In the present paper, the Navier-Stokes equations are studied in several axially symmetric cases. In them incompressible viscous fluids rotate about their axes and can change their shape. In the considered cases, three exact solutions to the Navier-Stokes equations are found. The first of these solutions describes rotating viscous fluids that are gradually cooling. The second of them describes nonstationary rotations with axial motions of viscous fluids. The third of the obtained solutions to the Navier-Stokes equations concerns rotating viscous fluids with stationary velocities. It is used to describe the observable phenomenon of differential rotation of the visible surfaces of stars and giant gas planets.