Linear stability of a shear-imposed heated falling film

A linear stability analysis is carried out for a thin liquid film flowing down a uniformly heated inclined plate under the action of gravity. The free surface of the liquid film is subjected to a constant imposed shear stress. The long-wave approximation is used to capture the hydrodynamic H mode instability analytically. The imposed shear stress exerts a destabilizing effect on the hydrodynamic H mode instability. The Padé approximation predicts the temporal growth rate more accurately compared to that determined from the third order and fifth order long wave approximations. To capture the hydrodynamic shear mode and thermo-capillary S and P modes numerically, the Chebyshev spectral method is employed. Imposed shear is found to have a destabilizing effect on the S mode and shear mode instabilities, whereas it stabilizes the P mode instability. It has been observed that after a certain critical value of the imposed shear stress, the P mode instability disappears. The P mode instability gets destabilized with increasing spanwise wavenumber, whereas the shear mode, H mode and S mode instabilities get stabilized with increasing spanwise wavenumber. The Marangoni number exerts a destabilizing effect on all the instability modes, namely, H, S, P and shear mode.