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]

Flow description

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Near the corner, the flow can be assumed to be Stokes flow. Describing the two-dimensional planar problem by the cylindrical coordinates   with velocity components   defined by a stream function such that

 

the governing equation can be shown to be simply the biharmonic equation  . The equation has to be solved with homogeneous boundary conditions (conditions taken for two walls separated by angle  )

 

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

 

where   are constants and the real part of the eigenvalues are always greater than unity. The eigenvalues   will be function of the angle  , but regardless eigenfunctions can be written down for any  ,

 

For antisymmetrical solution, the eigenfunction is even and hence   and the boundary conditions demand  . The equations admits no real root when  °. These complex eigenvalues indeed correspond to the moffatt eddies. The complex eigenvalue if given by   where

 

Here  .

See also

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References

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  1. ^ Moffatt, H. K. (1964). "Viscous and resistive eddies near a sharp corner". Journal of Fluid Mechanics. 18 (1): 1–18. Bibcode:1964JFM....18....1M. doi:10.1017/S0022112064000015. S2CID 123251976.
  2. ^ Dean, W. R.; Montagnon, P. E. (1949). "On the steady motion of viscous liquid in a corner". Mathematical Proceedings of the Cambridge Philosophical Society. 45 (3). Cambridge University Press: 389–394. Bibcode:1949PCPS...45..389D. doi:10.1017/S0305004100025019. S2CID 122817160.
  3. ^ Rayleigh, L. (1911). XXIII. Hydrodynamical notes. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 21(122), 177-195.
  4. ^ Shankar, P. N. (2005). "Moffatt eddies in the cone". Journal of Fluid Mechanics. 539: 113–135. Bibcode:2005JFM...539..113S. doi:10.1017/S0022112005005458. S2CID 58910487.
  5. ^ Shankar, P. N. (2007). Slow Viscous Flows: Qualitative Features and Quantitative Analysis Using Complex Eigenfunction Expansions (With CD-ROM). World Scientific.