On the Yih-Marangoni instability of a two-phase plane Poiseuille flow in a hydrophobic channel

The linear stability analysis of a plane Poiseuille flow of two immiscible, incompressible fluids of different viscosities and densities in a hydrophobic channel, in the presence of an insoluble surfactant at the interface is examined, within the framework of Orr-Sommerfeld system. The walls of the channel are modeled as walls with velocity slip/no-slip. The equations governing the flow system are solved numerically by a Chebyshev collocation method for a wide range of dimensionless parameters describing the flow system. The effects of slip on the neutral stability boundaries for the interface modes in the presence/absence of an insoluble surfactant at the interface are examined for different thickness ratios of the two layers, with density and/or viscosity stratification. Slip conditions at the wall show a promise for control of the Yih-Marangoni instability of the corresponding flow system in a rigid channel. The influence of the parameters on the critical Reynolds number for the shear mode is assessed. The study reveals that it is possible to control instabilities in interface dominated rigid channel flows by designing the walls of the channel as hydrophobic/rough/porous or undulated surfaces as these can be modeled as one with slip at the substrates.