Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-12-01T02:32:27.398Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  06 July 2012

K. R. Rajagopal ,
G. Saccomandi  and
L. Vergori
K. R. Rajagopal
Affiliation:
Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA
G. Saccomandi
Affiliation:
Dipartimento di Ingegneria Industriale, Università di Perugia, via G. Duranti, 06125, Italy
L. Vergori*
Affiliation:
Dipartimento di Matematica, Università del Salento, Strada Prov. Lecce-Arnesano, 73100 Lecce, Italy
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

JFM classification

Low-Reynolds-number flows: Lubrication theory Geophysical and Geological Flows: Shallow water flows Waves/Free-surface Flows: Wave breaking
Type
Papers
Information
Journal of Fluid Mechanics , Volume 706 , 10 September 2012 , pp. 173 - 189
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Ancey, C. 2007 Plasticity and geophysical flows: a review. J. Non-Newtonian Fluid Mech. 142, 4–35.CrossRefGoogle Scholar
2. Andrade, E. C. 1934 Theory of viscosity of liquids. Phil. Mag. 17, 497–698.CrossRefGoogle Scholar
3. Barus, C. 1893 Isotherms, isopiestics and isometrics relative to viscosity. Am. J. Sci. 45, 87–96.CrossRefGoogle Scholar
4. Benjamin, T. B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid. Mech. 2, 554–574.CrossRefGoogle Scholar
5. Bertozzi, A. L. & Shearer, M. 2000 Existence of undercompressive traveling waves in thin film equations. SIAM J. Math. Anal. 32, 194–213.CrossRefGoogle Scholar
6. Binnie, A. M. 1957 Experiments on onset of wave formation on films of water flowing down a vertical plane. J. Fluid. Mech. 2, 551–553.CrossRefGoogle Scholar
7. Bridgman, P. W. 1931 The Physics of High Pressure. Macmillan.Google Scholar
8. Bulicek, M., Malek, J. & Rajagopal, K. R. 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, 665–707.CrossRefGoogle Scholar
9. Carasso, A. & Shen, M.-C. 1977 On viscous fluid flow down an inclined plane and the development of roll waves. SIAM J. Appl. Math. 33, 399–426.CrossRefGoogle Scholar
10. Dowson, D. & Higginson, G. R. 1966 Elastohydrodynamic Lubrication: The Fundamentals of Roller and Gear Lubrication. Pergamon.Google Scholar
11. Dressler, R. F. 1949 Mathematical solution of the problem of roll-waves in inclined open channels. Commun. Pure Appl. Math. 2, 149–194.CrossRefGoogle Scholar
12. Dukler, A. E. & Berkelin, O. P. 1952 Characteristics of flow in falling fluid films. Chem. Engng Prog. 48, 557–563.Google Scholar
13. Friedman, S. J. & Miller, C. O. 1941 Liquid films in the viscous flow region. Ind. Engng Chem. 33, 885–891.CrossRefGoogle Scholar
14. Gauss, C. F. 1829 Ueber ein allgemeines Grundgesetz der Mechanik. J. Reine Angew. Math. 4, 232–235.Google Scholar
15. Grimley, S. S. 1945 Liquid flow conditions in packed towers. Trans. Inst. Chem. Engrs 23, 228–235.Google Scholar
16. Gupta, A. S. & Rai, L. 1967 Stability of an elastico-viscous liquid film flowing down an inclined plane. Proc. Camb. Phil. Soc. Math. Phys. Sci. 63, 527–536.CrossRefGoogle Scholar
17. Huppert, H. E. 1982a Flow and instability of a viscous current down a slope. Nature 300, 427–429.CrossRefGoogle Scholar
18. Huppert, H. E. 1982b The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.CrossRefGoogle Scholar
19. Ivanilov, Y. P. 1962 Rolling waves in an inclined channel. USSR Comput. Math. Math. Phys. 1, 1235–1252.CrossRefGoogle Scholar
20. Jeffreys, H. 1925 The flow of water in an inclined channel of rectangular section. Phil. Mag. 49, 793–807.CrossRefGoogle Scholar
21. Jones, S. J. & Chew, H. A. M. 1983 Creep of ice as a function of hydrostatic pressure. J. Phys. Chem. 87 (21), 4064–4066.CrossRefGoogle Scholar
22. Keulegan, G. H. & Patterson, G. W. 1940 A criterion for instability in steep channels. Trans. AGU Part II 594–596.Google Scholar
23. Kirkbride, C. G. 1934a Heat transfer by condensing vapor on vertical tubes. Ind. Engng Chem. 26, 425–428.CrossRefGoogle Scholar
24. Kirkbride, C. G. 1934b Heat transfer by condensing vapours on vertical tubes. Trans. Am. Inst. Chem. Engrs 30, 170–186.Google Scholar
25. Kondic, L. & Diez, J. 2001 Pattern formation in gravity driven flow of thin films: constant flux flow. Phys. Fluids 13, 3168–3184.CrossRefGoogle Scholar
26. Man, C. S. & Sun, Q. X. 1987 On the significance of normal stresses effects in the flow of glaciers. J. Glaciol. 33, 268–273.CrossRefGoogle Scholar
27. Mayer, P. G. 1959 Roll waves and slug flows in inclined open channels. Proc. Am. Soc. Civ. Engrs 85, 99–141.Google Scholar
28. McTigue, D. F., Passman, S. L. & Jones, S. J. 1985 Normal stress effect in the creep of ice. J. Glaciol. 31, 120–126.CrossRefGoogle Scholar
29. Nusselt, W. 1916 Die Oberflachenkondensation des Wasserdampfes. Z. Verein. Deutsch. Ing. 60, 541–546, 569–575.Google Scholar
30. Paterson, W. S. B. 1994 The Physics of Glaciers, third edition. Pergamon.Google Scholar
31. Perazzo, C. A. & Gratton, J. 2003 Thin film of non-Newtonian fluid on an incline. Phys. Rev. E 67, 016307 1–6.CrossRefGoogle Scholar
32. Rajagopal, K. R. 2006 On implicit constitutive theories for fluids. J. Fluid Mech. 550, 243–249.CrossRefGoogle Scholar
33. Rajagopal, K. R. & Saccomandi, G. 2006 On internal constraints in continuum mechanics. Differ. Equ. Nonlinear Mech. 2006, 18572.Google Scholar
34. Rajagopal, K. R. & Srinivasa, A. R. 2005 On the nature of constraints for continua undergoing dissipative processes. Proc. R. Soc. Lond. A 461, 2785–2795.Google Scholar
35. Rajagopal, K. R. & Szeri, A. Z. 2003 On an inconsistency in the derivation of the equations of elastohydrodynamic lubrication. Proc. R. Soc. A 459, 2771–2786.CrossRefGoogle Scholar
36. Saccomandi, G. & Vergori, L. 2010 Piezo-viscous flows over an inclined surface. Q. Appl. Math. 68, 747–763.CrossRefGoogle Scholar
37. Schoof, C. & Hindmarsh, R. C. A. 2010 Thin-film flows with wall slip: an asymptotic analysis of higher order glacier flow models. Q. J. Mech. Appl. Math. 63, 73–114.CrossRefGoogle Scholar
38. Silvi, N. & Dussan, E. B. 1985 On the rewetting of an inclined solid surface by a liquid. Phys. Fluids 28, 5–7.CrossRefGoogle Scholar
39. Stokes, G. G. 1845 On the theories of the internal friction of fluids in motion, and motion of elastic solids. Trans. Camb. Phil. Soc. 8, 287–305.Google Scholar
40. Szeri, A. Z. 1998 Fluid Film Lubrication: Theory and Design. Cambridge University Press.CrossRefGoogle Scholar
41. Yih, C. S. 1955 Stability of parallel laminar flow with a free surface. (ed. Naghdi, P. M. ). In Proceedings of the Second U.S. National Congress of Applied Mechanics 623–628.Google Scholar
42. Yih, C. S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334.CrossRefGoogle Scholar
43. Yih, C. S. 1965 Stability of a non-Newtonian liquid film flowing down an inclined plane. Phys. Fluids 8, 1257–1262.CrossRefGoogle Scholar