Matrix approach to Lagrangian fluid dynamics (Q2755596)

The authors suggest a new approach to hydrodynamics. It is Lagrangian (material) in its nature, but differs from conventional Lagrangian approach as follows. In the latter one looks for the trajectory of a particle, while here the fluid motion is considered as continuous deformation of infinitesimal material element \(d{\mathbf X}\) from its initial value \(d{\mathbf X}_0\), so that the evolution is described by the equation \(d{\mathbf X}={\mathbf R}d{\mathbf X}_0\), and the matrix \({\mathbf R}\) as a function of time and spatial coordinates provides a complete description of the flow. The equations describing the evolution of \({\mathbf R}\) are written down for the case of ideal incompressible fluid. A class of exact solutions, that describes rotational non-stationary motions, is found and investigated.