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We use the vorticity transportation equation as the start point--with thehelp of stream function for two-dimensional planar incompressible flows--toobtain exact solutions that characterize evolution and dynamics of the flows.These traveling wave solutions, including both continuous and solitary waves,also recover the results existing in the theory of potential flows. We furtheranalyze the flow patterns regarding symmetry solutions generated by Lietransformation groups. Some of the symmetry solutions have not been discussedin the current literature. They could display flow patterns very different fromthat of the original exact solutions, suggesting the symmetry solutions are nottrivial. Particularly, the symmetry solutions to Lame-Oseen vortex and doubletexhibit more complicated dynamics.

Characterizing two-dimensional incompressible flows through traveling wave and symmetry solutions of the vorticity transportation equation