Stokes Stream Function - Cylindrical Coordinates

Cylindrical Coordinates

Consider a cylindrical coordinate system ( ρ, φ, z ), with the z–axis the line around which the incompressible flow is axisymmetrical, φ the azimuthal angle and ρ the distance to the z–axis. Then the flow velocity components u_ρ and u_z can be expressed in terms of the Stokes stream function by:

$\begin{align} u_\rho &= - \frac{1}{\rho}\, \frac{\partial \Psi}{\partial z}, \\ u_z &= + \frac{1}{\rho}\, \frac{\partial \Psi}{\partial \rho}. \end{align}$

The azimuthal velocity component u_φ does not depend on the stream function. Due to the axisymmetry, all three velocity components ( u_ρ, u_φ, u_z ) only depend on ρ and z and not on the azimuth φ.

The volume flux, through the surface bounded by a constant value ψ of the Stokes stream function, is equal to 2π ψ.

Read more about this topic: Stokes Stream Function