Waves on a film of power-law fluid flowing down an inclined plane at moderate Reynolds number

Waves that occur at the surface of a power-law fluid film flowing down an i nclined plane are investigated. Using the method of integral relations, an evolution equation is derived for two types of wave equations which are pos sible under long wave approximation. This equation is valid for moderate Re ynolds numbers and reveals the presence of both kinematic and dynamic wave processes which may either act together or singularly dominate the wave fie ld depending on the order of different parameters. It is shown that, at a s mall flow rate, kinematic waves dominate the flow field and it acquires ene rgy from the mean flow, while, for high flow rate, inertial waves dominate and the energy comes from the kinematic waves. This energy transfer from ki nematic waves to inertial waves depends on the power-law index n. Linear st ability analysis predicts the contribution of different terms in the wave m echanism. Further, it is found that surface tension plays a double role, fo r the kinematic wave process, it exerts dissipative effects so that a finit e amplitude case may be established, but for the dynamic wave process it yi elds dispersion. The evolution equation is capable of predicting amplitudes , shapes, and interaction at the finite amplitude level. It is also shown t hat the results of the interaction may lead either to forward breaking wave s or solitary waves with dark soliton depending on the flow rate, Weber num ber 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.