Steady-State Solutions to the Navier–Stokes Equation

The concept of steady-state solutions to the Navier–Stokes equation is defined. Such solutions extend the notion of stationary ones, diminish exponentially over time, and have a fixed spatial field of velocities and constant pressure in the absence of external fields. A way of constructing these solutions is considered, and the problem of Taylor vortices is solved. A mathematical model of a tornado is proposed, within which a steady-state solution is obtained as an eigenfunction of the problem in the form of a vortex. A model for the formation of the structure of a gas cloud is proposed, based on the Navier–Stokes equation. It is shown that spiral arms arise from flows of gas moving outward, due to the Coriolis force. It is proved that the number of arms is even and their structure is independent of the angular velocity of rotation. We obtain a formula for the twist angle of the spirals, depending on the cloud parameters for when .