Changes between Version 12 and Version 13 of ThermalConduction


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Timestamp:
12/20/21 12:31:58 (3 years ago)
Author:
Jonathan
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  • ThermalConduction

    v12 v13  
    2525where the c's are coefficients usually given as some number in references.
    2626
     27We will use $\kappa_{\parallel} = c_{\parallel} T^{\lambda_{\parallel}}$ and the same for $\kappa_{\perp}$ where $\lambda_{\parallel}=5/2$ and $\lambda_{\perp}=-1/2$
     28
     29
    2730== Collecting power's of T ==
    2831
    2932So we can rewrite the equations as
    3033
    31 $\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \left [ \hat{b} \left ( \kappa_\parallel T^{\lambda_\parallel} - \frac{n^2 \kappa_\perp}{B^2} T^{\lambda_\perp} \right ) \left ( \hat{b} \cdot \nabla T \right ) + \frac{n^2 \kappa_\perp}{B^2} T^{\lambda_\perp} \nabla T \right ]$
     34$\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \left [ \hat{b} \left ( \kappa_\parallel - \frac{n^2 \kappa_\perp}{B^2} \right ) \left ( \hat{b} \cdot \nabla T \right ) + \frac{n^2 \kappa_\perp}{B^2} \nabla T \right ]$
     35
     36
     37== Einstein simplification ==
     38
     39or in Einstein notation
     40
     41$\rho c_v \partial_t T = \partial_i \left [ b_i \left (\kappa_\parallel \left ( b_j \partial_j T\right )  - \frac{n^2 \kappa_\perp}{B^2 } \left ( b_j \partial_j  T \right )  \right ) + \frac{n^2 \kappa_\perp}{B^2 } \partial_i T \right ]$
     42
    3243
    3344or
    3445
    35 $\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \left [ \hat{b} \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( \hat{b} \cdot \nabla T^{\lambda_\parallel+1} \right )  - \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( \hat{b} \cdot \nabla T^{\lambda_\perp + 1} \right )  \right ) + \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \nabla T^{\lambda_\perp+1} \right ]$
    36 
    37 == Einstein simplification ==
    38 
    39 or in Einstein notation
    40 
    41 $\rho c_v \partial_t T = \partial_i \left [ b_i \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( b_j \partial_j T^{\lambda_\parallel+1} \right )  - \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( b_j \partial_j  T^{\lambda_\perp + 1} \right )  \right ) + \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \partial_i T^{\lambda_\perp+1} \right ]$
    42 
    43 or
    44 
    45 $\rho c_v  \partial_t T = \partial_i \left [ b_i \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( b_j \partial_j T^{\lambda_\parallel+1} \right )  - \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( b_j \partial_j  T^{\lambda_\perp + 1} \right )  \right ) + \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \delta_{ij} \partial_j T^{\lambda_\perp+1} \right ]$
    46 
    47 or
    48 
    49 $\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 \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} $
    50 
    51 or
    52 
    53 $\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} $
     46$\rho c_v  \partial_t T = \partial_i \kappa_\parallel  b_i  b_j \partial_j T +   \partial_i \kappa_\perp \left ( \delta_{ij} - b_i b_j \right )  \partial_j T$
    5447
    5548
     
    6255Now we can rewrite the equation
    6356
    64 $\partial_t T = \displaystyle \sum_{\parallel, \perp} A \partial_i  B_{ij} \partial_j T^{\lambda+1}$
     57$\partial_t T = \displaystyle \sum_{\parallel, \perp} A \partial_i  B_{ij} \partial_j T$
    6558
    6659where the $\parallel$ or $\perp$ subscript on $A$, $B_{ij}$, and $\lambda$ is implied. 
    6760
    68 $A_\parallel = \frac{\kappa_\parallel}{\rho c_v  \left ( \lambda_\parallel + 1\right ) }$ and $A_\perp = \frac{\kappa_\perp}{\rho c_v  \left (\lambda_\perp + 1 \right )}$
    69 
    70 and $B_{\parallel, ij} = b_i b_j$ and $B_{\perp, ij} = n^2 B^{-2} \left ( \delta_{ij}-b_i b_j \right )$
     61$A_\parallel = \frac{c_\parallel}{\rho c_v }$ and $A_\perp = \frac{c_\perp}{\rho c_v }$
     62
     63and $B_{\parallel, ij} = T^{\lambda_{\parallel}}b_i b_j$ and $B_{\perp, ij} = n^2 B^{-2} T^{\lambda_\perp} \left ( \delta_{ij}-b_i b_j \right )$
    7164
    7265Now to solve this implicitly, we need to replace $T$ with $T_*$ where
     
    7669Note for Backward Euler, $\phi = 1$ and for Crank Nicholson, $\phi = 1/2$
    7770
    78 $ \partial_t T = \displaystyle \sum_{\parallel, \perp} A \partial_i  B_{ij} \partial_j T_*^{\lambda+1}$
    79 
    80 It is simpler to linearize the equation in terms of $T_*$ and then subsitute then vice-versa - though both give the same answer
    81 
    82 $T_*^{\lambda + 1} \approx T^{\lambda + 1} +  \left ( \lambda + 1 \right ) T^{\lambda} \left ( T_{*} - T \right ) =  T^{\lambda + 1} +  \left ( \lambda + 1 \right ) T^{\lambda} \phi \left ( T' - T \right )$
    83 $= \left(1 - \phi \left ( \lambda + 1\right ) \right ) T^{\lambda + 1}+  \phi \left ( \lambda + 1 \right ) T^{\lambda} T' $
    84 
    85 
    86 So we have
    87 
    88 $\partial_t T = \displaystyle \sum_{\parallel, \perp}{A \partial_i  B_{ij} \partial_j \left [ \left ( 1 - \phi \left ( \lambda + 1 \right ) \right )T^{\lambda + 1} + \phi \left ( \lambda + 1 \right ) T^{\lambda} T' \right ]}$
    89 
    90 $\partial_t T = \displaystyle \sum_{\parallel, \perp}{A \partial_i  B_{ij} \left [ \left ( 1 - \phi \left ( \lambda + 1 \right ) \right ) \partial_j T^{\lambda + 1} + \phi \left ( \lambda + 1 \right ) \partial_j T^{\lambda} T' \right ]}$
    91 
    92 $\partial_t T = \displaystyle \sum_{\parallel, \perp}{A \partial_i  B_{ij} \left [ C \partial_j T^{\lambda + 1} + D \partial_j T^{\lambda} T' \right ]}$
    93 
    94 where $ C= \left ( 1 - \phi \left ( \lambda + 1 \right ) \right )$ and $ D = \phi \left ( \lambda + 1 \right )$
     71$ \partial_t T = \displaystyle \sum_{\parallel, \perp} A \partial_i  B_{ij} \partial_j T_*$
     72
     73$ \partial_t T = \displaystyle \sum_{\parallel, \perp} A \partial_i  B_{ij} \left [ C \partial_j T + D \partial_j T' \right]$
     74
     75
     76where $ C= \left ( 1 - \phi \right )$ and $ D = \phi $
    9577
    9678Now we can expand the derivatives and get
     
    10284$ \partial_t T = \frac{ T'_0 - T_0}{\Delta t}$
    10385
    104 $ \partial_i B_{ij} \partial_j T^{\lambda+1} = \left [ \frac{B^{ij}_{\hat{i}} T^{\lambda+1}_{\hat{i}+\hat{j}} - B^{ij}_{\hat{i}}T^{\lambda+1}_{\hat{i}-\hat{j}}-B^{ij}_{-\hat{i}} T^{\lambda+1}_{-\hat{i}+\hat{j}} + B^{ij}_{-\hat{i}}T^{\lambda+1}_{-\hat{i}-\hat{j}}}{4 \Delta x^2} \right ] \left ( 1-\delta_{ij} \right )$
    105 $ + \left [ \frac{B^{ij}_{\hat{i/2}} T^{\lambda+1}_{\hat{i}} - B^{ij}_{\hat{i/2}}T^{\lambda+1}_{0}-B^{ij}_{-\hat{i/2}} T^{\lambda+1}_{0} + B^{ij}_{-\hat{i/2}}T^{\lambda+1}_{\hat{-i}}}{\Delta x^2} \right ] \delta_{ij} $
    106 
    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 )  $
    108 $+ \left [ \frac{B^{ij}_{\hat{i/2}} T^{\lambda}_{\hat{i}}T'_{\hat{i}} - B^{ij}_{\hat{i/2}}T^{\lambda}_{0}T'_{0}-B^{ij}_{-\hat{i/2}} T^{\lambda}_{0}T'_{0} + B^{ij}_{-\hat{i/2}}T^{\lambda}_{-\hat{i}}T'_{-\hat{i}}}{\Delta x^2} \right ] \delta_{ij}$
    109 
    110 
    111 $ \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}$
     86$ \partial_i B_{ij} \partial_j T = \left [ \frac{B^{ij}_{\hat{i}} T_{\hat{i}+\hat{j}} - B^{ij}_{\hat{i}}T_{\hat{i}-\hat{j}}-B^{ij}_{-\hat{i}} T_{-\hat{i}+\hat{j}} + B^{ij}_{-\hat{i}}T_{-\hat{i}-\hat{j}}}{4 \Delta x^2} \right ] \left ( 1-\delta_{ij} \right )$
     87$ + \left [ \frac{B^{ij}_{\hat{i/2}} T_{\hat{i}} - B^{ij}_{\hat{i/2}}T_{0}-B^{ij}_{-\hat{i/2}} T_{0} + B^{ij}_{-\hat{i/2}}T_{\hat{-i}}}{\Delta x^2} \right ] \delta_{ij} $
    11288
    11389
     
    11591
    11692
    117 $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}} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} \right ] = $
    118 $T'_0 - \Delta t \displaystyle \sum_{\parallel, \perp} AD \left [  \alpha_0 T^{\lambda}_0T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda}_{\pm \hat{j}} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda}_{\pm \hat{i} \pm \hat{j}} T'_{\pm \hat{i} \pm \hat{j}} \right ]$
     93$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}} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T_{\pm \hat{i} \pm \hat{j}} \right ] = $
     94$T'_0 - \Delta t \displaystyle \sum_{\parallel, \perp} AD \left [  \alpha_0 T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T'_{\pm \hat{i} \pm \hat{j}} \right ]$
    11995
    12096where
     
    140116
    141117||  ||   $\parallel$   ||   $\perp$   ||
    142 || $A$ || $\frac{\kappa_\parallel}{\rho c_v  \left ( \lambda_\parallel + 1 \right ) }$ || $\frac{\kappa_\perp}{\rho c_v  \left (\lambda_\perp + 1\right )}$ ||
    143 || $B_{ij} $ || $b_i b_j$ || $n^2 B^{-2} \left (\delta_{ij} - b_i b_j \right )$ ||
    144 || $C$ ||  $\left ( 1 - \phi \left ( \lambda_\parallel + 1 \right ) \right )$ ||  $ \left ( 1 - \phi \left ( \lambda_\perp + 1 \right ) \right )$ ||
    145 || $D$ ||  $\phi \left ( \lambda_\parallel + 1 \right )$ ||  $ \phi \left ( \lambda_\perp + 1 \right ) $||
     118|| $A$ || $\frac{c_\parallel}{\rho c_v   }$ || $\frac{c_\perp}{\rho c_v }$ ||
     119|| $B_{ij} $ || $b_i b_j T^{\lambda_{\parallel}}$ || $n^2 B^{-2} T^{\lambda_{\perp}} \left (\delta_{ij} - b_i b_j \right )$ ||
     120|| $C$ ||  $\left ( 1 - \phi \right )$ ||  $ \left ( 1 - \phi \right )$ ||
     121|| $D$ ||  $\phi $ ||  $ \phi $||
    146122
    147123||$\alpha_0$ || $- \displaystyle \sum_{i} \frac{B^{ii}_{\hat{i}} + B^{ii}_{-\hat{i}}}{\Delta x^2}$ ||
     
    165141For the Isotropic Case, the diffusion equation looks like
    166142
    167 $\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \kappa T^\lambda \nabla T$
    168 
    169 or
    170 
    171 $\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \frac{\kappa}{\lambda+1} \nabla T^{\lambda+1}$
     143$\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \kappa \nabla T$
    172144
    173145or in Einstein notation