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Three-dimensional capillary waves due to a submerged source with small surface tension

Published online by Cambridge University Press:  28 January 2019

Christopher J. Lustri Open the ORCID record for Christopher J. Lustri [Opens in a new window] ,
Ravindra Pethiyagoda Open the ORCID record for Ravindra Pethiyagoda [Opens in a new window]  and
S. Jonathan Chapman Open the ORCID record for S. Jonathan Chapman [Opens in a new window]
Christopher J. Lustri*
Affiliation:
Department of Mathematics and Statistics, Macquarie University, Sydney NSW 2109, Australia
Ravindra Pethiyagoda
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Brisbane QLD 4001, Australia
S. Jonathan Chapman
Affiliation:
Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, Oxford OX1 3LB, UK
*
†Email address for correspondence: christopher.lustri@mq.edu.au
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Abstract

Steady and unsteady linearised flow past a submerged source are studied in the small-surface-tension limit, in the absence of gravitational effects. The free-surface capillary waves generated are exponentially small in the surface tension, and are determined using the theory of exponential asymptotics. In the steady problem, capillary waves are found to extend upstream from the source, switching on across curves on the free surface known as Stokes lines. Asymptotic predictions are compared with computational solutions for the position of the free surface. In the unsteady problem, transient effects cause the solution to display more complicated asymptotic behaviour, such as higher-order Stokes lines. The theory of exponential asymptotics is applied to show how the capillary waves evolve over time, and eventually tend to the steady solution.

JFM classification

Waves/Free-surface Flows: Capillary waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 863 , 25 March 2019 , pp. 670 - 701
Copyright
© 2019 Cambridge University Press 

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References

Aoki, T., Koike, T. & Takei, Y. 2002 Vanishing of Stokes curves. In Microlocal Analysis and Complex Fourier Analysis (ed. Kawai, T. & Fujita, K.). World Scientific.Google Scholar
Batchelor, G. K. 1953 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bennett, T., Howls, C. J., Nemes, G. & Olde Daalhuis, A. B. 2018 Globally exact asymptotics for integrals with arbitrary order saddles. SIAM J. Math. Anal. 50 (2), 2144–2177.10.1137/17M1154217Google Scholar
Berk, H. L., Nevis, W. M. & Roberts, K. V. 1982 New Stokes lines in WKB theory. J. Math. Phys. 23 (6), 988–1002.10.1063/1.525467Google Scholar
Berry, M. V. 1991 Asymptotics, superasymptotics, hyperasymptotics. In Asymptotics Beyond All Orders (ed. Segur, H., Tanveer, S. & Levine, H.), pp. 1–14. Plenum.Google Scholar
Berry, M. V. & Howls, C. J. 1990 Hyperasymptotics. Proc. R. Soc. Lond. A 430 (1880), 653–668.10.1098/rspa.1990.0111Google Scholar
Blyth, M. G. & Vanden-Broeck, J.-M. 2004 New solutions for capillary waves on fluid sheets. J. Fluid Mech. 507, 255–264.10.1017/S0022112004009012Google Scholar
Body, G. L., King, J. R. & Tew, R. H. 2005 Exponential asymptotics of a fifth-order partial differential equation. Eur. J. Appl. Maths 16 (5), 647–681.10.1017/S0956792505006224Google Scholar
Boyd, J. P. 1991 Weakly non-local solitons for capillary-gravity waves: fifth-degree Korteweg–de Vries equation. Physica D 48 (1), 129–146.Google Scholar
Boyd, J. P. 1998 Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics: Generalized Solitons and Hyperasymptotic Perturbation Theory, Mathematics and its Applications, vol. 442. Kluwer.10.1007/978-1-4615-5825-5Google Scholar
Boyd, J. P. 1999 The devils invention: asymptotic, superasymptotic and hyperasymptotic series. Acta Appl. Math. 56 (1), 1–98.10.1023/A:1006145903624Google Scholar
Boyd, J. P. 2005 Hyperasymptotics and the linear boundary layer problem: why asymptotic series diverge. SIAM Rev. 47 (3), 553–575.10.1137/S003614450444436XGoogle Scholar
Chapman, S. J. 1996 On the non-universality of the error function in the smoothing of Stokes discontinuities. Proc. R. Soc. Lond. A 452 (1953), 2225–2230.Google Scholar
Chapman, S. J., King, J. R., Ockendon, J. R. & Adams, K. L. 1998 Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations. Proc. R. Soc. Lond. A 454 (1978), 2733–2755.10.1098/rspa.1998.0278Google Scholar
Chapman, S. J. & Mortimer, D. B. 2005 Exponential asymptotics and Stokes lines in a partial differential equation. Proc. R. Soc. Lond. A 461, 2385–2421.10.1098/rspa.2005.1475Google Scholar
Chapman, S. J. & Vanden-Broeck, J.-M. 2002 Exponential asymptotics and capillary waves. SIAM J. Appl. Maths 62 (6), 1872–1898.Google Scholar
Chapman, S. J. & Vanden-Broeck, J.-M. 2006 Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299–326.10.1017/S0022112006002394Google Scholar
Chepelianskii, A. D., Chevy, F. & Raphael, E. 2008 Capillary-gravity waves generated by a slow moving object. Phys. Rev. Lett. 100 (7), 074504.10.1103/PhysRevLett.100.074504Google Scholar
Crapper, G. D. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2 (6), 532–540.10.1017/S0022112057000348Google Scholar
Crowdy, D. 2001 Steady nonlinear capillary waves on curved sheets. Eur. J. Appl. Maths 12 (6), 689–708.10.1017/S0956792501004612Google Scholar
Crowdy, D. G. 1999 Exact solutions for steady capillary waves on a fluid annulus. J. Nonlinear Sci. 9 (6), 615–640.10.1007/s003329900080Google Scholar
Dias, F. & Kharif, C. 1999 Nonlinear gravity and capillary-gravity waves. Annu. Rev. Fluid Mech. 31 (1), 301–346.10.1146/annurev.fluid.31.1.301Google Scholar
Dingle, R. B. 1973 Asymptotic Expansions: Their Derivation and Interpretation. Academic Press.Google Scholar
DLMF2018 Nist Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.20 of 2018-09-15 (ed. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller & B. V. Saunders).Google Scholar
Fork, D. K., Anderson, G. B., Boyce, J. B., Johnson, R. I. & Mei, P. 1996 Capillary waves in pulsed excimer laser crystallized amorphous silicon. Appl. Phys. Lett. 68 (15), 2138–2140.10.1063/1.115610Google Scholar
Grimshaw, R. 2011 Exponential asymptotics and generalized solitary waves. In Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances (ed. Steinrück, H., Pfeiffer, F., Rammerstorfer, F. G., Salençon, J., Schrefler, B. & Serafini, P.), CISM Courses and Lectures, vol. 523, pp. 71–120. Springer.Google Scholar
Grimshaw, R. & Joshi, N. 1995 Weakly nonlocal solitary waves in a singularly perturbed Korteweg–de Vries equation. SIAM J. Appl. Maths 55 (1), 124–135.10.1137/S0036139993243825Google Scholar
Hogan, S. J. 1979 Some effects of surface tension on steep water waves. J. Fluid Mech. 91 (1), 167–180.10.1017/S0022112079000094Google Scholar
Hogan, S. J. 1984 Particle trajectories in nonlinear capillary waves. J. Fluid Mech. 143, 243–252.10.1017/S0022112084001336Google Scholar
Hogan, S. J. 1986 Highest waves, phase speeds and particle trajectories of nonlinear capillary waves on sheets of fluid. J. Fluid Mech. 172, 547–563.10.1017/S0022112086001866Google Scholar
Howls, C. J., Langman, P. J. & Olde Daalhuis, A. B. 2004 On the higher-order Stokes phenomenon. Proc. R. Soc. Lond. A 460 (2121), 2285–2303.10.1098/rspa.2004.1299Google Scholar
Kinnersley, W. 1976 Exact large amplitude capillary waves on sheets of fluid. J. Fluid Mech. 77 (2), 229–241.10.1017/S0022112076002085Google Scholar
Lustri, C. J. & Chapman, S. J. 2013 Steady gravity waves due to a submerged source. J. Fluid Mech. 732, 660–686.10.1017/jfm.2013.425Google Scholar
Lustri, C. J. & Chapman, S. J. 2014 Unsteady gravity waves due to a submerged source. Eur. J. Appl. Maths 25, 655–680.10.1017/S0956792514000217Google Scholar
Lustri, C. J., McCue, S. W. & Binder, B. J. 2012 Free surface flow past topography: a beyond-all-orders approach. Eur. J. Appl. Maths 23 (4), 441–467.10.1017/S0956792512000022Google Scholar
Lustri, C. J., McCue, S. W. & Chapman, S. J. 2013 Exponential asymptotics of free surface flow due to a line source. IMA J. Appl Maths 78 (4), 697–713.10.1093/imamat/hxt016Google Scholar
Ockendon, J. R., Howison, S., Lacey, A. & Movchan, A. 1999 Applied Partial Differential Equations. Oxford University Press.Google Scholar
Olde Daalhuis, A. B., Chapman, S. J., King, J. R., Ockendon, J. R. & Tew, R. H. 1995 Stokes phenomenon and matched asymptotic expansions. SIAM J. Appl. Maths 55 (6), 1469–1483.10.1137/S0036139994261769Google Scholar
Pomeau, Y., Ramani, A. & Grammaticos, B. 1988 Structural stability of the Korteweg–de Vries solitons under a singular perturbation. Physica D 31 (1), 127–134.Google Scholar
Regan, M. J., Pershan, P. S., Magnussen, O. M., Ocko, B. M., Deutsch, M. & Berman, L. E. 1996 Capillary-wave roughening of surface-induced layering in liquid gallium. Phys. Rev. B 54, 9730–9733.10.1103/PhysRevB.54.9730Google Scholar
Stokes, G. G. 1864 On the discontinuity of arbitrary constants which appear in divergent developments. Trans. Camb. Phil. Soc. 10, 106–128.Google Scholar
Stone, J. T., Self, R. H. & Howls, C. J. 2017 Aeroacoustic catastrophes: upstream cusp beaming in Lilley’s equation. Proc. R. Soc. Lond. A 473 (2201), 20160880.Google Scholar
Trinh, P. H. 2011 Exponential asymptotics and Stokes line smoothing for generalized solitary waves. In Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances (ed. Steinrück, H., Pfeiffer, F., Rammerstorfer, F. G., Salençon, J., Schrefler, B. & Serafini, P.), CISM Courses and Lectures, vol. 523, pp. 121–126. Springer.Google Scholar
Trinh, P. H. & Chapman, S. J. 2013a New gravity–capillary waves at low speeds. Part 1. Linear geometries. J. Fluid Mech. 724, 367–391.10.1017/jfm.2013.110Google Scholar
Trinh, P. H. & Chapman, S. J. 2013b New gravity–capillary waves at low speeds. Part 2. Nonlinear geometries. J. Fluid Mech. 724, 392–424.10.1017/jfm.2013.129Google Scholar
Trinh, P. H. & Chapman, S. J. 2014 The wake of a two-dimensional ship in the low-speed limit: results for multi-cornered hulls. J. Fluid Mech. 741, 492–513.10.1017/jfm.2013.589Google Scholar
Trinh, P. H., Chapman, S. J. & Vanden-Broeck, J.-M. 2011 Do waveless ships exist? Results for single-cornered hulls. J. Fluid Mech. 685, 413–439.10.1017/jfm.2011.325Google Scholar
Tucker, V. A. 1969 Wave-making by whirligig beetles (gyrinidae). Science 166 (3907), 897–899.10.1126/science.166.3907.897Google Scholar
Vanden-Broeck, J.-M. 1996 Capillary waves with variable surface tension. Z. Angew. Math. Phys. 47 (5), 799–808.10.1007/BF00915276Google Scholar
Vanden-Broeck, J.-M. 2004 Nonlinear capillary free-surface flows. J. Engng Maths 50 (4), 415–426.10.1007/s10665-004-1769-2Google Scholar
Vanden-Broeck, J.-M. 2010 Gravity-Capillary Free-Surface Flows. Cambridge University Press.10.1017/CBO9780511730276Google Scholar
Vanden-Broeck, J.-M. & Keller, J. B. 1980 A new family of capillary waves. J. Fluid Mech. 98 (1), 161–169.10.1017/S0022112080000080Google Scholar
Vanden-Broeck, J.-M., Miloh, T. & Spivack, B. 1998 Axisymmetric capillary waves. Wave Motion 27 (3), 245–256.10.1016/S0165-2125(97)80078-9Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley.Google Scholar
Yang, T.-S. & Akylas, T. R. 1996 Weakly nonlocal gravity–capillary solitary waves. Phys. Fluids 8 (6), 1506–1514.10.1063/1.868926Google Scholar