Moffatt eddies

Near the corner, the flow can be assumed to be Stokes flow. Describing the two-dimensional planar problem by the cylindrical coordinates ${\displaystyle (r,\theta )}$ with velocity components ${\displaystyle (u_{r},u_{\theta })}$ defined by a stream function such that

${\displaystyle u_{r}={\frac {1}{r}}{\frac {\partial \psi }{\partial \theta }},\quad u_{\theta }=-{\frac {\partial \psi }{\partial r}}}$

the governing equation can be shown to be simply the biharmonic equation ${\displaystyle \nabla ^{4}\psi =0}$. The equation has to be solved with homogeneous boundary conditions (conditions taken for two walls separated by angle ${\displaystyle 2\alpha }$)

${\displaystyle {\begin{aligned}r>0,\ \theta =-\alpha$ :&\quad u_{r}=0,\ u_{\theta }=0\\r>0,\ \theta =\alpha :&\quad u_{r}=0,\ u_{\theta }=0.\end{aligned}}}

The Taylor scraping flow is similar to this problem but driven inhomogeneous boundary condition. The solution is obtained by the eigenfunction expansion,^[5]

${\displaystyle \psi =\sum _{n=1}^{\infty }A_{n}r^{\lambda _{n}}f_{\lambda _{n}}(\theta )}$

where ${\displaystyle A_{n}}$ are constants and the real part of the eigenvalues are always greater than unity. The eigenvalues ${\displaystyle \lambda _{n}}$ will be function of the angle ${\displaystyle \alpha }$, but regardless eigenfunctions can be written down for any ${\displaystyle \lambda }$,

${\displaystyle {\begin{aligned}f_{0}&=A+B\theta +C\theta ^{2}+D\theta ^{3},\\f_{1}&=A\cos \theta +B\sin \theta +C\theta \cos \theta +D\theta \sin \theta ,\\f_{2}&=A\cos 2\theta +B\sin 2\theta +C\theta +D,\\f_{\lambda }&=A\cos \lambda \theta +B\sin \lambda \theta +C\cos(\lambda -2)\theta +D\sin(\lambda -2)\theta ,\quad \lambda \geq 2.\end{aligned}}}$

For antisymmetrical solution, the eigenfunction is even and hence ${\displaystyle B=D=0}$ and the boundary conditions demand ${\displaystyle \sin 2(\lambda -1)\alpha =-(\lambda -1)\sin 2\alpha }$. The equations admits no real root when ${\displaystyle 2\alpha <146}$°. These complex eigenvalues indeed correspond to the moffatt eddies. The complex eigenvalue if given by ${\displaystyle \lambda _{n}=1+(2\alpha )^{-1}(\xi _{n}+i\eta _{n})}$ where

${\displaystyle {\begin{aligned}\sin \xi \cosh \eta &=-k\xi ,\\\cos \xi \sinh \xi &=-k\eta .\end{aligned}}}$

Here ${\displaystyle k=\sin 2\alpha /2\alpha }$.

• Taylor scraping flow