Exact coordinate-dependent Beltrami coefficient for unsteady axially symmetric viscous vortices

Exact radial-coordinate dependent Beltrami coefficients for unsteady axially symmetric viscous Beltrami vortices under external forces are sought in cylindrical coordinate system. Two types of solutions are presented. One breaks the scaling invariance and decays temporally exponentially. The other one keeps the scaling invariance and asymptotically exhibits a power-law decay. As a byproduct, the exact generalized Beltrami vortex solutions are found and are shown to have lower energy densities.