# zbMATH — the first resource for mathematics

Continuum methods of physical modeling. Continuum mechanics, dimensional analysis, turbulence. (English) Zbl 1058.76001
Berlin: Springer (ISBN 3-540-20619-1/hbk). xv, 635 p. (2004).
What is needed to set up continuum thermodynamical models that are capable of accounting for the enormous complexity of problems belonging to the large class of “environmentally related systems”, i.e. problems arising in the context of soil, ice and snow mechanics, fluid dynamics, oceanography, limnology and climate dynamics? The answer(s!) can be found in this book: starting with a presentation of the foundations of continuum thermodynamics (Part I, 7 chapters), the reader is introduced into basic kinematics, balance- and jump conditions and constitutive theory. To properly set the stage for later applications, individual chapters are then devoted to phase transition processes (chapter 6) and to the theory of mixtures (of homogeneous as well as heterogeneous nature, chapter 7). Moreover, the second law of thermodynamics is throughout the book exploited rather in the spirit of Müller and Liu than by an application of Coleman-Noll approach – the latter is known to be too restrictive and is likely to yield erroneous results for many environmentally related systems. In doing so, the authors go beyond what is usually treated in introductory continuum mechanics courses.
Part II on dimensional analysis (with chapters 8 and 9, “Theoretical foundations” and “Similitude and model experiments”) provides guidelines on how to perform laboratory experiments (usually performed at scales different from 1:1) and culminates in the proof of Buckingham’s theorem. Great care has been taken to illustrate this rather abstract algebraic procedure with many examples.
Finally, Part III deals with turbulence theory (4 chapters) and discusses (after an introduction of fundamental concepts, chapter 10) the $$k-\varepsilon$$ model for density-preserving and Boussinesq fluids (chapter 11), and a number of its applications (chapter 13). Algebraic Reynolds stress models are treated in chapter 12. Since the closure conditions of turbulence theory can be based essentially on the same concepts as the formulation of constitutive quantities in continuum mechanical modeling, the joint presentation of continuum mechanics (Part I) and turbulence (Part III) facilitates the learning and understanding of the latter considerably.
The methods presented in this book can be used not only for environmentally related problems, but rather for a majority of problems arising in the modeling of complex, heterogeneous media ranging from natural to artificial materials used in today’s mechanical engineering. The book appeals to both researchers and students from mechanics, mechanical engineering, physics and mathematics, and can be used as a reference as well as a textbook (for self-teaching as well as for teaching). Only basic mathematics is required as prerequisite. The book contains many carefully selected exercises (with worked-out solutions), illustrations and suggested further reading – it is excellent reading and highly recommended!

##### MSC:
 76-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to fluid mechanics 74-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids 76F60 $$k$$-$$\varepsilon$$ modeling in turbulence 86-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geophysics