Moffatt eddies are sequences of eddies that develop in corners bounded by plane walls (or sometimes between a wall and a free surface) due to an arbitrary disturbance acting at asymptotically large distances from the corner. Although the source of motion is the arbitrary disturbance at large distances, the eddies develop quite independently and thus solution of these eddies emerges from an eigenvalue problem, a self-similar solution of the second kind.
The eddies are named after Keith Moffatt, who discovered these eddies in 1964,[1] although some of the results were already obtained by William Reginald Dean and P. E. Montagnon in 1949.[2] Lord Rayleigh also studied the problem of flow near the corner with homogeneous boundary conditions in 1911.[3] Moffatt eddies inside cones are solved by P. N. Shankar.[4]
Near the corner, the flow can be assumed to be Stokes flow. Describing the two-dimensional planar problem by the cylindrical coordinates
(r,\theta)
(ur,u\theta)
ur=
| 1 | |
| r |
| \partial\psi | |
| \partial\theta |
,
| u | ||||
|
\nabla4\psi=0
2\alpha
\begin{align} r>0, \theta=-\alpha:& ur=0, u\theta=0\\ r>0, \theta=\alpha:& ur=0, u\theta=0. \end{align}
The Taylor scraping flow is similar to this problem but driven inhomogeneous boundary condition. The solution is obtained by the eigenfunction expansion,[5]
\psi=
| infty | |
| \sum | |
| n=1 |
An
| λn | |
| r |
| f | |
| λn |
(\theta)
where
An
λn
\alpha
λ
\begin{align} f0&=A+B\theta+C\theta2+
| 3,\\ f | |
| D\theta | |
| 1 |
&=A\cos\theta+B\sin\theta+C\theta\cos\theta+D\theta\sin\theta,\\ f2&=A\cos2\theta+B\sin2\theta+C\theta+D,\\ fλ&=A\cosλ\theta+B\sinλ\theta+C\cos(λ-2)\theta+D\sin(λ-2)\theta, λ\geq2. \end{align}
For antisymmetrical solution, the eigenfunction is even and hence
B=D=0
\sin2(λ-1)\alpha=-(λ-1)\sin2\alpha
2\alpha<146
λn=1+(2\alpha)-1(\xin+iηn)
\begin{align} \sin\xi\coshη&=-k\xi,\\ \cos\xi\sinh\xi&=-kη. \end{align}
Here
k=\sin2\alpha/2\alpha