Vorticity: $\tilde{\omega}=\nabla \times V$ Navier-Stokes Equation: $\frac{ \partial V }{ \partial t }+(V\cdot \nabla)V=-\frac{1}{\rho}\nabla P+\nu \nabla^{2}V$ Curl of Navier Stokes $\frac{D\tilde{\omega}}{Dt}=(\tilde{\omega}\cdot \nabla)V + \nu \nabla^{2}\tilde{\omega}$ Instead of $u$ and $v$, we use the streamfunction ${} \psi {}$ and $\omega$: $u = \frac{ \partial \psi }{ \partial y }$ $v=-\frac{ \partial \psi }{ \partial x }$ $\omega = \frac{ \partial v }{ \partial x }-\frac{ \partial u }{ \partial y}$ $\omega = \nabla \times v$ - streamlines (tangent to velocity): constant $\psi$ - vorticity vector orthogonal to plane of 2D flow - 2D continuity automatically satisfied: $\frac{ \partial u }{ \partial x }+\frac{ \partial v }{ \partial y }=0$ This gives us the coupled PDEs: $\frac{ \partial^{2}\psi }{ \partial x^{2} }+\frac{ \partial^{2}\psi }{ \partial y^{2} }=-\omega$ ${} \rho \frac{ \partial \omega }{ \partial t } + \rho u\frac{ \partial \omega }{ \partial x } + \rho v \frac{ \partial \omega }{ \partial y } = \mu\left( \frac{ \partial^{2}\omega }{ \partial x^{2} }+\frac{ \partial^{2}\omega }{ \partial y^{2} } \right){}$ Boundary conditions are complicated… Vorticity-streamfunction is good for 2D, but not as good in 3D because you have 3 components of vorticity and it becomes as expensive as NS