The long wave fluid flows on inclined porous media with nonlinear Forchheimer's law

The surface fluid flows coupled with porous media flows in substrates occur in many circumstances in industry and natural settings. In this paper, we investigate the long wave solutions for the surface flows on inclined porous media. The important feature is that such flows are derived by the Navier-Stokes equations governing the clear flows in the surface fluids and the nonlinear Forchheimer's equations for the porous media flows in substrates. The problem is reduced to a corresponding Orr-Sommerfeld problem by linearizing the infinitesimal perturbations in the system of coupled equations for analyzing long wave solutions of surface flows. Numerical analysis is taken by using Chebyshev collocation numerical method to the eigenvalue problems of the Orr-Sommerfeld systems for analyzing critical condition and stable region of long wave solutions. We compare the result with that for very small drag constant by Darcy's law and study numerically the effects of parameters including various drag constants on the long wave solutions with Forchheimer's law.