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

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.