Stability of surfactant-laden double-layered viscoelastic fluids flowing over an inclined plane

TL;DR

This work presents a linear stability analysis of a surfactant-laden, double-layer viscoelastic film flowing down an inclined plane, modeled with Walters' $B''$ constitutive equations. Using an Orr–Sommerfeld framework and Chebyshev spectral collocation, the authors identify three primary unstable modes (SM, IM, ISM) and a shear mode (SHM), and map how top- and bottom-layer elasticity, viscosity/density stratification, and interfacial Marangoni stresses govern stability across parameter spaces. Key findings reveal that top-layer elasticity can destabilize surface waves near onset but may be counteracted by top-surface surfactants, while bottom-layer elasticity tends to destabilize interfacial modes in certain regimes; the interfacial Marangoni force can suppress or promote instability depending on the viscosity ratio $m$ and Marangoni numbers. Overall, elasticity and surfactant effects interact nontrivially, with ISM and SHM showing pronounced sensitivity to $\gamma_i$ and $m$, providing insights for coating processes and physiological flows where viscoelasticity and surface-tactive agents coexist.

Abstract

The linear dynamics and instability mechanisms 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 analysis detects three distinct types of instabilities: surface, interface, and interface surfactant modes. 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 wave instability depends on both viscosity and density stratification. Stronger top-layer viscoelasticity suppresses interfacial instability, while increased bottom-layer viscoelasticity amplifies it, provided the viscosity ratio $m>1$. However, in the case of $m<1$, 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 effect on the interfacial wave in the longwave/shortwave regions. Meanwhile, top-layer viscoelasticity stabilizes the interfacial surfactant mode. However, this mode can be destabilized in the vicinity of the instability threshold but is effectively stabilized far away from the onset of instability by higher bottom-layer viscoelasticity.

Figures (22)

• Figure 1: Sketch of a double-layered viscoelastic fluid flow with insoluble surfactant at the top surface and interface.
• Figure 2: The variation of space convergence with the order of Chebyshev polynomials for moderate values of $Re_1$ when the flow parameters are $\gamma_1=0.1$, $\gamma_2=0.1$, $k=0.3$, $r=1$, $m=1.5$, $\delta=1$, $\mathrm{Ca}_1=1$, $\mathrm{Ca}_2=1$, $\mathrm{Ma}_1=0.1$, $\mathrm{Ma}_2=3.0$, $\theta=0.2~\textrm{rad}$. (b) The variation of space convergence with the order of Chebyshev polynomials for vary high values of $Re_1$ when the flow parameters are $\gamma_1=\gamma_2=10^{-5}$, $k=0.6$, $r=6$, $m=6$, $\delta=1$, $\mathrm{Ca}_1=1$, $\mathrm{Ca}_2=1$, $\mathrm{Ma}_1=1$, $\mathrm{Ma}_2=1$, and $\theta=1^{\circ}$.
• Figure 3: The effect of top-layer viscoelasticity $\gamma_1$ on the eigenvalues in $(c_r,c_i)$ plane when (a) $k=0.1$, $Re_1=5$, $Ma_1=0.5$, $Ma_2=3$, $\gamma_2=0.1$, $r=1$, $m=1.5$, and $\theta=0.2~\textrm{rad}$, (b) $k=0.5$, $Re_1=30$, $Ma_1=0.5$, $Ma_2=0.005$, $\gamma_2=0.001$, $r=1$, $m=1.5$, and $\theta=0.2~\textrm{rad}$, (c) $k=0.7$, $Re_1=50$, $Ma_1=0.01$, $Ma_2=0.01$, $\gamma_2=0.001$, $r=1$, $m=1.5$, $Pe_1=1000$, $Pe_2=500$, and $\theta=0.2~\textrm{rad}$, and (d) $k=1$, $Re_1=30000$, $Ma_1=1$, $Ma_2=1$, $\gamma_2=1\times 10^{-5}$, $r=6$, $m=6$, and $\theta=0.017~\textrm{rad}$. The remaining parameters are $\delta=1$, $\mathrm{Ca}_1=1$ and $\mathrm{Ca}_2=1$. Here $\gamma_1\in[0,~0.4]$ in (a), $\gamma_1\in[0,~0.004]$ in (b) and (c), and $\gamma_1\in[0,~4\times 10^{-5}]$ in (d). The solid blue circular shapes mark the optimal complex wave speed $c$.
• Figure 4: (a) The unstable boundary lines ($\omega_i=0$) of the SM in ($Re_1-k$) plane for varying top-layer viscoelasticity $\gamma_1$, (b) the corresponding temporal growth rate curves when $Re_1=5$ (top), and $Re_1=15$ (bottom). Here, the fixed value $\gamma_2=0.1$ with the remaining parameters $\gamma_2=0.1$, $r=1$, $m=1.5$, $\delta=1$, $Ca_1=1$, $Ca_2=1$, $Ma_1=0.5$, $Ma_2=3.0$, $\theta=0.2~\textrm{rad}$, $Pe_1=\infty$ and $Pe_2=\infty$. The magenta solid line is the stability boundary line for $Ma_1=\gamma_1=\gamma_2=0$, and the blue solid circular symbols represent the results of samanta2014effect.
• Figure 5: (a) The unstable boundary lines ($\omega_i=0$) of the SM in ($Re_1-k$) plane for varying bottom-layer viscoelasticity $\gamma_2$, (b) the corresponding temporal growth rate curves when $Re_1=5$. Here, the fixed value $\gamma_1=0.1$ with the remaining parameters as in Fig. \ref{['f4']}.