Mathematical and numerical multi-scale modelling of multiphysics problems

In this paper we discuss two multi-scale procedures, both of mathematical nature as opposed to purelynumerical ones. Examples are shown for the two cases. Attention is also devoted to thermodynamical aspects such as thermodynamic consistency and non-equilibrium thermodynamics. Advances for the firstaspect are obtained by adopting the thermodynamically constrained averaging theory TCAT as shown in the case of a stress tensor for multi-component media. The second aspect has allowed to solve numerically,with relative ease, the case of non-isothermal leaching. The absence of proofs of thermodynamic consistencyin case of asymptotic theory of homogenization with ?nite size of the unit cell is also pointed out.