Abstract
An exponential traveling-wave solution of the free surface equation for a viscous film flowing down an inclined plane is presented. The free surface equation, valid for long waves of finite amplitude, is obtained from the general equation for waves of arbitrary amplitude, which is developed by the method of multiple-scales and considering a series solution expanded at two orders higher than the series used by Lin in 1974. Thus, confining ourselves to the case of weakly nonlinear wave motion, we use the Ince transformation method to find an exponential travelling-wave solution that depend of the physical parameters of the viscous fluid , the angle θ of the inclined plane and the wave parameters, which are subjected to three constraints between them. This solution show a free surface with a dynamic gap of thickness between the top and the bottom of the inclined plane. Also, we find a particular numerical example, for laminar flow, that may represent the evolution of the free surface h(x,t) of a viscous film flowing down an inclined plane, when a small quantity, of the same fluid, is added constantly to the primary fluid in the top of the inclined plane. An interesting result of this case show that the final state, when t → ∞, is also stationary and has the same behavior as the unperturbed primary flow, however the final layer of fluid has a greater thickness.
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