Modeling film flows down inclined planes

Abstract

A new model of film flow down an inclined plane is derived by a method combining results of the classical long wavelength expansion to a weighted-residuals technique. It can be expressed as a set of three coupled evolution equations for three slowly varying fields, the thickness h, the flow-rate q, and a new variable τ that measures the departure of the wall shear from the shear predicted by a parabolic velocity profile. Results of a preliminary study are in good agreement with theoretical asymptotic properties close to the instability threshold, laboratory experiments beyond threshold and numerical simulations of the full Navier-Stokes equations.

Inertial effects on thin-film wave structures with imposed surface shear on an inclined plane

“…Taking the surface of the film to have surface tension , a balance of forces on the interface between liquid and gas, when written in component form [17], leads to the normal stress boundary condition, hxx 1 +h 2…”

“…In some cases, breakdown of the numerical results can occur in the simulation of Shkadov models due to over-prediction of capillary wave amplitudes, resulting in very thin film thickness. The accuracy of the Shkadov model in predicting capillary ripples is shown to decrease with increasing Reynolds number (see figures 4 and 5 of [18]), with larger than actual capillary waves being predicted. Indeed, an essential assumption of this model concerns the number of degrees of freedom of polynomials used to approximate the crosswise distribution of the streamwise velocity.…”

“…Previous studies, for example [17,23], for shear driven flow on an inclined plane describe wave structures depending on their initial profiles as well as uniform upstream and downstream boundary conditions. Initial profiles were chosen to be of a two-tier configuration, with either the upstream or downstream film height the higher of the two.…”

“…It does not predict the Hopf bifurcation, necessary to foresee the formation of periodic waves on uniform thickness film flow on inclined planes [14]. Ruyer-Quil et al [17] claim that fluid films in a moving frame of reference flowing down an inclined plane are unstable against waves when their thickness becomes larger than a specified threshold value h Nc = (3 R c ) 1 3 , R c being a critical Reynolds number. The critical Reynolds number is given by Cheng and Chang [6] as R c = cot ✓ who claim that periodic forcing applied at the inlet leads to disturbances propagating downstream whilst growing in amplitude -this happens at Reynolds numbers larger than critical [16].…”

“…An example of sinusoidal waves at the inlet which have evolved into much larger amplitude solitary pulses is given in [5]. Ruyer-Quil et al [18] also claim that there are limitations to the Shkadov model deriving from the lack of freedom in the description of the hydrodynamic fields, due to the nature of the depth-averaging technique. Notably, the simplification has the benefit of reduced computational cost on reducing the dimensionality of the Navier-Stokes equation.…”