Changes between Version 13 and Version 14 of ThermalConduction


Ignore:
Timestamp:
12/20/21 12:34:33 (3 years ago)
Author:
Jonathan
Comment:

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  • ThermalConduction

    v13 v14  
    145145or in Einstein notation
    146146
    147 $\rho c_v \partial_t T = \partial_i \frac{\kappa}{\lambda+1} \partial_i T^{\lambda+1}$
     147$\rho c_v \partial_t T = \partial_i \kappa \partial_i T$
    148148
    149149which we can rewrite as
    150150
    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$\partial_t T = A \partial_i B \delta_{ij} \partial j T$
    154154
    155155This is the exact same form as before, but now
     
    165165So in summary we have
    166166
    167 || $A$ || $\frac{1}{\rho c_v  \left ( \lambda + 1 \right ) }$ ||
     167|| $A$ || $\frac{1}{\rho c_v  }$ ||
    168168|| $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 $ ||
    171171
    172172and
     
    179179And our equation becomes
    180180
    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$T'_0 - \Delta t \displaystyle \sum_{\parallel, \perp} AD \left [  \alpha_0 T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T'_{\pm \hat{j}}  \right ]$
    183183
    184184