Abstract
Abstract The relation between heat flux and temperature gradient has been considered as a constitutive structure or as a balance law in different approaches. Both views may allow a description of heat conduction characterized by finite speed propagation of temperature disturbances. Such a result, which overcomes Fourier’s drawback of infinite speed propagation, can be obtained also by considering insufficient the representation of a conductor, even when it is considered to be rigid, rather than the sole relation between heat flux and temperature gradient. We comment this last view and describe the intersection with previous proposals. Eventually, we show how under Fourier’s law we can have traveling-wave-type temperature propagation when thermal microstructures are accounted for.
Highlights
A well-known problem in the classical theory of heat conduction is that Fourier’s relation between heat flux q and temperature gradient ∇T, namely q = −κ∇T, with κ being the material conductivity, implies an infinite propagation speed for temperature variations
The relation between heat flux and temperature gradient has been considered as a constitutive structure or as a balance law in different approaches
The way we describe the conductor is too skeletal: matter in nature has a microstructure, and it is exactly the influence of microstructural interactions that is the source of the hyperbolic character in the phenomenology of heat conduction
Summary
A well-known problem in the classical theory of heat conduction is that Fourier’s relation between heat flux q and temperature gradient ∇T, namely q = −κ∇T, with κ being the material conductivity, implies an infinite propagation speed for temperature variations. Notwithstanding this aspect, Fourier’s law is largely used in technical applications as it appears to be a reasonable approximation of very wide phenomenology. Attempts at overcoming the physical drawback for the propagation speed of temperature variations in rigid conductors seem to follow distinct— not mutually excluding, but rather intersecting— viewpoints They may be summarized in this way: 1. As a specific technical result of the present work, we show how an approach falling within item 3 above, and introduced in reference [16], leads to traveling-wave-type classical real solutions; thermal waves are foreseen in another view falling under item 3, the one pertaining to heat-driven microstructures, as introduced in reference [17]; the analysis of this last type of waves is in reference [18]
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