Modulation Equations for Roll Waves on Vertically Falling Films of a Power-Law Fluid

Affiliation(s)

Laboratoire de Mecanique de Lille, UMR CNRS 8107, Villeneuve d’Ascq, France.

Lavrentyev Institute of Hydrodynamics, Novosibirsk State University, Novosibirsk, Russia.

Laboratoire de Mecanique de Lille, UMR CNRS 8107, Villeneuve d’Ascq, France.

Lavrentyev Institute of Hydrodynamics, Novosibirsk State University, Novosibirsk, Russia.

ABSTRACT

Waves of finite amplitude on a thin layer of non-Newtonian fluid modelled as a power-law fluid are considered. In the long wave approximation, the system of equations taking into account the viscous and nonlinear effects has the hyper- bolic type. For the two-parameter family of periodic waves in the film flow on a vertical wall the modulation equations for nonlinear wave trains are derived and investigated. The stability criterium for roll waves based on the hyperbolicity of the modulation equations is suggested. It is shown that the evolution of stable roll waves can be described by self-similar solutions of the modulation equations.

Waves of finite amplitude on a thin layer of non-Newtonian fluid modelled as a power-law fluid are considered. In the long wave approximation, the system of equations taking into account the viscous and nonlinear effects has the hyper- bolic type. For the two-parameter family of periodic waves in the film flow on a vertical wall the modulation equations for nonlinear wave trains are derived and investigated. The stability criterium for roll waves based on the hyperbolicity of the modulation equations is suggested. It is shown that the evolution of stable roll waves can be described by self-similar solutions of the modulation equations.

KEYWORDS

Power-Law Fluid; Thin Film Flow; Vertical Wall; Modulation Equations; Nonlinear Stability; Roll Wave

Power-Law Fluid; Thin Film Flow; Vertical Wall; Modulation Equations; Nonlinear Stability; Roll Wave

Cite this paper

nullA. Boudlal and V. Liapidevskii, "Modulation Equations for Roll Waves on Vertically Falling Films of a Power-Law Fluid,"*World Journal of Mechanics*, Vol. 2 No. 1, 2012, pp. 1-8. doi: 10.4236/wjm.2012.21001.

nullA. Boudlal and V. Liapidevskii, "Modulation Equations for Roll Waves on Vertically Falling Films of a Power-Law Fluid,"

References

[1] C. O. Ng and C. C. Mei, “Roll Waves on Shallow Layer of Mud Modelled as a Power-Law Fluid,” Journal of Fluid Mechanics, Vol. 263, 1994, pp. 151-1834.

[2] T. Jeffreys, “The Flow of Water in Inclined Channel of Rectangular Section,” Philosophical Magazine, Vol. 49, No. 6, 1925, pp. 793-807.

[3] F. Dressler, “Mathematical Solution of the Problem of Roll Waves in Inclined Open Channels,” Communications on Pure and Applied Mathematics, Vol. 2, No. 2-3, 1949, pp. 149-194. doi:10.1002/cpa.3160020203

[4] A. Boudlal and V. Yu. Liapidevskii “Stability of Roll Waves in Open Channels,” Comptes Rendus Mécanique, Vol. 330, No. 4, 2002, pp. 209-295. doi:10.1016/S1631-0721(02)01461-4

[5] S. V. Alekseenko, V. E. Nakoryakov and B. G. Pokusaev, “Wave Flow in Liquid Films,” Begell House, New York, 1994.

[6] V. A. Buchin and G. A. Shaposhnikova, “Flow of Shallow Water with a Periodic System of Jumps over a Vertical Surface,” Doklady Physics, Vol. 54, No. 5, 2009, pp. 248-251. doi:10.1134/S1028335809050073

[7] P. Y. Julien and D. M. Hartley, “Formation of Roll Waves in Laminar Sheet Flow,” Journal of Hydraulic Research, Vol. 24, No. 1, 1986, pp. 5-17. doi:10.1080/00221688609499329

[8] A. Boudlal and V. Yu. Liapidevskii, “Stability of Roll Waves on a Vertical Wall,” International Conference “Fluxes and Structures in Fluids: Physics of Geospheres”, Moscow, 2010, pp. 48-53.

[9] A. Boudlal and V. Yu. Liapidevskii, “Stability of Regular Roll Waves,” Computation Technologies, Vol. 10, No. 2, 2004, pp. 3-14.

[10] B. L. Rozhdestvenskii and N. N. Janenko, “Systems of Quasilinear Equations and Their Application to Gaz Dynamics,” American Mathematical Society Translations: Series 2, Vol. 55, 1983.

[11] G. B. Whitham, “Linear and Nonlinear Waves,” John Wiley and Sons, New York, 1974.

[12] V. Ya. Shkadov, “Theory of Wave Flows of a Thin Layer of a Viscous Liquid,” Izvestiya Akademii Nauk SSSR Mekhanika Zhidkosti I Gaza, Vol. 2, 1968, pp. 20-25.

[13] C. E. Mesa and V. Balakotaiah, “Modelling and Experimental Studies of Large Amplitude Waves on Vertically Falling Films,” Chemical Engineering Science, Vol. 63, No. 19, 2008, pp. 4704-4734. doi:10.1016/j.ces.2007.12.030

[1] C. O. Ng and C. C. Mei, “Roll Waves on Shallow Layer of Mud Modelled as a Power-Law Fluid,” Journal of Fluid Mechanics, Vol. 263, 1994, pp. 151-1834.

[2] T. Jeffreys, “The Flow of Water in Inclined Channel of Rectangular Section,” Philosophical Magazine, Vol. 49, No. 6, 1925, pp. 793-807.

[3] F. Dressler, “Mathematical Solution of the Problem of Roll Waves in Inclined Open Channels,” Communications on Pure and Applied Mathematics, Vol. 2, No. 2-3, 1949, pp. 149-194. doi:10.1002/cpa.3160020203

[4] A. Boudlal and V. Yu. Liapidevskii “Stability of Roll Waves in Open Channels,” Comptes Rendus Mécanique, Vol. 330, No. 4, 2002, pp. 209-295. doi:10.1016/S1631-0721(02)01461-4

[5] S. V. Alekseenko, V. E. Nakoryakov and B. G. Pokusaev, “Wave Flow in Liquid Films,” Begell House, New York, 1994.

[6] V. A. Buchin and G. A. Shaposhnikova, “Flow of Shallow Water with a Periodic System of Jumps over a Vertical Surface,” Doklady Physics, Vol. 54, No. 5, 2009, pp. 248-251. doi:10.1134/S1028335809050073

[7] P. Y. Julien and D. M. Hartley, “Formation of Roll Waves in Laminar Sheet Flow,” Journal of Hydraulic Research, Vol. 24, No. 1, 1986, pp. 5-17. doi:10.1080/00221688609499329

[8] A. Boudlal and V. Yu. Liapidevskii, “Stability of Roll Waves on a Vertical Wall,” International Conference “Fluxes and Structures in Fluids: Physics of Geospheres”, Moscow, 2010, pp. 48-53.

[9] A. Boudlal and V. Yu. Liapidevskii, “Stability of Regular Roll Waves,” Computation Technologies, Vol. 10, No. 2, 2004, pp. 3-14.

[10] B. L. Rozhdestvenskii and N. N. Janenko, “Systems of Quasilinear Equations and Their Application to Gaz Dynamics,” American Mathematical Society Translations: Series 2, Vol. 55, 1983.

[11] G. B. Whitham, “Linear and Nonlinear Waves,” John Wiley and Sons, New York, 1974.

[12] V. Ya. Shkadov, “Theory of Wave Flows of a Thin Layer of a Viscous Liquid,” Izvestiya Akademii Nauk SSSR Mekhanika Zhidkosti I Gaza, Vol. 2, 1968, pp. 20-25.

[13] C. E. Mesa and V. Balakotaiah, “Modelling and Experimental Studies of Large Amplitude Waves on Vertically Falling Films,” Chemical Engineering Science, Vol. 63, No. 19, 2008, pp. 4704-4734. doi:10.1016/j.ces.2007.12.030