Fluid Dynamics Research 29 (2001) 199–220 Waves on a lm of power-law uid owing down an inclined plane at moderate Reynolds number Bhabani Shankar Dandapat ∗ , Asim Mukhopadhyay Physics and Applied Mathematics Unit, Indian Statistical Institute, 203, B. T. Road, Calcutta-700 035, India Received 18 October 2000; received in revised form 10 March 2001; accepted 30 June 2001 Abstract Waves that occur at the surface of a power-law uid lm owing down an inclined plane are investigated. Using the method of integral relations, an evolution equation is derived for two types of wave equations which are possible under long wave approximation. This equation is valid for moderate Reynolds numbers and reveals the presence of both kinematic and dynamic wave processes which may either act together or singularly dominate the wave eld depending on the order of dierent parameters. It is shown that, at a small ow rate, kinematic waves dominate the ow eld and it acquires energy from the mean ow, while, for high ow rate, inertial waves dominate and the energy comes from the kinematic waves. This energy transfer from kinematic waves to inertial waves depends on the power-law index n. Linear stability analysis predicts the contribution of dierent terms in the wave mechanism. Further, it is found that surface tension plays a double role, for the kinematic wave process, it exerts dissipative eects so that a nite amplitude case may be established, but for the dynamic wave process it yields dispersion. The evolution equation is capable of predicting amplitudes, shapes, and interaction at the nite amplitude level. It is also shown that the results of the interaction may lead either to forward breaking waves or solitary waves with dark soliton depending on the ow rate, Weber number and the angle of inclination with the horizon. Power-law index n plays a vital role in the wave mechanism. c 2001 Published by The Japan Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved. Keywords: Power-law uid lm; Waves on falling lm; Stability of power-law uid lm; KdV waves; Forward breaking waves 1. Introduction Flow of thin liquid lm on an inclined plane has drawn the attention of studies since the last ve decades due to its various applications in the technological development of modern science. Linear stability of long waves on a layer of viscous uid owing down an inclined plane was investigated by Yih (1963), who found the critical Reynolds number by a regular perturbation method. Prior to this study, Benjamin (1957) approximated the eigenfunction in the Orr–Sommerfeld equation governing * Corresponding author. Fax: +91-33-577-6680. E-mail addresses: dandapat@isical.ac.in (B.S. Dandapat); asim tf@isical.ac.in (A. Mukhopadhyay). 0169-5983/01/$20.00 c 2001 Published by The Japan Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved. PII:S0169-5983(01)00024-7