New classes of exact solutions of three-dimensional Navier-Stokes equations

New classes of exact solutions of the three-dimensional unsteady Navier-Stokes equations containing arbitrary functions and parameters are described. Various periodic and other solutions, which are expressed through elementary functions are obtained. The general physical interpretation and classification of solutions is given.

💡 Research Summary

The paper presents several new families of exact solutions to the three‑dimensional unsteady Navier‑Stokes (N‑S) equations, expanding the limited set of known analytical results for this notoriously nonlinear system. The authors begin by recalling that classical exact solutions—such as the Lamb, Poiseuille, and Strah­nov flows—are highly constrained by symmetry or special forcing, and therefore cannot capture the full richness of three‑dimensional fluid dynamics. To overcome this limitation, they adopt a structured ansatz for the velocity field that separates spatial variables in a way that still satisfies the incompressibility condition.

Specifically, the velocity is written as
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