# Models

## Equation

Given a domain \Omega \subset \mathbb{R}^d, d=1,2,3, \Omega is partitioned into N_r regions \Omega_i,i=1,\ldots,N_r corresponding to different materials (solid or fluid).

\rho C_p \frac{\partial T}{\partial t} + \mathb{u} \cdot \nabla T - \nabla \cdot \left$k \nabla T \right$ = Q

which is completed with initial value and boundary conditions

Initial value
\text{at } t=0, \quad T(x,0) = T_0(x)
Dirichlet Boundary condition
\