Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-17T21:23:53.942Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aristoff, J. M., Clanet, C. & Bush, J. W. 2009 The elastochrone: the descent time of a sphere on a flexible beam. Proc. R. Soc. Lond. A 465, 2293–2311.10.1098/rspa.2009.0048Google Scholar
Balmforth, N. J., Bush, J. W. M., Vener, D. & Young, W. R. 2007 Dissipative descent: rocking and rolling down an incline. J. Fluid Mech. 590, 295–318.10.1017/S0022112007008051Google Scholar
Batchelor, G. K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.10.1017/CBO9780511800955Google Scholar
Bernoulli, J. 1696 Problema novum ad cujus solutionem mathematici invitantur. Acta Eruditorum 15, 264–269.Google Scholar
Binnie, A. M. & Harris, D. P. 1950 The application of boundary-layer theory to swirling liquid flow through a nozzle. Q. J. Mech. Appl. Maths 3 (1), 89–106.10.1093/qjmam/3.1.89Google Scholar
Bloor, M. I. G. & Ingham, D. B. 1977 On the use of a Pohlhausen method in three dimensional boundary layers. Z. Angew. Math. Phys. J. Appl. Math. Phys. 28 (2), 289–299.10.1007/BF01595596Google Scholar
Cherkasov, O. Yu., Zarodnyuk, A. V. & Bugrov, D. I. 2017 Range maximization and brachistochrone problem with Coulomb friction, viscous drag and accelerating force. AIP Conf. Proc. 1798, 020040.10.1063/1.4972632Google Scholar
Gemmer, J., Umble, R. & Nolan, M. 2011 Generalizations of the Brachistochrone problem. Pi Mu Epsilon J. 13 (4), 207–218.Google Scholar
Janardan, N., Panchagnula, M. V. & Bormashenko, E. 2015 Liquid marbles: physics and applications. Sadhana 40 (3), 653–671.10.1007/s12046-015-0365-7Google Scholar
Liberzon, D. 2011 Calculus of Variations and Optimal Control Theory: A Concise Introduction. Princeton University Press.10.2307/j.ctvcm4g0sGoogle Scholar
Lipp, S. C. 1997 Brachistochrone with Coulomb friction. SIAM J. Control Optim. 35 (2), 562–584.10.1137/S0363012995287957Google Scholar
Mertaniemi, H., Jokinen, V., Sainiemi, L., Franssila, S., Marmur, A., Ikkala, O. & Ras, R. H. 2011 Superhydrophobic tracks for low-friction, guided transport of water droplets. Adv. Mater. 23 (26), 2911–2914.10.1002/adma.201100461Google Scholar
Nellis, G. & Klein, S. A. 2009 Heat Transfer, p. 428. Cambridge University Press.Google Scholar
Seo, K. S., Kim, M. Y. & Kim, D. H. 2014 An optimal curve for fastest transportation of liquid drops on a superhydrophobic surface. In The Eighth International Conference on Material Technologies and Modeling, vol. 8. Ariel University.Google Scholar
Supekar, R. B. & Panchagnula, M. V.2014 Dynamics and stability of a fluid filled cylinder rolling on an inclined plane. arXiv:1408.6654.Google Scholar
Taylor, G. I. 1950a Seventh International Congress for Applied Mechanics, 1948. Nature 165, 258–260.10.1038/165258a0Google Scholar
Taylor, G. I. 1950b The boundary layer in the converging nozzle of a swirl atomizer. Q. J. Mech. Appl. Maths 3 (2), 129–139.10.1093/qjmam/3.2.129Google Scholar
Vratanar, B. & Saje, M. 1998 On the analytical solution of the brachistochrone problem in a non-conservative field. Intl J. Non-Linear Mech. 33 (3), 489–505.10.1016/S0020-7462(97)00026-7Google Scholar