Axisymmetric flow of a viscous fluid near the vertex of a body

Abstract

Axially symmetric motion of a viscous fluid in a cone is considered on the basis of the Stokes assumption. Near the apex of the cone the solution obtained reveals features quite similar to those of that near a sharp corner in two dimensions, which has been discussed already. An infinite sequence of eddies is induced near the apex for values less than about 80·9° of the semi-angle of the cone, which is measured from the symmetry axis lying in the fluid. The solution found by Pell & Payne for a spindle in a uniform stream offers a good illustration of the general discussion. Special attention is paid to the angle 120° for the spindle as well as the cone. The limiting case of zero angle of the cone corresponds to the flow occurring in a circular cylinder.

Nonaxisymmetric stokes flow between concentric cones

“…Both Wakiya (11) and Liu & Joseph (12) have studied this geometry and demonstrated the formation of axisymmetric eddies. Malyuga (13) and Shankar (14) have extended these results to consider fully three-dimensional flows.…”

Bifurcations of a creeping air–water flow in a conical container

“…Wakiya [13] revealed that the eddies exists in a cone whose half-angle  (i.e., the angle between the axis and the sidewall) is less than 80.9. This feature of conical flows is important for our study, where  = 30, 45 and 60.…”

“…Fig. 2(a) It is known that a single-fluid flow (e.g., with no water) has a cascade of eddies with alternating circulation directions [13]. The dimensions and strengths of eddies tend to zero as the cone tip is approached.…”

“…Even a slow ("creeping") motion has an unbounded set of eddies near the tip. Such sets were first discovered in a creeping flow between two 3 inclined walls [12] and then in a cone, where axisymmetric [13][14][15] and three-dimensional [16][17] eddies were found.…”

Confined axisymmetric laminar jets with large expansion ratios

“…Apart from (6.4) and (6.5), the rescaled velocity must satisfyŨ t = −ŪX and V t =V atX = −∞ andŨ t = V t = 0 atX = ∞, the leading-order boundary conditions obtained by neglecting in (6.6) and (6.7) terms that vanish when Re a → 0. With a zero mass flux, the leading-order solution must show forX → ∞ a velocity that decays withX at an exponential rate, eventually giving rise to the Stokes toroidal vortices described by Wakiya (1976). This leading-order description can be expected to fail asŨ t decays to values of order Re a , where the mass flux should be taken into account, eventually leading to either the Poiseuille profile or a uniform velocity profile.…”