Changes between Version 14 and Version 15 of u/johannjc/scratchpad5

Timestamp:

10/02/15 13:35:31 (10 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad5

@@ -142 +142 @@
 
 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
+
+$\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \frac{\kappa}{\lambda+1} \nabla T^{\lambda+1}$
+
+or in Einstein notation
+
+$\rho c_v \partial_t T = \partial_i \frac{\kappa}{\lambda+1} \partial_i T^{\lambda+1}$
+
+which we can rewrite as
+
+$\partial_t T = A \partial_i B \partial i T^{\lambda+1}$
+
+$\partial_t T = A \partial_i B \delta_{ij} \partial j T^{\lambda+1}$
+
+This is the exact same form as before, but now
+
+$B_{ij}=\kappa \delta{ij}$