The stability of a gravity-driven film flow on a porous inclined substrate is considered, when the film is contaminated by an insoluble surfactant, in the frame work of Orr-Sommerfeld analysis. The classical long-wave asymptotic expansion for small wave numbers reveals the occurrence of two modes, the Yih mode and the Marangoni mode for a clean/a contaminated film over a porous substrate and this is confirmed by the numerical solution of the Orr-Sommerfeld system using the spectral-Tau collocation method. The results show that the Marangoni mode is always stable and dominates the Yih mode for small Reynolds numbers; as the Reynolds number increases, the growth rate of the Yih mode increases, until, an exchange of stability occurs, and after that the Yih mode dominates. The role of the surfactant is to increase the critical Reynolds number, indicating its stabilizing effect. The growth rate increases with an increase in permeability, in the region where the Yih mode dominates the Marangoni mode. Also, the growth rate is more for a film (both clean and contaminated) over a thicker porous layer than over a thinner one. From the neutral stability maps, it is observed that the critical Reynolds number decreases with an increase in permeability in the case of a thicker porous layer, both for a clean and a contaminated film over it. Further, the range of unstable wave number increases with an increase in the thickness of the porous layer. The film flow system is more unstable for a film over a thicker porous layer than over a thinner one. However, for small wave numbers, it is possible to find the range of values of the parameters characterizing the porous medium for which the film flow can be stabilized for both a clean film/a contaminated film as compared to such a film over an impermeable substrate; further, it is possible to enhance the instability of such a film flow system outside of this stability window, for appropriate choices of the porous substrate characteristics. C

This manuscript provides a theoretical stability analysis for the configuration of two-layer immiscible flow down an inclined plane with velocity slip along the incline in the limit of zero Reynolds number. Surfactants may be present at the air-liquid interface, liquid-liquid interface, or both. In addition to an Orr-Sommerfeld analysis (again at zero Reynolds number), a long wavelength stability analysis is performed and the results are shown to be consistent. The interface mode, namely the mode of instability that arises because of viscosity stratification, is examined. Stability results (growth rates as a function of slip parameter and neutral stability boundaries) for various configurations of viscosity, surfactant placement, and layer thickness are compared with those of the previous literature and found to agree. It is found that velocity slip along the inclined plane reduces the maximum growth rate of instabilities in configurations where they occur, and the range of unstable wave numbers shrinks as well, indicating that slip has a promise for stabilization. This suggests that there is a possibility of using this favourably as a control option for two-layer flows in the absence or presence of surfactants, in relevant applications by designing the substrate to be a porous substrate with small permeability or a slippery substrate or a rough substrate or a hydrophobic substrate which can be modelled as substrates with velocity slip.

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A mathematical model is developed to investigate the dynamics and rupture of a pre-lens tear film on a contact lens. The contact lens is modeled as a saturated porous medium of constant, finite thickness and is described by the Darcy-Brinkman equations with stress-jump condition at the interface. The model incorporates the influence of capillarity, gravitational drainage, contact lens properties such as the permeability, the porosity, and the thickness of the contact lens on the evolution and rupture of a pre-lens tear film, when the eyelid has opened after a blink. Two models are derived for the evolution of a pre-lens tear film thickness using lubrication theory and are solved numerically; the first uses shear-free surface condition and the second, the tangentially immobile free surface condition. The results reveal that life span of a pre-lens tear film is longer on a thinner contact lens for all values of permeability and porosity parameter considered. An increase in permeability of contact lens, porosity or stress-jump parameter increases the rate of thinning of the film and advances the rupture time. The viscous-viscous interaction between the porous contact lens and the pre-lens tear film increases the resistance offered by the frictional forces to the rate of thinning of pre-lens tear film. This slows down the thinning process and hence delays the rupture of the film as compared to that predicted by the models of Nong and Anderson [SIAM. J. Appl. Math. 70, 2771–2795 (2010)] derived in the framework of Darcy model.