Theoretical studies of film flows began with the classical work of Nusselt where he obtained exact solutions for Navier-Stokes equations for a thin layer of viscous liquid free falling down a smooth vertical wall. Further theoretical and experimental investigations have demonstrated that the Nusselt solution is not achieved in practice and, as a rule, the film surface is covered with waves. The problem of nonlinear waves in the film falling down a smooth plate has much in common with that of a steady-state viscous layer flow along a corrugated surface. In both cases the equations are significantly non-linear, the free surface is previously unknown, the surface tension forces play great role and there exists a spatial period. At the present time, there are several theoretical [1-3] and only one experimental [4] work specifically devoted to film flows along corrugated surfaces. The authors of [1] investigated film flow over a surface with small corrugation amplitude in compare with the Nusselt thickness. The authors of [2] and [3] considered a creeping flow when the inertia and surface tension forces were ignored.

The purpose of the present paper is to study hydrodynamics of the film flow down vertically over both one-dimensional and two-dimensional corrugated surfaces. Viscosity, inertia and surface tension forces were taken into account. The calculations have been carried out on base of both Navier-Stokes and integral equations. Different shape of corrugations was considered and their amplitude was comparable with the Nusselt’s film thickness. In the case of two-dimensional vertical corrugated surface the inclination angle of ribs is additional parameter and at zero value of the parameter we have one-dimensional flow. As a result it has been obtained that, for example, at small Reynolds numbers area of "thick" film forms in the surface cavity and area of "thin" film forms on the top. While increasing the Reynolds number the free surface "straightens" and becomes parallel to gravity vector. Average film thickness is always greater than Nusselt thickness and, as Re decreases, this difference increases due to the presence of "thick" films in the cavities. Stagnation zones are found and their transformation with increasing of Reynolds number is investigated. In the case of two-dimensional surface the critical inclination angle was obtained when the steady-state film flow solution with completely wetted wall surface existed.

It is well known that the viscous liquid film falling down a vertical smooth plate is unstable at any value of Reynolds number. Both in theory and experiment there are long-wavy disturbances of the film free surface which are increasing in time and lead to the formation of nonlinear structures. In the present paper both the linear and nonlinear stability of the film flow down the vertical corrugated surface with respect to the free surface disturbances is investigated. Amplitude and period of the corrugation were comparable with the Nusselt's film thickness and the capillary constant, respectively. Stability of Nusselt's smooth solution is the limit of our problem when amplitude of corrugation was zero. Linearization of the governing equations gives us the eigenvalue problem with periodic coefficients. We used Floke's theorem and numerical procedure for the calculations of the eigenvalues. As a result the range of the parameters of corrugations where we have no unstable disturbances has been obtained. Neutral curves are found and the most amplified disturbances are evaluated. As a result of unstable disturbances evolution the new regimes of flow over corrugated surface are formed. It is shown in the paper that the new regimes are represented by the double Fourier series and ones are calculated numerically. It is obtained that the corrugations stabilize the film flow and amplitude of waves is much smaller than that of in the case of wavy flow over vertical smooth surface at the same Reynolds number.

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REFERENCES

1. Wang, C. Y. 1981 Liquid Film Flowing Slowly Down a Wavy Incline, AlChE J. 27, 207.

2. Pozrikidis, C. 1988 The Flow of a Liquid Film Along a Periodic Wall, J. Fluid Mech, 188, 275.

3. Shetty, S., Cerro, R. L. 1993 Flow of a Thin Film Over a Periodic Surface, Int. J. Multiphase Flow, 19, #6,1013.

4. Zhao, L., Cerro, R. L. 1992 Experimental Characterization of Viscous Film Flows Over Complex Surfaces, Int. J. Multiphase Flow, 18 #6, 495.