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 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 field depending on the order of different parameters. It is shown that, at a small flow rate, kinematic waves dominate the flow field and it acquires energy from the mean flow, while, for high flow 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 different terms in the wave mechanism. Further, it is found that surface tension plays a double role, for the kinematic wave process, it exerts dissipative effects so that a finite 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 finite 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 flow rate, Weber number and the angle of inclination with the horizon. Power-law index n plays a vital role in the wave mechanism.