The 3D stream function ${\bf \Psi}$ for a steady flow field can be defined as:
$\rho{\bf u}={\bf \nabla}\times {\bf \Psi}$.
Where, ${\bf u}$ = velocity, $\rho$ = density.
Now, this ${\bf \Psi}$ can be in turn represented as: ${\bf \Psi}=\chi{\bf \nabla}\psi $. Where, $\chi$ and $\psi$ are stream surfaces. What is the physical significance of $\chi$ and $\psi$?
Best Answer
The 3D stream function ${\bf \Psi}$ exists such that $\rho{\bf u}={\bf \nabla}\times {\bf \Psi}$ in a simply connected region for an incompressible flow ${\bf \nabla}\cdot(\rho{\bf u})=0$ due to Poincare Lemma.
Any 3D vector field ${\bf \Psi}={\bf \nabla}\varphi + \chi{\bf \nabla}\psi $ can be represented by 3 scalar Clebsch potentials.
We can remove the $\varphi$ potential due to the gauge symmetry ${\bf \Psi}\to{\bf \Psi}+{\bf \nabla}\varphi$.
The flow $\rho{\bf u}={\bf \nabla}\chi \times {\bf \nabla}\psi$ is along the 1D intersection of the two 2D equipotential surfaces for $\chi$ and $\psi$.
$(\chi,\psi)$ is a canonical pair: The flow is invariant under canonical transformations $(\chi,\psi)\to (\chi^{\prime},\psi^{\prime})$.