Drag-minimising bodies in confined Stokes flows

Slow viscous flow around a fixed body generates a shape-dependent drag. We explore the drag-minimising shapes of bodies centred between two parallel plates in two-dimensional viscous flow. The channel width introduces a length scale so that the optimal profile is area-dependent. We solve the shape optimisation problem numerically over a wide range of areas. We also compute the optimal elliptical shapes and this identifies how these shapes should be slightly altered to reduce the drag with reductions of up to $3.8\,\%$ attained at high areas. More broadly, we derive two properties of general optimal shapes within the confined flow: the magnitude of the surface vorticity is approximately (but not exactly) constant and the noses have sharp angles that are independent of area. For relatively small bodies, the optimal shape becomes identical to that in an unconfined geometry, but the drag is qualitatively different owing to the influence of confinement; within a channel, it is proportional to the inverse of the logarithm of the body area. At relatively large areas, the optimal body becomes long and its surface is approximately parallel to the channel boundaries, except in the vicinity of the noses. Using a lubrication approximation, we recast the optimisation problem as an Euler–Lagrange equation that is solved to determine the drag-minimising shape, finding that the drag is proportional to the body area in this regime.