Hostname: page-component-88dd8db54-7p5v5 Total loading time: 0 Render date: 2024-03-05T08:49:15.902Z Has data issue: false hasContentIssue false

A reciprocal theorem for Marangoni propulsion

Published online by Cambridge University Press:  11 February 2014

Hassan Masoud*
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Applied Mathematics Laboratory, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
Howard A. Stone
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Email address for correspondence:


We study the Marangoni propulsion of a spheroidal particle located at a liquid–gas interface. The particle asymmetrically releases an insoluble surface-active agent and so creates and maintains a surface tension gradient leading to the self-propulsion. Assuming that the surface tension has a linear dependence on the concentration of the released agent, we derive closed-form expressions for the translational speed of the particle in the limit of small capillary, Péclet and Reynolds numbers. Our derivations are based on the Lorentz reciprocal theorem, which eliminates the need to develop the detailed flow field.

© 2014 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.)


Acree, W. E. 1984 Empirical expression for predicting surface tension of liquid mixtures. J. Colloid Interface Sci. 101, 575576.CrossRefGoogle Scholar
Adamson, A. W. 1990 Physical Chemistry of Surfaces. Wiley.Google Scholar
Bush, J. W. M. & Hu, D. L. 2006 Walking on water: biolocomotion at the interface. Annu. Rev. Fluid Mech. 38, 339369.Google Scholar
Crowdy, D. G. 2013 Wall effects on self-diffusiophoretic Janus particles: a theoretical study. J. Fluid Mech. 735, 473498.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics, with Special Applications to Particulate Media. Prentice-Hall.Google Scholar
Lauga, E. & Davis, A. M. J. 2012 Viscous Marangoni propulsion. J. Fluid Mech. 705, 120133.Google Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics. McGraw-Hill.Google Scholar
Nadim, A., Haj-Hariri, H. & Borhan, A. 1990 Thermocapillary migration of slightly deformed droplets. Part. Sci. Technol. 8, 191198.Google Scholar
Nakata, S., Iguchi, Y., Ose, S., Kuboyama, M., Ishii, T. & Yoshikawa, K. 1997 Self-rotation of a camphor scraping on water: new insight into the old problem. Langmuir 13, 44544458.CrossRefGoogle Scholar
Rayleigh, L. 1889 Measurements of the amount of oil necessary in order to check the motions of camphor upon water. Proc. R. Soc. Lond. 47, 364367.Google Scholar
Stone, H. A. & Samuel, A. D. T. 1996 Propulsion of microorganisms by surface distortions. Phys. Rev. Lett. 77, 41024104.Google Scholar
Zhang, H., Duan, W., Liu, L. & Sen, A. 2013 Depolymerization-powered autonomous motors using biocompatible fuel. J. Am. Chem. Soc. 135, 1573415737.Google Scholar