For any incompressible flow, with or without viscosity, the velocity vector $\vec{v}$ must satisfy the condition that its divergence is equal to zero; a well-known result of vector analysis is that any divergence-free vector can be written in the form


\begin{displaymath}\vec{v} = \nabla\times\vec{\psi}, \notag
\end{displaymath}  

where $\vec{\psi}$ is the vector stream function. In general, this is not a very useful concept, since the new unknown function is also a vector. However, in the special case of two-dimensional plane flow and three-dimensional axisymmetric flow, the vector stream function has only one component; thus it becomes a scalar unknown, with the same resulting simplification as the velocity potential.

Timothy S Choe
2003-03-15