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Scraping of a thin layer of viscoplastic fluid

J. J. Taylor-West* and A. J. Hogg

  • School of Mathematics, University of Bristol, Bristol BS8 1QU, United Kingdom

  • *j.taylor-west@bristol.ac.uk
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Phys. Rev. Fluids 9, 053301 – Published 14 May, 2024

DOI: https://doi.org/10.1103/PhysRevFluids.9.053301

Abstract

We analyze the scraping of a thin layer of viscoplastic fluid residing on a horizontal surface by a translating rigid scraper. This motion generates a mound of fluid upstream of the scraper and a residual layer behind it, both of which are modelled using a shallow layer theory for a Bingham or Herschel-Bulkley fluid. The flows ahead of and behind the scraper are coupled by the motion in the gap under the scraper, which is driven both by the translation of the scraper and by the induced pressure gradient due to the difference in flow thickness upstream and downstream. When the gap between the scraper and the underlying surface is sufficiently small, we find that a steady state emerges after a relatively long transient and that en route to this state, the unsteady dynamics exhibit a variety of regimes that are self-similar to leading order. We construct these solutions explicitly and derive key scalings for the temporal development of the flowing viscoplastic layer, as well as identifying the timescales at which there are transitions between the regimes. These predictions are confirmed by comparison with results from the numerical integration of the full system. Finally, we report results from preliminary laboratory experiments, which are compared with predictions from the shallow-layer theory, obtaining reasonable agreement once a slip boundary condition is included in the model, as motivated by experimental observations.

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References (40)

  1. C. Ancey and S. Cochard, The dam-break problem for Herschel-Bulkley viscoplastic fluids down steep flumes, J. Non-Newton. Fluid Mech. 158, 18 (2009).
  2. C. C. Mei, K.-F. Liu, and M. Yuhi, Mud flow—Slow and fast, in Geomorphological Fluid Mechanics, edited by N. J. Balmforth and A. Provenzale (Springer, Berlin, 2001), pp. 548–577.
  3. G. Sant, C. F. Ferraris, and J. Weiss, Rheological properties of cement pastes: A discussion of structure formation and mechanical property development, Cement Concrete Res. 38, 1286 (2008).
  4. A. B. Ross, S. K. Wilson, and B. R. Duffy, Blade coating of a power-law fluid, Phys. Fluids 11, 958 (1999).
  5. C. Xu and N. Willenbacher, How rheological properties affect fine-line screen printing of pastes: a combined rheological and high-speed video imaging study, J. Coat. Technol. Res. 15, 1401 (2018).
  6. M. Maillard, C. Mézière, P. Moucheront, C. Courrier, and P. Coussot, Blade-coating of yield stress fluids, J. Non-Newton. Fluid Mech. 237, 16 (2016).
  7. I. J. Hewitt and N. J. Balmforth, Viscoplastic lubrication theory with application to bearings and the washboard instability of a planing plate, J. Non-Newton. Fluid Mech. 169-170, 74 (2012).
  8. S. H. Emerman and D. Turcotte, A fluid model for the shape of accretionary wedges, Earth Planet. Sci. Lett. 63, 379 (1983).
  9. C. A. Perazzo and J. Gratton, Asymptotic regimes of ridge and rift formation in a thin viscous sheet model, Phys. Fluids 20, 043103 (2008).
  10. T. V. Ball, C. E. Penney, J. A. Neufeld, and A. C. Copley, Controls on the geometry and evolution of thin-skinned fold-thrust belts, and applications to the Makran accretionary prism and Indo–Burman Ranges, Geophys. J. Int. 218, 247 (2019).
  11. G. S. Stockmal, Modeling of large-scale accretionary wedge deformation, J. Geophys. Res.: Solid Earth 88, 8271 (1983).
  12. D. Davis, J. Suppe, and F. A. Dahlen, Mechanics of fold-and-thrust belts and accretionary wedges, J. Geophys. Res.: Solid Earth 88, 1153 (1983).
  13. G. S. Stockmal, C. Beaumont, M. Nguyen, and B. Lee, Mechanics of thin-skinned fold-and-thrust belts: Insights from numerical models, in Whence the Mountains? Inquiries into the Evolution of Orogenic Systems: A Volume in Honor of Raymond A. Price, edited by J. W. Sears, T. A. Harms, and C. A. Evenchick (Geological Society of America, Boulder, CO, 2007).
  14. G. Simpson, Mechanics of non-critical fold–thrust belts based on finite element models, Tectonophysics 499, 142 (2011).
  15. J. B. Ruh, B. J. P. Kaus, and J.-P. Burg, Numerical investigation of deformation mechanics in fold-and-thrust belts: Influence of rheology of single and multiple décollements, Tectonics 31, TC3005 (2012).
  16. J. R. Lister and E. M. Hinton, Using a squeegee on a layer of viscous or viscoplastic fluid, Phys. Rev. Fluids 7, 104101 (2022).
  17. N. J. Balmforth and R. V. Craster, A consistent thin-layer theory for Bingham plastics, J. Non-Newton. Fluid Mech. 84, 65 (1999).
  18. N. J. Balmforth, R. V. Craster, A. C. Rust, and R. Sassi, Viscoplastic flow over an inclined surface, J. Non-Newton. Fluid Mech. 142, 219 (2007).
  19. J. J. Taylor-West, Flows of Viscoplastic Fluids, Ph.D. thesis, University of Bristol, 2023.
  20. A. Lawal, D. M. Kalyon, and U. Yilmazer, Extrusion and lubrication flows of viscoplastic fluids with wall slip, Chem. Eng. Commun. 122, 127 (1993).
  21. N. J. Balmforth, R. V. Craster, and R. Sassi, Shallow viscoplastic flow on an inclined plane, J. Fluid Mech. 470, 1 (2002).
  22. J. F. Nye and G. I. Taylor, The flow of glaciers and ice-sheets as a problem in plasticity, Proc. R. Soc. Lond. A 207, 554 (1951).
  23. H. A. Barnes, A review of the slip (wall depletion) of polymer solutions, emulsions and particle suspensions in viscometers: its cause, character, and cure, J. Non-Newton. Fluid Mech. 56, 221 (1995).
  24. J. M. Piau, Carbopol gels: Elastoviscoplastic and slippery glasses made of individual swollen sponges: Meso- and macroscopic properties, constitutive equations and scaling laws, J. Non-Newton. Fluid Mech. 144, 1 (2007).
  25. M. Cloitre and R. T. Bonnecaze, A review on wall slip in high solid dispersions, Rheol Acta 56, 283 (2017).
  26. M. Daneshi, A. Pourzahedi, D. Martinez, and D. Grecov, Characterising wall-slip behaviour of carbopol gels in a fully-developed Poiseuille flow, J. Non-Newton. Fluid Mech. 269, 65 (2019).
  27. N. J. Balmforth, Viscoplastic asymptotics and other analytical methods, in Lectures on Visco-Plastic Fluid Mechanics, edited by G. Ovarlez and S. Hormozi (Springer International, Cham, 2019), pp. 41–82.
  28. A. P. Dufresne, T. V. Ball, and N. J. Balmforth, Viscoplastic Saffman–Taylor fingers with and without wall slip, J. Non-Newton, Fluid Mech. 312, 104970 (2023).
  29. R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids (Wiley, New York, 1987).
  30. L.-H. Luu and Y. Forterre, Drop impact of yield-stress fluids, J. Fluid Mech. 632, 301 (2009).
  31. J. van der Kolk, D. Tieman, and M. Jalaal, Viscoplastic lines: printing a single filament of yield stress material on a surface, J. Fluid Mech. 958, A34 (2023).
  32. J. Boujlel and P. Coussot, Measuring the surface tension of yield stress fluids, Soft Matter 9, 5898 (2013).
  33. B. Géraud, L. Jørgensen, L. Petit, H. Delanoë-Ayari, P. Jop, and C. Barentin, Capillary rise of yield-stress fluids, Europhys. Lett. 107, 58002 (2014).
  34. L. Jørgensen, M. Le Merrer, H. Delanoë-Ayari, and C. Barentin, Yield stress and elasticity influence on surface tension measurements, Soft Matter 11, 5111 (2015).
  35. A. N. Campbell, Activity coefficients, densities, dipole moments, and surface tensions of the system triethylamine–methylethylketone–water, Can. J. Chem. 59, 127 (1981).
  36. B. A. Noskov, A. V. Akentiev, and R. Miller, Dynamic surface properties of poly(vinylpyrrolidone) solutions, J. Colloid Interface Sci. 255, 417 (2002).
  37. Y. Liu, N. J. Balmforth, S. Hormozi, and D. R. Hewitt, Two-dimensional viscoplastic dambreaks, J. Non-Newton. Fluid Mech. 238, 65 (2016).
  38. N. J. Balmforth, R. V. Craster, P. Perona, A. C. Rust, and R. Sassi, Viscoplastic dam breaks and the Bostwick consistometer, J. Non-Newton. Fluid Mech. 142, 63 (2007).
  39. P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey et al., SciPy 1.0: fundamental algorithms for scientific computing in python, Nature Methods 17, 261 (2020).
  40. G. P. Matson and A. J. Hogg, Two-dimensional dam break flows of Herschel–Bulkley fluids: The approach to the arrested state, J. Non-Newton. Fluid Mech. 142, 79 (2007).

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Phys. Rev. Fluids 9, 053301– Published 14 May, 2024
  • Received 16 January 2024
  • Accepted 26 April 2024
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