Optimal bodies in viscoplastic and power-law fluids

Abstract

In a seminal paper, Pironneau (1973 J. Fluid Mech., vol. 59, pp. 117–128) showed that the lowest-drag shape of fixed volume in Stokes flow has a surface vorticity with constant magnitude over the entire body. In this paper, the viscoplastic version of the problem is analysed. The first result is that the surface vorticity on the optimal body in a Herschel– Bulkley fluid cannot vanish or become singular (in both two- and three-dimensional geometries). For the special cases of power-law fluids and high-yield-strength fluids, the change in drag following a small change in body shape is directly related to the surface vorticity, which is then shown to be constant on the optimal body. These resul..