Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 1 / 5A (Fluid Dynamics)

The displayed evolution law uses the usual assumptions of incompressible flow, constant density and conservative body force. With $\mathbf{\omega }=\nabla \times \mathbf{u}$, the Euler equation isUse $(\mathbf{u}\cdot \nabla )\mathbf{u}=\nabla (|\mathbf{u}{|}^2/2)-\mathbf{u}\times \mathbf{\omega }$ and take a curl. Gradients have zero curl, andHere $\nabla \cdot \mathbf{\omega }=0$ because it is a curl, and $\nabla \cdot \mathbf{u}=0$ by incompressibility. The vorticity equation is thereforeThe material derivative combines local change with advection by fluid particles. The right-hand side is vortex stretching and tilting: spatial variation of velocity along a vortex line changes its strength and direction. Along each particle path this is a homogeneous linear equation for vorticity. For a smooth velocity field its solution with zero initial vorticity remains zero by uniqueness. Thus an initially irrotational flow remains irrotational while these assumptions and smoothness hold.

For a plane flow $\mathbf{u}=(u(x,y,t),v(x,y,t),0)$, the vorticity is $\mathbf{\omega }=(0,0,\zeta )$ and $(\mathbf{\omega }\cdot \nabla )\mathbf{u}=\zeta {\partial }_z\mathbf{u}=0$. Hence $D\zeta /Dt=0$. Every particle retains its initial value, so an initially uniform value givesInviscid motion alone is insufficient for the printed simplified law: compressible variable-density motion generally adds $-\mathbf{\omega }\nabla \cdot \mathbf{u}+\nabla \rho \times \nabla p/{\rho }^2$. Those terms vanish under the assumptions used above.