Stability dynamics of surfactant-contaminated double-layered viscoelastic fluid flowing down an inclined plane

The linear stability dynamics of double-layered weakly viscoelastic fluid flowing over an inclined plane are analyzed in the presence of insoluble surfactant at both the free surface and interface. The constitutive equation of the non-Newtonian flow field follows the rheological property of Walters' B ′′ model. The Orr-Sommerfeld-type boundary value problem is derived using the classical normal-mode approach and numerically solved within the framework of the Chebyshev spectral collocation method. Numerical outcomes detect three distinct types of instabilities: surface, interface, and interface surfactant. The viscoelasticity of both the top and bottom layers strengthens the surface wave instability in the longwave region. On the other hand, the behavior of interfacial instability depends on both viscosity and density stratification. The stronger top-layer viscoelasticity suppresses the interfacial wave instability, while increased bottom-layer viscoelasticity amplifies it, provided the viscosity ratio m > 1. However, for m < 1, top-layer viscoelasticity provides strong interfacial wave stabilization in the longwave region but becomes comparatively weak in the shortwave regime. The viscoelasticity of the bottom layer has a destabilizing/stabilizing influence on it in the longwave/shortwave regions. Meanwhile, the top-layer viscoelasticity stabilizes the interfacial surfactant mode. However, this mode can be destabilized near the instability onset but is effectively stabilized far away from it by higher bottom-layer viscoelasticity. Additionally, at high Reynolds numbers with low inclination angles, a new instability, named shear mode instability, corresponds to both layers emerges when the viscosity and density of the bottom layer become much stronger than the top layer. Both top and bottom layer viscoelasticity intensifies the top shear layer instability, while the bottom-layer shear wave instability is suppressed/amplified with an increase in the viscoelasticity of the top/bottom layered fluid. The shear wave instability exhibits significantly stronger sensitivity to the viscoelastic coefficients of both layers than all the aforementioned unstable modes, while the interface and interface surfactant modes are more sensitive than the surface mode