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Viscoelastic effects in circular edge waves

Published online by Cambridge University Press:  25 May 2021

X. Shao ,
P. Wilson ,
J.B. Bostwick Open the ORCID record for J.B. Bostwick [Opens in a new window]  and
J.R. Saylor
X. Shao
Affiliation:
Department of Mechanical Engineering, Clemson University, Clemson, SC29634, USA
P. Wilson
Affiliation:
Department of Mechanical Engineering, Clemson University, Clemson, SC29634, USA
J.B. Bostwick*
Affiliation:
Department of Mechanical Engineering, Clemson University, Clemson, SC29634, USA
J.R. Saylor
Affiliation:
Department of Mechanical Engineering, Clemson University, Clemson, SC29634, USA
*
†Email address for correspondence: [email protected]
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Abstract

Surface waves are excited at the boundary of a mechanically vibrated cylindrical container and are referred to as edge waves. Resonant waves are considered, which are formed by a travelling wave formed at the edge and constructively interfering with its centre reflection. These waves exhibit an axisymmetric spatial structure defined by the mode number $n$. Viscoelastic effects are investigated using two materials with tunable properties; (i) glycerol/water mixtures (viscosity) and (ii) agarose gels (elasticity). Long-exposure white-light imaging is used to quantify the magnitude of the wave slope from which frequency-response diagrams are obtained via frequency sweeps. Resonance peaks and bandwidths are identified. These results show that for a given $n$, the resonance frequency decreases with viscosity and increases with elasticity. The amplitude of the resonance peaks are much lower for gels and decrease further with mode number, indicating that much larger driving amplitudes are needed to overcome the elasticity and excite edge waves. The natural frequencies for a viscoelastic fluid in a cylindrical container with a pinned contact-line are computed from a theoretical model that depends upon the dimensionless Ohnesorge number ${Oh}$, elastocapillary number ${Ec}$ and Bond number ${Bo}$. All show good agreement with experimental observations. The eigenvalue problem is equivalent to the classic damped-driven oscillator model on linear operators with viscosity appearing as a damping force and elasticity and surface tension as restorative forces, consistent with our physical interpretation of these viscoelastic effects.

JFM classification

Waves/Free-surface Flows: Capillary waves Non-Newtonian Flows: Viscoelasticity Waves/Free-surface Flows: Elastic waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 919 , 25 July 2021 , A18
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Benjamin, T.B. 1967 Instability of periodic wavetrains in nonlinear dispersive systems. Proc. R. Soc. Lond. A 299 (1456), 59–76.Google Scholar
Benjamin, T.B. & Feir, J.E. 1967 The disintegration of wave trains on deep water. J. Fluid Mech. 27 (3), 417–430.10.1017/S002211206700045XCrossRefGoogle Scholar
Benjamin, T.B. & Scott, J.C. 1979 Gravity-capillary waves with edge constraints. J. Fluid Mech. 92 (2), 241–267.10.1017/S0022112079000616CrossRefGoogle Scholar
Benjamin, T.B. & Ursell, F.J. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225 (1163), 505–515.Google Scholar
Bico, J., Reyssat, É. & Roman, B. 2018 Elastocapillarity: when surface tension deforms elastic solids. Annu. Rev. Fluid Mech. 50, 629–659.10.1146/annurev-fluid-122316-050130CrossRefGoogle Scholar
Bostwick, J.B. & Steen, P.H. 2009 Capillary oscillations of a constrained liquid drop. Phys. Fluids 21 (3), 032108.10.1063/1.3103344CrossRefGoogle Scholar
Bostwick, J.B. & Steen, P.H. 2013 a Coupled oscillations of deformable spherical-cap droplets. Part 1. Inviscid motions. J. Fluid Mech. 714, 312–335.10.1017/jfm.2012.483CrossRefGoogle Scholar
Bostwick, J.B. & Steen, P.H. 2013 b Coupled oscillations of deformable spherical-cap droplets. Part 2. Viscous motions. J. Fluid Mech. 714, 336–360.10.1017/jfm.2012.480CrossRefGoogle Scholar
Bostwick, J.B. & Steen, P.H. 2015 Stability of constrained capillary surfaces. Annu. Rev. Fluid Mech. 47, 539–568.10.1146/annurev-fluid-010814-013626CrossRefGoogle Scholar
Bostwick, J.B. & Steen, P.H. 2016 Response of driven sessile drops with contact-line dissipation. Soft Matt. 12 (43), 8919–8926.10.1039/C6SM01928ECrossRefGoogle ScholarPubMed
Case, K.M. & Parkinson, W.C. 1957 Damping of surface waves in an incompressible liquid. J. Fluid Mech. 2 (2), 172–184.10.1017/S0022112057000051CrossRefGoogle Scholar
Chen, P., Güven, S., Usta, O.B., Yarmush, M.L. & Demirci, U. 2015 Biotunable acoustic node assembly of organoids. Adv. Healthc. Mater. 4 (13), 1937–1943.10.1002/adhm.201500279CrossRefGoogle ScholarPubMed
Christiansen, B., Alstrøm, P. & Levinsen, M.T. 1992 Ordered capillary-wave states: quasicrystals, hexagons, and radial waves. Phys. Rev. Lett. 68 (14), 2157–2160.10.1103/PhysRevLett.68.2157CrossRefGoogle Scholar
Davis, S.H. 1980 Moving contact lines and rivulet instabilities. Part 1. The static rivulet. J. Fluid Mech. 98 (2), 225–242.10.1017/S0022112080000110CrossRefGoogle Scholar
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383–409.10.1017/S0022112090003603CrossRefGoogle Scholar
Douady, S. & Fauve, S. 1988 Pattern selection in Faraday instability. Europhys. Lett. 6 (3), 221.10.1209/0295-5075/6/3/006CrossRefGoogle Scholar
Edwards, W.S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123–148.10.1017/S0022112094003642CrossRefGoogle Scholar
Fan, R., Piou, M., Darling, E., Cormier, D., Sun, J. & Wan, J. 2016 Bio-printing cell-laden matrigel–agarose constructs. J. Biomater. Appl. 31 (5), 684–692.10.1177/0885328216669238CrossRefGoogle ScholarPubMed
Faraday, M. 1831 XVII. On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 299–340.Google Scholar
Graham-Eagle, J. 1983 A new method for calculating eigenvalues with applications to gravity-capillary waves with edge constraints. In Mathematical Proceedings of the Cambridge Philosophical Society, vol. 94, pp. 553–564. Cambridge University Press.10.1017/S0305004100000943CrossRefGoogle Scholar
Grzelka, M., Bostwick, J.B. & Daniels, K.E. 2017 Capillary fracture of ultrasoft gels: variability and delayed nucleation. Soft Matt. 13 (16), 2962–2966.10.1039/C7SM00257BCrossRefGoogle ScholarPubMed
Guven, S., Chen, P., Inci, F., Tasoglu, S., Erkmen, B. & Demirci, U. 2015 Multiscale assembly for tissue engineering and regenerative medicine. Trends Biotechnol. 33 (5), 269–279.10.1016/j.tibtech.2015.02.003CrossRefGoogle ScholarPubMed
Harden, J.L., Pleiner, H. & Pincus, P.A. 1991 Hydrodynamic surface modes on concentrated polymer solutions and gels. J. Chem. Phys. 94 (7), 5208–5221.10.1063/1.460525CrossRefGoogle Scholar
Henderson, D.M. & Miles, J.W. 1994 Surface-wave damping in a circular cylinder with a fixed contact line. J. Fluid Mech. 275, 285–299.10.1017/S0022112094002363CrossRefGoogle Scholar
Hocking, L.M. 1987 The damping of capillary-gravity waves at a rigid boundary. J. Fluid Mech. 179, 253–266.10.1017/S0022112087001514CrossRefGoogle Scholar
Huang, Y., Wolfe, C.L.P., Zhang, J. & Zhong, J.-Q. 2020 Streaming controlled by meniscus shape. J. Fluid Mech. 895, A1.10.1017/jfm.2020.281CrossRefGoogle Scholar
Joseph, D., Funada, T. & Wang, J. 2007 Potential Flows of Viscous and Viscoelastic Liquids. Cambridge University Press.10.1017/CBO9780511550928CrossRefGoogle Scholar
Kidambi, R. 2009 a Capillary damping of inviscid surface waves in a circular cylinder. J. Fluid Mech. 627, 323–340.10.1017/S0022112009005898CrossRefGoogle Scholar
Kidambi, R. 2009 b Meniscus effects on the frequency and damping of capillary-gravity waves in a brimful circular cylinder. Wave Motion 46 (2), 144–154.10.1016/j.wavemoti.2008.10.001CrossRefGoogle Scholar
Kidambi, R. 2013 Inviscid faraday waves in a brimful circular cylinder. J. Fluid Mech. 724, 671–694.10.1017/jfm.2013.178CrossRefGoogle Scholar
Kumar, K. & Bajaj, K.M.S. 1995 Competing patterns in the Faraday experiment. Phys. Rev. E 52 (5), R4606.10.1103/PhysRevE.52.R4606CrossRefGoogle ScholarPubMed
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Lyubimov, D.V., Lyubimova, T.P. & Shklyaev, S.V. 2006 Behavior of a drop on an oscillating solid plate. Phys. Fluids 18, 012101.CrossRefGoogle Scholar
Martel, C., Nicolas, J.A. & Vega, J.M. 1998 Surface-wave damping in a brimful circular cylinder. J. Fluid Mech. 360, 213–228.CrossRefGoogle Scholar
McKinley, G.H. 2005 Dimensionless groups for understanding free surface flows of complex fluids. Rheol. Bull. 74 (2).Google Scholar
Mei, C.C. & Liu, L.F. 1973 The damping of surface gravity waves in a bounded liquid. J. Fluid Mech. 59 (2), 239–256.CrossRefGoogle Scholar
Mezger, T.G. 2006 The Rheology Handbook: For Users of Rotational and Oscillatory Rheometers. Vincentz Network GmbH & Co KG.Google Scholar
Michel, G., Pétrélis, F. & Fauve, S. 2016 Acoustic measurement of surface wave damping by a meniscus. Phys. Rev. Lett. 116 (17), 174301.CrossRefGoogle ScholarPubMed
Murphy, S.V. & Atala, A. 2014 3d bioprinting of tissues and organs. Nat. Biotechnol. 32 (8), 773–785.CrossRefGoogle ScholarPubMed
Nicolás, J.A. 2005 Effects of static contact angles on inviscid gravity-capillary waves. Phys. Fluids 17 (2), 022101.CrossRefGoogle Scholar
Padrino, J.C., Funada, T. & Joseph, D.D. 2007 Purely irrotational theories for the viscous effects on the oscillations of drops and bubbles. Intl J. Multiphase Flow 34, 61–75.CrossRefGoogle Scholar
Perlin, M. & Schultz, W.W. 2000 Capillary effects on surface waves. Annu. Rev. Fluid Mech. 32 (1), 241–274.CrossRefGoogle Scholar
Picard, C. & Davoust, L. 2006 Dilational rheology of an air–water interface functionalized by biomolecules: the role of surface diffusion. Rheol. Acta 45 (4), 497–504.CrossRefGoogle Scholar
Picard, C. & Davoust, L. 2007 Resonance frequencies of meniscus waves as a physical mechanism for a dna biosensor. Langmuir 23 (3), 1394–1402.CrossRefGoogle ScholarPubMed
Prosperetti, A. 2012 Linear oscillations of constrained drops, bubbles, and plane liquid surfaces. Phys. Fluids 24 (3), 032109.CrossRefGoogle Scholar
Saylor, J.R. & Kinard, A.L. 2005 Simulation of particle deposition beneath faraday waves in thin liquid films. Phys. Fluids 17 (4), 047106.CrossRefGoogle Scholar
Saylor, J.R., Szeri, A.J. & Foulks, G.P. 2000 Measurement of surfactant properties using a circular capillary wave field. Exp. Fluids 29 (6), 509–518.CrossRefGoogle Scholar
Shao, X., Fredericks, S.A., Saylor, J.R. & Bostwick, J.B. 2019 Elastocapillary transition in gel drop oscillations. Phys. Rev. Lett. 123 (18), 188002.10.1103/PhysRevLett.123.188002CrossRefGoogle ScholarPubMed
Shao, X., Fredericks, S.A., Saylor, J.R. & Bostwick, J.B. 2020 A method for determining surface tension, viscosity, and elasticity of gels via ultrasonic levitation of gel drops. J. Acoust. Soc. Am. 147 (4), 2488–2498.CrossRefGoogle ScholarPubMed
Shmyrov, A., Mizev, A., Shmyrova, A. & Mizeva, I. 2019 Capillary wave method: an alternative approach to wave excitation and to wave profile reconstruction. Phys. Fluids 31 (1), 012101.CrossRefGoogle Scholar
Strickland, S.L., Shearer, M. & Daniels, K.E. 2015 Spatiotemporal measurement of surfactant distribution on gravity–capillary waves. J. Fluid Mech. 777, 523–543.CrossRefGoogle Scholar
Style, R.W., Jagota, A., Hui, C.-Y. & Dufresne, E.R. 2017 Elastocapillarity: surface tension and the mechanics of soft solids. Annu. Rev. Conden. Ma. P. 8, 99–118.CrossRefGoogle Scholar
Takamura, K., Fischer, H. & Morrow, N.R. 2012 Physical properties of aqueous glycerol solutions. J. Petrol. Sci. Engng 98, 50–60.CrossRefGoogle Scholar
Tokita, M. & Hikichi, K. 1987 Mechanical studies of sol-gel transition: universal behavior of elastic modulus. Phys. Rev. A 35 (10), 4329.CrossRefGoogle ScholarPubMed
Westra, M.-T., Binks, D.J. & Van De Water, W. 2003 Patterns of Faraday waves. J. Fluid Mech. 496, 1–32.CrossRefGoogle Scholar
Wright, P.H. & Saylor, J.R. 2003 Patterning of particulate films using Faraday waves. Rev. Sci. Instrum. 74 (9), 4063–4070.CrossRefGoogle Scholar
Zakharov, V.E. & Ostrovsky, L.A. 2009 Modulation instability: the beginning. Physica D 238 (5), 540–548.CrossRefGoogle Scholar