Effect of surfactant on the long-wave instability of a shear-imposed liquid flow down an inclined plane

Abstract

The effect of an insoluble surfactant on the linear stability of a shear-imposed flow down an inclined plane is examined in the long-wavelength limit. It has been known that a free falling film flow with surfactant is stable to long-wavelength disturbances at sufficiently small Reynolds numbers. Imposing an additional interfacial shear, however, could cause instability due to the shear-induced Marangoni effect. Two modes of the stability are identified and the corresponding growth rates are derived. The underlying mechanisms of the stability are also elucidated in detail.

Shear-imposed falling film

“…We + 7675 672 cot θ τ which recovers the critical Reynolds number very well for a shear-imposed falling film Re c = 5 cot θ /[2(1 + τ )], obtained by Smith (1990) and Wei (2005a) in the limit k → 0, i.e. when the streamwise viscous dissipation terms are ignored into the model.…”

“…The reduced equations normally consist of a coupled system of two equations in terms of local film thickness h(x, t) and local flow rate q(x, t). The motivation is to extend the previous model, studied by Smith (1990) and Wei (2005a), in detail up to moderate values of the Reynolds number when the free surface is uncontaminated. We propose a two-equation model that is consistent up to order O( ) in inertial terms and up to order O( 2 ) in viscous diffusion terms, where ∼ (1/λ) 1 and the wavelength λ of the disturbance is large compared to the depth of the liquid film.…”

“…(2.1c) (ii) At the free surface y = h(x, t), the tangential stress is balanced by the imposed shear stress, whereas the normal stress is balanced by surface tension times curvature and yields the dynamic condition (Smith 1982(Smith , 1990Wei 2005aWei ,b, 2007 …”

“…He reported that the unsteadiness associated with perturbation of a base flow is the dominant cause of a long-wave instability. Later, Wei (2005a) extended this study by including an insoluble surfactant at the top of a film surface. In general, a surfactant exhibits a stabilizing effect on a surface wave by suppressing the long-wave instability.…”

Liquid layer characteristics in gas–liquid flow in slightly inclined pipes: Effect of non-ionic surfactant additives

“…Thenceforth the drag reduction phenomenon has been extensively studied by many researchers in different pipe or plane geometries (e.g. Warholic et al, 1999;Kawaguchi et al, 2002;Duangprasert et al, 2008;Rozenblit et al, 2006;Wei, 2005).…”