Mandre, Shreyas (2022) Brachistochronous motion of a flat plate parallel to its surface immersed in a fluid. Journal of Fluid Mechanics, 939 . A27. doi:10.1017/jfm.2022.217 ISSN 0022-1120.

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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 ∝t1/4 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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics T Technology > TJ Mechanical engineering and machinery
Divisions:	Faculty of Science, Engineering and Medicine > Engineering > Engineering Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Fluid mechanics, Kinematics, Aerodynamics, Mathematical optimization
Journal or Publication Title:	Journal of Fluid Mechanics
Publisher:	Cambridge University Press
ISSN:	0022-1120
Official Date:	25 May 2022
Dates:	Date Event 25 May 2022 Published 30 March 2022 Available 3 March 2022 Accepted
Volume:	939
Article Number:	A27
DOI:	10.1017/jfm.2022.217
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access (Creative Commons)
Date of first compliant deposit:	3 May 2022
Date of first compliant Open Access:	3 May 2022

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