(752g) Hyperbolicity of Heat Transport Processes

Ozorio Cassol, G. Sr. - Presenter, University of Alberta
Dubljevic, S., University of Alberta
Heat transport models are ubiquitously present in chemical, petrochemical, manufacturing and process industry as mathematical models of distributed parameter systems given by parabolic partial differential equations (PDEs). The conductive (diffusive) transport over macroscopic length scale is characterized by the well known Fourier's law and a large number of research efforts have been realized in the realm of modelling, control and optimization of such models. However, the salient characteristic of these models is that any initial disturbance in the material body is propagated instantly due to the parabolic nature of the partial differential equation (PDE) model. To eliminate this unphysical feature, we revisit the Fourier law to take into account the thermal inertia, resulting in a hyperbolic heat equation which avoids the phenomenon of infinite speed of propagation of initial data [1,2].

The main conservation laws are embedded in a modelling variety provided by the hyperbolicity of the transport systems which is physically relevant and desired property as action at distance is precluded and physically meaningful finite speed of phenomena propagation is ensured. Furthermore, the hyperbolicity mathematically ensures the well-posedness of local Cauchy problems [3]. In addition, a Stefan problem is considered, which is a specific type of heat transport boundary value problem for a partial differential equation describing the heat distribution evolution in a phase changing medium [4]. Since the moving interface is unknown a priori, the solution needs to take into account determination of the moving boundary position as well as accurate heat propagation. Finally, we provide derivation of the hyperbolic heat transport equations, the stability analysis and the numerical results for simple heat diffusion problem on fixed domain and Stefan problem.

Literature Cited

  1. Cattaneo C. On a form of heat equation which eliminates the paradox of instantaneous propagation. C R Acad Sci Paris. 1958;pp. 431–433.

  2. Vernotte P. Les paradoxes de la theorie continue de l’equation de la chaleur. C R Acad Sci Paris. 1958;246:3154–3155.

  3. Fischer AE, Marsden JE. The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system, I. Communications in Mathematical Physics. 1972;28(1):1–38.

  4. Gupta S. The Classical Stefan Problem. Basic Concepts, Modelling and Analysis. North-Holland: Applied mathematics and Mechanics. 2003.