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Theoretical and Computational Fluid Dynamics

, Volume 33, Issue 2, pp 215–234 | Cite as

Shallow flow of an inhomogeneous incompressible fluid down an inclined plane

  • Lorenzo FusiEmail author
Original Article

Abstract

In this paper, we study a model for the shallow flow of an inhomogeneous incompressible fluid down an inclined plane. The constitutive response of this class of fluids has been discussed in Málek and Rajagopal (Mech Mater 38(3):233–242, 2006). We consider a planar geometry in which the length scale ratio is small so that the lubrication approximation can be applied. We analyze two different flow regimes: in the first one, the viscous and the gravitational forces are balanced and the system is governed by a hyperbolic equation for the fluid film thickness. In the second regime, the frictional forces are balanced by the pressure gradient and the system is governed by a parabolic equation for the film thickness. For both regimes, we perform numerical simulations that show the evolution of the film thickness using some general data that are well suited for highlighting the main features of the flow.

Keywords

Incompressible fluids Inhomogeneous fluids Variable viscosity Lubrication approximation 

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Notes

References

  1. 1.
    Chandran, K.B., Yoganathan, A.P., Rittgers, S.E.: Biofluid Mechanics. The Human Circulation. CRC Press, Boca Raton (2007)Google Scholar
  2. 2.
    Cristescu, N.D., Cazacu, O., Cristescu, C.: A model for slow motion of natural slopes. Can. Geotech. J. 39, 924–937 (2002)CrossRefGoogle Scholar
  3. 3.
    Fowler, A.C., Larson, D.A.: On the flow of polythermal glaciers, I. Model and preliminary analysis. Proc. R. Soc. Lond. A 363(1713), 217–242 (1978)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Fusi, L.: Two-dimensional thin-film flow of an incompressible inhomogeneous fluid in a channel. J. Non-Newton. Fluid Mech. 260, 87–100 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fusi, L., Farina, A., Rosso, F., Rajagopal, K.: Thin-film flow of an inhomogeneous fluid with density-dependent viscosity. Fluids 4(1), 30 (2019).  https://doi.org/10.3390/fluids4010030
  6. 6.
    Hutter, K.: Time-dependent surface elevation of an ice slope. J. Glaciol. 25, 247–266 (1980)CrossRefGoogle Scholar
  7. 7.
    Korteweg, D.J.: Sur la forme que prenent les équations du mouvements des fluides si l’on tient compte des forces capilaires causées par des variations de densité considérables mains continues et sur la théorie de la capillarité dans l’hypothése dd’une varation continue de la densité. Arch. Néerl Sci. Exactes Nat. Ser. II(6), 1–24 (1901)zbMATHGoogle Scholar
  8. 8.
    Knoch, D., Malcherek, A.: A numerical model for simulation of fluid mud with different rheological behaviors. Ocean Dyn. 61, 245–256 (2011)CrossRefGoogle Scholar
  9. 9.
    Málek, J., Rajagopal, K.R.: On the modeling of inhomogeneous incompressible fluid-like bodies. Mech. Mater. 38(3), 233–242 (2006)CrossRefGoogle Scholar
  10. 10.
    Nye, J.F.: The response of glaciers and ice-sheets to seasonal and climatic changes. Proc. R. Soc. Lond. A 256(1287), 559–584 (1960)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Oron, A., Davis, S.H., Bankoff, S.G.: Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980 (1997)CrossRefGoogle Scholar
  12. 12.
    Otsuki, M., Hayakawa, H., Luding, S.: Behavior of pressure and viscosity at high densities for two-dimensional hard and soft granular materials. Prog. Theor. Phys. Suppl. 184, 110–133 (2010)CrossRefzbMATHGoogle Scholar
  13. 13.
    Pritchard, D., Duffy, B.R., Wilson, S.K.: Shallow flows of generalised Newtonian fluids on an inclined plane. J. Eng. Math. 94(1), 115–133 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ruyer-Quil, C., Manneville, P.: Modeling film flows down inclined planes. Eur. Phys. J. B 6, 277–292 (1998)CrossRefzbMATHGoogle Scholar
  15. 15.
    Rajagopal, K.R., Srinivasa, A.: On the thermomechanics of shape memory wires, Zeitschrift für angewandte Mathematik und Physik. ZAMP 50(3), 140–459-496 (1999)Google Scholar
  16. 16.
    Rajagopal, K.R., Srinivasa, A.: A thermodynamic frame work for rate type fluid models. J. Non-Newton. Fluid Mech. 88(3), 207–227 (2000)CrossRefzbMATHGoogle Scholar
  17. 17.
    Rajagopal, K.R., Srinivasa, A.: Modeling anisotropic fluids within the frame-work of bodies with multiple natural configurations. J. Non-Newton. Fluid Mech. 99(2), 109–124 (2001)CrossRefzbMATHGoogle Scholar
  18. 18.
    Rao, I., Rajagopal, K.R.: A study of strain-induced crystallization of polymers. Int. J. Solids Struct. 38(6), 1149–1167 (2001)CrossRefzbMATHGoogle Scholar
  19. 19.
    Saccomandi, G., Vergori, L.: Piezo-viscous flow over an inclined surface. Q. Appl. Math. 68(4), 747–763 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics. Springer, Berlin (1965)zbMATHGoogle Scholar
  21. 21.
    Truesdell, C.: A First Course in Rational Mechanics. Academic Press, Boston (1991)zbMATHGoogle Scholar
  22. 22.
    Wu, W.T., Aubry, N., Massoudi, M.: Channel flow of a mixture of granular materials and a fluid. In: Fluids Engineering Systems and Technologies, vol. 7A (2013)Google Scholar
  23. 23.
    Cannon, J.R.: The One-Dimensional Heat Equation. Cambridge University Press, Cambridge (1984)CrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Informatica “U. dini”Università degli Studi di FirenzeFirenzeItaly

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