A new elastic instability in gravity-driven viscoelastic film flow

We examine the linear stability of the gravity-driven flow of a viscoelastic fluid film down an inclined plane. The viscoelastic fluid is modeled using the Oldroyd-B constitutive equation and, therefore, exhibits a constant shear viscosity and a positive first normal stress difference in viscometric shearing flows; the latter class of flows includes the aforesaid film-flow configuration. We show that the film-flow configuration is susceptible to two distinct purely elastic instabilities in the inertialess limit. The first instability owes its origin entirely to the existence of a free surface and has been examined earlier [Shaqfeh et al., “The stability of gravity driven viscoelastic film-flow at low to moderate Reynolds number,” J. Non-Newtonian Fluid Mech. 31, 87–113 (1989)]. The second one is the analog of the centermode instability recently discovered in plane Poiseuille flow [Khalid et al., “Continuous pathway between the elasto-inertial and elastic turbulent states in viscoelastic channel flow,” Phys. Rev. Lett. 127, 134502 (2021)] and owes its origin to the base-state shear; it is an example of a purely elastic instability of shearing flows with rectilinear streamlines. One may draw an analogy of the aforesaid pair of unstable elastic modes with the inertial free-surface and shear-driven instabilities known for the analogous flow configuration of a Newtonian fluid. While surface tension has the expected stabilizing effect on the Newtonian and elastic free-surface modes, its effect on the corresponding shear modes is, surprisingly, more complicated. For both the Newtonian shear mode and the elastic centermode, surface tension plays a dual role, with there being parameter regimes where it acts as a stabilizing and destabilizing influence. While the Newtonian shear mode remains unstable in the limit of vanishing surface tension, the elastic centermode becomes unstable only when the appropriate non-dimensional surface tension parameter exceeds a threshold. In the limit of surface tension being infinitely dominant, the free-surface boundary conditions for the film-flow configuration reduce to the centerline symmetry conditions satisfied by the elastic centermode in plane Poiseuille flow. As a result, the regime of instability of the film-flow centermode becomes identical to that of the original channel-flow centermode. At intermediate values of the surface tension parameter, however, there exist regimes where the film-flow centermode is unstable even when its channel-flow counterpart is stable. We end with a discussion of the added role of inertia on the aforementioned elastic instabilities.