Changes between Version 12 and Version 13 of ThermalConduction

Timestamp:

12/20/21 12:31:58 (4 years ago)

Author:

Jonathan

Comment:

—

ThermalConduction

@@ -25 +25 @@
 where the c's are coefficients usually given as some number in references.
 
+We will use $\kappa_{\parallel} = c_{\parallel} T^{\lambda_{\parallel}}$ and the same for $\kappa_{\perp}$ where $\lambda_{\parallel}=5/2$ and $\lambda_{\perp}=-1/2$
+
+
 == Collecting power's of T ==
 
 So we can rewrite the equations as 
 
-$\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \left [ \hat{b} \left ( \kappa_\parallel T^{\lambda_\parallel} - \frac{n^2 \kappa_\perp}{B^2} T^{\lambda_\perp} \right ) \left ( \hat{b} \cdot \nabla T \right ) + \frac{n^2 \kappa_\perp}{B^2} T^{\lambda_\perp} \nabla T \right ]$
+$\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \left [ \hat{b} \left ( \kappa_\parallel - \frac{n^2 \kappa_\perp}{B^2} \right ) \left ( \hat{b} \cdot \nabla T \right ) + \frac{n^2 \kappa_\perp}{B^2} \nabla T \right ]$
+
+
+== Einstein simplification ==
+
+or in Einstein notation
+
+$\rho c_v \partial_t T = \partial_i \left [ b_i \left (\kappa_\parallel \left ( b_j \partial_j T\right )  - \frac{n^2 \kappa_\perp}{B^2 } \left ( b_j \partial_j  T \right )  \right ) + \frac{n^2 \kappa_\perp}{B^2 } \partial_i T \right ]$
+
 
 or 
 
-$\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \left [ \hat{b} \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( \hat{b} \cdot \nabla T^{\lambda_\parallel+1} \right )  - \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( \hat{b} \cdot \nabla T^{\lambda_\perp + 1} \right )  \right ) + \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \nabla T^{\lambda_\perp+1} \right ]$
-
-== Einstein simplification ==
-
-or in Einstein notation
-
-$\rho c_v \partial_t T = \partial_i \left [ b_i \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( b_j \partial_j T^{\lambda_\parallel+1} \right )  - \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( b_j \partial_j  T^{\lambda_\perp + 1} \right )  \right ) + \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \partial_i T^{\lambda_\perp+1} \right ]$
-
-or 
-
-$\rho c_v  \partial_t T = \partial_i \left [ b_i \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( b_j \partial_j T^{\lambda_\parallel+1} \right )  - \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( b_j \partial_j  T^{\lambda_\perp + 1} \right )  \right ) + \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \delta_{ij} \partial_j T^{\lambda_\perp+1} \right ]$
-
-or 
-
-$\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} $
-
-or 
-
-$\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} $
+$\rho c_v  \partial_t T = \partial_i \kappa_\parallel  b_i  b_j \partial_j T +   \partial_i \kappa_\perp \left ( \delta_{ij} - b_i b_j \right )  \partial_j T$
 
 
@@ -62 +55 @@
 Now we can rewrite the equation 
 
-$\partial_t T = \displaystyle \sum_{\parallel, \perp} A \partial_i  B_{ij} \partial_j T^{\lambda+1}$
+$\partial_t T = \displaystyle \sum_{\parallel, \perp} A \partial_i  B_{ij} \partial_j T$
 
 where the $\parallel$ or $\perp$ subscript on $A$, $B_{ij}$, and $\lambda$ is implied.  
 
-$A_\parallel = \frac{\kappa_\parallel}{\rho c_v  \left ( \lambda_\parallel + 1\right ) }$ and $A_\perp = \frac{\kappa_\perp}{\rho c_v  \left (\lambda_\perp + 1 \right )}$
-
-and $B_{\parallel, ij} = b_i b_j$ and $B_{\perp, ij} = n^2 B^{-2} \left ( \delta_{ij}-b_i b_j \right )$
+$A_\parallel = \frac{c_\parallel}{\rho c_v }$ and $A_\perp = \frac{c_\perp}{\rho c_v }$
+
+and $B_{\parallel, ij} = T^{\lambda_{\parallel}}b_i b_j$ and $B_{\perp, ij} = n^2 B^{-2} T^{\lambda_\perp} \left ( \delta_{ij}-b_i b_j \right )$
 
 Now to solve this implicitly, we need to replace $T$ with $T_*$ where 
@@ -76 +69 @@
 Note for Backward Euler, $\phi = 1$ and for Crank Nicholson, $\phi = 1/2$
 
-$ \partial_t T = \displaystyle \sum_{\parallel, \perp} A \partial_i  B_{ij} \partial_j T_*^{\lambda+1}$
-
-It is simpler to linearize the equation in terms of $T_*$ and then subsitute then vice-versa - though both give the same answer
-
-$T_*^{\lambda + 1} \approx T^{\lambda + 1} +  \left ( \lambda + 1 \right ) T^{\lambda} \left ( T_{*} - T \right ) =  T^{\lambda + 1} +  \left ( \lambda + 1 \right ) T^{\lambda} \phi \left ( T' - T \right )$
-$= \left(1 - \phi \left ( \lambda + 1\right ) \right ) T^{\lambda + 1}+  \phi \left ( \lambda + 1 \right ) T^{\lambda} T' $
-
-
-So we have
-
-$\partial_t T = \displaystyle \sum_{\parallel, \perp}{A \partial_i  B_{ij} \partial_j \left [ \left ( 1 - \phi \left ( \lambda + 1 \right ) \right )T^{\lambda + 1} + \phi \left ( \lambda + 1 \right ) T^{\lambda} T' \right ]}$
-
-$\partial_t T = \displaystyle \sum_{\parallel, \perp}{A \partial_i  B_{ij} \left [ \left ( 1 - \phi \left ( \lambda + 1 \right ) \right ) \partial_j T^{\lambda + 1} + \phi \left ( \lambda + 1 \right ) \partial_j T^{\lambda} T' \right ]}$
-
-$\partial_t T = \displaystyle \sum_{\parallel, \perp}{A \partial_i  B_{ij} \left [ C \partial_j T^{\lambda + 1} + D \partial_j T^{\lambda} T' \right ]}$
-
-where $ C= \left ( 1 - \phi \left ( \lambda + 1 \right ) \right )$ and $ D = \phi \left ( \lambda + 1 \right )$
+$ \partial_t T = \displaystyle \sum_{\parallel, \perp} A \partial_i  B_{ij} \partial_j T_*$
+
+$ \partial_t T = \displaystyle \sum_{\parallel, \perp} A \partial_i  B_{ij} \left [ C \partial_j T + D \partial_j T' \right]$
+
+
+where $ C= \left ( 1 - \phi \right )$ and $ D = \phi $
 
 Now we can expand the derivatives and get
@@ -102 +84 @@
 $ \partial_t T = \frac{ T'_0 - T_0}{\Delta t}$
 
-$ \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 )$
-$ + \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} $
-
-$ \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 )  $
-$+ \left [ \frac{B^{ij}_{\hat{i/2}} T^{\lambda}_{\hat{i}}T'_{\hat{i}} - B^{ij}_{\hat{i/2}}T^{\lambda}_{0}T'_{0}-B^{ij}_{-\hat{i/2}} T^{\lambda}_{0}T'_{0} + B^{ij}_{-\hat{i/2}}T^{\lambda}_{-\hat{i}}T'_{-\hat{i}}}{\Delta x^2} \right ] \delta_{ij}$
-
-
-$ \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}$
+$ \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 )$
+$ + \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} $
 
 
@@ -115 +91 @@
 
 
-$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 ] = $
-$T'_0 - \Delta t \displaystyle \sum_{\parallel, \perp} AD \left [  \alpha_0 T^{\lambda}_0T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda}_{\pm \hat{j}} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda}_{\pm \hat{i} \pm \hat{j}} T'_{\pm \hat{i} \pm \hat{j}} \right ]$
+$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 ] = $
+$T'_0 - \Delta t \displaystyle \sum_{\parallel, \perp} AD \left [  \alpha_0 T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T'_{\pm \hat{i} \pm \hat{j}} \right ]$
 
 where
@@ -140 +116 @@
 
 ||  ||   $\parallel$   ||   $\perp$   ||
-|| $A$ || $\frac{\kappa_\parallel}{\rho c_v  \left ( \lambda_\parallel + 1 \right ) }$ || $\frac{\kappa_\perp}{\rho c_v  \left (\lambda_\perp + 1\right )}$ ||
-|| $B_{ij} $ || $b_i b_j$ || $n^2 B^{-2} \left (\delta_{ij} - b_i b_j \right )$ ||
-|| $C$ ||  $\left ( 1 - \phi \left ( \lambda_\parallel + 1 \right ) \right )$ ||  $ \left ( 1 - \phi \left ( \lambda_\perp + 1 \right ) \right )$ ||
-|| $D$ ||  $\phi \left ( \lambda_\parallel + 1 \right )$ ||  $ \phi \left ( \lambda_\perp + 1 \right ) $||
+|| $A$ || $\frac{c_\parallel}{\rho c_v   }$ || $\frac{c_\perp}{\rho c_v }$ ||
+|| $B_{ij} $ || $b_i b_j T^{\lambda_{\parallel}}$ || $n^2 B^{-2} T^{\lambda_{\perp}} \left (\delta_{ij} - b_i b_j \right )$ ||
+|| $C$ ||  $\left ( 1 - \phi \right )$ ||  $ \left ( 1 - \phi \right )$ ||
+|| $D$ ||  $\phi $ ||  $ \phi $||
 
 ||$\alpha_0$ || $- \displaystyle \sum_{i} \frac{B^{ii}_{\hat{i}} + B^{ii}_{-\hat{i}}}{\Delta x^2}$ ||
@@ -165 +141 @@
 For the Isotropic Case, the diffusion equation looks like
 
-$\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \kappa T^\lambda \nabla T$
-
-or
-
-$\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \frac{\kappa}{\lambda+1} \nabla T^{\lambda+1}$
+$\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \kappa \nabla T$
 
 or in Einstein notation