Role of slip on Marangoni instability in two-layer creeping flow

The current study presents the effect of slip on the linear stability analysis of two immiscible Newtonian fluids flowing under the conditions of creeping flow in two layer configuration within an infinitely long 2-D channel. This two-layer flow system involves the mass transfer of a soluble surfactant from the upper fluid layer to the lower fluid layer, with the upper wall maintaining a more constant concentration of soluble surfactant compared to the lower wall. This mass transfer of the soluble surfactant across the liquid-liquid interface induces variations in the surface tensions of the fluids, leading to the development of Marangoni stresses near the interface, which can cause interfacial instability known as Marangoni instability. In this setup, the upper wall of the channel is treated as a slippery surface. The analysis focuses on the effect of upper slippery surface on the Marangoni instability by implementing a slip condition on the upper wall of the channel. Orr-Sommerfeld and species transport equations are formulated and solved using Chebyshev spectral collocation method. For the typical values of depth ratio, viscosity ratio of the fluid layers and slip length of upper slippery wall, the results reveals the Marangoni instability can effectively regulated by the introducing the slippery effect. In cases where the depth ratio is unity, increasing the slip length of the upper wall destabilizes Marangoni instability when the viscosity ratio is greater than unity, while it stabilizes when the viscosity ratio is less than unity. These findings underscore the significant role of slip at the upper wall in regulating the stability of the two-layer flow system in the presence of soluble species.