Shear-imposed falling film

Abstract

The study of a film falling down an inclined plane is revisited in the presence of imposed shear stress. Earlier studies regarding this topic (Smith, on the basis of a low Reynolds number, are extended up to moderate values of the Reynolds number. The mechanism of the primary instability is provided under the framework of a two-wave structure, which is normally a combination of kinematic and dynamic waves. In general, the primary instability appears when the kinematic wave speed exceeds the speed of dynamic waves. An equality criterion between their speeds yields the neutral stability condition. Similarly, it is revealed that the nonlinear travelling wave solutions also depend on the kinematic and dynamic wave speeds, and an equality criterion between the speeds leads to an analytical expression for the speed of a family of travelling waves as a function of the Froude number. This new analytical result is compared with numerical prediction, and an excellent agreement is achieved. Direct numerical simulations of the low-dimensional model have been performed in order to analyse the spatiotemporal behaviour of nonlinear waves by applying a constant shear stress in the upstream and downstream directions. It is noticed that the presence of imposed shear stress in the upstream (downstream) direction makes the evolution of spatially growing waves weaker (stronger).

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

“…A profile initiated by h l = h 3 (figures [16][17][18][19] produces an undercompressive front at Re ⇤ = 0, also reported by Levy and Shearer [12]. We observe commonality of solution exhibited by changes in initial profile with equivalent boundary conditions developing into identically shaped wave structures.…”

“…These capillary ripples are mesh-independent, indicating this is a model feature. They are also present in profiles obtained in other numerical studies which include the e↵ects of inertia [19], as well as observed in experimental work by Kofman et al [10]. Further investigations were carried out by varying the surface shear.…”

“…Samanta [19] investigates the mechanism of instability in shear imposed flow on an inclined plane. The geometry of the problem studied is very similar to the model we propose but with disturbances generated at the inlet which evolve into periodic travelling waves.…”

“…Another important comparison is the magnitude of the Reynolds number which di↵ers from what we term the reduced Reynolds number by ". The avenue of exploration in [19] is predominantly concerning the linear instability threshold and recovering the critical Reynolds and Froude numbers for the shear-imposed falling film. Although we will not be conducting any formal stability analysis, section 5 contains some discussion about the evolution of perturbations in order to characterise waves structures.…”

Optimal disturbance growth in shear‐imposed falling film

“…The mechanism of primary instability for the surface wave was nicely addressed by Smith when additional constant shear stress is exerted at the free surface. The model proposed by Smith was further revisited by Samanta to take into account the nonlinear inertial effect on the traveling wave solutions. It was reported that the surface instability exclusively occurs due to the competition between the kinematic wave and the dynamic wave.…”