Optimization

We determine the globally minimum time 𝑇 needed to translate a thin submerged flat plate a given distance parallel to its surface within a work budget. The Reynolds number for the flow is assumed to be large so that the drag on the plate arises from skin friction in a thin viscous boundary layer. The minimum is determined computationally using a steepest descent, where an adjoint formulation is used to compute the gradients.

We consider the time-dependent speed of an infinitely long cylinder that minimizes the net work done on the surrounding fluid to travel a given distance perpendicular to its axis in a fixed amount of time. The flow that develops is two-dimensional. An analytical solution is possible using calculus of variations for the case that the distance travelled and the viscous boundary layer thickness that develops are much smaller than the circle radius.

How should a given amount of material be moulded into a cantilevered beam clamped at one end, so that it will have the furthest horizontal reach? Here, we formulate and solve this variational problem for the optimal variation of the cross-section area of a heavy cantilevered beam with a given volume V, Young’s modulus E, and density ρ, subject to gravity g. We find that the cross-sectional area should vary according a universal profile that is independent of material parameters, with both the length and maximum reach-out distance of the branch that scale as $(EV/ρg)^1/4$, with a universal self-similar shape at the tip with the area of cross-section $a∼s^3$, s being the distance from the tip, consistent with earlier observations of tree branches, but with a different local interpretation than given before.