Work-minimizing kinematics for small displacement of an infinitely long cylinder J. Fluid Mech. 893 R4 (2020) PDF

by Mandre

Jun 25, 2020 Fluid mechanics Optimization Boundary Layers

Abstract:

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. If 𝑡 represents the time since the commencement of motion and 𝑇 the final time, then the optimum speed profile is 𝐶𝑡^1/4(𝑇−𝑡)^1/4, where 𝐶 is determined by the distance travelled. The result also holds for rigid-body translations and rotation of cylinders formed by extrusion of smooth but otherwise arbitrary curves.