Modal analysis of a viscous fluid falling over a compliantwall

We study the modal instability of a three-dimensional Newtonian viscous fluid falling over a compliant wall under the framework of coupled evolution equations for normal velocity and normal vorticity, respectively. The Chebyshev spectral collocation numerical technique is applied in exploring unstable modes for the three-dimensional disturbances. The unstable zones pertaining to surface mode, wall mode and shear mode reduce as long as the spanwise wavenumber amplifies and corroborates the stabilizing influence of spanwise wavenumber. However, the onset of instability for the wall mode decays as long as the capillary number amplifies and ensures the destabilizing influence of the capillary number. Furthermore, the critical Reynolds number corresponding to surface mode found in the long-wave zone shifts towards the finite wavelength zone with increasing spanwise wavenumber and confirms the extinction of long-wave instability. But the inertialess instability created due to the wall mode disappears with increasing spanwise wavenumber. Moreover, the shear mode emerges in the high Reynolds number zone and is stabilized in the presence of spanwise wavenumber. In addition, the analytical derivation of Squire's theorem is provided in appendix B for the flow over a compliant wall.