A falling film down a slippery inclined plane

Abstract

A gravity-driven film flow on a slippery inclined plane is considered within the framework of long-wave and boundary layer approximations. Two coupled depth-averaged equations are derived in terms of the local flow rate $q(x, t)$ and the film thickness $h(x, t)$. Linear stability analysis of the averaged equations shows good agreement with the Orr–Sommerfeld analysis. The effect of a slip at the wall on the primary instability has been found to be non-trivial. Close to the instability onset, the effect is destabilising whereas it becomes stabilising at larger values of the Reynolds number. Nonlinear travelling waves are amplified by the presence of the slip. Comparisons to direct numerical simulations show a remarkable agreement for all tested values of parameters. The averaged equations capture satisfactorily the speed, shape and velocity distribution in the waves. The Navier slip condition is observed to significantly enhance the backflow phenomenon in the capillary region of the solitary waves with a possible effect on heat and mass transfer.

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

“…Computed velocity profiles with backflow are depicted in figure 7 of [13], with potential inflexion points which could indicate that a cubic profile may be a more suitable approximation in this region. This is in agreement with work by Samanta et al [20], who conclude that in a nonlinear regime, the backflow phenomenon is shown to be intensified by a no-slip condition enforced between the fluid and the plane. Backflow in the capillary region is investigated experimentally by Dietze et al [7].…”

“…Often used for describing falling films, the long wave equation, and other similar models, are derived using a perturbation expansion method for the stream function, determined to O("), and then substituted into the surface kinematic boundary condition. An early publication using this method by Benney [2] gives a single evolution equation for film thickness, however this has been criticised as predicting unrealistic wave profiles as the wave amplitude increases [14], as well as leading to finite-time singularities [20,21].…”

Stability of viscous film flow coating the interior of a vertical tube with a porous wall

“…This originates from the inclusion of the curvature term 2 S zz in Eq. (24). Being similar to the exterior case, the cutoff wave number corresponds to the classical Rayleigh-Plateau mode for the capillary instability of a viscous jet.…”

“…The one-sided model has been widely used in many recent papers on the stability of falling films on porous inclines [21][22][23]. In mathematical form, this one-sided model is identical to that of the problem of a film falling down a slippery inclined plane [24]. Liu and Liu [25] proposed a two-sided model by giving up the assumptions in Ref.…”

Effects of velocity slip on the inertialess instability of a contaminated two-layer film flow

“…It is also worth mentioning here that flow over rough or textured substrates at the microscale and/or superhydrophobic substrates can be examined by modelling these substrates as smooth substrates with an effective slip at the interface [33]. Further, Blake [34], Vinogradova [35,36], and Voronov and Papavassiliou [37] have pointed out that large slip lengths of the order of 50 µm can be observed in the case of grooved substrates when one considers the combination of the above two effects, and therefore, flow over grooved substrates can be again analysed using flow over a slippery substrate.…”