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**Fluid-structure interactions.
(Interactions fluides-structures.)**
*(French)*
Zbl 0754.73071

Recherches en Mathématiques Appliquées. 23. Paris etc.: Masson. xi, 212 p. (1992).

In chapter 1 the basic variational approach to linearized dynamics of elastic bodies is formulated, the equivalent finite element discretization indicated and the appropriateness of the Ritz-Galerkin technique to the resolution of the boundary value problems highlighted.

The Fredholm alternatives leading to unique solutions when the excitation and response are sinusoidally periodic, are quoted, leading on to a discussion of proper modes of the appropriate spectral problem. Properties of proper modes and values are discussed and comparison theorems surveyed: in particular it is emphasized that the proper frequencies (values) of the discretized problem are greater than those of the corresponding continuous problem in any range. A further theorem — principle of separation — relating to similar structures, one of which is subjected to linear constraints, is also quoted. Finally, systems dependent on one or two parameters and the ‘crossing of modes’ are discussed. As is evident from the above, the fluids in the discussion are of necessity elastic, and in chapter 2 the classical equations for small motions are reviewed.

The remainder of the monograph is devoted to specific classes of problems: (i) Chapter 3 deals with ‘sloshing’ in tanks or reservoirs of heavy fluids in the absence of surface tension. (ii) Chapter 4 enlarges the discussion by introducing surface tension. (iii) Chapter 5 introduces the effects of elasticity in the reservoir. (iv) Chapter 6 reintroduces the effects of gravity neglected in chapter 5. (v) In chapter 7 the topic changes application; the acoustics of a compressible fluid filled cavity is treated. (vi) In chapter 8 the subject is the harmonic vibrations of an elastic structure containing a weightless compressible fluid, with or without free surface. (vii) The final chapter develops the subject of modal reduction for fluid-solid interactions.

The work which forms an element in a series of research monographs has copious references to the relevant literature and the fields of application are identified in the introductions to the individual chapters. The treatment of the specific topics is purely analytical throughout, dealing only in the generalities that can be derived from the main theorems applied to the specific classes of problems mentioned above.

The Fredholm alternatives leading to unique solutions when the excitation and response are sinusoidally periodic, are quoted, leading on to a discussion of proper modes of the appropriate spectral problem. Properties of proper modes and values are discussed and comparison theorems surveyed: in particular it is emphasized that the proper frequencies (values) of the discretized problem are greater than those of the corresponding continuous problem in any range. A further theorem — principle of separation — relating to similar structures, one of which is subjected to linear constraints, is also quoted. Finally, systems dependent on one or two parameters and the ‘crossing of modes’ are discussed. As is evident from the above, the fluids in the discussion are of necessity elastic, and in chapter 2 the classical equations for small motions are reviewed.

The remainder of the monograph is devoted to specific classes of problems: (i) Chapter 3 deals with ‘sloshing’ in tanks or reservoirs of heavy fluids in the absence of surface tension. (ii) Chapter 4 enlarges the discussion by introducing surface tension. (iii) Chapter 5 introduces the effects of elasticity in the reservoir. (iv) Chapter 6 reintroduces the effects of gravity neglected in chapter 5. (v) In chapter 7 the topic changes application; the acoustics of a compressible fluid filled cavity is treated. (vi) In chapter 8 the subject is the harmonic vibrations of an elastic structure containing a weightless compressible fluid, with or without free surface. (vii) The final chapter develops the subject of modal reduction for fluid-solid interactions.

The work which forms an element in a series of research monographs has copious references to the relevant literature and the fields of application are identified in the introductions to the individual chapters. The treatment of the specific topics is purely analytical throughout, dealing only in the generalities that can be derived from the main theorems applied to the specific classes of problems mentioned above.

Reviewer: J.Dunwoody (Belfast)

### MSC:

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

76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74P10 | Optimization of other properties in solid mechanics |