Stability of a layer of liquid flowing down an inclined plane

Summary The stability of the flow down an inclined plane is studied for small angles of inclination β. The same problem has been studied by S. P. Lin, however using an incorrect boundary condition. The correctly formulated eigenvalueproblem is solved by a numerical integration of the Orr-Sommerfeld equation employing the orthonormalization technique. It is shown that in the range 3′<β<1° a decrease in β means a decrease in the critical Reynolds number for the hard mode, which is a shear wave modified by the presence of the free surface. In that range the stability is still more or less governed by the stability of the soft waves, which are essentially surface waves modified by the presence of shear. For values of β<1′ the stability is governed by the hard mode, contradictory to Lin's statements. In that case instability occurs at high Reynolds numbers.