On a class of Beltrami vector fields and associated exact solutions to the Navier–Stokes equations

In this paper, we derive a class of exact solutions to the incompressible, unsteady Navier–Stokes equations using vector-valued stream functions of a special form. We show that certain known solutions to the Navier–Stokes equations generated by Beltrami vector fields belong to this class. The energy dissipation rate of the obtained solutions and the boundary conditions are studied in some detail. In particular, we derive exact solutions to the Navier–Stokes equations that satisfy physically relevant boundary conditions in regions bounded by impermeable plane walls.