Linear stability of a falling film over a heated slippery plane

A detailed parametric study on the linear stability analysis of a three-dimensional thin liquid film flowing down a uniformly heated slippery inclined plane is carried out for disturbances of arbitrary wavenumbers, where the liquid film satisfies Newton's law of cooling at the film surface. A coupled system of boundary value problems is formulated in terms of the amplitudes of perturbation normal velocity and perturbation temperature, respectively. Analytical solution of the boundary value problems demonstrates the existence of three dominant modes, the so-called H mode, S mode, and P mode, where the S mode and P mode emerge due to the thermocapillary effect. It is found that the onset of instabilities for the H mode, S mode, and P mode reduces in the presence of wall slip and leads to a destabilizing influence. Numerical solution based on the Chebyshev spectral collocation method unveils that the finite wavenumber H-mode instability can be stabilized, but the S-mode instability and the finite wavenumber P-mode instability can be destabilized by increasing the value of the Marangoni number. On the other hand, the Biot number shows a dual role in the H-mode and S-mode instabilities. But the P-mode instability can be made stable with the increasing value of the Biot number and the decreasing values of the Marangoni number and the Prandtl number. Furthermore, the H-mode and S-mode instabilities become weaker, but the P-mode instability becomes stronger, with the increasing value of the spanwise wavenumber. In addition, the shear mode emerges in the numerical simulation when the Reynolds number is large, which can be destabilized slightly with the increasing value of the Marangoni number; however, it can be stabilized with the increasing value of the slip length and introducing the spanwise wavenumber to the infinitesimal perturbation.