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On angles of separation in Stokes flows. (English) Zbl 0545.76039

This interesting and elegant paper clarifies the problem of the angles of separation in Stokes flows in two and three dimensions.

The flow separation accompanied by eddy formation was discovered by H. K. Moffatt [J. Fluid Mech. 18, 1-18 (1964; Zbl 0118.205)] who showed that the flow between planes which intersect at an angle of less than about 146.3$\circ$ has an eddy structure with the infinite sequence of line vortices close to the corner. The presence of similar structures in two-dimensional linear shear or stagnation-point flows over a cylinder touching a plane was shown by A. M. J. Davis and the author [Q. J. Mech. Appl. Math. 30, 355-368 (1977; Zbl 0384.76018) and J. Fluid Mech. 81, 551-564 (1977; Zbl 0379.76026)], and in three dimensional case by A. M. J. Davis, the author, J. M. Dorrepaal and K. B. Ranger [J. Fluid Mech. 77, 625-644 (1976; Zbl 0346.76022)]. These studies suggested that the angle at which the separating streamlines detach from the boundaries tends to 58.61$\circ$ as the point of contact is approached.

The studies of Stokes flow in the axisymmetric flow past a closed torus performed by J. M. Dorrepaal, S. R. Majumdar, the author and K. B. Ranger [Q. J. Mech. Appl. Math. 29, 381-397 (1976; Zbl 0355.76024)] showed the presence of separation and formation of an infinite set of nested ring vortices from within the central cusps. This analysis suggested that there exists an angle of separation close to 45.25$\circ ·$

The present paper proves that these conjectures are true. The proof bases on the analysis of the Stokes flow close to the touching point of cylindrical and spherical bodies. The local solution in the vicinity of the separation point depends of eigenvalues of a consistency condition of two homogeneous equations describing the boundary conditions.

Two particular points of the analysis should be mentioned. It is shown that an arbitrary Stokes flow between parallel planes decying at infinity (e.g. flow generated by the moving particle) eventually becomes antisymmetric far from the body and cells are formed. The length of these cells is constant and the limiting angle with the boundaries is again 48.15$\circ ·$

A similar analysis for the flow in a circular cylinder shows that, for example for a slow motion of a sphere, there is an infinite system of closed eddies both far ahead and behind the sphere.

Reviewer: R.Herczynski
##### MSC:
 76D07 Stokes and related (Oseen, etc.) flows 76D10 Boundary-layer theory, separation and reattachment, etc. (incompressible viscous fluids)