Continuum mechanics of media with inner structures (Q1995662)

This very interesting paper gives us a more accurate description of the possible motions (or deformations) of continuous media. The new element is the inner structure of media, described by an interaction of two Riemann manifolds equipped with two different metrics and a smooth bundle. A connection of this bundle exists. The parallel transport of fibers along this connection is an isometry of fibers with respect to the metric on the first manifold. The movement of the medium is described by a particular a vector space which preserves this bundle, with some additional hypothesis. This approach allows us consider the internal structure of the studied media: for example, different manifolds occur for homonuclear or heteronuclear diatomic gases. The thermodynamic theory of such media with internal structure is based on some extensive quantities: mass, volume, internal energy, deformation -- see [\textit{V. V. Lychagin}, in: Nonlinear PDEs, their geometry, and applications. Proceedings of the Wisła 18 summer school, Wisła, Poland, August 20--30, 2018. Cham: Birkhäuser. 3--52 (2019; Zbl 1428.53111)]. The Navier-Stokes equations and the conservation laws for mass and energy are obtained in the last part. The authors highlight the coordinate-free method for obtaining these equations in a more explicit and transparent way.