Stokes Stream Function - Spherical Coordinates

Spherical Coordinates

In spherical coordinates ( r, θ, φ ), r is the radial distance from the origin, θ is the zenith angle and φ is the azimuthal angle. In axisymmetric flow, with θ = 0 the rotational symmetry axis, the quantities describing the flow are again independent of the azimuth φ. The flow velocity components u_r and u_θ are related to the Stokes stream function through:

$\begin{align} u_r &= + \frac{1}{r^2\, \sin(\theta)}\, \frac{\partial \Psi}{\partial \theta}, \\ u_\theta &= - \frac{1}{r\, \sin(\theta)}\, \frac{\partial \Psi}{\partial r}. \end{align}$

Again, the azimuthal velocity component u_φ is not a function of the Stokes stream function ψ. The volume flux through a stream tube, bounded by a surface of constant ψ, equals 2π ψ, as before.

Read more about this topic: Stokes Stream Function