- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Lagrangian versus Eulerian view on the mean drift and streaming flows in orbital sloshing

A. Bongarzone and F. Gallaire*

  • *Contact author: francois.gallaire@epfl.ch
PDF

Phys. Rev. Fluids 9, 124803 – Published 20 December, 2024

DOI: https://doi.org/10.1103/PhysRevFluids.9.124803

Abstract

Orbital sloshing is a technique to gently mix a container's liquid content and it is commonly used in fermentation and cell cultivation processes. Besides the rich wave dynamics observed at the interface, Bouvard et al. [Phys. Rev. Fluids 2, 084801 (2017)] unveiled the structure of the Lagrangian mean flow hiding in the fluid bulk. The latter flow shows a global toroidal (azimuthal) rotation codirected with the wave and nontrivial poloidal vortices near the contact line. Rotating sloshing waves are known to induce a net motion of fluid particles and hence a wave-averaged difference between the Eulerian flow—viscous streaming—and the Lagrangian flow, that is commonly referred to as Stokes drift. Nevertheless, discerning these two components in an experiment is challenging as they scale similarly with the forcing amplitude and frequency. Their relative contributions remain therefore unquantified, particularly for highly viscous fluids, for which prior analysis, based on inviscid arguments, fails. In this work, we construct a truncated asymptotic approximation of the problem, where the solution at each order is computed numerically to describe as accurately as possible the contact line region, otherwise analytically untractable. The results of this weakly nonlinear analysis in terms of first-order wave and second-order mean flow are then thoroughly compared with the experiments by Bouvard et al. [Phys. Rev. Fluids 2, 084801 (2017)], showing a remarkable agreement for off-resonance frequencies. When viscosity matters, our findings suggest that it is incorrect to attribute the poloidal patterns solely to the Eulerian streaming flow and that viscous corrections to the Stokes drift are equally important in the resulting mean Lagrangian flow.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (88)

  1. L. E. McDaniel and E. G. Bailey, Effect of shaking speed and type of closure on shake flask cultures, Appl. Microbiol. 17, 286 (1969).
  2. F. M. Wurm, Production of recombinant protein therapeutics in cultivated mammalian cells, Nat. Biotechnol. 22, 1393 (2004).
  3. C.-M. Liu and L.-N. Hong, Development of a shaking bioreactor system for animal cell cultures, Biochem. Eng. J. 7, 121 (2001).
  4. M. J. De Jesus, P. Girard, M. Bourgeois, G. Baumgartner, B. Jacko, H. Amstutz, and F. M. Wurm, Tubespin satellites: A fast track approach for process development with animal cells using shaking technology, Biochem. Eng. J. 17, 217 (2004).
  5. N. Muller, M. Derouazi, F. Van Tilborgh, S. Wulhfard, D. L. Hacker, M. Jordan, and F. M. Wurm, Scalable transient gene expression in Chinese hamster ovary cells in instrumented and non-instrumented cultivation systems, Biotechnol. Lett. 29, 703 (2007).
  6. A. Handa-Corrigan, A. N. Emery, and R. E. Spier, Effect of gas–liquid interfaces on the growth of suspended mammalian cells: Mechanisms of cell damage by bubbles, Enzyme Microb. Technol. 11, 230 (1989).
  7. G. Kretzmer and K. Schügerl, Response of mammalian cells to shear stress, Appl. Microbiol. Biotechnol. 34, 613 (1991).
  8. E. T. Papoutsakis, Fluid-mechanical damage of animal cells in bioreactors, Trends Biotechnol. 9, 427 (1991).
  9. H. M. Kim and J. P. Kizito, Stirring free surface flows due to horizontal circulatory oscillation of a partially filled container, Chem. Eng. Commun. 196, 1300 (2009).
  10. J. Büchs, U. Maier, C. Milbradt, and B. Zoels, Power consumption in shaking flasks on rotary shaking machines: I. Power consumption measurement in unbaffled flasks at low liquid viscosity, Biotechnol. Bioeng. 68, 589 (2000).
  11. J. Büchs, U. Maier, C. Milbradt, and B. Zoels, Power consumption in shaking flasks on rotary shaking machines: II. Nondimensional description of specific power consumption and flow regimes in unbaffled flasks at elevated liquid viscosity, Biotechnol. Bioeng. 68, 594 (2000).
  12. J. Büchs, Introduction to advantages and problems of shaken cultures, Biochem. Eng. J. 7, 91 (2001).
  13. U. Maier, M. Losen, and J. Büchs, Advances in understanding and modeling the gas–liquid mass transfer in shake flasks, Biochem. Eng. J. 17, 155 (2004).
  14. N. Muller, P. Girard, D. L. Hacker, M. Jordan, and F. M. Wurm, Orbital shaker technology for the cultivation of mammalian cells in suspension, Biotechnol. Bioeng. 89, 400 (2005).
  15. M. Micheletti, T. Barrett, S. D. Doig, F. Baganz, M. S. Levy, J. M. Woodley, and G. J. Lye, Fluid mixing in shaken bioreactors: Implications for scale-up predictions from microlitre-scale microbial and mammalian cell cultures, Chem. Eng. Sci. 61, 2939 (2006).
  16. X. Zhang, C. Bürki, M. Stettler, D. De Sanctis, M. Perrone, M. Discacciati, N. Parolini, M. DeJesus, D. L. Hacker, A. Quarteroni, and F. M. Wurm, Efficient oxygen transfer by surface aeration in shaken cylindrical containers for mammalian cell cultivation at volumetric scales up to 1000 l, Biochem. Eng. J. 45, 41 (2009).
  17. S. Tissot, M. Farhat, D. L. Hacker, T. Anderlei, M. Kühner, C. Comninellis, and F. M. Wurm, Determination of a scale-up factor from mixing time studies in orbitally shaken bioreactors, Biochem. Eng. J. 52, 181 (2010).
  18. R.-K. Tan, W. Eberhard, and J. Büchs, Measurement and characterization of mixing time in shake flasks, Chem. Eng. Sci. 66, 440 (2011).
  19. S. Tissot, A. Oberbek, M. Reclari, M. Dreyer, D. L. Hacker, L. Baldi, M. Farhat, and F. M. Wurm, Efficient and reproducible mammalian cell bioprocesses without probes and controllers?, New Biotechnol. 28, 382 (2011).
  20. W. Klöckner and J. Büchs, Advances in shaking technologies, Trends Biotechnol. 30, 307 (2012).
  21. H. N. Abramson, The dynamic behavior of liquids in moving containers, with applications to space vehicle technology, NASA Tech. Rep. SP-106 (NASA, Washington, 1966).
  22. H. N. Abramson, W.-H. Chu, and D. D. Kana, Some studies of nonlinear lateral sloshing in rigid containers, NASA Contractor Rep. NASA CR-375 (NASA, 1966).
  23. W.-H. Chu, Subharmonic oscillations in an arbitrary tank resulting from axial excitation, Trans. ASME J. Appl. Mech. 35, 148 (1968).
  24. A. Royon-Lebeaud, E. J. Hopfinger, and A. Cartellier, Liquid sloshing and wave breaking in circular and square-base cylindrical containers, J. Fluid Mech. 577, 467 (2007).
  25. E. J. Hopfinger and V. Baumbach, Liquid sloshing in cylindrical fuel tanks, Prog. Propul. Phys. 1, 279 (2009).
  26. M. Reclari, Hydrodynamics of orbital shaken bioreactors, Technical Report (EPFL, 2013).
  27. M. Reclari, M. Dreyer, S. Tissot, D. Obreschkow, F. M. Wurm, and M. Farhat, Surface wave dynamics in orbital shaken cylindrical containers, Phys. Fluids 26, 052104 (2014).
  28. F. Moisy, J. Bouvard, and W. Herreman, Counter-rotation in an orbitally shaken glass of beer, Europhys. Lett. 122, 34002 (2018).
  29. O. M. Faltinsen, I. A. Lukovsky, and A. N. Timokha, Resonant sloshing in an upright annular tank, J. Fluid Mech. 804, 608 (2016).
  30. A. N. Timokha and I. Raynovskyy, The damped sloshing in an upright circular tank due to an orbital forcing, Dopov. Nac. Akad. Nauk Ukr. 10, 48 (2017).
  31. I. Raynovskyy and A. N. Timokha, Steady-state resonant sloshing in an upright cylindrical container performing a circular orbital motion, Math. Probl. Eng. 2018, 5487178 (2018).
  32. I. A. Raynovskyy and A. N. Timokha, Damped steady-state resonant sloshing in a circular base container, Fluid Dyn. Res. 50, 045502 (2018).
  33. I. A. Raynovskyy and A. N. Timokha, Sloshing in Upright Circular Containers: Theory, Analytical Solutions, and Applications (CRC Press, Boca Raton, FL, 2020).
  34. G. M. Horstmann, W. Herreman, and T. Weier, Linear damped interfacial wave theory for an orbitally shaken upright circular cylinder, J. Fluid Mech. 891, A22 (2020).
  35. G. M. Horstmann, S. Anders, D. H. Kelley, and T. Weier, Formation of spiral waves in cylindrical containers under orbital excitation, J. Fluid Mech. 925, A28 (2021).
  36. B.-F. Chen, C.-H. Wu, and O. M. Faltinsen, The mechanism of switching direction of swirling sloshing waves, J. Fluid Mech. 954, A2 (2023).
  37. A. Bongarzone, M. Guido, and F. Gallaire, An amplitude equation modelling the double-crest swirling in orbital-shaken cylindrical containers, J. Fluid Mech. 943, A28 (2022).
  38. A. Marcotte, F. Gallaire, and A. Bongarzone, Super-harmonically resonant swirling waves in longitudinally forced circular cylinders, J. Fluid Mech. 966, A41 (2023).
  39. A. Marcotte, F. Gallaire, and A. Bongarzone, Swirling against the forcing: Evidence of stable counterdirected sloshing waves in orbital-shaken reservoirs, Phys. Rev. Fluids 8, 084802 (2023).
  40. H. Lamb, Hydrodynamics (Cambridge University Press, Cambridge, UK, 1993).
  41. O. M. Faltinsen and A. N. Timokha, Sloshing (Cambridge University Press, Cambridge, UK, 2009).
  42. R. A. Ibrahim, Liquid Sloshing Dynamics: Theory and Applications (Cambridge University Press, Cambridge, UK, 2005).
  43. R. E. Hutton, An investigation of nonlinear, nonplanar oscillations of fluid in cylindrical container, NASA Tech. Rep. D-1870 (NASA, 1963).
  44. J. Bouvard, W. Herreman, and F. Moisy, Mean mass transport in an orbitally shaken cylindrical container, Phys. Rev. Fluids 2, 084801 (2017).
  45. G. G. Stokes, On the theory of oscillatory waves, Trans. Cambridge Philos. Soc. 8, 441 (1847).
  46. T. S. Van Den Bremer and Ø. Breivik, Stokes drift, Philos. Trans. R. Soc. A 376, 20170104 (2018).
  47. J. Vanneste and W. R. Young, Stokes drift and its discontents, Philos. Trans. R. Soc. A 380, 20210032 (2022).
  48. G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, UK, 1967).
  49. A. D. D. Craik, The drift velocity of water waves, J. Fluid Mech. 116, 187 (1982).
  50. A. D. D. Craik and S. Leibovich, A rational model for Langmuir circulations, J. Fluid Mech. 73, 401 (1976).
  51. N. Périnet, P. Gutiérrez, H. Urra, N. Mujica, and L. Gordillo, Streaming patterns in Faraday waves, J. Fluid Mech. 819, 285 (2017).
  52. M. Higuera, E. Knobloch, and J. M. Vega, Dynamics of nearly inviscid Faraday waves in almost circular containers, Physica D 201, 83 (2005).
  53. N. Riley, Steady streaming, Annu. Rev. Fluid Mech. 33, 43 (2001).
  54. Odd M. Faltinsen and Alexander N. Timokha, An inviscid analysis of the Prandtl azimuthal mass transport during swirl-type sloshing, J. Fluid Mech. 865, 884 (2019).
  55. R. E. Hutton, Fluid-particle motion during rotary sloshing, Trans. ASME J. Appl. Mech. 31, 123 (1964).
  56. J. A. Nicolás and J. M. Vega, Three-dimensional streaming flows driven by oscillatory boundary layers, Fluid Dyn. Res. 32, 119 (2003).
  57. C. Huh and L. E. Scriven, Hydrodynamic model of steady movement of a solid/liquid/fluid contact line, J. Colloid Interface Sci. 35, 85 (1971).
  58. S. H. Davis and E. B. Dussan V., On the motion of a fluid-fluid interface along a solid surface, J. Fluid Mech. 65, 71 (1974).
  59. C. L. M. H. Navier, Mémoire sur les lois du mouvement des fluides, Mém. Acad. Sci. 6, 389 (1823).
  60. E. Lauga, M. Brenner, and H. Stone, Microfluidics: The No-Slip Boundary Condition (Springer, Berlin, Heidelberg, 2007), pp. 1219–1240.
  61. F. Viola and F. Gallaire, Theoretical framework to analyze the combined effect of surface tension and viscosity on the damping rate of sloshing waves, Phys. Rev. Fluids 3, 094801 (2018).
  62. J. W. Miles, Capillary-viscous forcing of surface waves, J. Fluid Mech. 219, 635 (1990).
  63. C. L. Ting and M. Perlin, Boundary conditions in the vicinity of the contact line at a vertically oscillating upright plate: An experimental investigation, J. Fluid Mech. 295, 263 (1995).
  64. A. Bongarzone and F. Gallaire, Numerical estimate of the viscous damping of capillary-gravity waves: A macroscopic depth-dependent slip-length model, arXiv:2207.06907.
  65. A. Bongarzone and F. Gallaire, Stick-slip-to-stick transition of liquid oscillations in a U-shaped tube, Phys. Rev. Fluids 9, 034401 (2024).
  66. A. Bongarzone, F. Viola, and F. Gallaire, Relaxation of capillary-gravity waves due to contact line nonlinearity: A projection method, Chaos 31, 123124 (2021).
  67. A. Bongarzone, F. Viola, S. Camarri, and F. Gallaire, Subharmonic parametric instability in nearly brimful circular cylinders: A weakly nonlinear analysis, J. Fluid Mech. 947, A24 (2022).
  68. F. Viola, C. Arratia, and F. Gallaire, Mode selection in trailing vortices: Harmonic response of the nonparallel Batchelor vortex, J. Fluid Mech. 790, 523 (2016).
  69. W. Heinrichs, Spectral collocation schemes on the unit disc, J. Comput. Phys. 199, 66 (2004).
  70. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Springer Science & Business Media, Cham, 2007).
  71. A. Sommariva, Fast construction of Fejér and Clenshaw–Curtis rules for general weight functions, Comput. Math. Appl. 65, 682 (2013).
  72. F. Viola, P.-T. Brun, and F. Gallaire, Capillary hysteresis in sloshing dynamics: A weakly nonlinear analysis, J. Fluid Mech. 837, 788 (2018).
  73. O. M. Phillips, The Dynamics of the Upper Ocean (Cambridge University Press, Cambridge, UK, 1977).
  74. M. P. Escudier, Observations of the flow produced in a cylindrical container by a rotating endwall, Exp. Fluids 2, 189 (1984).
  75. J. Thomson, Collected Papers in Physics and Engineering (Cambridge University Press, Cambridge, UK, 2016).
  76. G. S. Narimanov, Movement of a tank partly filled by a fluid: The taking into account of non-smallness of amplitude, Prikl. Math. Mech. (in Russian) 21, 513 (1957).
  77. N. N. Moiseev, On the theory of nonlinear vibrations of a liquid of finite volume, J. Appl. Math. Mech. 22, 860 (1958).
  78. F. T. Dodge, D. D. Kana, and H. N. Abramson, Liquid surface oscillations in longitudinally excited rigid cylindrical containers, AIAA J. 3, 685 (1965).
  79. O. M. Faltinsen, A nonlinear theory of sloshing in rectangular tanks, J. Ship Res. 18, 224 (1974).
  80. G. S. Narimanov, L. V. Dokuchaev, and I. A. Lukovsky, Nonlinear Dynamics of Flying Apparatus with Liquid (Mashinostroenie, Moscow, 1977) (in Russian).
  81. I. A. Lukovsky, Nonlinear Dynamics, Mathematical Models for Rigid Bodies with a Liquid (De Gruyter, Berlin, München, Boston, 2015).
  82. O. M. Faltinsen, O. F. Rognebakke, and A. N. Timokha, Resonant three-dimensional nonlinear sloshing in a square-base basin. Part 2. Effect of higher modes, J. Fluid Mech. 523, 199 (2005).
  83. M. Higuera, J. M. Vega, and E. Knobloch, Coupled amplitude-streaming flow equations for nearly inviscid Faraday waves in small aspect ratio containers, J. Nonlin. Sci. 12, 505 (2002).
  84. E. Martín and J. M. Vega, The effect of surface contamination on the drift instability of standing Faraday waves, J. Fluid Mech. 546, 203 (2006).
  85. M. Higuera, J. Porter, F. Varas, and J. M. Vega, Nonlinear dynamics of confined liquid systems with interfaces subject to forced vibrations, Adv. Colloid Interface Sci. 206, 106 (2014).
  86. A. Bongarzone, B. Jouron, F. Viola, and F. Gallaire, A revised gap-averaged floquet analysis of Faraday waves in Hele-Shaw cells, J. Fluid Mech. 977, A45 (2023).
  87. T. Fullana, Simulation and optimization of complex phenomena in multiphase flows, Ph.D. thesis, Sorbonne Université, 2022.
  88. W. Weheliye, M. Yianneskis, and A. Ducci, On the fluid dynamics of shaken bioreactors—Flow characterization and transition, AIChE J. 59, 334 (2013).

Outline

Information

Phys. Rev. Fluids 9, 124803– Published 20 December, 2024
  • Received 3 July 2024
  • Accepted 2 December 2024
Crossmark - Check for updates button

©2024 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation