Long interfacial waves inside an inclined permeable channel

The time evolution of superposed layers of fluid flowing down inside an inclined permeable channel is investigated. Using the Kármán-Pohlhausen approximation, the problem is reduced to the study of the evolution equation for the liquid–liquid interface of the liquids film derived through a long-wave approximation. A linear stability analysis of the base flow is performed. The solutions and stability of the non-linear stationary long waves are investigated. A special form of the stationary long waves (say Shkadov waves) is introduced.

► We have studied the stability and evolution of long waves among superposed Newtonian films inside an inclined permeable channel.
► Using Kármán-Pohlhausen approximation, an evolution equation for the lower film thickness profile was derived.
► The Hopf bifurcation is a dominate state in some cases. We have named some cases of a stationary profile as the Shkadov wave.
► The wave is a homoclinic trajectory in the phase space in some cases.