Stability analysis of viscoelastic film flows over an inclined substrate with rectangular trenches

We consider the hydrodynamic stability of viscoelastic films flowing over inclined structured substrates with rectangular trenches. This topography allows investigating independently the effects of trench unit length, depth, width and inclination angle and complements the earlier work by Pettas et al. (Phys. Rev. Fluids, vol. 4, 2019, 33). We account for material rheology by employing the ePTT model. We perform a parametric study of the steady flow and its linear stability, assuming two-dimensional perturbations along the streamwise direction of arbitrary wavelength via the Floquet–Bloch theory. Our predictions for Newtonian liquids are in excellent agreement with previous results. We demonstrate that even for Newtonian liquids, the trench depth has a non-trivial effect on the flow stability. In viscoelastic solutions, the interaction of fluid elasticity with substrate morphology may have a significant impact on the flow dynamics, leading to either the enhancement or suppression of instabilities. Topography characteristics combined with enough material elasticity stabilize the flow. However, beyond a specific trench depth, this effect saturates by eddy formation inside the cavity. Moreover, the stability is also affected by the aspect ratio and shape of the trenches: flows over substrates with a pillar-like configuration are stabilized significantly. On the other hand, flows are destabilized by material shear-thinning. This study helps identifying the shape of the substrate that maximizes/minimizes the viscoelastic mechanisms. This is impossible when considering substrates with sinusoidal topography, but could be of utmost importance for several technological applications, by providing the potential for instability control through the development of appropriately tailored substrates.