Evolution of Thin Liquid Film for Newtonian and Power-Law Non-Newtonian Fluids

Document Type : Article


1 Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran

2 Foolad Institute of Technology, Fooladshahr, Isfahan, 84916-63763, Iran


Analytical relation for slow motion thin liquid films bounded by a fixed wall and free surface for Newtonian and non-Newtonian fluids are obtained in this work. Assuming long-wave approximation, the momentum and continuity equations for thin liquid films of power-law fluids are simplified and solved analytically to derive the evolution equation of thin liquid films. The evolution equation is derived for two- and three-dimensional cases. A relation for evolution of thin films is obtained for a simple case in which the liquid film is supported from below by a solid surface and subjected to gravity and constant surface tension forces. This evolution equation of thin film has been solved numerically in order to compare the behavior of Newtonian and non-Newtonian liquids for different Bond numbers. It is shown that the power-law model at low and high strain rates is invalid and it affects the results. The Rayleigh-Taylor instability is another subject that is studied in this work. This interesting phenomena is investigated by solving the evolution equation numerically for different Bond numbers. The results show that the evolution of the free surface thin film for pseudo plastic fluids is different from that of Newtonian and dilatant fluids


Main Subjects


1. Gad-el-Hak, M., The MEMS Handbook, CRC Press (2001).
2. Oron, A., Davis, S.H. and Banko , S.G. \Long-scale
evolution of thin liquid lms", Rev. Modern Phys., 68,
pp. 931-980 (1997).
3. Yiantsios, S.G. and Higgins, B.G. \Rayleigh-Taylor
instability in thin viscous lms", Phys. Fluids, 1, pp.
1484-1501 (1989).
4. Yiantsios, S.G. and Higgins, B.G. \Rapture of thin
lms: nonlinear stability analysis", J. Colloid Interface
Sci., 147, pp. 341-350 (1991).
5. Oron, A. and Rosenau, P. \Formation of patterns
induced by thermocapillarity and gravity", J. Phy.
(France), 2, pp. 131-146 (1992).
6. Fermigier, M., Limat, L., Wesfreid, J.E., Boudient,
P. and Quilliet, C. \Two-dimensional patterns in
Rayleigh-Taylor instability of a thin layer", J. Fluid
Mech., 236, pp. 349-383 (1992).
7. Ruyer-Quil, C. and Manneville, P. \Improved modeling
ows down inclined planes", Eur. Phys. J. B,
15, pp. 357-369 (2000).
8. Houseman, G.A. and Molnar, P. \Gravitational
(Rayleigh-Taylor) instability of a layer with non-linear
viscosity and convective thinning of continental lithosphere",
Int. J. Geophys., 128, pp. 125-150 (1997).
278 R. Nasehi and E. Shirani/Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 266{279
9. Fomin, S., Watterson, J. and Raghunathan, S. \The
run-o condition for rimming
ow of a power-law

uid", Theoret. Comput. Fluid Dynamics, 15, pp. 83-
94 (2001).
10. Perazzo, C.A. and Gratton, J. \Thin lm of non-
uid on an incline", Phusical Review, 67,
pp. 1-6 (2003).
11. Perazzo, C.A. and Gratton, J. \Steady and traveling

ows of a power-law liquid over an incline", J. Non-
Newtonian Fluid Mech., 118, pp. 57-64 (2004).
12. Balmforth, N., Ghadge, Sh. and Myers, T. \Surface
tension driven ngering of a viscoplastic lm", J. Non-
Newtonian Fluid Mech., 142, pp. 143-149 (2007).
13. Miladinova, S., Lebon, G. and Toshev, E. \Thin- lm

ow of a power-law liquid falling down an inclined
plate", J. Non-Newtonian Fluid Mech., 122, pp. 69-
78 (2004).
14. Myers, T.G. \Application of non-Newtonian models to
thin lm
ow", Physical Review, 72, pp. 1-11 (2005).
15. Perazzo, C.A. and Gratton, J. \Exact solutions for
two-dimensional steady
ows of a power-law liquid on
an incline", Phys. Fluids, 17, pp. 1-8 (2005).
16. Heining, C. and Aksel, N. \E ects of inertia and
surface tension on a power-law
owing down a
wavy incline", Int. J. Multiphase Flow, 36, pp. 847-
857 (2010).
17. Hu, B. and Kieweg, S.L. \The e ect of surface tension
on the gravity-driven thin lm
ow of Newtonian and
uids", Computers & Fluids, 64, pp. 83-90