Stability of gravity-driven free-surface flow past a deformable solid: The role of depth-dependent modulus

The linear stability of a Newtonian liquid layer flowing down an inclined plane lined with a deformable linear elastic solid characterized by a continuously varying modulus is analyzed in this study. A low-wave-number asymptotic analysis is performed to obtain an analytical expression for the complex wavespeed which shows striking similarity with the earlier results of Sahu and Shankar [Sahu and Shankar, Phys. Rev. E 94, 013111 (2016)] for gravity-driven flow of Newtonian fluid past a solid bilayer having constant shear modulus (in each layer) lined on a rigid inclined plane. This shows that a deformable solid layer having a continuously varying shear modulus can be treated as a generalization of a system having multiple solid layers of constant shear modulus. Also, in the low-wave-number limit, we show that the stability of the free surface is governed by the value of effective shear modulus $G_{\mathrm{eff}}$, and not by the detailed spatial variation of the modulus. Here the effective shear modulus ($H/G_{\mathrm{eff}}={\int }_1^{1+H}1/[E_0+\overline{E}\left(z\right)]dz$, where $[E_0+\overline{E}\left(z\right)]$ represents the modulus gradient function) characterizes the overall modulus of the elastic solid, which is obtained by treating the continuous variation to be the limit of the arrangement of solid layers of infinitesimal thickness (each having a constant shear modulus) in a series. At finite wave numbers, we show that the free-surface and the liquid-solid interface become unstable as we increase the value of $\mathrm{\Gamma }$, where $\mathrm{\Gamma }$ indicates the ratio of viscous stresses in the fluid to elastic stresses in the solid. When the system is analysed for different types of spatial modulus variations, we find results similar to those of Gkanis and Kumar [Gkanis and Kumar, Phys. Rev. E 73, 026307 (2006)], i.e., when we have two different configurations of the shear modulus function that have the same spatially averaged modulus, but have different values at the interface, the system is more stable for the configuration having higher shear modulus at the liquid-solid interface. In a similar manner, when we examined systems having the same shear modulus at the liquid-solid interface and same average modulus, the more stable case is the one which has a higher value of shear modulus just below the interface. Thus the use of deformable solids with a depth-dependent modulus potentially offers more control in the passive manipulation of the instabilities.

Figure 1

Schematic diagram showing the system under consideration with nondimensional coordinates.

Figure 2

Comparison of numerical results for continuous and constant modulus to low $k$ asymptotic results: $c_i$ vs $k$ for $\beta =\pi /4, \text{Re}=1.0, H=0.5, {\mathrm{\Gamma }}_{\mathrm{eff}}=1$. The horizontal line at $c_i=0$ acts as a reference to differentiate between stable and unstable regions.

Figure 3

$c_i$ vs $k$ variation for different solid configurations: $H=0.5, \text{Re}=1, {\mathrm{\Gamma }}_{\mathrm{eff}}=1, \beta =\pi /4, b=0.5$.

Figure 4

$c_i$ vs $k$ variation for the GL mode for different values of ${\mathrm{\Gamma }}_{\mathrm{eff}}$: Data for $a_1=16, a_2=1, \beta =\pi /4, \text{Re}=1.0, H=0.5, b=0.5$.

Figure 5

$c_i$ vs $k$ variation for different values of $\delta$: Data for $\beta =\pi /4, \text{Re}=1.0, H=0.5, {\mathrm{\Gamma }}_{\mathrm{eff}}=2.5, b=0.5, a_1=16, a_2=1$, and $\delta$ as mentioned in the figure.

Figure 6

Variation of the solid modulus with $z$: Data for $H=0.5, b=0.5, a_1=16, a_2=1$, and $\delta$ as mentioned in the figure.

Figure 7

Variation of the solid modulus with $z$: Data for $H=0.5, b=0.5, a_1=1, a_2=16$, and $\delta$ as mentioned in the figure.

Figure 8

$c_i$ vs $k$ variation for the different values of $\delta$: Data for $\beta =\pi /4, \text{Re}=1.0, H=0.5, {\mathrm{\Gamma }}_{\mathrm{eff}}=2.5, b=0.5, a_1=1, a_2=16$, and $\delta$ as mentioned in the figure.

Figure 9

Effect of $\delta$ on neutral stability curves in the ${\mathrm{\Gamma }}_{\mathrm{eff}}-k$ plane: $\beta =\pi /4, b=0.5, \text{Re}=1.0, H=0.5, a_1=16$, and $a_2=1$ and $\delta$ as mentioned in the figure.

Figure 10

Effect of $\delta$ on neutral stability curves in the ${\mathrm{\Gamma }}_{\mathrm{eff}}-k$ plane: $\beta =\pi /4, b=0.5, \text{Re}=1.0, H=0.5, a_1=1$, and $a_2=16$ and $\delta$ as mentioned in the figure.

Figure 11

Variation of the solid modulus with $z$: Data for $H=0.5$ and $b=0.5$.

Figure 12

Neutral stability curves in the ${\mathrm{\Gamma }}_{\mathrm{eff}}$- $k$ plane showing the stability zones for the different solid modulus functions: Data for $\text{Re}=1, H=0.5$ and $b=0.5, \beta =\pi /4$.