The other optimal Stokes drag profile

TL;DR

This paper identifies a new optimal shape of fixed surface area in Stokes flow with significantly lower drag than a sphere, characterized by a specific tangent angle at its ends, with implications for particle design.

Contribution

It introduces and characterizes a novel optimal shape of fixed surface area in Stokes flow, expanding understanding beyond the known fixed-volume optimal shape.

Findings

The shape has 11.3% lower drag than a sphere of equal surface area.

The shape's ends are tangent to a cone of approximately 30.8 degrees.

Surface vorticity is proportional to mean surface curvature.

Abstract

The lowest drag shape of fixed volume in Stokes flow has been known for some 40 years. It is front-back symmetric and similar to an American football with ends tangent to a cone of 60 degrees. The analogous convex axisymmetric shape of fixed surface area, which may be of interest for particle design in chemistry and colloidal science, is characterized in this paper. This "other" optimal shape has a surface vorticity proportional to the mean surface curvature, which is used with a local analysis of the flow near the tip to show that the front and rear ends are tangent to a cone of angle 30.8 degrees. Using the boundary element method, we numerically represent the shape by expanding its tangent angle in terms decaying odd Legendre modes, and show that it has 11.3% lower drag than a sphere of equal surface area, significantly more pronounced than for the fixed-volume optimal.