A finite element formulation for a convection–diffusion equation based on Cattaneo’s law

Abstract

A number of contributions have been made during the last decades to model pure-diffusive transport problems by using the so-called hyperbolic diffusion equations [M. Zakari, D. Jou, Equations of state and transport equations in viscous cosmological models, Phys. Rev. D 48 (1993) 1597–1601; T. Ruggeri, A. Muracchini, L. Seccia, Shock waves and second sound in a rigid heat conductor: a critical temperature for NaF and Bi, Phys. Rev. Lett. 64 (1990) 2640–2643]. These equations are used for both mass and heat transport. The hyperbolic diffusion equations are obtained by substituting the classic time-independent constitutive equation (Fick’s [A. Fick, Uber diffusion, Poggendorff’s Annal. Phys. Chem. 94 (1855) 59–86] and Fourier’s [J.B. Fourier, Théorie analytique de la chaleur, Jacques Gabay, 1822] law, respectively) by a more general time-dependent equation, due to Cattaneo [M.C. Cattaneo, Sur une forme de l’equation de la chaleur liminant le paradoxe d’une propagation instantane, Comptes Rendus de L’Academie des Sciences: Series I-Mathematics 247 (1958) 431–433]. In some applications the use of a parabolic model for diffusive processes is accurate enough in spite of predicting an infinite speed of propagation (Cattaneo, 1958). However, the use of a wave-like equation that predicts a finite velocity of propagation is necessary in many other calculations [A. Compte, The generalized Cattaneo equation for the description of anomalous transport processes, J. Phys. A: Math. General 30 (1997) 7277–7289]. The studies of heat or mass transport with finite velocity of propagation have been limited, so far, to pure-diffusive situations. This paper proposes a formulation for convection–diffusion problems based on a Cattaneo-type law. The finite element solution of the proposed equations is addressed. The performance of the algorithm is verified by solving some 2D test cases. Some interesting features of the proposed model can be observed from these examples and we conclude that the proposal is a feasible alternative to the parabolic model for real engineering simulations.