for incompressible flow, without external forces:
Here is the velocity field written in the cylindrical coordinate system , where is the radius and is the angle in the plane, is the pressure, and is the viscosity coefficient.
The continuity equation reads:
NB: is given by the formula , and the velocity field is:
Cylindrical coordinates are probably most convenient for the study of axisymmetric solutions, i.e. when the velocity field does not depend on the angle coordinate . Then we get the following equations:
The above equations still look too complicated and unintuitive. Instead of the variables , we will use the new triple of variables , where
(such a exists because of the continuity equation), and is the the phi-component of the vorticity vector: