Effect of a soluble surfactant on the linear stability of two-phase flows in a finite-length channel

We study numerically the effect of a soluble surfactant on the stability of two-phase flow in a finite-length microchannel. We calculate the steady base flow and its global eigenmodes for experimentally relevant choices of material, kinetic, and flow parameters. The results show that the system is unstable for capillary numbers above a critical value. The surfactant surface concentration takes values of the order of the maximum packing density over the whole interface, even for very small volume concentrations. The two streams drag the surfactant molecules toward the downstream end of the interface against the action of the Marangoni stress. The sharp reduction of the interfacial tension at that end enhances the interface deformation and considerably destabilizes the system, translating into a sharp reduction of the critical capillary number.

Figure 1

Sketch of the fluid configuration.

Figure 2

Growth rate ${\omega }_i$ and frequency ${\omega }_r$ of the eigenvalues with ${\omega }_i\ge -8.42\times {10}^{-4}$ for $Ca=0.00681$ (solid symbols) and $Ca=0.00690$ (open symbols). The results were calculated for $\{\rho =1.72$, $\mu =0.5$, Oh = 0.00141, $q=1$; $\mathrm{Pe}={\mathrm{Pe}}_s=3.54\times {10}^6$, $k_a=5.96\times {10}^{-6}$, $k_d=5.37\times {10}^{-9}$, Ma = 0.225, $c_{\infty }=0.0055\}$.

Figure 3

Interface deformation and surfactant volume concentration $c(x,y)$ of the base flow for $\{\rho =1.72$, $\mu =0.5$, Oh = 0.00141, $Ca=0.00681$, $q=1$; $\mathrm{Pe}={\mathrm{Pe}}_s=3.54\times {10}^6$, $k_a=5.96\times {10}^{-6}$, $k_d=5.37\times {10}^{-9}$, Ma = 0.225, $c_{\infty }=0.0055\}$. The dashed line is the interface deformation in the absence of surfactant for the same values of $\rho$, $\mu$, Oh, $q$, and $Ca$.

Figure 4

(a) Surfactant volume concentration $c(x,y)$ at $x=15.16$. (b) Interface contour $h\left(x\right)$ and surfactant surface concentration $\mathrm{\Gamma }\left(x\right)$. The results were calculated for $\mathrm{Pe}={\mathrm{Pe}}_s=3.54\times {10}^6$ (solid line) and $3.54\times {10}^4$ (dashed line) and for $\{\rho =1.72$, $\mu =0.5$, Oh = 0.00141, $Ca=0.00681$, $q=1$; $k_a=5.96\times {10}^{-6}$, $k_d=5.37\times {10}^{-9}$, Ma = 0.225, $c_{\infty }=0.0055\}$.

Figure 5

Perturbation of (a) the interface contour Re[$\delta h\left(x\right)]$ and (b) the surfactant surface concentration Re[$\delta \mathrm{\Gamma }\left(x\right)]$ for $\mathrm{Pe}={\mathrm{Pe}}_s=3.54\times {10}^6$ (solid blue lines) and $3.54\times {10}^4$ (dashed black lines). The results were calculated for $\{\rho =1.72$, $\mu =0.5$, Oh = 0.00141, $Ca=0.00681$, $q=1$; $k_a=5.96\times {10}^{-6}$, $k_d=5.37\times {10}^{-9}$, Ma = 0.225, $c_{\infty }=0.0055\}$.

Figure 6

(a) Surfactant surface concentration $\mathrm{\Gamma }\left(x\right)$ and (b) interface deformation $h\left(x\right)$ for $\{\rho =1.72$, Oh = 0.00141; $Ca=0.00703$, $\mathrm{Pe}={\mathrm{Pe}}_s=3.54\times {10}^4$, $k_a=5.96\times {10}^{-6}$, $k_d=5.37\times {10}^{-9}$, Ma = 0.225, $c_{\infty }=0.0055\}$ and $(\mu =0.9,q=0.8)$ (solid lines) and $(\mu =0.72,q=1)$ (dashed lines).

Figure 7

Maximum interface deformation, $\max \left(\right|h\left(x\right)\left|\right)$, as a function of the capillary number $Ca$ for different values of the viscosity ratio $\mu$ for $\{\rho =1.72$, Oh = 0.00141, $q=1$; $\mathrm{Pe}={\mathrm{Pe}}_s=3.54\times {10}^4$, $k_a=5.96\times {10}^{-6}$, $k_d=5.37\times {10}^{-9}$, Ma = 0.225, $c_{\infty }=0.0055\}$ (circles). The solid and open symbols correspond to stable and unstable base flows, respectively.

Figure 8

Shape of the velocity profile $u\left(y\right)$ at $x=8$ for $c_{\infty }=0$ (solid line) and $c_{\infty }=0.0055$ (dashed line). The results were calculated for $\{\rho =1.72$, $\mu =0.5$, Oh = 0.00141, $Ca=0.00681$, $q=1$; $\mathrm{Pe}={\mathrm{Pe}}_s=3.54\times {10}^4$, $k_a=5.96\times {10}^{-6}$, $k_d=5.37\times {10}^{-9}$, Ma = 0.225$\}$. The blue lines correspond to the interface positions.

Figure 9

Growth rate ${\omega }_i$ and frequency ${\omega }_r$ of the eigenvalues with ${\omega }_i\ge -0.0164$ for ($c_{\infty }=0$, $Ca=0.0248$) (solid symbols) and ($c_{\infty }=0.0055$, $Ca=0.00728$) (open symbols). The results were calculated for $\{\rho =1.72$, $\mu =0.8$, Oh = 0.00141, $q=1$; $\mathrm{Pe}={\mathrm{Pe}}_s=3.54\times {10}^4$, $k_a=5.96\times {10}^{-6}$, $k_d=5.37\times {10}^{-9}$, Ma = 0.225$\}$.

Figure 10

Interface perturbation Re[$\delta h\left(x\right)]$ for the critical conditions ($c_{\infty }=0$, $Ca=0.0248$) (solid line) and ($c_{\infty }=0.0055$, $Ca=0.00728$) (dashed line). The results were calculated for $\{\rho =1.72$, $\mu =0.8$, Oh = 0.00141, $q=1$; $\mathrm{Pe}={\mathrm{Pe}}_s=3.54\times {10}^4$, $k_a=5.96\times {10}^{-6}$, $k_d=5.37\times {10}^{-9}$, Ma = 0.225$\}$. Re[$\delta h\left(x\right)]$ has been normalized so that the area enclosed in the two cases is the same.

Figure 11

Critical capillary number $C{a}^{*}$ as a function of the viscosity ratio $\mu$ for $\{\rho =1.72$, Oh = 0.00141, $q=1$; $\mathrm{Pe}={\mathrm{Pe}}_s=3.54\times {10}^4$, $k_a=5.96\times {10}^{-6}$, $k_d=5.37\times {10}^{-9}$, Ma = 0.225$\}$ and for $c_{\infty }=0$ (solid symbols) and 0.0055 (open symbols).