Skip to main content
×
Home
    • Aa
    • Aa

Flow of fluids with pressure- and shear-dependent viscosity down an inclined plane

  • K. R. Rajagopal (a1), G. Saccomandi (a2) and L. Vergori (a3)
Abstract
Abstract

In this paper we consider a fluid whose viscosity depends on both the mean normal stress and the shear rate flowing down an inclined plane. Such flows have relevance to geophysical flows. In order to make the problem amenable to analysis, we consider a generalization of the lubrication approximation for the flows of such fluids based on the development of the generalization of the Reynolds equation for such flows. This allows us to obtain analytical solutions to the problem of propagation of waves in a fluid flowing down an inclined plane. We find that the dependence of the viscosity on the pressure can increase the breaking time by an order of magnitude or more than that for the classical Newtonian fluid. In the viscous regime, we find both upslope and downslope travelling wave solutions, and these solutions are quantitatively and qualitatively different from the classical Newtonian solutions.

Copyright
Corresponding author
Email address for correspondence: luigi.vergori@unisalento.it
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

1. C. Ancey 2007 Plasticity and geophysical flows: a review. J. Non-Newtonian Fluid Mech. 142, 435.

2. E. C. Andrade 1934 Theory of viscosity of liquids. Phil. Mag. 17, 497698.

5. A. L. Bertozzi & M. Shearer 2000 Existence of undercompressive traveling waves in thin film equations. SIAM J. Math. Anal. 32, 194213.

8. M. Bulicek , J. Malek & K. R. Rajagopal 2009 Mathematical analysis of unsteady flows of fluids with pressure, shear-rate and temperature dependent material moduli, that slip at solid boundaries. SIAM J. Math. Anal. 41, 665707.

9. A. Carasso & M.-C. Shen 1977 On viscous fluid flow down an inclined plane and the development of roll waves. SIAM J. Appl. Math. 33, 399426.

11. R. F. Dressler 1949 Mathematical solution of the problem of roll-waves in inclined open channels. Commun. Pure Appl. Math. 2, 149194.

13. S. J. Friedman & C. O. Miller 1941 Liquid films in the viscous flow region. Ind. Engng Chem. 33, 885891.

17. H. E. Huppert 1982a Flow and instability of a viscous current down a slope. Nature 300, 427429.

19. Y. P. Ivanilov 1962 Rolling waves in an inclined channel. USSR Comput. Math. Math. Phys. 1, 12351252.

20. H. Jeffreys 1925 The flow of water in an inclined channel of rectangular section. Phil. Mag. 49, 793807.

21. S. J. Jones & H. A. M. Chew 1983 Creep of ice as a function of hydrostatic pressure. J. Phys. Chem. 87 (21), 40644066.

22. G. H. Keulegan & G. W. Patterson 1940 A criterion for instability in steep channels. Trans. AGU Part II 594596.

23. C. G. Kirkbride 1934a Heat transfer by condensing vapor on vertical tubes. Ind. Engng Chem. 26, 425428.

25. L. Kondic & J. Diez 2001 Pattern formation in gravity driven flow of thin films: constant flux flow. Phys. Fluids 13, 31683184.

30. W. S. B. Paterson 1994 The Physics of Glaciers, third edition. Pergamon.

31. C. A. Perazzo & J. Gratton 2003 Thin film of non-Newtonian fluid on an incline. Phys. Rev. E 67, 016307 1–6.

34. K. R. Rajagopal & A. R. Srinivasa 2005 On the nature of constraints for continua undergoing dissipative processes. Proc. R. Soc. Lond. A 461, 27852795.

35. K. R. Rajagopal & A. Z. Szeri 2003 On an inconsistency in the derivation of the equations of elastohydrodynamic lubrication. Proc. R. Soc. A 459, 27712786.

36. G. Saccomandi & L. Vergori 2010 Piezo-viscous flows over an inclined surface. Q. Appl. Math. 68, 747763.

37. C. Schoof & R. C. A. Hindmarsh 2010 Thin-film flows with wall slip: an asymptotic analysis of higher order glacier flow models. Q. J. Mech. Appl. Math. 63, 73114.

38. N. Silvi & E. B. Dussan 1985 On the rewetting of an inclined solid surface by a liquid. Phys. Fluids 28, 57.

42. C. S. Yih 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321334.

43. C. S. Yih 1965 Stability of a non-Newtonian liquid film flowing down an inclined plane. Phys. Fluids 8, 12571262.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 10 *
Loading metrics...

Abstract views

Total abstract views: 127 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 27th May 2017. This data will be updated every 24 hours.