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On the brachistochrone of a fluid-filled cylinder

Published online by Cambridge University Press:  26 February 2019

Srikanth Sarma Gurram Open the ORCID record for Srikanth Sarma Gurram [Opens in a new window] ,
Sharan Raja ,
Pallab Sinha Mahapatra  and
Mahesh V. Panchagnula Open the ORCID record for Mahesh V. Panchagnula [Opens in a new window]
Srikanth Sarma Gurram
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036, India
Sharan Raja
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036, India
Pallab Sinha Mahapatra
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036, India
Mahesh V. Panchagnula*
Affiliation:
Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai 600036, India
*
†Email address for correspondence: [email protected]
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Abstract

We discuss a fluid dynamic variant of the classical Bernoulli’s brachistochrone problem. The classical brachistochrone for a non-dissipative particle is governed by maximization of the particle’s kinetic energy, resulting in a cycloid. We consider a variant where the particle is replaced by a cylinder (bottle) filled with a viscous fluid and attempt to identify the shape of the curve connecting two points along which the bottle would move in the shortest time. We derive the system of integro-differential equations governing system dynamics for a given shape of the curve. Using these equations, we pose the brachistochrone problem by invoking an optimal control formalism and show that (in general) the curve deviates from a cycloid. This is due to the fact that increasing the rate of change of the bottle’s kinetic energy is accompanied by increased viscous dissipation. We show that the bottle motion is governed by a balance between the desire to minimize travel time and the need to reach the end point in the face of increased dissipation. The trade-off between these two physical forces plays a vital role in determining the brachistochrone of a fluid-filled cylinder. We show that in the two limits of either vanishing or high viscosity, the brachistochrone for this problem reduces to a cycloid. An intermediate viscosity range is identified where the fluid brachistochrone is non-cycloidal. Finally, we show the relevance of these results to the dynamics of a rolling liquid marble.

JFM classification

Mathematical Foundations: General fluid mechanics Mathematical Foundations: Variational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 865 , 25 April 2019 , pp. 775 - 789
Copyright
© 2019 Cambridge University Press 

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