Linear stability analysis of a surfactant-laden shear-imposed falling film

Abstract

A study of the linear stability analysis of a shear-imposed fluid flowing down an inclined plane is performed when the free surface of the fluid is covered by an insoluble surfactant. The purpose is to extend the earlier work [H. H. Wei, “Effect of surfactant on the long-wave instability of a shear-imposed liquid flow down an inclined plane,” Phys. Fluids 17, 012103 (2005)] for disturbances of arbitrary wavenumbers. The Orr-Sommerfeld boundary value problem is formulated and solved numerically based on the Chebyshev spectral collocation method. Two temporal modes, the so-called surface mode and surfactant mode, are detected in the long-wave regime. The surfactant mode becomes unstable when the Péclet number exceeds its critical value. In fact, the instability of the surfactant mode occurs on account for the imposed shear stress. Energy budget analysis predicts that the kinetic energy of the infinitesimal disturbance grows with the imposed shear stress. On the other hand, the numerical results reveal that both surface and surfactant modes can be destabilized by increasing the value of the imposed shear stress. Similarly, it is demonstrated that the shear mode becomes more unstable in the presence of the imposed shear stress. However, it can be stabilized by incorporating the insoluble surfactant at the free surface. Apparently, it seems that inertia does not play any role in the surfactant mode in the moderate Reynolds number regime. Furthermore, the competition between surface and shear modes is discussed.

Thermodynamics of a Liquid Film in the Presence of External Shear Stress

“…The solution to (59) that satisfies (61)-(63) is taken the form As a special case of our model, when no heat transfer, the system is reduced to Eq. (67) only, in which a similar equation is in the model given by Bhat and Samanta (2019). In the following the surface and the surfactant modes will be discussed in terms of the first-order wave speed c1…”

“…In the absence of heat transfer, the above linear system is closed to the problem considered by Bhat and Samanta (2019). For dealing with this system, we have two temporal modes as illustrated by (Kwak and Pozrikidis (2001); Wei (2004)), the first is the interface mode (ĥ 0 ≠ 0, Γ 0 ≠ 0), this mode is caused by the surface deflections via the validity of leadingorder kinematic relation (54) gives c0 (as in freefalling films).…”

“…Their results show that there are two roles found for the sur-face surfactants; one is stabilized the system when it concentrated and the other is unstable behavior as the diffusion of the surfactant increased. The second, Bhat and Samanta (2019) have recently per-formed the linear stability behavior of a heated liquid film flowing down an inclined substrate the free surface of the layer is subjected to shear-imposed stress in which an insoluble surfactant has covered the interface.…”

Optimal disturbance growth in shear‐imposed falling film

“…Despite the prevalence of surface instability commonly dominates in low to moderate Reynolds number regime, another instability driven by the shear mode often emerges in the falling film when the Reynolds number is very large and the inclination angle is sufficiently small . Indeed, the flow parameters play a substantial role in the competition between the surface mode and the shear mode to trigger the primary instability.…”

“…29 Despite the prevalence of surface instability commonly dominates in low to moderate Reynolds number regime, another instability driven by the shear mode often emerges in the falling film when the Reynolds number is very large and the inclination angle is sufficiently small. [30][31][32][33][34][35][36] Indeed, the flow parameters play a substantial role in the competition between the surface mode and the shear mode to trigger the primary instability. As discussed by Bruin, 31 the primary instability is dominated by the shear mode once the inclination angle is less than 0.5 0 , because in this case the resulting critical Reynolds number for the shear mode is less than that for the surface mode.…”