Stability of soluble surfactant-laden falling film over a hydrophobic incline in the presence of external shear

TL;DR

This work analyzes the linear stability of gravity-driven, soluble-surfactant-laden thin film flowing down a slippery inclined plane under external shear by formulating an Orr–Sommerfeld eigenproblem for coupled flow and surfactant transport. It combines analytic longwave asymptotics ($k\ll1$) and Chebyshev collocation for finite wavelengths to identify a surface mode and a surfactant (Marangoni) mode, along with a finite-wavelength shear mode at high modified Reynolds numbers, and it uses an energy-budget framework to dissect energy transfer among shear, surface tension, and surfactant effects. The base flow is modulated by slip length $\delta$ and imposed shear $\tau$, while soluble-surfactant parameters $\beta_a$ and $R_a$ stabilize the surfactant mode; the surface-mode neutral stability is captured by $Re_{c_s}$ with contributions from $\delta$, $\tau$, and Marangoni stresses via $E_0$, whereas the surfactant mode exhibits a $Pe_{ca,m}$ threshold dependent on $Ma$, $R_a$, $\beta_a$, and kinetic parameters. Overall, slip and external shear yield dual effects on the surface mode, whereas soluble-surfactant properties tend to stabilize the surfactant mode; the energy-budget results highlight the dominant roles of work against shear and surface tension in driving or damping instability, with implications for coating flows and lung-fluid dynamics.

Abstract

The hydrodynamic stability analysis of gravity-driven, soluble surfactant-laden fluid streaming down a slippery, slanted plane in the presence of external shear force is being explored in this article. The Navier-Stokes equations are considered for the fluid flow along with the appropriate advection-diffusion equations for the concentration of different surfactant species. The monomers considered here are anticipated to dissolve in the bulk flow and can be adsorbed at the interface of air-liquid as well. Also, the adsorption-desorption kinetics of the surfactants at the free space is taken into consideration. The motivation behind this work is to extend the work of Karapetsas and Bontozoglou[1] for flow over a slippery bottom and in the presence of externally imposed shear forces and observe their impact on the flow dynamics. The Orr-Sommerfeld eigensystem is obtained, then it is solved analytically using the longwave approximation method in the longwave regime ($k \ll 1$) and subsequently, the Chebyshev spectral collocation method is employed for numerical evaluation in the arbitrary wave regime. Using the analytical method, two longwave modes, viz, surface mode and surfactant mode, are detected. Alternatively, the numerical analysis substantiated the existence of temporal surface and surfactant modes. Moreover, another temporal mode named shear mode arises in the high modified Reynolds number region at a low inclination angle. Thereafter, the modified Reynolds-Orr energy equation is deduced under the normal mode conditions, and the behaviour of different energy components is investigated for various slip parameters and imposed shear force.

Figures (21)

• Figure 1: The depiction of the problem with the coordinate system in the two-dimensional frame. The soluble surfactant is present in the bulk as monomers, and the size shown is not to scale. Also, the moieties of one surfactant molecule are shown in close view.
• Figure 2: The streamwise base velocity profile $U(y)$ as a function of $y$ for varying (a) imposed shear $\tau$, (b) slippery bottom $\delta$. For (a) $\delta = 0.04$ and (b) $\tau = 0.5$ are chosen as fixed values during computing the above results.
• Figure 3: The comparison of the phase speed between the surface mode and surfactant mode against (a) external shear force $\tau$, (b) slip parameter $\delta$. The other parameters are $\beta_a = 0.01$, $R_a = 1.0$, $k_a = 0.01$, $M_{tot} = 0.1$.
• Figure 4: The variation of critical Reynolds number against (a) external shear force $\tau$, (b) slip parameter $\delta$. In the first set of figures, the solid lines, dashed lines, and dash-dot lines represent the values of external shear force $\tau = 0.0, 0.5, 1.0$ respectively. Similarly, in the second set of figures, the solid, dashed, and dash-dot lines represent the values of the slip parameter $\delta = 0.00, 0.04, 0.08$ respectively. Also, the Marangoni number $Ma$ for the corresponding figure is shown within the figure. The other parameters are $\beta_a = 0.01$, $R_a = 1.0$, $M_{tot} = 0.1$.
• Figure 5: The variation of critical Péclet number against (a) external shear force $\tau$, (b) Marangoni number $Ma$. In the first set of figures, the solid lines, dashed lines, and dash-dot lines represent the values of the slip parameter $\delta = 0.00, 0.04, 0.08$ respectively. Similarly, in the second set of figures, the top figure represents the critical Péclet number against the Marangoni number $Ma$ for varying external shear force $\tau$ with $\delta = 0.04$ and the bottom one for varying slip parameter $\delta$ with $\tau = 0.5$. The other parameters are $\xi = 90^{\circ}$, $\chi = 1$, $\Sigma = 2.0$, $Sc_b = 10$,$M_{tot} = 0.1$, $\beta_a = 0.01$, $R_a = 1.0$ and $k_a = 1.0$ .