Stability of inclined flow of a liquid film with soluble surfactants and variable mass density

In this paper we carry out a theoretical study of the stability of gravity-driven flow of liquid layers doped with a soluble surfactant. We develop a model for the flow that accounts for both the concentration of surfactant adsorbed to the surface and that dissolved into the bulk of the layer. Furthermore, the mass density of the surfactant-laden liquid is assumed to increase with the concentration of surfactant in the bulk. A linear stability analysis is performed on the steady flow of a layer with uniform thickness. A nonlinear reduced dimensionality model is also constructed by means of a weighted-residuals approach and used to study the instability of the equilibrium flow and the development of permanent surface waves that arise from the instability.

Figure 1

Schematic representation of a thin film flowing down an inclined plane.

Figure 2

Neutral stability curves with $cot\beta =1$, $\text{Ka}=100$, ${\text{Sc}}_s={\text{Sc}}_b=700$, ${\xi }_a=k_d=1$, $\mathrm{\Sigma }=2$, and $M_{\mathrm{tot}}=1$.

Figure 3

${\mathrm{Re}}_{\mathrm{crit}}$ as a function of $M_{\mathrm{tot}}$ with ${\alpha }_1=0$, $cot\beta =1$, $\text{Ka}=100$, and $\mathrm{\Sigma }=2$.

Figure 4

${\text{Re}}_{\mathrm{crit}}$ as a function of $M_{\mathrm{tot}}$ with $cot\beta =1$, $\text{Ka}=100$, $\mathrm{\Sigma }=2$, ${\text{Sc}}_s={\text{Sc}}_b=700$, and ${\xi }_a=k=1$.

Figure 5

${\text{Re}}_{\mathrm{crit}}$ as a function of $M_{\mathrm{tot}}$ with $cot\beta =1$, $\text{Ka}=100$, $\mathrm{\Sigma }=2$, ${\text{Sc}}_s=700$, ${\alpha }_1=0.1$, and ${\xi }_a=k_d=1$.

Figure 6

${\text{Re}}_{\mathrm{crit}}$ as a function of $M_{\mathrm{tot}}$ with $cot\beta =1$, $\text{Ka}=100$, $\mathrm{\Sigma }=2$, ${\alpha }_1=0.1$, ${\text{Sc}}_s={\text{Sc}}_b=700$, and $k_d=1$.

Figure 7

${\text{Re}}_{\mathrm{crit}}$ as a function of $M_{\mathrm{tot}}$ with $cot\beta =1$, $\text{Ka}=100$, $\mathrm{\Sigma }=2$, ${\alpha }_1=0.1$, ${\text{Sc}}_s=700$, and ${\xi }_a=10$ and $k_d=1$.

Figure 8

${\text{Re}}_{\mathrm{crit}}$ as a function of $M_{\mathrm{tot}}$ with $cot\beta =1$, $\text{Ka}=100$, $\mathrm{\Sigma }=2$, ${\text{Sc}}_s={\text{Sc}}_b=700$, ${\xi }_a=10$, and $k_d=1$.

Figure 9

${\text{Re}}_{\mathrm{crit}}$ as a function of $M_{\mathrm{tot}}$ with $cot\beta =1$, ${\alpha }_1=0.1$, $\mathrm{\Sigma }=2$, ${\text{Sc}}_s={\text{Sc}}_b=700$, ${\xi }_a=10$, and $k_d=1$.

Figure 10

${\text{Re}}_{\mathrm{crit}}$ as a function of $M_{\mathrm{tot}}$ with $cot\beta =1$, $\text{Ka}=100$, ${\alpha }_1=0.1$, ${\text{Sc}}_s={\text{Sc}}_b=700$, ${\xi }_a=10$, and $k_d=1$.

Figure 11

${\text{Re}}_0$ versus ${\mathrm{\Gamma }}_E$ for various values of ${\xi }_s$ with $cot\beta =1$ and $M=0.25$.

Figure 12

Comparison in ${\text{Re}}_{\mathrm{crit}}$ for $0<{\mathrm{\Gamma }}_E<1$.

Figure 13

Time evolution of the flow rate.

Figure 14

Time evolution of the surface surfactant concentration.

Figure 15

Time evolution of the flow rate.

Figure 16

Time evolution of the surface surfactant concentration.