Fluids and periodic structures.

*(English)*Zbl 0910.76002
Research in Applied Mathematics. Chichester: Wiley. Paris: Masson. xii, 333 p. (1995).

This book is devoted to mathematical foundations of different formulations of problems on vibrations of periodic structures in fluids. The authors accentuate that “the principal tools we use to understand the fluid-solid structures are borrowed from functional analysis and, more specifically, from the spectral theory of linear operators”. The authors study the fluid-structure problems from a qualitative point of view, namely they study problems on solution existence, eigenvalue spectrum, spectrum distribution, estimations of spectrum bounds and some other problems. So they construct a strong background for numerical methods for studying frequencies of fluid-solid systems.

The authors formulate some directions in the investigation of asymptotic properties of fluid-solid structures. As the main approach they use the Bloch wave decomposition method. For studying asymptotic properties of the spectrum, the homogenization method is applied. The Bloch wave decomposition method, which is often used in the solid state physics, is transformed and enhanced for the homogenization procedure by using the two-scale convergence technique. Some approaches are used for numerical examples which illustrate general possibilities of proposed methods.

On the whole, the book is oriented to mathematicians, but also to engineers, who are interested in mathematical foundations of numerical approaches and in some estimations of spectra for different problems of fluid-structure vibrations. This book also may be used as a special course for graduate and post-graduate students.

The authors formulate some directions in the investigation of asymptotic properties of fluid-solid structures. As the main approach they use the Bloch wave decomposition method. For studying asymptotic properties of the spectrum, the homogenization method is applied. The Bloch wave decomposition method, which is often used in the solid state physics, is transformed and enhanced for the homogenization procedure by using the two-scale convergence technique. Some approaches are used for numerical examples which illustrate general possibilities of proposed methods.

On the whole, the book is oriented to mathematicians, but also to engineers, who are interested in mathematical foundations of numerical approaches and in some estimations of spectra for different problems of fluid-structure vibrations. This book also may be used as a special course for graduate and post-graduate students.

Reviewer: O.Limarchenko (Kyïv)

##### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76D99 | Incompressible viscous fluids |

76B99 | Incompressible inviscid fluids |

74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |

35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |

35P20 | Asymptotic distributions of eigenvalues in context of PDEs |