Suppression of instability in liquid flow down an inclined plane by a deformable solid layer

Abstract

The linear stability of a liquid layer flowing down an inclined plane lined with a deformable, viscoelastic solid layer is analyzed in order to determine the effect of the elastohydrodynamic coupling between the liquid flow and solid deformation on the free-surface instability in the liquid layer. The stability of this two-layer system is characterized by two qualitatively different interfacial instability modes: In the absence of the deformable solid layer, the free surface of the liquid film undergoes a long-wave instability due to fluid inertia. With the presence of the deformable solid layer, the interface between the fluid and the solid undergoes a finite-wavelength instability when the deformable solid becomes sufficiently soft. The effect of the solid layer deformability on the free-surface instability of the liquid film flow is analyzed using a long-wave asymptotic analysis. The asymptotic results show that for a fixed Reynolds number and inclination angle, the free-surface instability is completely suppressed in the long-wave limit when the nondimensional (inverse) solid elasticity parameter Gamma=Va eta/(GR)increases beyond a critical value. Here, Va is the average velocity of the liquid film flow, eta is the viscosity of the liquid, G is the shear modulus of the solid layer, and is R the thickness of the liquid layer. The predictions of the asymptotic analysis are verified and extended to finite wavelengths using a numerical solution, and this indicates that the suppression of the free-surface instability indeed continues to finite wavelength disturbances. Further increase of Gamma is found to have two consequences: first, the interface between the liquid and the deformable solid layer could become unstable at finite wavelengths; second, the free-surface interfacial mode could also become unstable at finite wavelengths due to an increase in solid layer deformability. However, our numerical results demonstrate that, for a given average velocity, there exists a sufficient window in the value of shear modulus G where both the unstable modes are absent at all wavelengths. Our study therefore suggests that soft solid layer coatings could potentially provide a passive method of suppressing free-surface instabilities in liquid film flows.

Stability of gravity-driven free-surface flow past a deformable solid at zero and finite Reynolds number

“…These studies, however, employed the linear elastic model 15 to describe the deformation in the solid, which is valid only when the deformation gradients in the solid are small. In a recent paper, Gkanis and Kumar 16 used the more rigorous neo-Hookean model 17,18 and demonstrated in the creeping-flow limit that the predictions from the two solid models could be significantly different for the stability of flow down an inclined plane covered with a deformable solid, and noted that the earlier predictions of instability suppression 11,12 must be reexamined using the neo-Hookean solid model. The freesurface instability is absent in the creeping-flow limit, so it is necessary to consider nonzero Reynolds number in order to analyze the suppression of free-surface instability by the deformable solid.…”

“…More recently, Gkanis and Kumar 16 studied the same configuration as in Ref. 12, but used the neo-Hookean model to describe the solid deformation, and considered the creeping-flow limit where the GL mode instability is absent in rigid surfaces. Their results showed that the solid deformability destabilizes only one of the two interfacial modes, which they identified as the LS mode ͑in the present nomenclature͒.…”

“…To this end, we undertake here a comprehensive study of stability of liquid flow down an inclined plane lined with a deformable solid, at zero and finite Reynolds number. The primary objective of the present work is to critically evaluate earlier predictions 11,12 concerning instability suppression obtained using the linear elastic model by comparing them with the results from the neo-Hookean solid model. Before proceeding, we briefly discuss related previous work in order to place our work in a proper context.…”

“…Clearly, this mode will be present even when there is more than one liquid layer adjacent to the solid. For two-layer or freesurface flow past a deformable solid, it was shown using the linear viscoelastic solid model 11,12 that the deformability of the solid could lead to complete suppression of the interfacial instabilities under appropriate conditions, without destabilizing the LS mode. Specifically, Shankar and Sahu 12 considered the stability of liquid flow down an inclined plane lined with a linear viscoelastic solid layer at finite Reynolds number, and showed that beyond a critical value of a solid deformability parameter ͑which is proportional to the ratio of viscous forces in the liquid to elastic forces in the solid͒, the inertia-driven GL mode instability can be suppressed at all wave numbers.…”

“…Previous studies 7-10 have suggested "active" methods such as imposed wall oscillations or heating as possible strategies for suppressing interfacial instabilities. Recently, the possibility of using deformable solid coatings as a "passive" method to suppress interfacial instabilities in twolayer and free-surface flows was explored for both Newtonian 11,12 and viscoelastic 13,14 fluids. These studies, however, employed the linear elastic model 15 to describe the deformation in the solid, which is valid only when the deformation gradients in the solid are small.…”