Viscous and resistive eddies near a sharp corner

TLDR

The study presents similarity solutions for viscous flow near a sharp corner between two planes, covering various boundary conditions and asymptotic regimes, including the zero‑angle limit and large‑distance behavior. The authors derive the scaling ratios of successive eddies across all admissible corner angles and show that an applied magnetic field from a line current expands the angle range that produces eddies. For rigid boundaries below a critical corner angle, the flow inevitably forms a cascade of diminishing eddies, whereas with free surfaces no eddies arise.

Abstract

Some simple similarity solutions are presented for the flow of a viscous fluid near a sharp corner between two planes on which a variety of boundary conditions may be imposed. The general flow near a corner between plane boundaries at rest is then considered, and it is shown that when either or both of the boundaries is a rigid wall and when the angle between the planes is less than a certain critical angle, any flow sufficiently near the corner must consist of a sequence of eddies of decreasing size and rapidly decreasing intensity. The ratios of dimensions and intensities of successive eddies are determined for the full range of angles for which the eddies exist. The limiting case of zero angle corresponds to the flow at some distance from a two-dimensional disturbance in a fluid between parallel boundaries. The general flow near a corner between two plane free surfaces is also determined; eddies do not appear in this case. The asymptotic flow at a large distance from a corner due to an arbitrary disturbance near the corner is mathematically similar to the above, and has comparable properties. When the fluid is electrically conducting, similarity solutions may be obtained when the only applied magnetic field is that due to a line current along the intersection of the two planes; it is shown that the effect of such a current is to widen the range of corner angles for which eddies must appear.