Abstract
The linearized stability problem for a steady parallel flow of a thin viscous fluid layer flowing down an inclined plane has been investigated by several authors. When the volume flux exceeds a critical value, the steady parallel flow is known to be unstable. In order to explain the development of steady surface waves on such a flow, finite-amplitude effects must be considered. The growth of a linear unstable periodic perturbation and its nonlinear interaction with higher harmonics are studied. For small volume fluxes and fluids with sufficiently strong surface tension, such as water and alcohol, it is shown that for a flow down a vertical plane a steady finite-amplitude surface wave will develop. The transient development of the finite-amplitude waves is investigated by numerical integration. Numerical values for wave velocities and amplitudes are given. The dependence of the wave form on the wavelength of the initial perturbation is also discussed.
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