Wave-flow theory for a thin viscous liquid layer

On the basis of a simplified system of equations we study the process of development and stability of wave flows in a thin layer of a viscous liquid. Any unstable disturbance of the laminar flow grows and leads to the establishment of the wave regime. The time to establish the flow changes little for large flow rates, but increases sharply with reduction of the flow rate. Given the same amplitudes of the initial disturbances, the optimum regimes which provide the greatest flow rate in a layer of given average thickness develop more rapidly than the other regimes. All the wave regimes are unstable to disturbances in the form of traveling waves. With moderate flow rates, the optimum regimes will be most stable to near-by disturbances.Strictly periodic wave flows in a thin layer of a viscous liquid under the influence of the gravity force were calculated in [1], Various flow wave regimes which differ in wavelength can theoretically be established for a given liquid flow rate. In particular, there is a wavelength for which the flowing layer exhibits minimum average thickness (and maximum flow rate for a given average thickness). These optimum regimes correspond closely to the experimental data [2]; however, with regard to calculation technique these regimes are no different from the others. In the following we make a comparison of the wave regimes on the basis of the nature of their development and stability.