Axisymmetric slow viscous flow past an arbitrary convex body of revolution

Abstract

Considerable advances have been made in the past few years in treating a variety of problems in slender-body Stokes flow (Taylor 1969; Batchelor 1970; Cox 1970, 1971; Tillett 1970). However, the problem of treating the creeping motion past bluff objects, whose boundaries do not conform to a constant co-ordinate surface of one of the special orthogonal co-ordinate systems for which the Stokes slow-flow equation is simply separable, is still largely unsolved. In the slender-body Stokes flow studies mentioned above, the viscous-flow boundary-value problem is formulated approximately as an integral equation for an unknown distribution of Stokeslets over a line enclosed by the body. The theory is valid for only very extended shapes, since the error in drag decays inversely as the logarithm of the aspect ratio of the object. By contrast, the present authors show that the boundary-value problem for the axisymmetric flow past an arbitrary convex body of revolution can be formulated exactly as an integral equation for an unknown distribution of ring-like singularities over the surface of the body. The kernel in this integral equation is closely related to the fundamental separable solutions of the Stokes slow-flow equation when written in an oblate spheroidal co-ordinate system of vanishing aspect ratio. The two lowest-order appropriate spheroidal singularities are found to provide a complete description for all surface elements, except those perpendicular to the axis. Higher-order singularities of all orders are required to describe axially perpendicular surfaces, such as the ends of a cylinder or the blunt base of an object. The newly derived integral equation is solved numerically to provide the first theoretical solutions for low aspect ratio cylinders and cones. The theoretically predicted drag results are in excellent agreement with experimentally measured values.

Numerical Multipole and Boundary Integral Equation Techniques in Stokes Flow

“…The first generalizations of the multipole collocation technique were to axisymmetric bounded flows and the representation of an arbitrary convex body of revolution using a distribution of oblate spheroidal singularities whose foci coincided with the surface of the body (Gluckman et al 1972). …”

Application of Boundary Collocation Method in Fluid Mechanics to Stokes Flow Problems

“…Third, we truncate the infinite series, which appears in the formulas defining the velocity and the microrotation. In order to obtain a unique solution, the boundary conditions on the sphere (7)(8) are applied at a finite number of discrete points on the sphere. Then we solve a derived linear set of equations by numerical method to find B n , A n , and D n .…”

“…in Stokes flow. A few illustrative examples include an arbitrary convex body of revolution in [7], multiple spheres in a cylinder [17] and two spheroids in a uniform stream [18]. Ganatos and coworkers made major modification to the theory and extended it to handle variety of non-axisymmetric creeping flow problems with planar symmetry where the boundaries conform to more than a single orthogonal co-ordinate system.…”

Drag and wakes of freely falling 60° cones at intermediate Reynolds numbers

“…The experimentally determined drag coefficients are available 1 only for cones with vertex angles of 30°and 60°at a Reynolds number of 2.7ϫ 10 5 . On the other hand, Gluckman et al 2 calculated approximately the drag of a 60°cone in a Stokes flow. As for the wakes behind cones, Goldburg and Florsheim 3 measured the Strouhal number for cones with vertex angles of 20°and 40°falling through an aqueous glycerin solution and observed a staggered array of two rows of ring-shaped hairpin vortices behind the 20°cone.…”