New classes of exact solutions and some transformations of the Navier-Stokes equations

New classes of exact solutions of three-dimensional nonstationary Navier-Stokes equations are described. These solutions contain arbitrary functions. Many periodic solutions (both with respect to the spatial coordinate and with respect to time) and aperiodic solutions are obtained, which can be expressed in terms of elementary functions. A Crocco-type transformation is presented, which reduces the order of the equation for the longitudinal component of the velocity. Problems concerning the nonlinear stability/instability of the solutions thus obtained are investigated. It turns out that a specific feature of many solutions of the Navier-Stokes equations is their instability. It is shown that instability can take place not only for rather large Reynolds numbers but also for arbitrarily small ones (and can be independent of the velocity profile of the fluid). A general physical interpretation and classification of solutions is given.