# 3D Stream function in fluid mechanics

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 $\psi$?

• The author has directly defined the stream function as ψ = χ grad ψ. I am having trouble understanding why he has defined stream it in this manner. Specifically, I cannot figure out intuition behind that definition? – suraj kulkarni Sep 2 '18 at 15:23
• Which author? Which page? – Qmechanic Sep 2 '18 at 20:29
• Are you comfortable with the 2D version of this, where ##chi## is a constant? Does that make sense to you from the standpoint of physical interpretation? – Chet Miller Sep 2 '18 at 22:06
• You should mention the book's name. As far as I know, if $\psi$ is the scalar stream-function for 2d or axisymmetric flow, then velocity field is given by the curl of the vector $[0,0,\psi]$. – Deep Sep 3 '18 at 6:02

## 1 Answer

1. 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.

2. Any 3D vector field ${\bf \Psi}={\bf \nabla}\varphi + \chi{\bf \nabla}\psi$ can be represented by 3 scalar Clebsch potentials.

3. We can remove the $\varphi$ potential due to the gauge symmetry ${\bf \Psi}\to{\bf \Psi}+{\bf \nabla}\varphi$.

4. 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 $\psi$.

5. $(\chi,\psi)$ is a canonical pair: The flow is invariant under canonical transformations $(\chi,\psi)\to (\chi^{\prime},\psi^{\prime})$.

• +1 Great answer. I just realized that a divergence-free vector field allows it to be written as curl of another vector field. – Deep Sep 4 '18 at 5:26