Thermally modulated cross-stream migration of a surfactant-laden deformable drop in a Poiseuille flow

A theoretical model is developed to study the cross-stream migration of a deformable surfactant-laden droplet suspended in a nonisothermal Poiseuille flow. In addition to the thermocapillary migration of the droplet, presence of shape deformation due to the imposed flow redistributes the surfactants along the interface that has a significant effect on its dynamics, which has yet remained unexplored. Owing to the nonlinearity present in the system of governing equations, an asymptotic approach is adopted to capture the intricate and nontrivial coupling between the various influencing parameters. Assuming negligible fluid inertia and thermal convection, the droplet migration velocity is obtained through small-deformation analysis in the limits of convective and diffusive surfactant transport. For the former limiting case, the droplet migrates towards the flow centerline when the temperature of the suspending medium increases in the direction of the imposed flow; however, the direction of its cross-stream migration reverses for a system with a high viscosity ratio of droplet to carrier phase. Under the same limiting scenario and for systems with low viscosity ratio, when the temperature linearly decreases in the direction of the imposed flow, the cross-stream migration velocity reduces with the increase in the applied temperature gradient till a critical point is reached at which there occurs no cross-stream migration. Beyond the critical point, there is a gradual increase in the magnitude of the cross-stream velocity with further rise in the imposed temperature gradient. The droplet, below the critical temperature gradient, migrates towards the flow centerline; however, above it the droplet moves away from the centerline. For the other limiting case where surfactant transport is dominated by surface convection, the magnitude of the cross-stream velocity is found to be significantly larger and at the same time independent of the droplet-carrier phase viscosity ratio.

Figure 1

Schematic of a surfactant-laden droplet of radius $a$ suspended in a Poiseuille flow field. A linearly increasing temperature field (${\overline{T}}_{\infty }$) in the direction of the imposed flow ($\overline{z}$ direction) is shown. The droplet is placed at an off-center position, $\overline{e}$ being the eccentricity. $\overline{R}$ is the distance from centerline of flow to the point of zero velocity. Both spherical ($\overline{r},\theta ,\phi$) as well as Cartesian coordinate system ($\overline{x},\overline{y},\overline{z}$) are shown. The $\overline{x}$ axis is directed away from the centerline of the droplet.

Figure 2

Variation of cross-stream migration velocity of the droplet ($U_x$) with λ for different values of $\mathrm{M}{\mathrm{a}}_T$. (a) Here the temperature increases in the direction of imposed Poiseuille flow ($\zeta =1$), whereas in (b) the temperature decreases in the direction of the imposed flow ($\zeta =-1$). The other parameters are $\delta =1$, $\beta =0.5$, $\kappa =3$, $R=5$, $e=1$, and Ca = 0.1.

Figure 3

Contour plot of the surfactant distribution ($\tilde{\mathrm{\Gamma }}$) on the droplet surface for two cases: (a) the droplet is suspended in an isothermal flow field ($\mathrm{M}{\mathrm{a}}_T=0$) and (b) the droplet is suspended in a nonisothermal flow field with the temperature increasing in the direction of the imposed flow ($\mathrm{M}{\mathrm{a}}_T=1$). The other parameter values are $\delta =1,\beta =0.5,\lambda =0.1,\kappa =3,R=5,e=1$, and $\mathrm{Ca}=0.1$.

Figure 4

Variation of surface tension along two different planes which are parallel to and are located on either sides of the axial plane, for two cases: (a) the droplet is suspended in an isothermal flow field and (b) the droplet is suspended in a nonisothermal flow field with the temperature increasing in the direction of the imposed flow ($\mathrm{M}{\mathrm{a}}_T=1$). The parameter values are $\delta =1,\beta =0.5,\lambda =0.1,\kappa =3,R=5,e=1$, and $\mathrm{Ca}=0.1$.

Figure 5

Contour plot of the surfactant distribution ($\tilde{\mathrm{\Gamma }}$) on the droplet surface for the case when the temperature linearly decreases along the direction of the imposed flow ($\zeta =-1$). The plot is been shown for two cases: (a) $\mathrm{M}{\mathrm{a}}_T<\mathrm{M}{{\mathrm{a}}_T}^{*}$ and (b) $\mathrm{M}{\mathrm{a}}_T>\mathrm{M}{{\mathrm{a}}_T}^{*}$. The parameter values are $\delta =1,\beta =0.5,\lambda =0.1,\kappa =3,R=5,e=1$, and $\mathrm{Ca}=0.1$.

Figure 6

Variation of surface tension along two planes ($\theta =\pi /4,3\pi /4$) parallel to and located on either sides of the axial plane for the case when the temperature decreases in the direction of the imposed flow. In panel (a) $\mathrm{M}{\mathrm{a}}_T<\mathrm{M}{{\mathrm{a}}_T}^{*}$, and in panel (b) $\mathrm{M}{\mathrm{a}}_T<\mathrm{M}{{\mathrm{a}}_T}^{*}$. The parameter values are $\delta =1,\beta =0.5,\lambda =0.1,\kappa =3,R=5,e=1$, and $\mathrm{Ca}=0.1$.

Figure 7

Variation of cross-stream migration velocity ($U_x$) with β for different values of $\mathrm{M}{\mathrm{a}}_T$. (a) Here the applied temperature gradient increases in the direction of imposed Poiseuille flow ($\zeta =1$). The values of $\mathrm{M}{\mathrm{a}}_T$ are 0, 0.5, and 1. (b) Here the applied temperature gradient decreases in a direction of the imposed flow ($\zeta =1$). The values of $\mathrm{M}{\mathrm{a}}_T$ are 0, 0.05, 0.1, and 0.2. The other parameters are $\delta =1,\beta =0.5,R=5,e=1$, and $\mathrm{Ca}=0.1$.

Figure 8

Contour plot of the surfactant distribution ($\tilde{\mathrm{\Gamma }}$) on the droplet surface for two cases: (a) isothermal flow field ($\mathrm{M}{\mathrm{a}}_T=0$) and (b) nonisothermal flow field with the temperature increasing in the direction of the imposed flow ($\mathrm{M}{\mathrm{a}}_T=1$). The contour plot is shown for the limitng case of $\mathrm{P}{\mathrm{e}}_s\to \infty$. The parameter values are $\delta =1$, $\beta =0.5$, $\lambda =0.1$, $R=5$, $e=1$, and Ca = 0.1.

Figure 9

Contour plot of the surfactant distribution ($\tilde{\mathrm{\Gamma }}$) on the droplet surface for two scenarios: (a) $\mathrm{M}{\mathrm{a}}_T=0.01$ and (b) $\mathrm{M}{\mathrm{a}}_T=0.5$. The figures are shown for the case of linearly decreasing temperature in the direction of imposed flow and for the limiting case of high surface Péclet number ($\mathrm{P}{\mathrm{e}}_s\to \infty$). The parameter values are δ = 1, β = 0.5, λ = 0.1, R = 5, $e=1$, and Ca = 0.1.