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Heat waves by Brian Straughan PDF Stampa E-mail
Sabato 14 Luglio 2012 21:17

Re: Heat waves

By Brian Straughan

Springer Verlag


J.-B. J. Fourier’s description (1822) of heat conduction at macroscopic scale implies as a paradoxical consequence the instantaneous propagation at infinity of temperature disturbances – the remark is common matter of standard textbooks. Since the first attempts by J. C. Maxwell and, later, by C. Cattaneo (the so-called Maxwell-Cattaneo model), several researchers have proposed modifications of the original Fourier’s law with the aim of eliminating that undesirable consequence. Analyses have been developed even coupling heat conduction with standard elasticity or its evolution toward the description of material complexity. Essentially the attention has been focused on the determination of appropriate modifications of the relationship between heat flux and temperature, the latter being interpreted as the macroscopic phenomenological one in most cases. The list of approaches includes also views accounting for memory effects in material behaviour, and leading to schemes where techniques of integral differential equations find a playground. Hyperbolic-type representations of heat propagation may arise also from a kinetic-theory approach based on an evolution of H. Grad truncation method, as proposed by I. M\"{u}ller and T. Ruggeri.

“With the advent of micro-scale technology there is increasing evidence that thermal motion is via a wave mechanism as opposed to by diffusion.” So B. Straughan writes in beginning his book on heat waves, a wide overview on several proposals of models foreseeing finite-speed heat propagation, enriched by detailed comments and a wide bibliography.

Chapter 1 deals with rigid conductors modelling. It includes Maxwell-Cattaneo, Guyer-Krumhansl, and Green-Laws theories, models involving heat flux history, other schemes related with temperature dependent conductivities, and schemes accounting for low spatial scales temperatures.

Thermoelasticity appears in Chapter 2 where the author lists a number of approaches trying to include the rate of temperature in the set of state variables – with a related (necessary in some sense) variation of the second law – or imagining internal (non-observable) variables measuring the detachment from thermodynamic equilibrium and satisfying their own kinetics, or involving the so-called thermal displacement. The latter is defined formally, to within a constant, as the integral of the phenomenological temperature, distinguished by the absolute one. The exact physical meaning of the thermal displacement would emerge once we would have a clear definition of temperature in deformable bodies, based on microscopic mechanisms, as it is available in the case of monoatomic gases. In this direction a tentative could be made at least in the case of crystals by thinking in terms of phonons, the ‘particles’ emerging from the quantization of elastic waves. Chapter 2 includes discussions of heat propagation in elastic solids with diffused microvoids, affine microstructures, and in elastic bodies with piezoelectric effects, proposing various variations on the theme, all falling within the general model-building framework of the mechanics of complex materials, a framework not discussed in the book.

Variations of the schemes introduced in the first chapter appear in Chapter 3 where the focus is on heat propagation in fluids.

Chapter 4 deals with acceleration waves. The topic goes as follows: Consider a standard thermoelastic body. A discontinuity surface through which the acceleration suffers jump while deformation, velocity, and temperature are continuous is called acceleration wave. The definition can be naturally extended to the case of complex materials, by imagining that the conditions above on deformation and velocity apply to the field describing the morphology of the material microstructure. Such waves are also homothermal when the temperature time rate and its spatial gradient are continuous through them. For acceleration waves it is possible to determine the propagation of the amplitude of the acceleration jump. This question is discussed in different settings in Chapter 4. Shock waves appear instead in Chapter 5 with some numerical aspects related with them. Chapter 6 deals with qualitative estimates in different setting with related uniqueness results. The chapter contains also a description of explosive instabilities in heat transfer.

Various types of spatial decay involving temperature and heat flux are discussed in Chapter 7. The issue has prominent physical interest.

Chapter 8 is dedicated to a rather specific topic: thermal convection in nanofluids. Different possible relevant models are discussed. Finally, possible applications of some concepts presented elsewhere in the book are included in Chapter 9. The list involves problems appearing in disparate fields such as biology, cosmology, and the description of traffic flow.

In presenting the various specific models commented in the book, the author prefers component wise notation instead of the tensor intrinsic one. Moreover, foundational questions in continuum thermodynamics are not discussed (I am thinking of issues concerning the existence of a temperature manifold and the abstract foundations of thermodynamics on the basis of the concepts of state space, state transformations, and actions). Of course, in writing a book, choices have to be made.

In any case, the presentation of the material is clear and consequential, reading is pleasant. The book can be useful for scholars working in continuum thermodynamics and applied mathematics. It can be chosen as textbook for some graduate courses or in PhD programs in applied mathematics, or mathematical engineering. The presence of exercises at the end of Chapters 1, 2, 4, 5, 6, 9 are useful didactic additions.

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