Stability of gravity-driven particle-laden flows – roles of shear-induced migration and normal stresses

Abstract

In this paper, we study the role of shear-induced migration and particle-induced normal stresses in the formation and stability of a particle-laden, gravity-driven shallow flow. We first examine the modification of the base-state Nusselt flow due to the underlying microstructure, how shear-induced migration leads to viscosity stratification. We inspect the development of the base state via the boundary layer formation in the ‘shallow’ limit and find a reduction in entrance length with increasing bulk particle concentration and an increase in entrance length with increasing Péclet number ( $Pe_p = \dot {\gamma } a^2 / D_0$ , where $\dot{\gamma}$ is the average shear rate, a is the particle size and $D_0$ is the single particle diffusivity). A linear stability analysis is then performed on the fully developed state to identify two modes of instability typically found in gravity-driven falling films – the long-wave surface and the short-wave shear modes. We find that when the associated Péclet number is $Pe_p \ll 1$ , increasing bulk particle volume fraction delays the onset of instability for both the surface mode and shear mode. However, with $Pe_p = {O}(1)$ , we find an enhancement in both modes of instability. We also find that, beyond a critical Péclet number, for a fixed particle volume fraction, the surface mode is unstable even in the absence of fluid inertia. The enhanced destabilisation is attributed to the combined effects of base-state viscosity stratification and momentum forcing via particle concentration perturbations. We also show that the physics behind the enhancement of instability is independent of the choice of the constitutive model used to describe the dynamics of the particle phase, provided the chosen model has elements of shear-induced migration.

A particularly unstable film

“…One touchstone for this study in prior analysis is the stability of a flowing film on a heated substrate (Goussis & Kelly 1985). The base flow studied by DR resembles this flow as both exhibit a viscosity stratification, with a lower viscosity near the plane wall. For the suspension, this is a result of migration towards the free surface and viscosity dependence on of the typical form , with maximum packing fraction .…”

“…Here, is the uniform base state thickness seen in figure 1( b ). The work by DR predicts not only destabilizing effects of the particles to the classical (Benjamin 1957; Yih 1963) inertial instabilities of a film, but also a previously unknown instability at vanishing inertia.

Figure 1.( a ) A particle-laden film and ( b ) the predicted development of the height and profile for at , from a uniformly concentrated film at .

…”

“…Dhas & Roy (2022) (DR) recently presented a stability analysis for flow of a Brownian suspension down an inclined plane, as illustrated by figure 1( a ). The flow is considered in the shallow limit, with the flow length and film depth satisfying , allowing the simplifications of lubrication and boundary layer analyses.…”

“…The axial coordinate is scaled by , and the final thickness in ( b ) is the base-state . From Dhas & Roy (2022). …”

“…An issue that features prominently in the work of DR is shear-induced particle migration, which has long been a theme in suspension studies (Leighton & Acrivos 1987), and is associated with the normal stresses attributable to the particle phase (Morris & Boulay 1999). Due to variation in particle volume fraction caused by migration, the base state for the stability analysis is altered from the half-parabola velocity profile of the pure liquid.…”