#### Brachistochronous motion of a flat plate parallel to its surface immersed in a fluid J. Fluid Mech., 939 A27 (2022) PDF

##### by Mandre

Fluid mechanics Optimization Boundary Layers**Abstract:**

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. Because the equations governing
fluid mechanics for this problem are nonlinear, multiple local minima could exist. Exploiting
the quadratic nature of the objective function and the constraining differential equations, we
derive and apply a “spectral condition” to show the converged local optimum to be a global
one. The condition states that the optimum is global if the Hessian of the Lagrangian in the
state variables constructed using the converged adjoint field is positive semi-definite at every
instance. The globally optimum kinematics of the plate starts from rest with speed ∝ 𝑡 ^{1/4} and
comes to rest with speed ∝ (𝑇 − 𝑡) ^{1/4} as a function of time 𝑡. For distances much longer than
the plate, the work-minimizing kinematics consists of an optimum startup, a constant-speed
cruising, and an optimum stopping.