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An Isoperimetric Sloshing Problem in a Shallow Container with Surface Tension

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Abstract

In 1965, B. A. Troesch solved the isoperimetric sloshing problem of determining the container shape that maximizes the fundamental sloshing frequency among two classes of shallow containers: symmetric canals with a given free surface width and cross-sectional area, and radially symmetric containers with a given rim radius and volume (Commun Pure Appl Math 18(1–2):319–338, 1965, https://doi.org/10.1002/cpa.3160180124). Here, we extend these results in two ways: (i) we consider surface tension effects on the fluid free surface, assuming a flat equilibrium free surface together with a pinned contact line, and (ii) we consider sinusoidal waves traveling along the canal with wavenumber \(\alpha \ge 0\) and spatial period \(2\pi /\alpha \); two-dimensional sloshing corresponds to the case \(\alpha = 0\). Generalizing our recent variational characterization of fluid sloshing with surface tension to the case of a pinned contact line, we derive the pinned-edge linear shallow sloshing problem, which is an eigenvalue problem for a generalized Sturm-Liouville system. In the case without surface tension, we show that the optimal shallow canal is a rectangular canal for any \(\alpha > 0\). In the presence of surface tension, we solve for the maximizing cross-section explicitly for shallow canals with any given \(\alpha \ge 0\) and shallow radially symmetric containers with m azimuthal nodal lines, \(m = 0, 1\). Our results reveal that the squared maximal sloshing frequency increases considerably as surface tension increases. Interestingly, both the optimal shallow canal for \(\alpha = 0\) and the optimal shallow radially symmetric container are not convex. As a consequence of our explicit solutions, we establish convergence of the maximizing cross-sections, as surface tension vanishes, to the maximizing cross-sections without surface tension.

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References

  1. Anderson, G., Vamanamurthy, M., Vuorinen, M.: Monotonicity rules in calculus. Am. Math. Mon. 113(9), 805–816 (2006). https://doi.org/10.1080/00029890.2006.11920367

    Article  MathSciNet  MATH  Google Scholar 

  2. Bauer, H.F.: Liquid oscillations in a circular cylindrical container with “sliding’’ contact line. Forschung im Ingenieurwesen 58(10), 240–251 (1992). https://doi.org/10.1007/BF02574547

    Article  Google Scholar 

  3. Benjamin, T.B., Graham-Eagle, J.: Long gravity-capillary waves with edge constraints. IMA J. Appl. Math. 35(1), 91–114 (1985). https://doi.org/10.1093/imamat/35.1.91

    Article  MathSciNet  MATH  Google Scholar 

  4. Benjamin, T.B., Scott, J.C.: Gravity-capillary waves with edge constraints. J. Fluid Mech. 92(2), 241–267 (1979). https://doi.org/10.1017/S0022112079000616

    Article  MathSciNet  MATH  Google Scholar 

  5. Bernardi, C., Dauge, M., Maday, Y.: Spectral Methods for Axisymmetric Domains. Gauthier-Villars, Paris (1999)

    MATH  Google Scholar 

  6. Budiansky, B.: Sloshing of liquids in circular canals and spherical tanks. J. Aerosp. Sci. 27(3), 161–173 (1960). https://doi.org/10.2514/8.8467

    Article  MathSciNet  Google Scholar 

  7. Chu, W.H.: Fuel sloshing in a spherical tank filled to an arbitrary depth. AIAA J. 2(11), 1972–1979 (1964). https://doi.org/10.2514/3.2713

    Article  MATH  Google Scholar 

  8. Evans, D.V., Linton, C.M.: Sloshing frequencies. Q. J. Mech. Appl. Math. 46(1), 71–87 (1993). https://doi.org/10.1093/qjmam/46.1.71

    Article  MathSciNet  MATH  Google Scholar 

  9. Faltinsen, O.M., Timokha, A.N.: Sloshing, vol. 577. Cambridge University Press, Cambridge (2009)

    MATH  Google Scholar 

  10. Fontelos, M., López-Rıos, J.: Gravity waves oscillations at semicircular and general 2D containers: an efficient computational approach to 2D sloshing problem. Z. Angew. Math. Phys 71, 75 (2020). https://doi.org/10.1007/s00033-020-01299-4

    Article  MathSciNet  MATH  Google Scholar 

  11. Fox, D.W., Kuttler, J.R.: Sloshing frequencies. Z. Angew. Math. Phys. (ZAMP) 34(5), 668–696 (1983). https://doi.org/10.1007/BF00948809

    Article  MathSciNet  MATH  Google Scholar 

  12. Gaunt, R.E.: Inequalities for modified Bessel functions and their integrals. J. Math. Anal. Appl. 420(1), 373–386 (2014). https://doi.org/10.1016/j.jmaa.2014.05.083

    Article  MathSciNet  MATH  Google Scholar 

  13. Gaunt, R.E.: Inequalities for integrals of the modified Struve function of the first kind. Results Math. 73, 1–10 (2018). https://doi.org/10.1007/s00025-018-0827-4

    Article  MathSciNet  MATH  Google Scholar 

  14. Graham-Eagle, J.: 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, Cambridge (1983). https://doi.org/10.1017/S0305004100000943

  15. Groves, M.D.: Theoretical aspects of gravity-capillary waves in non-rectangular channels. J. Fluid Mech. 290, 377–404 (1995). https://doi.org/10.1017/S0022112095002552

    Article  MathSciNet  MATH  Google Scholar 

  16. Henderson, D., Miles, J.: Surface-wave damping in a circular cylinder with a fixed contact line. J. Fluid Mech. 275, 285–299 (1994). https://doi.org/10.1017/S0022112094002363

    Article  MathSciNet  MATH  Google Scholar 

  17. Hocking, L.M.: The damping of capillary-gravity waves at a rigid boundary. J. Fluid Mech. 179, 253–266 (1987). https://doi.org/10.1017/S0022112087001514

    Article  MATH  Google Scholar 

  18. Hocking, L.M.: Waves produced by a vertically oscillating plate. J. Fluid Mech. 179, 267–281 (1987). https://doi.org/10.1017/S0022112087001526

    Article  MATH  Google Scholar 

  19. Ibrahim, R.A.: Liquid Sloshing Dynamics: Theory and Applications. Cambridge University Press, Cambridge (2005). https://doi.org/10.1017/CBO9780511536656

    Book  MATH  Google Scholar 

  20. Kidambi, R.: Meniscus effects on the frequency and damping of capillary-gravity waves in a brimful circular cylinder. Wave Motion 46(2), 144–154 (2009). https://doi.org/10.1016/j.wavemoti.2008.10.001

    Article  MathSciNet  MATH  Google Scholar 

  21. Kolaei, A., Rakheja, S., Richard, M.J.: Effects of tank cross-section on dynamic fluid slosh loads and roll stability of a partly-filled tank truck. Eur. J. Mech. B Fluids 46, 46–58 (2014). https://doi.org/10.1016/j.euromechflu.2014.01.008

    Article  MathSciNet  MATH  Google Scholar 

  22. Kulczycki, T., Kuznetsov, N.: ‘High spots’ theorems for sloshing problems. Bull. Lond. Math. Soc. 41(3), 494–505 (2009). https://doi.org/10.1112/blms/bdp021

    Article  MathSciNet  MATH  Google Scholar 

  23. Kulczycki, T., Kuznetsov, N.: On the ‘high spots’ of fundamental sloshing modes in a trough. Proc. R. Soc. A Math. Phys. Eng. Sci. 467(2129), 1491–1502 (2011). https://doi.org/10.1098/rspa.2010.0258

    Article  MathSciNet  MATH  Google Scholar 

  24. Kulczycki, T., Kwaśnicki, M.: On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains. Proc. Lond. Math. Soc. 105(5), 921–952 (2012). https://doi.org/10.1112/plms/pds015

    Article  MathSciNet  MATH  Google Scholar 

  25. Kuzanek, J.F.: Existence and uniqueness of solutions to a fourth order nonlinear eigenvalue problem. SIAM J. Appl. Math. 27(2), 341–354 (1974). https://doi.org/10.1137/0127025

    Article  MathSciNet  MATH  Google Scholar 

  26. Kuzanek, J.F.: An isoperimetric problem with a nonlinear side condition for sloshing in a symmetric shallow canal. Z. Angew. Math. Phys. (ZAMP) 25(6), 753–763 (1974). https://doi.org/10.1007/BF01590261

    Article  MathSciNet  MATH  Google Scholar 

  27. Kuznetsov, N.G.: A variational method of determining the eigenfrequencies of a liquid in a channel. J. Appl. Math. Mech. 54(4), 458–465 (1990). https://doi.org/10.1016/0021-8928(90)90056-G

    Article  MathSciNet  MATH  Google Scholar 

  28. Lamb, H.: Hydrodynamics. Cambridge University Press, Cambridge (1932)

    MATH  Google Scholar 

  29. Lawrence, H.R., Wang, C.J., Reddy, R.B.: Variational solution of fuel sloshing modes. J. Jet Propul. 28(11), 729–736 (1958). https://doi.org/10.2514/8.7443

    Article  Google Scholar 

  30. Li, Y., Wang, Z.: An approximate analytical solution of sloshing frequencies for a liquid in various shape aqueducts. Shock Vib. (2014). https://doi.org/10.1155/2014/672648

  31. McIver, P.: Sloshing frequencies for cylindrical and spherical containers filled to an arbitrary depth. J. Fluid Mech. 201, 243–257 (1989). https://doi.org/10.1017/S0022112089000923

    Article  MathSciNet  MATH  Google Scholar 

  32. McIver, P., McIver, M.: Sloshing frequencies of longitudinal modes for a liquid contained in a trough. J. Fluid Mech. 252, 525–541 (1993). https://doi.org/10.1017/S0022112093003866

    Article  MathSciNet  MATH  Google Scholar 

  33. Mercier, B., Raugel, G.: Résolution d’un problème aux limites dans un ouvert axisymétrique par éléments finis en \( r, z\) et séries de Fourier en \(\theta \). ESAIM Math. Model. Numer. Anal. 16(4), 405–461 (1982). https://doi.org/10.1051/m2an/1982160404051

    Article  MATH  Google Scholar 

  34. Moiseev, N.N.: Introduction to the theory of oscillations of liquid-containing bodies. Adv. Appl. Mech. 8, 233–289 (1964). https://doi.org/10.1016/S0065-2156(08)70356-9

    Article  MathSciNet  MATH  Google Scholar 

  35. Moiseev, N., Petrov, A.: The calculation of free oscillations of a liquid in a motionless container. Adv. Appl. Mech. 9, 91–154 (1966). https://doi.org/10.1016/S0065-2156(08)70007-3

    Article  Google Scholar 

  36. Myshkis, A.D., Babskii, V.G., Kopachevskii, N.D., Slobozhanin, L.A., Tyuptsov, A.D.: Low-Gravity Fluid Mechanics. Springer, Berlin (1987)

    Book  Google Scholar 

  37. Nicolás, J.A.: Effects of static contact angles on inviscid gravity-capillary waves. Phys. Fluids 17(2), 022101 (2005). https://doi.org/10.1063/1.1829111

    Article  MathSciNet  MATH  Google Scholar 

  38. Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V., Cohl, H.S., McClain, M.A. (eds.) NIST Handbook of Mathematical Functions. Release 1.1.0 of 2020-12-15 (2010). http://dlmf.nist.gov/

  39. Otremba, F., Romero Navarrete, J.A., Lozano Guzmán, A.A.: Modelling of a partially loaded road tanker during a braking-in-a-turn maneuver. Actuators 7, 45 (2018). https://doi.org/10.3390/act7030045

    Article  Google Scholar 

  40. Reynolds, W.C., Satterlee, H.M.: Liquid propellant behavior at low and zero g. NASA Spec. Publ. 106, 387–439 (1966)

    Google Scholar 

  41. Shankar, P.: A simple method for studying low-gravity sloshing frequencies. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 459(2040), 3109–3130 (2003). https://doi.org/10.1098/rspa.2003.1154

    Article  MathSciNet  MATH  Google Scholar 

  42. Shankar, P.: Frequencies of gravity-capillary waves on highly curved interfaces with edge constraints. Fluid Dyn. Res. 39(6), 457 (2007). https://doi.org/10.1016/j.fluiddyn.2006.12.002

    Article  MathSciNet  MATH  Google Scholar 

  43. Shen, M.: Nonlinear capillary waves under gravity with edge constraints in a channel. Phys. Fluids 26(6), 1417–1421 (1983). https://doi.org/10.1063/1.864311

    Article  MathSciNet  MATH  Google Scholar 

  44. Simpson, H.C., Spector, S.J.: Some monotonicity results for ratios of modified Bessel functions. Q. Appl. Math. 42(1), 95–98 (1984). https://doi.org/10.1090/qam/736509

    Article  MathSciNet  MATH  Google Scholar 

  45. Stoker, J.J.: Water Waves: The Mathematical Theory with Applications, vol. 36. Wiley, New York (2011). https://doi.org/10.1002/9781118033159

    Book  MATH  Google Scholar 

  46. Tan, C.H., Hohenegger, C., Osting, B.: A variational characterization of fluid sloshing with surface tension. SIAM J. Appl. Math. 77(3), 995–1019 (2017). https://doi.org/10.1137/16M1104330

    Article  MathSciNet  MATH  Google Scholar 

  47. Troesch, B.A.: An isoperimetric sloshing problem. Commun. Pure Appl. Math. 18(1–2), 319–338 (1965). https://doi.org/10.1002/cpa.3160180124

    Article  MathSciNet  MATH  Google Scholar 

  48. Troesch, B.A.: Fluid motion in a shallow trapezoidal container. SIAM J. Appl. Math. 15(3), 627–636 (1967). https://doi.org/10.1137/0115054

    Article  MATH  Google Scholar 

  49. Troesch, B.A.: Integral inequalities for two functions. Arch. Ration. Mech. Anal. 24(2), 128–140 (1967). https://doi.org/10.1007/BF00281444

    Article  MathSciNet  MATH  Google Scholar 

  50. Vera, C., Paulin, J., Suarez, B., Gutierrez, M.: Simulation of freight trains equipped with partially filled tank containers and related resonance phenomenon. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit 219(4), 245–259 (2005). https://doi.org/10.1243/095440905X8916

    Article  Google Scholar 

  51. Willis, N., Tan, C.H., Hohenegger, C., Osting, B.: High spots for the ice-fishing problem with surface tension. SIAM J. Appl. Math. (to appear) (2022)

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Acknowledgements

We would like to thank Emma Coates, Emily Dryden, Calvin Khor, Robert Viator, and Nathan Willis for stimulating discussions. We are also grateful to the referees for their valuable comments and suggestions which greatly improved the manuscript.

Funding

The work of C. H. Tan and B. Osting was partially funded by NSF DMS 17-52202.

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  1. Department of Mathematics, Wake Forest University, Winston-Salem, NC, 27109, USA

    Chee Han Tan

  2. Department of Mathematics, University of Utah, Salt Lake City, 84112, USA

    Christel Hohenegger & Braxton Osting

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Tan, C.H., Hohenegger, C. & Osting, B. An Isoperimetric Sloshing Problem in a Shallow Container with Surface Tension. Appl Math Optim 87, 33 (2023). https://doi.org/10.1007/s00245-022-09936-2

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