Christov–Morro theory for non-isothermal diffusion

We propose a theory for diffusion of a substance in a body allowing for changes in temperature. The key aspect is that the body is allowed to deform although we restrict our attention to the case where the velocity field is known. In accordance with recent developments in the literature, we concentrate on a situation where diffusion and temperature diffusion are governed by equations which have more of a hyperbolic nature than parabolic. Since this involves relaxation time equations for both the heat flux and the solute flux the fact that the body can deform necessitates the use of appropriate objective time derivatives. In this regard our work is based on recent work of Christov and Morro on heat transport in a moving body. An analysis of well posedness of the theory is commenced in that we establish the uniqueness of a solution to the boundary–initial value problem, and continuous dependence on the initial data for the same.