Hostname: page-component-7c8c6479df-94d59 Total loading time: 0 Render date: 2024-03-28T20:42:59.527Z Has data issue: false hasContentIssue false

Oscillatory Marangoni flows with inertia

Published online by Cambridge University Press:  19 August 2016

Orest Shardt*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Hassan Masoud
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Department of Mechanical Engineering, University of Nevada, Reno, NV 89557, USA
Howard A. Stone
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
Email address for correspondence: oshardt@princeton.edu

Abstract

When the surface of a liquid has a non-uniform distribution of a surfactant that lowers surface tension, the resulting variation in surface tension drives a flow that spreads the surfactant towards a uniform distribution. We study the spreading dynamics of an insoluble and non-diffusing surfactant on an initially motionless liquid. We derive solutions for the evolution over time of sinusoidal variations in surfactant concentration with a small initial amplitude relative to the average concentration. In this limit, the coupled flow and surfactant transport equations are linear. In contrast to exponential decay when the inertia of the flow is negligible, the solution for unsteady Stokes flow exhibits oscillations when inertia is sufficient to spread the surfactant beyond a uniform distribution. This oscillatory behaviour exhibits two properties that distinguish it from that of a simple harmonic oscillator: the amplitude changes sign at most three times, and the decay at late times follows a power law with an exponent of $-3/2$. As the surface oscillates, the structure of the subsurface flow alternates between one and two rows of counter-rotating vortices, starting with one row and ending with two during the late-time monotonic decay. We also examine numerically the evolution of the surfactant distribution when the system is nonlinear due to a large initial amplitude.

Type
Papers
Copyright
© 2016 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

Bush, J. W. M. & Hu, D. L. 2006 Walking on water: Biolocomotion at the interface. Annu. Rev. Fluid Mech. 38, 339369.Google Scholar
DLMF 2015 NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/, Release 1.0.10 of 2015-08-07, online companion to Olver et al. (2010).Google Scholar
Dussaud, A. D. & Troian, S. M. 1998 Dynamics of spontaneous spreading with evaporation on a deep fluid layer. Phys. Fluids 10, 2338.Google Scholar
Edmonstone, B. D., Craster, R. V. & Matar, O. K. 2006 Surfactant-induced fingering phenomena beyond the critical micelle concentration. J. Fluid Mech. 564, 105138.Google Scholar
Erdélyi, A., Magnus, W., Oberhettinger, F. & Tricomi, F. G. 1954 Tables of Integral Transforms. McGraw-Hill.Google Scholar
Fernandez, J. M. & Homsy, G. M. 2004 Chemical reaction-driven tip-streaming phenomena in a pendant drop. Phys. Fluids 16, 25482555.CrossRefGoogle Scholar
Grotberg, J. B. 1994 Pulmonary flow and transport phenomena. Annu. Rev. Fluid Mech. 26, 529571.Google Scholar
Jensen, O. E. 1994 Self-similar, surfactant-driven flows. Phys. Fluids 6, 10841094.Google Scholar
Jensen, O. E. 1995 The spreading of insoluble surfactant at the free surface of a deep fluid layer. J. Fluid Mech. 293, 349378.Google Scholar
Jensen, O. E. & Grotberg, J. B. 1992 Insoluble surfactant spreading on a thin viscous film: shock evolution and film rupture. J. Fluid Mech. 240, 259288.Google Scholar
Jensen, O. E. & Naire, S. 2006 The spreading and stability of a surfactant-laden drop on a prewetted substrate. J. Fluid Mech. 554, 524.CrossRefGoogle Scholar
Kohira, M. I., Hayashima, Y., Nagayama, M. & Nakata, S. 2001 Synchronized self-motion of two camphor boats. Langmuir 17, 71247129.Google Scholar
Kovalchuk, V. I., Kamusewitz, H., Vollhardt, D. & Kovalchuk, N. M. 1999 Auto-oscillation of surface tension. Phys. Rev. E 60, 20292036.Google Scholar
Lauga, E. & Davis, A. M. J. 2012 Viscous Marangoni propulsion. J. Fluid Mech. 705, 120133.CrossRefGoogle Scholar
Lavrenteva, O. M. & Nir, A. 2001 Spontaneous thermocapillary interaction of drops: unsteady convective effects at high Peclet numbers. Phys. Fluids 13, 368381.Google Scholar
Lucassen, J. 1968a Longitudinal capillary waves. Part 1: theory. Trans. Faraday Soc. 64, 22212229.CrossRefGoogle Scholar
Lucassen, J. 1968b Longitudinal capillary waves. Part 2: experiments. Trans. Faraday Soc. 64, 22302235.Google Scholar
Masoud, H. & Shelley, M. J. 2014 Collective surfing of chemically active particles. Phys. Rev. Lett. 112, 128304.Google Scholar
Masoud, H. & Stone, H. A. 2014 A reciprocal theorem for Marangoni propulsion. J. Fluid Mech. 741, R4.CrossRefGoogle Scholar
Matar, O. K. & Troian, S. M. 1999 Spreading of a surfactant monolayer on a thin liquid film: onset and evolution of digitated structures. Chaos 9, 141153.Google Scholar
Nakata, S., Doi, Y. & Kitahata, H. 2005 Synchronized sailing of two camphor boats in polygonal chambers. J. Phys. Chem. B 109, 17981802.CrossRefGoogle ScholarPubMed
Nakata, S. & Hayashima, Y. 1998 Spontaneous dancing of a camphor scraping. J. Chem. Soc. Faraday Trans. 94, 36553658.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.Google Scholar
Olver, F. W. J., Lozier, D. W., Boisvert, R. F. & Clark, C. W.(Eds) 2010 NIST Handbook of Mathematical Functions, Cambridge University Press; print companion to (DLMF 2015).Google Scholar
Pimienta, V. & Antoine, C. 2014 Self-propulsion on liquid surfaces. Curr. Opin. Colloid Interface Sci. 19, 290299.CrossRefGoogle Scholar
Rayleigh, Lord 1890 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
Scriven, L. E. & Sternling, C. V. 1960 The Marangoni effects. Nature 187, 186188.Google Scholar
Sternling, C. V. & Scriven, L. E. 1959 Interfacial turbulence: hydrodynamic instability and the Marangoni effect. AIChE J. 5, 514523.CrossRefGoogle Scholar
Stocker, R. & Bush, J. W. M. 2007 Spontaneous oscillations of a sessile lens. J. Fluid Mech. 583, 465475.Google Scholar
Thess, A. 1996 Stokes flow at infinite Marangoni number: exact solutions for the spreading and collapse of a surfactant. Phys. Scr. T 67, 96100.Google Scholar
Thom, A. 1933 The flow past circular cylinders at low speeds. Proc. R. Soc. Lond. A 141, 651669.Google Scholar
Troian, S. M., Herbolzheimer, E. & Safran, S. A. 1990 Model for the fingering instability of spreading surfactant drops. Phys. Rev. Lett. 65, 333336.Google Scholar
Velarde, M. G. & Chu, X.-L. 1989 Dissipative hydrodynamic oscillators. I: Marangoni effect and sustained longitudinal waves at the interface of two liquids. Nuovo Cimento D 11, 709716.CrossRefGoogle Scholar