Rotational dynamics of a deformable medium

The aim of the present investigation has been to derive from the fundamental Cauchy's first law of continuum mechanics the explicit form of the Eulerian general equation which governs the three-axial generalized rotation about the centre of mass of a self-gravitating deformable finite material continuum, viscolinear (i.e., Newtonian) or not, consisting of compressible fluid of arbitrary viscosity, in an external field of force. The generalized rotation is a superposition of the so-called rigid-body (i.e., time dependent only) rotation of the continuum plus a nonrigidbody (i.e., position-time dependent) rotation of its configurations.In Section 2, which follows brief introductory remarks outlining the problem, we develop a mathematical theory which describes the whole phenomenon in terms of two rotation tensors corresponding, respectively, to the rigid-body and nonrigid-body rotation modes. In Section 3, we derive the differmation vectors of velocity and acceleration. The equations we have obtained are a very general version of Navier Stokes' equations, which were not given in previous investigations. In Section 4, we perform integration of the left-hand side of Cauchy's first law, cross-multiplied by the position vector, without any restriction. In Section 6, integration of the right-hand side of the same law, cross-multiplied by the position vector, is carried out, by taking account of actually simplifying assumptions stated in Section 5. All the integral terms occurring in both sides are expressed explicitly by quantities evaluated in terms of components of properly defined moments.Finally, in Section 7, the system of the general Eulerian equations is set up; and some easy modifications are given, which describe nicely physical models of special interest; while the concluding Section 8 contains a general discussion of the results.