Optimal bodies in viscoplastic and power-law fluids

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 results inform a local analysis of the flow near the sharp tips at either end of the optimal body, which determines the tip angle in different non-Newtonian fluids. In shear-thinning and viscoplastic fluids, the viscosity decreases with strain rate and so the fluid effectively self-lubricates in regions of high shear allowing for a sharper optimal body. Indeed, in a high-yield-strength fluid, the optimal body is entirely surrounded by a thin viscoplastic boundary layer and in a planar geometry, the interior tip angles converge to 90^∘ in the plastic limit (the tip angles are 102.6^∘ in a Newtonian fluid). In the other limit of a perfect shear-thickening fluid, any regions of high strain rate are heavily penalised and so the optimal body is much blunter with the two tip angles converging to 150^∘ .