## Thermodynamics.(English)Zbl 0637.73002

Interaction of Mechanics and Mathematics. Boston - London - Melbourne: Pitman Advanced Publishing Program. XVII, 521 p. (1985).
As thermodynamics must be considered as a branch of physics still being in development even in its basic features, any book in this field reflects the individual point of view of the author. This turns out to be especially valid in the case of Müller, who contributed to this topic a lot of new ideas, concepts, and suggestions, in the phenomenological theory as well as the kinetic or statistical approach to the matter.
The author took the change to lay down the fundamental concepts of modern thermodynamics and to apply them in various interesting topics. Before he starts to work out the basic concepts such as balance laws, constitutive equations, material frame indifference, the entropy principle, stability, and incompressibility, he exemplifies them within the familiar class of Navier-Stokes fluids with Fourier heat flux. Of course, there is a lot of discussion on the principle of material frame indifference and the objections on it published by the same author in former papers [e.g. Dynamical problems in continuum physics, Proc. Workshop IMA, Minneapolis/Minn. 1984/85, IMA Vol. Math. Appl. 4, 257-276 (1987; Zbl 0627.76084) and Lect. Notes Phys. 199, 32-71 (1984; Zbl 0575.76072)]. One should take care of the fact, that this principle is defined by different authors of rational mechanics in different ways. Müller’s form of the principle does not postulate the objectivity of certain variables, but instead the independence of the constitutive equations of the observer (which is sometimes called ‘principle of form invariance’). The constitutive theory, that the author still favours, is that of history functionals, although he introduces internal variables in some later parts of the book. This implies fading memory from the outset, as being appropriate for viscous fluids and gases, but excludes plastic and shape memory effects.
It is worth mentioning that Müller applies and evaluates the entropy principle in his specific form by means of Lagrangian multipliers, and there is much space dedicated to the presentation and discussion of other classical and modern forms of the second law.
In what follows, Müller describes - mixtures, §(surface thermodynamics, esp. membranes and droplets, - thermoelasticity, - electromagnetic fluids, - nematic liquid crystals, - relativistic thermodynamics, and what he calls ‘extended thermodynamics’, a generalization of the ‘ordinary’ thermodynamics in order to obtain finite speeds of temperature and shear waves and hyperbolicity of the ruling differential equations. The author uses a cartesian index notation. The reader misses a list of the used symbols, as the ‘extended’ use of F, M, and $$\Lambda$$ in the last part might lead to confusion.
The book gives a systematic introduction to modern thermodynamics and many interesting applications. It is written in a very carefull and didactical way and surely presents a huge amount of interesting new ideas. The viewer highly recommends this book as an exceptional and very important work.
Reviewer: A.Bertram

### MSC:

 74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids 74Axx Generalities, axiomatics, foundations of continuum mechanics of solids 74A15 Thermodynamics in solid mechanics 76A02 Foundations of fluid mechanics 80A05 Foundations of thermodynamics and heat transfer 74A20 Theory of constitutive functions in solid mechanics 74F05 Thermal effects in solid mechanics 80A17 Thermodynamics of continua 76W05 Magnetohydrodynamics and electrohydrodynamics

### Citations:

Zbl 0627.76084; Zbl 0575.76072