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Spreading of Lava as a Non-Newtonian Fluid in the Conditions of Partial Slip on the Underlying Surface

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Abstract

The effect of the boundary condition imposed on the underlying surface on the propagation of lava flows is studied in the case in which the non-Newtonian properties of the medium should be taken into account. The problem of lava spreading over a plane horizontal plane is solved. The no-slip condition on the underlying surface is replaced by the partial slip condition. The lava is modeled as a power-law fluid. The flow is assumed to be axisymmetric. In the thin layer approximation the problem reduces to the solution of one nonlinear partial differential second-order equation with an additional integral condition. In the case of the power-law time dependence of the lava flow rate a self-similar solution is obtained; however, it exists only under some constraints on the problem parameters. The non-self-similar solution is considered numerically. It is shown that the lava propagation velocity can be considerably higher, when the slip is taken into account.

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REFERENCES

  1. R. W. Griffiths, “The dynamics of lava flows,” Annu. Rev. Fluid Mech. 32, 477–518 (2000).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. H. E. Huppert, J. B. Shepherd, H. Sigurdsson, and R. S. J. Sparks, “On lava dome growth, with application to the 1979 lava extrusion of the Soufriere of St. Vincent,” J. Volc. Geotherm. Res. 14, 199–222 (1982).

    Article  ADS  Google Scholar 

  3. M. Dragoni, I. Borsari, and A. Tallarico, “A model for the shape of lava flow fronts,” J. Geophys. Res. 110, B09203 (2005).

    ADS  Google Scholar 

  4. A. A. Osiptsov, “Three-dimensional lava flows over a non-axisymmetric conical surface,” Fluid Dynamics 41(2), 198—210 (2006).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. A. Tallarico, M. Dragoni, M. Filippucci, A. Piombo, S. Santini, and A. Valerio, “Cooling of a channeled lava flow with non-Newtonian rheology: Crust formation and surface radiance,” Annals of Geophysics. 54(5), 510–520 (2011).

    Google Scholar 

  6. E. A. Vedeneeva, “Lava spreading during volcanic eruptions on the condition of partial slip along the underlying surface,” Fluid Dynamics 50(2), 203—214 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  7. N. J. Balmforth, A. S. Burbidge, R. V. Craster, J. Salzig, and A. Shen, “Viscoplastic models of isothermal lava domes,” J. Fluid Mech. 403, 37–65 (2000).

    Article  ADS  MATH  Google Scholar 

  8. N. V. Koronovskii, General Geology (Knizhnyi Dom Universitet, Moscow, 2006) [in Russian].

    Google Scholar 

  9. N. V. Koronovskii and A. F. Yakusheva, Fundamentals of Geology (Vysshaya Shkola, Moscow, 1991) [in Russian].

    Google Scholar 

  10. H. Sigurdsson, B. Houghton, S. McNutt, H. Rymer, and J. Stix, The Encyclopedia of Volcanoes (Elsevier, 2015).

    Google Scholar 

  11. M. Chevrel, C. Cimarelli, L. deBiasi, J. B. Hanson, Y. Lavallee, F. Arzilli, and D. B. Dingwell, “Viscosity measurements of crystallizing andesite from Tungurahua volcano (Ecuador),” Geochem. Geophys. Geosyst. 16(3), 870–889 (2015).

    Article  ADS  Google Scholar 

  12. M. Pistone, B. Cordonnier, P. Ulmer, and L. Caricchi, “Rheological flow laws for multiphase magmas: An empirical approach,” J. Volc. Geoth. Res. 321, 158–170, (2016).

    Article  ADS  Google Scholar 

  13. A. Castruccio, A. C. Rust, and R. S. J. Sparks, “Rheology and flow of crystal-bearing lavas: Insights from analogue gravity currents,” Earth Planet. Sci. Lett. 297(3–4), 471–480 (2010).

    Article  ADS  Google Scholar 

  14. A. Tran, M. L. Rudolph, and M. Manga, “Bubble mobility in mud and magmatic volcanoes,” J. Volcanol. Geotherm. Res. 294, 11–24 (2015).

    Article  ADS  Google Scholar 

  15. A. A. Kutepov, A. D. Polyanin, Z. D. Zapryanov, A. V. Vyaz’min, and D. A. Kazenin, Chemical Fluid Dynamics (Bureau Quantum, Moscow, 1996) [in Russian].

    Google Scholar 

  16. G. Ovarlez and S. Hormozi, Lectures on Visco-Plastic Fluid Mechanics (Springer, 2019).

    Book  MATH  Google Scholar 

  17. L. L. Ferras, J. M. Nobrega, and F. T. Pinho, “Analytical solutions for Newtonian and inelastic non-Newtonian flows with wall slip,” J. Non-Newtonian Fluid Mech. 175–176, 76–88 (2012).

    Article  Google Scholar 

  18. K. D. Housiadas, “Viscoelastic Poiseuille flows with total normal stress dependent, nonlinear Navier slip at the wall,” Phys. Fluids 25(4), 043105 (2013).

    Article  ADS  MATH  Google Scholar 

  19. C. G. Georgiou and G. Kaoullas, “Newtonian flow in a triangular duct with slip at the wall,” Meccanica 48, 2577–2583 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  20. Y. Damianou, C. G. Georgiou, and I. Moulitsas, “Combined effects of compressibility and slip in flows of a Herschel-Bulkley fluid,” J. Non-Newtonian Fluid Mech. 193, 89–102 (2013).

    Article  Google Scholar 

  21. Y. Damianou, M. Philippou, G. Kaoullas, and C. G. Georgiou, “Cessation of viscoplastic Poiseuille flow with wall slip,” J. Non-Newtonian Fluid Mech. 203, 24–37 (2014).

    Article  Google Scholar 

  22. H. E. Huppert, “The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface,” J. Fluid Mech. 21, 43–58 (1982).

    Article  ADS  Google Scholar 

  23. R. Sayag and M. Grae Worster, “Axisymmetric gravity currents of power-law fluids over a rigid horizontal surface,” J. Fluid Mech. 716(R5) (2013).

  24. V. Di Federico, S. Cintoli, and G. Bizzarri, “Viscous spreading of non-Newtonian gravity currents in radial geometry,” WIT Transactions on Engineering Sciences 52, 399–409 (2006).

    Article  MathSciNet  Google Scholar 

  25. S. Longo, V. Di Federico, R. Archetti, L. Chiapponi, V. Ciriello, and M. Ungarish, “On the axisymmetric spreading of non-Newtonian power-law gravity currents of time-dependent volume: An experimental and theoretical investigation focused on the inference of rheological parameters,” J. Non-Newtonian Fluid Mech. 201, 69–79 (2013).

    Article  Google Scholar 

  26. S. N. Neossi Nguetchue and E. Momoniat, “Axisymmetric spreading of a thin power-law fluid under gravity on a horizontal plane,” Nonlinear Dyn. 52(4), 361–366 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  27. S. N. Neossi Nguetchue, S. Abelman, and E. Momoniat, “Symmetries and similarity solutions for the axisymmetric spreading under gravity of a thin power-law liquid drop on a horizontal plane,” Appl. Math. Model. 33, 4364–4377 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  28. V. Di Federico, S. Malavasi, and S. Cintoli, “Viscous spreading of non-Newtonian gravity currents on a plane,” Meccanica 41, 207–217 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  29. J. Gratton, F. Minotti, and S. M. Mahajan, “Theory of creeping gravity currents of a non-Newtonian liquid,” Phys. Rev. E 60(6), 6960–6967 (1999).

    Article  ADS  Google Scholar 

  30. A. Costa, G. Wadge, and O. Melnik, “Cyclic extrusion of a lava dome based on a stick-slip mechanism,” Earth Planet. Sci. Lett. 337–338, 39–46 (2012).

    Article  ADS  Google Scholar 

  31. T. Sochi, “Slip at fluid-solid interface,” Polym. Rev. 51(4), 309–340 (2011).

    Article  Google Scholar 

  32. A. Ya. Malkin, S. A. Patlazhan, and V. G. Kulichikhin, “Physicochemical phenomena leading to slip of a fluid along a solid surface,” Russ. Chem. Rev. 88(3), 319–349 (2019).

    Article  ADS  Google Scholar 

  33. A. Ya. Malkin and S. A. Patlazhan, “Wall slip for complex liquids – Phenomenon and its causes,” Adv. Colloid Interface Sci. 257, 42–57 (2018).

    Article  Google Scholar 

  34. S. G. Hatzikiriakos, “Wall slip of molten polymers,” Prog. Polym. Sci. 37(4), 624–643 (2012).

    Article  Google Scholar 

  35. S. G. Hatzikiriakos, “Slip mechanisms in complex fluid flows,” Soft Matter 11(40), 7851–7856 (2015).

    Article  ADS  Google Scholar 

  36. M. Cloitre and R. T. Bonnecaze, “A review on wall slip in high solid dispersions,” Rheol. Acta 56(3), 283–305 (2017).

    Article  Google Scholar 

  37. A. I. Ageev and A. N. Osiptsov, “Self-similar regimes of liquid-layer spreading along a superhydrophobic surface,” Fluid Dynamics 49(3), 330—342 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  38. A. I. Ageev and A. N. Osiptsov, “Stokes flow in a microchannel with superhydrophobic walls,” Fluid Dynamics 54(2), 205—217 (2019).

    Article  MATH  Google Scholar 

  39. L. M. Hocking and A. D. Rivers, “The spreading of a drop by capillary action,” J. Fluid Mech. 121, 425–442 (1982).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. L. M. Hocking, “Rival contact-angle models and the spreading of drops,” J. Fluid Mech. 239, 671–681 (1992).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. A. Oron, S. H. Davis, and S. G. Bankoff, “Long-scale evolution of thin liquid films,” Rev. Mod. Phys. 69(3), 931–980 (1997).

    Article  ADS  Google Scholar 

  42. L. M. Hocking and S. H. Davis, “Inertial effects in time-dependent motion of thin films and drops,” J. Fluid Mech. 467, 1–17 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. S. N. Reznik and A. L. Yarin, “Spreading of an axisymmetric viscous drop due to gravity and capillarity on a dry horizontal wall,” Int. J. Multiph. Flow 28, 1437–1457 (2002).

    Article  MATH  Google Scholar 

  44. N. N. Kalitkin and P. V. Koryakin, Numerical Methods. Book 2. Methods of Mathematical Physics (Akademiya, Moscow, 2013) [in Russian].

    Google Scholar 

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ACKNOWLEDGMENTS

The author wishes to thank O.E. Melnik for attention to the study and useful discussions.

Funding

The study was carried out with the support of the Russian Science Foundation (project 19-17-00027).

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Authors and Affiliations

  1. Institute of Mechanics, Moscow State University, 119192, Moscow, Russia

    E. A. Vedeneeva

  2. Institute of Earthquake Prediction Theory and Mathematical Geophysics, Profsoyuznaya ul. 84/32, 117997, Moscow, Russia

    E. A. Vedeneeva

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Correspondence to E. A. Vedeneeva.

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Translated by M. Lebedev

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Vedeneeva, E.A. Spreading of Lava as a Non-Newtonian Fluid in the Conditions of Partial Slip on the Underlying Surface. Fluid Dyn 56, 18–30 (2021). https://doi.org/10.1134/S0015462821010158

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