Stokes streamfunction $\psi$ describes streamlines and flow velocity in three-dimensional incompressible flow with axisymmetry and is defined as: $ u =\frac{\partial\psi}{\partial y}\quad v=-\frac{\partial\psi}{\partial x} $ Using these in the [[Navier-Stokes equations|Navier-Stokes equations]] and eliminating pressure results in: $ \frac{\partial}{\partial t}\nabla^2\psi+\frac{\partial\psi}{\partial y}\frac{\partial}{\partial x}\nabla^2\psi-\frac{\partial\psi}{\partial x}\frac{\partial}{\partial y}\nabla^2\psi=\nabla^4\psi $ System is reduced to one dependent variable $\psi$ and one equation, but is now fourth-order Relationship between [[Vorticity|vorticity]] and streamfunction: $ \eta=\nabla^2\psi $