Thermocapillary effects on eccentric compound drops in Poiseuille flows

The dynamics of a neutrally buoyant, eccentric compound drop, suspended in a Poiseuille flow, in the presence of an external temperature gradient have been analyzed here. Assuming the energy and momentum transport to be diffusion-dominated, we derive approximate analytical solutions for the temperature and the velocity fields and subsequently deduce the deformation using asymptotic analysis in bispherical coordinates. We establish that depending on the direction, the presence of an imposed temperature gradient may speed up or slow down the drops. The temperature gradient has a stronger influence on the velocity of the inner drop at different eccentricities, which in turn alters the relative velocity between the drops. As a result, it is possible to achieve equilibrium configurations, where both the drops move with the same velocity. Two such equilibrium configurations exist for temperature gradients beyond a critical limit, one of which is stable. It is revealed that the imposed temperature gradient always increases the deformation as compared to an isothermal flow. Depending on the position of the inner core with respect to the outer drop center and the direction of the applied temperature gradient, the shape of the inner drop may change from prolate to oblate (or vice versa), as it travels within the outer drop. The deformation in the inner drop increases with eccentricity, while the outer drop sees very little deformation even for relatively large capillary numbers.

Figure 1

Schematic of a neutrally buoyant eccentric compound drop in a Poiseuille flow (${\mathbf{u}}_{\infty }^{\prime }$), with an imposed temperature gradient ($G^{\prime }$). A cylindrical coordinate system (${\rho }^{\prime },\phi ,z_c^{\prime }$) and a spherical coordinate system ($r^{\prime },\theta ,\phi$) are attached to the outer drop's center of mass. The two drops in general move with different velocities given by ${\mathbf{U}}_2^{\prime }$ and ${\mathbf{U}}_3^{\prime }$ for the outer and inner drop, respectively.

Figure 2

Schematic representation of the bispherical coordinates ($\xi ,\eta ,\phi$) to describe an eccentric compound drop. There are two possible configurations: (a) the “upstream configuration” ($\xi >0$) and (b) the “downstream configuration” ($\xi <0$). The outer and the inner drop surfaces are located at $\pm {\xi }_1$ and $\pm {\xi }_2$, respectively. The reference cylindrical coordinates ($\rho ,\phi ,z_c$), as discussed in Sec. 2 (see Fig. 1), have also been shown.

Figure 3

(a) Comparison of outer drop velocity ($U_2$) vs Ma for eccentric drops (present study, denoted by lines) with $e=0.001$ and concentric drops (circle), for different choices of inner drop size, $R_I=0.2$, 0.5, and 0.8. (b) Comparison of inner drop velocity ($U_3$) vs Ma under the same scenario. Other relevant parameters in (a) and (b) are $R=5$, ${\lambda }_2=0.2$, ${\lambda }_3=1$, ${\kappa }_2=3$, ${\kappa }_3=1$. (c) Comparison of $U_2$ vs Ma in the limit of single drop with $R_I=0.001$ and $e=0.001$, with the results of the work of Das et al. [30], for different choices of ${\lambda }_2=0$, 0.2, and 1, whereas $R=5$, ${\kappa }_2=3$.

Figure 4

Variation of normalized velocity (a) for outer drop ($U_2/U_{2,\mathrm{Ma}=0}$) and (b) inner drop ($U_3/U_{3,\mathrm{Ma}=0}$), as a function of Ma for different eccentricities $e=0.78$, 0.6, 0.1, and 0.001 (concentric configuration). (c) Plot of $U_2$ vs $e$ and (d) $U_3$ vs $e$, for various choices of $\mathrm{Ma}=0$, $\pm 0.5$, and $\pm 1$. Other relevant parameters are $R_I=0.2$, $R=5$, ${\lambda }_2=0.2$, ${\lambda }_3=1$, ${\kappa }_2=3$, ${\kappa }_3=1$.

Figure 5

Variations in ${\gamma }_{ij}$ with ${\theta }_1$ and ${\theta }_2$ (as appropriate), for the upstream configuration: (a) across outer drop interface (${\gamma }_{12}$) and (b) inner drop interface (${\gamma }_{23}$), for different eccentricities, $e=0$ (concentric configuration), 0.1, 0.6, and 0.78 and $\mathrm{Ma}=1$; (c) across outer drop interface (${\gamma }_{12}$) and (d) inner drop interface (${\gamma }_{23}$), for different $\mathrm{Ma}=0,\pm 0.5$, and $\pm 1$ values with $e=0.78$. Other relevant parameters are $R_I=0.2$, $R=5$, ${\lambda }_2=0.2$, ${\lambda }_3=1$, ${\kappa }_2=3$, ${\kappa }_3=1$, and $\mathrm{Ca}=0.1$.

Figure 6

Plot of $U_2$ vs ${\kappa }_3$ (inner drop's thermal conductivity) in panel (a) and of $U_3$ vs ${\kappa }_3$ in panel (b) for various ${\kappa }_2$ (outer drop conductivity) = 0, 1, 3, 1000. Other relevant parameters are $R_I=0.2$, $R=5$, ${\lambda }_2=0.2$, ${\lambda }_3=1$, $e=0.78$, $\mathrm{Ma}=1$.

Figure 7

(a) Variation in relative drop velocity ($U_2-U_3$) with Ma for different eccentricities. (b) Phase plot of $\mathrm{\Delta }\dot{z}$ vs $\mathrm{\Delta }z$ [from Eq. (22)] for different Ma values; see legend. (c) Evolution of eccentricity $e\left(t\right)$ with time ($t$), computed from Eq. (22), subject to the initial condition, $e(t=0)=0$, for various choices of $\mathrm{Ma}=-0.1$, $-0.07$, and $-0.03$. Other relevant parameters are $R_I=0.2$, $R=5$, ${\lambda }_2=0.2$, ${\lambda }_3=1$, ${\kappa }_2=3$, and ${\kappa }_3=1$.

Figure 8

Variation of ${\mathcal{AR}}_O$ in panel (a) and of ${\mathcal{AR}}_I$ in panel (b) as a function of $\mathrm{\Delta }z$, for $\mathrm{Ma}=0$, $\pm 1$, and $\pm 2$ and $R_I=0.2$. (c) Plot of ${\mathcal{AR}}_O$ vs Ma and (d) ${\mathcal{AR}}_I$ vs Ma, for different inner drop sizes: $R_I=0.2$, 0.3, and 0.5, at the extreme respective eccentricities, $\mathrm{\Delta }z=0.78$, 0.68, and 0.48. Other relevant parameters are $R=5$, ${\lambda }_2=0.2$, ${\lambda }_3=1$, ${\kappa }_2=3$, ${\kappa }_3=1$, $\mathrm{Ca}=0.4$.

Figure 9

Curvature of the outer drop ($\mathbf{\nabla }·{\widehat{\mathbf{n}}}_{12}$) in (a) and the inner drop ($\mathbf{\nabla }·{\widehat{\mathbf{n}}}_{23}$) in (b) vs the respective polar angles ${\theta }_1$ and ${\theta }_2$, for various positions of the inner drop, $\mathrm{\Delta }z=-0.23$, $-0.43$, $-0.63$, $-0.73$, and $-0.78$. Constant curvatures are represented by solid blue lines. Other relevant parameters are $R_I=0.2$, $R=5$, ${\lambda }_2=0.2$, ${\lambda }_3=1$, ${\kappa }_2=3$, ${\kappa }_3=1$, $\mathrm{Ca}=0.4$, $\mathrm{Ma}=1$.

Figure 10

Shape of the inner drop as it travels within the outer drop for $\mathrm{Ma}=1$ in (a) and $\mathrm{Ma}=-1$ in (b). The solid red line represents the undeformed outer drop and the direction of relative movement has been shown with an arrow in each panel. Other relevant parameters are $R_I=0.2$, $R=5$, ${\lambda }_2=0.2$, ${\lambda }_3=1$, ${\kappa }_2=3$, ${\kappa }_3=1$, $\mathrm{Ca}=0.6$.