183 184 184