# Optimization

### Brachistochronous motion of a flat plate parallel to its surface immersed in a fluid

Fluid mechanics Optimization Boundary LayersWe 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.

### Work-minimizing kinematics for small displacement of an infinitely long cylinder

Fluid mechanics Optimization Boundary LayersWe 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.

### The branch with the furthest reach

Biomechanics OptimizationHow 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.