On the non-linear Maxwell–Cattaneo equation with non-constant diffusivity: Shock and discontinuity waves

• Published in: International journal of heat and mass transfer Vol. 51; no. 21; pp. 5327 - 5332
• Main Authors: Reverberi, Andrea Pietro, Bagnerini, Patrizia, Maga, Luigi, Bruzzone, Agostino Giacinto
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.10.2008; Elsevier
• Subjects: Boson degeneracy and superfluidity of 4he; Condensed matter: structure, mechanical and thermal properties; Exact sciences and technology; Finite-differences; Fundamental areas of phenomenology (including applications); Heat conduction; Heat transfer; Hyperbolic partial differential equations; Non-Fickian diffusion; Non-Fourier heat conduction; Physics; Quantum fluids and solids; liquid and solid helium; Transport processes, second and other sounds, and thermal counterflow; kapitza resistance; Finite-differences; Non-Fourier heat conduction; Hyperbolic partial differential equations; Non-Fickian diffusion; Discontinuity; Partial differential equations; Digital simulation; Hyperbolic equations; Thermal wave; Shock waves; Wave propagation; Heat mass transfer; Second sound; Modelling; Diffusion coefficient; Thermal conduction; Finite difference method
• ISSN: 0017-9310; 1879-2189
• DOI: 10.1016/j.ijheatmasstransfer.2008.01.039

A numerical study on a non-linear hyperbolic diffusion equation is proposed. The Hartree hybrid method combining finite difference techniques with the method of characteristics is used in the presence of discontinuities between initial and boundary conditions. The technique proved to be an useful tool to overcome oscillation problems and spurious solutions in case of strong non-linearities related to both attractive or repulsive interactions between diffusing species. Two different expressions for the diffusion coefficient are used in order to compare our results with the ones obtained in previous studies relying upon the Laplace transform technique and the MacCormack predictor–corrector method. Finally, an analytic approach based on the singular surface theory is proposed to motivate the numerical results and to clarify some controversial aspects concerning the penetration depth of a diffusive front in the presence of interactions.