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Obstructed gravity-driven flow down an incline

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Abstract

This investigation solves the problem of three-dimensional steady gravity-driven flow down an incline and around a two-dimensional cylindrical obstruction using lubrication theory. Both circular and elliptical cylindrical obstructions are considered using specific coordinates suited for the cylinder geometry. A finite difference scheme is proposed, and an asymptotic solution is derived which is used in conjunction with the numerical solution procedure. Good agreement is found with previous results for the case of the circular cylinder, and new results are obtained for the case of the elliptic cylinder.

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References

  1. Kapitza, P.L., Kapitza, S.P.: Wave flow of thin layers of a viscous fluid: III. Experimental study of undulatory flow conditions. J. Exp. Theor. Phys. 19, 105–120 (1949)

    Google Scholar 

  2. Craster, R.V., Matar, O.K.: Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198 (2009)

    Article  Google Scholar 

  3. Barberi, F., Carapezza, M.L.: The Control of Lava Flows at Mt. Etna, pp. 357–369. American Geophysical Union, Washington, DC (2004)

    Google Scholar 

  4. Griffiths, R.W.: The dynamics of lava flows. Ann. Rev. Fluid Mech. 32, 477–518 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  6. Hákonardóttir, K.M., Hogg, A.J., Batey, J., Woods, A.W.: Flying avalanches. Geophys. Res. Lett. 30, 2191 (2003)

    Article  Google Scholar 

  7. Hákonardóttir, K.M., Hogg, A.J.: Oblique shocks in rapid granular flows. Phys. Fluids 17, 077101 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Benjamin, T.B.: Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  9. Yih, C.-S.: Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334 (1963)

    Article  MATH  Google Scholar 

  10. Benney, D.J.: Long waves on liquid films. J. Math. Phys. 45, 150–155 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kalliadasis, S., Ruyer-Quil, C., Scheid, B., Velarde, M.G.: Falling Liquid Films. Springer, London (2012)

    Book  MATH  Google Scholar 

  12. Shkadov, V.Y.: Wave conditions in flow of thin layer of a viscous liquid under the action of gravity. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza 1(1), 43–51 (1967)

    Google Scholar 

  13. Ruyer-Quil, C., Manneville, P.: Improved modeling of flows down inclined planes. Eur. Phys. J. B 15, 357–369 (2000)

    Article  MATH  Google Scholar 

  14. Ruyer-Quil, C., Manneville, P.: Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations. Phys. Fluids 14, 170–183 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kalliadasis, S., Bielarz, C., Homsy, G.: Steady free-surface thin film flows over topography. Phys. Fluids 12, 1889–1898 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bielarz, C., Kalliadasis, S.: Time-dependent free-surface thin film flows over topography. Phys. Fluids 15, 2512–2524 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mazouchi, A., Homsy, G.M.: 2001 Free surface Stokes flow over topography. Phys. Fluids 13, 2751–2761 (2001)

    Article  MATH  Google Scholar 

  18. Gaskell, P.H., Jimack, P.K., Sellier, M., Thompson, H.M., Wilson, M.C.T.: Gravity-driven flow of continuous thin liquid films on non-porous substrates with topography. J. Fluid Mech. 509, 253–280 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Baxter, S.J., Power, H., Cliffe, K.A., Hibberd, S.: Three-dimensional thin film flow over and around an obstacle on an inclined plane. Phys. Fluids 21, 032102 (2009)

    Article  MATH  Google Scholar 

  20. Aksel, N., Schörner, M.: Films over topography: from creeping flow to linear stability, theory, and experiments, a review. Acta Mech. 229, 1453–1482 (2018)

    Article  MathSciNet  Google Scholar 

  21. Hinton, E.M., Hogg, A.J., Huppert, H.E.: Interaction of viscous free-surface flows with topography. J. Fluid Mech. 876, 912–938 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hinton, E.M., Hogg, A.J., Huppert, H.E.: Shallow free-surface Stokes flow around a corner. Phil. Trans. R. Soc. A378, 20190515 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hinton, E.M., Hogg, A.J., Huppert, H.E.: Viscous free-surface flows past cylinders. Phys. Rev. Fluids 5, 084101 (2020)

    Article  Google Scholar 

  24. Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1965)

    Google Scholar 

  25. Leal, L.G.: Laminar Flow and Convection Transport Processes: Scaling Principles and Asymptotic Analysis. Butterworth-Heinemann, Boston (1992)

    Google Scholar 

  26. LeVeque, R.J.: Finite Difference Methods for Ordinary and Partial Differential Equations—Steady-State and Time-Dependent Problems. SIAM, Philadelphia (2007)

    Book  MATH  Google Scholar 

  27. Dennis, S.C.R.: The numerical solution of the vorticity transport equation. In: Proceedings of 3rd International Conference on Numerical Methods, Fluid Dynamics. Lecture Notes in Physics, vol. 19, pp. 120–129. Springer, Berlin (1973)

  28. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas. Graphs and Mathematical Tables, Dover, New York (1965)

  29. Imai, I.: On the asymptotic behaviour of viscous fluid flow at a great distance from a cylindrical body, with special reference to Filon’s paradox. Proc. R. Soc. Lond. A208, 487–516 (1951)

    MathSciNet  MATH  Google Scholar 

  30. D’Alessio, S.J.D., Dennis, S.C.R.: A vorticity model for viscous flow past a cylinder. Comp. Fluids 23, 279–293 (1994)

    Article  MATH  Google Scholar 

  31. Dennis, S.C.R., Young, P.J.S.: Steady flow past an elliptic cylinder inclined to the stream. J. Eng. Math. 47, 101–120 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Faculty of Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    S. J. D. D’Alessio

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D’Alessio, S.J.D. Obstructed gravity-driven flow down an incline. Acta Mech 234, 3575–3594 (2023). https://doi.org/10.1007/s00707-023-03591-6

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  • DOI: https://doi.org/10.1007/s00707-023-03591-6

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