Passive manipulation of free-surface instability by deformable solid bilayers

This study deals with the elastohydrodynamic coupling that occurs in the flow of a liquid layer down an inclined plane lined with a deformable solid bilayer and its consequences on the stability of the free surface of the liquid layer. The fluid is Newtonian and incompressible, while the linear elastic constitutive relation has been considered for the deformable solid bilayer, and the densities of the fluid and the two solids are kept equal. A temporal linear stability analysis is carried out for this coupled solid-fluid system. A long-wave asymptotic analysis is employed to obtain an analytical expression for the complex wavespeed in the low wave-number regime, and a numerical shooting method is used to solve the coupled set of governing differential equations in order to obtain the stability criterion for arbitrary values of the wave number. In a previous work on plane Couette flow past an elastic bilayer, Neelmegam et al. [Phys. Rev. E 90, 043004 (2014)] showed that the instability of the flow can be significantly influenced by the nature of the solid layer, which is adjacent to the liquid layer. In stark contrast, for free-surface flow past a bilayer, our long-wave asymptotic analysis demonstrates that the stability of the free-surface mode is insensitive to the nature of the solid adjacent to the liquid layer. Instead, it is the effective shear modulus of the bilayer $G_{\text{eff}}$ (given by $H/G_{\text{eff}}=H_1/G_1+H_2/G_2$, where $H=H_1+H_2$ is the total thickness of the solid bilayer, $H_1$ and $H_2$ are the thicknesses of the two solid layers, and $G_1$ and $G_2$ are the shear moduli of the two solid layers) that determines the stability of the free surface in the long-wave limit. We show that for a given Reynolds number, the free-surface instability is stabilized when $G_{\text{eff}}$ decreases below a critical value. At finite wave numbers, our numerical solution indicates that additional instabilities at the free surface and the liquid-solid interface can be induced by wall deformability and inertia in the fluid and solid. Interestingly, the onset of these additional instabilities is sensitive to the relative placements of the two solid layers comprising the bilayer. We show that it is possible to delay the onset of these additional instabilities, while still suppressing the free-surface instability, by manipulating the ratio of the shear moduli and the thicknesses of the two solid layers in the bilayer. At moderate Reynolds number and finite wave number, we demonstrate that an exchange of modes occurs between the gas-liquid and liquid-solid interfacial modes as the solid bilayer becomes more deformable. We demonstrate further that dissipative effects in the individual solid layers have an important bearing on the stability of the system, and they could also be exploited in suppressing the instability. This study thus shows that the ability to passively manipulate and control interfacial instabilities increases substantially with the use of solid bilayers.

Figure 1

Schematic diagram explaining the system under consideration and the nondimensional coordinate system used.

Figure 2

Comparison of low-$k$ asymptotic and numerical results for the GL mode: $c_i$ vs $k$ for $G_r=1$, $\theta =\pi /4$, $\text{Re}=1.0$, $H=0.5$, $\beta =0.5$, ${\mathrm{\Gamma }}_{\text{eff}}=1$, ${\eta }_{r1}=0$, ${\eta }_{r2}=0$, $\mathrm{\Sigma }=0$, ${\mathrm{\Sigma }}_1=0$, and ${\mathrm{\Sigma }}_2=0$. The horizontal line with $c_i=0$ serves as a visual aid to demarcate stable and unstable regions.

Figure 3

$c_i$ vs $k$ for the GL mode for different values of ${\mathrm{\Gamma }}_{\text{eff}}$ : Data for $G_r=16$, $\theta =\pi /4$, $\text{Re}=1.0$, $H=0.5$, and $\beta =0.5$. The remaining parameters, viz., ${\eta }_{r1}$, ${\eta }_{r2}$, $\mathrm{\Sigma }$, ${\mathrm{\Sigma }}_1$, and ${\mathrm{\Sigma }}_2$, are set to zero.

Figure 4

$c_i$ vs $k$ for the GL mode for ${\mathrm{\Gamma }}_{\text{eff}}=2.5$: Data for $G_r=16$ and $1/16$, $\theta =\pi /4$, $\text{Re}=1.0$, $H=0.5$, and $\beta =0.5$. The remaining parameters, viz., ${\eta }_{r1}$, ${\eta }_{r2}$, $\mathrm{\Sigma }$, ${\mathrm{\Sigma }}_1$, and ${\mathrm{\Sigma }}_2$, are set to zero.

Figure 5

Effect of solid-layer deformability on the inertial LS mode: $c_i$ vs $\text{Re}$ for $G_r=1/50$, $\theta =\pi /4$, $\beta =0.05$, $H=0.5$, $k=0.1$, and different values of ${\mathrm{\Gamma }}_{\text{eff}}$. The remaining parameters, viz., ${\eta }_{r1}$, ${\eta }_{r2}$, $\mathrm{\Sigma }$, ${\mathrm{\Sigma }}_1$, and ${\mathrm{\Sigma }}_2$, are set to zero.

Figure 6

Effect of solid-layer deformability on the two different interfacial modes present in the system at $\text{Re}=5.0$: $c_i$ vs $k$ for $G_r=1/50$, $\theta =\pi /4$, $\beta =0.05$, $H=0.5$, and different values of ${\mathrm{\Gamma }}_{\text{eff}}$. The remaining parameters, viz., ${\eta }_{r1}$, ${\eta }_{r2}$, $\mathrm{\Sigma }$, ${\mathrm{\Sigma }}_1$, and ${\mathrm{\Sigma }}_2$, are set to zero.

Figure 7

Mode exchange phenomenon at $\text{Re}=5.0$: $c_i$ vs $k$ for $G_r=1/50$, $\theta =\pi /4$, $\beta =0.05$, $H=0.5$, and different values of ${\mathrm{\Gamma }}_{\text{eff}}$. The remaining parameters, viz., ${\eta }_{r1}$, ${\eta }_{r2}$, $\mathrm{\Sigma }$, ${\mathrm{\Sigma }}_1$, and ${\mathrm{\Sigma }}_2$, are set to zero.

Figure 8

Neutral stability curves for the GL and LS modes of instability in the ${\mathrm{\Gamma }}_{\text{eff}}\text{−}k$ plane: $G_r=1/16$ and 16, $\theta =\pi /4$, $\beta =0.5$, $\text{Re}=1.0$, and $H=0.5$. The remaining parameters, viz., ${\eta }_{r1}$, ${\eta }_{r2}$, $\mathrm{\Sigma }$, ${\mathrm{\Sigma }}_1$, and ${\mathrm{\Sigma }}_2$, are set to zero.

Figure 9

Neutral stability curves for the GL and LS modes of instability in the ${\mathrm{\Gamma }}_{\text{eff}}\text{−}k$ plane: $G_r=1/50$ and 50, $\theta =\pi /4$, $\text{Re}=1.0$, and $H=0.25$. The remaining parameters, viz., ${\eta }_{r1}$, ${\eta }_{r2}$, $\mathrm{\Sigma }$, ${\mathrm{\Sigma }}_1$, and ${\mathrm{\Sigma }}_2$, are set to zero.

Figure 10

Neutral stability curves for GL and LS modes of instability in the ${\mathrm{\Gamma }}_{\text{eff}}\text{−}k$ plane: $G_r=1/50$ and 50, $\theta =\pi /4$, $\text{Re}=1.0$, $H=0.25$, $\beta =0.5$, ${\eta }_{r2}=0$, $\mathrm{\Sigma }=0$, ${\mathrm{\Sigma }}_1=0$, ${\mathrm{\Sigma }}_2=0$, and different ${\eta }_{r1}$ values.

Figure 11

Neutral stability curves for GL and LS modes of instability in the ${\mathrm{\Gamma }}_{\text{eff}}\text{−}k$ plane: $G_r=1/50$ and 50, $\theta =\pi /4$, $\text{Re}=1.0$, $H=0.25$, $\beta =0.5$, ${\eta }_{r1}=0$, $\mathrm{\Sigma }=0$, ${\mathrm{\Sigma }}_1=0$, ${\mathrm{\Sigma }}_2=0$, and different ${\eta }_{r2}$ values.

Figure 12

Neutral stability curves for GL and LS modes of instability in the ${\mathrm{\Gamma }}_{\text{eff}}\text{−}k$ plane: $G_r=1/100$ and $1/50$, $\theta =\pi /4$, $\text{Re}=1.0$, $H=0.25$, $\beta =0.05$, ${\eta }_{r1}={\eta }_{r2}=0$, and $\mathrm{\Sigma }={\mathrm{\Sigma }}_1={\mathrm{\Sigma }}_2=0$.