Circulating Marangoni flows within droplets in smectic films

We present a theoretical study and numerical simulation of Marangoni convection within ellipsoidal isotropic droplets embedded in free-standing smectic films (FSSFs). The thermocapillary flows are analyzed for both isotropic droplets spontaneously formed in FSSF overheated above the bulk smectic-isotropic transition and oil lenses deposited on the surface of the smectic film. The realistic model for which the upper drop interface is free from the smectic layers, while at the lower drop surface the smectic layering persists is considered in detail. For isotropic droplets and oil lenses this leads effectively to a sticking of fluid motion at the border with a smectic shell. The above mentioned asymmetric configuration is realized experimentally when the temperature of the upper side of the film is higher than at the lower one. The full set of stationary solutions for Stokes stream functions describing the Marangoni convection flows within the ellipsoidal drops are derived analytically. The temperature distribution in the ellipsoidal drop and the surrounding air is determined in the frame of the perturbation theory. As a result, the analytical solutions for the stationary thermocapillary convection are obtained for different droplet ellipticity ratios and the heat conductivity of the liquid crystal and air. In parallel, the numerical hydrodynamic calculations of the thermocapillary motion in drops are made. Both analytical and numerical simulations predict the axially symmetric circulatory convection motion determined by the Marangoni effect at the droplet-free surface. Due to a curvature of the drop interface a temperature gradient along its free surface always exists. Thus, the thermocapillary convection within the ellipsoidal droplets in overheated FSSF is possible for the arbitrarily small Marangoni numbers. Possible experimental observations enabling the checking of our predictions are proposed.

Figure 1

Schematic view of fluid droplets in free-standing smectic films (FSSFs): (a) isotropic drops formed in overheated FSSFs. The drop is connected with the FSSF of uniform thickness by a meniscus. The drop has a lenslike shape and is symmetric relative to the horizontal plane. The height of the drop and the base radius of the cap are designated as $H$, $R_{\mathrm{cap}}$, respectively. (b) Oil lenses deposited on the surface of FSSFs. In all cases the film thickness $\mathfrak{h}$ is much smaller than the droplet height.

Figure 2

Representation of the frontal profile of the lenslike isotropic droplet in the approximation of the oblate spheroid. The parameters characterizing the ellipsoidal drop shape are designated in Fig. 1. For convenience, we fixed the zero of the $z$-coordinate axes in the center of the drop (in the middle plane of the FSSF). The parameters: $a,b$ are the semiaxes of ellipsoid, $a=c\sqrt{1+{\xi }_0^2}\equiv R_b$, $b=c{\xi }_0\equiv H/2$, ${\xi }_0/\sqrt{1+{\xi }_0^2}=b/a\ll 1$ (the latter inequality is possible only for ${\xi }_0\ll 1$), $c$ is a focus distance (coordinate of the focal point). Here ${\mathbf{e}}_{\xi }$, ${\mathbf{e}}_u$ are the unit vectors in the oblate spheroidal coordinates in their meridional plane. Note that ${\mathbf{e}}_{\xi }$ is outward normal vector to the oblate spheroidal surface of constant $\xi ={\xi }_0$, unit vector ${\mathbf{e}}_{\phi }$ is the azimuthal unit vector, oriented beyond the page (sheet) plane, ${\mathbf{e}}_u$ lies in the tangent plane to the oblate spheroid surface and completes the right-handed basis set $\{{\mathbf{e}}_u,{\mathbf{e}}_{\xi },{\mathbf{e}}_{\phi }\}$.

Figure 3

Illustration of the quality of approximation of the shape of isotropic drop in FSSF by an oblate spheroid.

Figure 4

Sketch of the free-standing smectic film with isotropic drop embedded in it. The upper surface of the droplet is free of the smectic layers, while the lower interface is covered with a nonuniform smectic shell. The thermoelectric devices above and below the drop are used to preset a vertical temperature gradient across the drop, $T_{up}>T_{dn}$.

Figure 5

Streamlines corresponding to the basic stream function ${\psi }_3$ for ${\xi }_0=0.1$; the ${\xi }_0$ value determines the ellipticity ratio of the droplet and is equal to $H/\left(2c\right)\approx H/\left(2R_b\right)$ (see Sec. 2a). All lengths are shown in the dimensionless form.

Figure 6

Streamlines corresponding to the basic stream function ${\psi }_4$ for ${\xi }_0=0.1$.

Figure 7

Streamlines corresponding to the basic stream function ${\psi }_5$ for ${\xi }_0=0.1$.

Figure 8

Streamlines corresponding to the basic stream function ${\psi }_6$ for ${\xi }_0=0.2$.

Figure 9

Streamlines corresponding to the first stream function ${\psi }_{1,st}$, for ${\xi }_0=0.1$. The bold black line at the bottom part of the drop indicates that the sticking (no slip) conditions are fulfilled at this interface. The same symbol will be used in the following figures. In contrast to this, the upper part of the drop is free (in contact with the air).

Figure 10

Streamlines corresponding to the first stream function ${\psi }_{2,st}$, for ${\xi }_0=0.1$.

Figure 11

Streamlines corresponding to the first stream function ${\psi }_{3,st}$, for ${\xi }_0=0.1$.

Figure 12

Streamlines corresponding to the first stream function ${\psi }_{4,st}$, for ${\xi }_0=0.1$.

Figure 13

Illustration of the convergence of the expansion procedure for the stream function in dependence on the number of the basic functions. Dependence of the relative error $\varepsilon$ in deviations of norms $E{\left[\left\{c_j\right\}\right]}^{\mathrm{free}}$, $E{\left[\left\{c_j\right\}\right]}^{\mathrm{stick}}$ from zero ($\mathrm{Ma}=1$, ${\xi }_0=0.1$).

Figure 14

Coefficients in decomposition of the stream function $\psi$ for even and odd basic solutions ($\mathrm{Ma}=1$, ${\xi }_0=0.1$, $u_0=0.1$). Illustration of the convergence of the expansion of the solution for $\psi$ over basic functions in dependence on their number $i$ as a function of a current integer number $k$. Both even and odd basic solutions are shown; the convergence is reached for $k$ values about 25.

Figure 15

Convergence of the coefficient $c_3$ of the expansion of the final stream function over the basic functions in dependence on their $\mathrm{number}(\mathrm{Ma}=1$, ${\xi }_0=0.1$).

Figure 16

Streamlines and temperature distribution for the generalized thermocapillary convection within the drop (${\xi }_0=0.1$, $\kappa =0.2$, $u_0=0.25$, $\text{Ma}=1$, $N_r=110$; $\kappa ={\varkappa }_{\mathrm{air}}/\varkappa$ is a relative heat conductivity); the color scale for the temperature is characterized by a more saturated light-purple in a hot areas). The sticking boundary conditions are extended above the circular edge of a drop (bold black line).

Figure 17

Modulus of velocity for the generalized thermoconvection flow within the drop (${\xi }_0=0.1$, $u_0=0.25$, $\kappa =0.2$, $\text{Ma}=1$, $N_r=110$). The dimensional values of $v$ can be deduced using the corresponding scaling parameter $\chi /H=4\times {10}^{-3}$ m ${\mathrm{s}}^{-1}$. The circulation period $\mathrm{\Delta }t$ for certain velocities can be determined from the integral over the closed trajectory of the motion as shown in the Discussion. In dimensionless form $\mathrm{\Delta }t$ values are 124, 151, 206, and 484 as counted off from the center of the vortex to its periphery, respectively (from dark streamline to light one in Fig. 16). The dimensional values of $\mathrm{\Delta }t$ can be obtained using the corresponding scaling parameter $\mathrm{\Delta }t=\left((H^2/2){\left({\xi }_0\chi \right)}^{-1}\right)\mathrm{\Delta }\tilde{t}$, $(H^2/2){\left({\xi }_0\chi \right)}^{-1}$, which is about 0.0125 s for the typical geometrical and material characteristics of the drop. In seconds they constitute 1.55 s, 1.89 s, 2.575 s, and 6.05 s, respectively.

Figure 18

Streamlines and temperature distribution (color map) for the generalized movement of thermocapillary convection (${\xi }_0=0.1$, $u_0=0$, $\kappa =0.2$, $\text{Ma}=1$, $N_r=110$).

Figure 19

Modulus of velocity for the generalized thermoconvection flow within the drop (${\xi }_0=0.1$, $u_0=0$, $\text{Ma}=1$, $N_r=110$). The circulation period $\mathrm{\Delta }t$ for certain velocities in dimensionless form is 135, 156, 206, and 489 as counted off from the center of the vortex to its periphery (from dark streamline to light one in Fig. 18). The dimensional values of $\mathrm{\Delta }t$ constitute 1.69 s, 1.95 s, 2.58 s, and 6.11 s, respectively.

Figure 20

Difference of streamlines $\psi$ for the generalized thermoconvection flow within the drop for $u_0=0$ and $u_0=0.25$, (${\xi }_0=0.1$, $\kappa =0.2$, $\text{Ma}=1$, $N_r=110$).

Figure 21

Illustration that the image part of the Marangoni number, Ma, decreases with the increase of the number $N_r$ of the accounted basic functions (four lowest values of Ma spectrum for each $N_r$ are presented, ${\xi }_0=0.1$, $\kappa =0.2$).

Figure 22

Phase diagram for the critical Marangoni number ${\mathrm{Ma}}_c$ as a function of the ellipticity ratio ${\xi }_0$ for different values of the relative heat conductivity $\kappa$.

Figure 23

Critical Marangoni flows in the ellipsoidal drop (${\xi }_0=0.1$, $\kappa =0.2$, ${\mathrm{Ma}}_c=-81.2$). The dashed and solid lines indicate the opposite direction of the fluid velocity in the neighboring vortices.

Figure 24

Critical Marangoni flows in the ellipsoidal drop (${\xi }_0=0.05$, $\kappa =0.2$, ${\mathrm{Ma}}_c=-75.9$). The dashed and solid lines indicate the opposite direction of the fluid velocity in the neighboring vortices.

Figure 25

Streamlines and temperature distribution (the logarithmic color scale for the temperature is characterized by more saturated light purple in hot areas) for the critical Marangoni flows in the ellipsoidal drop (${\xi }_0=0.05$, $\kappa =0.2$, ${\mathrm{Ma}}_c=-75.9$). The dashed and solid lines indicate the opposite direction of the fluid velocity in the neighboring vortices.

Figure 26

Temperature distribution within the isotropic droplets; the smectic shell is in contact with the bottom surface of isotropic droplet (a, b, c). The droplet shapes: ellipse (a), a biconvex lens (b), and a lens with the cut end at the edge (c).

Figure 27

Distribution of the fluid flow velocity within the isotropic droplets; the smectic shell is in contact with the bottom free surface of isotropic droplet (a, b, c). The droplet shapes: ellipse (a), biconvex lens (b), and a lens with the cut end at the edge (c).

Figure 28

Stream functions for the ellipsoidal drop with a smectic film on the bottom free surface: (a) $t=0.1t_{\mathrm{rel}}$, (b) $t=0.5t_{\mathrm{rel}}$, and (c) $t=20t_{\mathrm{rel}}$.

Figure 29

Sketch for the problem statement: smectic film is at the bottom drop surface.

Figure 30

Discretization of the spatial domain (the mesh used).