More around Cattaneo equation to describe heat transfer processes

Abstract

We revisit Cattaneo's derivation of the equation describing heat transfer and bearing his name, finding a more elaborated model equation, that we will compare with the time-lagged model. This equation does not comply with the second law of thermodynamics, as well as Cattaneo equation. The latter, however, is known to have provided meaningful results in certain experiments conducted in some liquids and solids at very low temperature. These experiments also match the results given by a nonlinear model proposed by A. Morro and T. Ruggeri in 1988, which instead does meet the laws of thermodynamics. A comparison is made with such a model too. KEYWORDS Cattaneo-Maxwell-Vernotte equation, heat equation, heat transfer, heat waves, second sound, thermal lag MSC CLASSIFICATION 80A20 Math Meth Appl Sci. 2020;43:5953-5962.wileyonlinelibrary.com/journal/mma

Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary

“…The parameter τ > 0 corresponds to time relaxation. For more about the physical interpretation of (1.1), its derivation and overall discussion see [6,19,10,11,42,13,20].This also includes an analysis of asymptotic behavior of solutions when the parameter of relaxation tends to zero [26,27,4,3] The issues of wellposedness and stability of solutions under homogeneous Dirichlet and Neumann boundary data were first addressed for both nonlinear and linearized (k = 0) dynamics around 2010 with the works of I. Lasiecka, R. Triggiani and B. Kaltenbacher [23,24,36,25]. For the analysis of the long-time dynamics of (1.1) for both linear and nonlinear cases, the function…”

Boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticity

“…Although not unique, one interesting way of obtaining (1.1) from a similar procedure as the one to obtain (1.2) is simply to use Maxwell-Cattaneo law [11,4,5] in place of Fourier's law. The advantage of this strategy (which is by no means physics-proof [30,7]) is that it provides a suitable model for studying relaxation effects and, since waves now propagate at finite speed, it allows the construction of optimal policies for controlling the HIFU field, for example. Overall, in its simplicity, (1.1) catches most of the key features that would be present in a more rigorous model, modulus the stringency of nonlinearities.…”

Thermal Bragg Shields for Thermal Waves.

“…An important difference is that the heat flux is not derived from a quadratic quantity unlike the case of acoustic or electrodynamics where the energy flow is described from quadratic quantitities of the wave amplitude 19 . Several authors have discussed the differences and misconceptions of heat waves [20][21][22][23] . The wave-like behavior can be used to control the heat flux with systems that are similar to those used in optics or acoustics, for example designing thermal crystals 24 and metamaterials where concepts such as thermal cloaking have been proposed 25 .…”