The effect of viscosity stratification on the stability of a free surface flow at low Reynolds number

Abstract

A zero Reynolds number approximation to the Orr–Sommerfeld equation is used to assess the effects that viscosity stratification has on the stability of a very viscous flow on an incline when surface tension is negligible. Results indicate that for a two-layer system with uniform density, the flow is always unstable when the viscosity of the upper layer is greater than that of the lower layer, regardless of the thickness of the upper layer. The wavenumber of the fastest growing mode is on the order of the inverse of the thickness of the upper layer, implying that the instability is manifested in waves having finite wavelength, whereas previous research on this topic has focused on a long wavelength approximation. It is further shown that neutral stability is independent of the angle of inclination of the underlying slope, although the growth rate of any instability is not. The results suggest that the transverse surficial ridges, which commonly occur on the surfaces of rock glacier forms, may be the product of a flow instability arising from the differing viscosities of the layers that comprise such features.

Inertialess multilayer film flow with surfactant: Stability and traveling waves

“…Working in the longwave limit in which interfacial tension and surface tension are both negligible, Kao [13] showed that for clean twolayer flow the basic state is always unstable when there is no inertia and when the upper layer is more viscous than the lower layer. This result was extended to the case of arbitrary wavenumber in the Stokes flow limit, and in the absence of interfacial and surface tension, by Loewenherz and Lawrence [14]. Further analysis by Chen [15] allowed for surface tension and inertia and concluded that the clean two-layer flow is always unstable at any Reynolds number if the less viscous fluid is next to the wall.…”

“…They also identified a new instability which is not present in the longwave regime, which occurs when the middle layer is thin and highly viscous. Furthermore, they showed that the largest growth rates found for the three-layer system are orders of magnitude larger than those found for the two-layer system by Loewenherz and Lawrence [14]. The physical mechanisms behind the three-layer instability was investigated by Jiang et al [28].…”

Two-Layer Flow with One Viscous Layer in Inclined Channels

“…The dynamics in the nonlinear regime have also been probed using one-dimensional averaged equations [11,12] and direct numerical simulations [13]- [16]. Previous work has also examined the dynamics of multi-layer flows in the context of industrial coating flows and geophysical flows [17,18] and focused on situations wherein the upper interface is a free surface.…”

Nonlinear interfacial dynamics in stratified multilayer channel flows

“…It has been established that two-layer flows in inclined or pressure-driven channels and single-layer free-surface flows down inclined planes, require fluid inertia for destabilisation, at least when the inclination to the horizontal is less than ninety degrees (see Chen 1995;Benjamin 1957;Yih 1963). However, in the case of two-layer free-surface flows, Kao (1968), Loewenherz & Lawrence (1989) and Chen (1993) showed that when the less viscous fluid is adjacent to the wall, then a long-wave instability can appear in the absence of inertia (zero Reynolds number); this instability has been termed inertialess instability. Chen (1993) argues that the instability arises from an interaction between the free surface and the interface, while an interpretation of the underlying mechanism has been given recently by Gao & Lu (2008).…”