^{1}and Roman O. Grigoriev

^{1}

### Abstract

We study the thermocapillary driven motion of a droplet suspended at an interface of two fluid layers subjected to an imposed temperature gradient parallel to the interface. We compute the temperature and velocity fields inside and outside of the droplet using a boundary collocation numerical scheme in the limit of small capillary and thermal Péclet numbers and compare the results with the classical problem of thermocapillary migration of a droplet in the bulk. In particular, we find that, for typical values of parameters, interfacial droplets migrate in the direction opposite to the temperature gradient, while in the classical problem migration is always in the direction of the gradient. Furthermore, we find that a rich variety of flow structures can emerge inside interfacial droplets. We also confirm that for parameters matching a recent experimental study of mixing inside interfacial microdroplets [R. O. Grigoriev, V. Sharma, and M. F. Schatz, Lab Chip6, 1369 (2006)] the interior flow can be approximated with reasonable accuracy by assuming the droplet to be completely submerged in the bottom layer.

We would like to thank Peter Mucha for his input in the early stages of this investigation and Michael Schatz, Vivek Sharma, and Daniel Borrero for sharing the experimental results. This material is based upon work supported by the National Science Foundation under Grant No. 0400370. Acknowledgment was also made to the donors of the American Chemical Society Petroleum Research Fund for partial support of this research.

I. INTRODUCTION

II. PROBLEM DESCRIPTION

A. Governing equations

B. External flow and temperature fields far from the droplet

C. Nondimensional parameters

D. Droplet shape

E. Thermocapillary migration speed

III. NUMERICAL METHOD

A. Temperature and velocity fields

B. Force on the droplet

C. Numerical solution

D. Code validation

IV. RESULTS AND DISCUSSION

A. Temperature field

B. Thermocapillary migration velocity

C. Interior flow field

1. Comparison with the model

2. Comparison with experiment

V. CONCLUSION

### Key Topics

- Fluid drops
- 201.0
- Liquid surfaces
- 32.0
- Surface tension
- 28.0
- Numerical solutions
- 24.0
- Boundary value problems
- 21.0

## Figures

A view of the interfacial drop in the plane (not to scale). We take the axis to be vertical and the axis to point in the direction of the imposed temperature gradient. The droplet is symmetric with respect to rotation about the axis (see Sec. II D).

A view of the interfacial drop in the plane (not to scale). We take the axis to be vertical and the axis to point in the direction of the imposed temperature gradient. The droplet is symmetric with respect to rotation about the axis (see Sec. II D).

Interfacial droplet shape dependence on surface tensions. A cross section is shown for (a) and , (b) and , (c) and , and (d) and .

Interfacial droplet shape dependence on surface tensions. A cross section is shown for (a) and , (b) and , (c) and , and (d) and .

The dependence of the residual (a) and condition number (b) on the truncation order without preconditioning (filled circles) and with preconditioning (open circles). The dimensionless parameters of Table I are all set to one except (the corresponding droplet shape is shown in Fig. 7).

The dependence of the residual (a) and condition number (b) on the truncation order without preconditioning (filled circles) and with preconditioning (open circles). The dimensionless parameters of Table I are all set to one except (the corresponding droplet shape is shown in Fig. 7).

The dependence of the residual (a) and condition number (b) on the droplet aspect ratio ( corresponds to spherical droplet and to a slender film).

The dependence of the residual (a) and condition number (b) on the droplet aspect ratio ( corresponds to spherical droplet and to a slender film).

The dependence of the residual (a) and condition number (b) on the truncation order. Filled circles correspond to a slender symmetrical droplet shown in Fig. 2(b). Open circles correspond to a strongly asymmetric droplet shown in Fig. 2(c).

The dependence of the residual (a) and condition number (b) on the truncation order. Filled circles correspond to a slender symmetrical droplet shown in Fig. 2(b). Open circles correspond to a strongly asymmetric droplet shown in Fig. 2(c).

The migration velocity dependence on with (a) and on with (b). The solid curves correspond to analytical solution (43) and the symbols represent solutions found numerically.

The migration velocity dependence on with (a) and on with (b). The solid curves correspond to analytical solution (43) and the symbols represent solutions found numerically.

Temperature field in the . The darker shading corresponding to cooler fluid with 20 isotherms evenly spaced over the range of temperature . The parameters are , and (a) , , , (b) , , , (c) , , , and (d) , , .

Temperature field in the . The darker shading corresponding to cooler fluid with 20 isotherms evenly spaced over the range of temperature . The parameters are , and (a) , , , (b) , , , (c) , , , and (d) , , .

The mobility function dependence on the bulk material parameters. The symbols show the numerical results for the interfacial droplet and the solid curve estimate (48).

The mobility function dependence on the bulk material parameters. The symbols show the numerical results for the interfacial droplet and the solid curve estimate (48).

The mobility function dependence on the interfacial material parameters. The symbols show the numerical results for the interfacial droplet and the solid curve estimate (48).

The mobility function dependence on the interfacial material parameters. The symbols show the numerical results for the interfacial droplet and the solid curve estimate (48).

The droplet shape and streamlines of the flow in the plane. The value of the surface tension ratio corresponds to a spherical droplet (a) and results in a slender droplet (b). All other parameters are fixed at unity.

The droplet shape and streamlines of the flow in the plane. The value of the surface tension ratio corresponds to a spherical droplet (a) and results in a slender droplet (b). All other parameters are fixed at unity.

The droplet shape and streamlines in the plane. The droplet is almost completely encapsulated by the substrate for (a) and almost completely expelled for (b). In both cases and all other parameters are set to unity.

The droplet shape and streamlines in the plane. The droplet is almost completely encapsulated by the substrate for (a) and almost completely expelled for (b). In both cases and all other parameters are set to unity.

The polar angle of the fixed points on the surface of the submerged (solid line) and the interfacial drop (circles) restricted to the plane.

The polar angle of the fixed points on the surface of the submerged (solid line) and the interfacial drop (circles) restricted to the plane.

The polar angle of the fixed points on the surface of the submerged (solid line) and the interfacial droplet (circles) restricted to the plane. The fixed points at the contact line, , are not shown.

The polar angle of the fixed points on the surface of the submerged (solid line) and the interfacial droplet (circles) restricted to the plane. The fixed points at the contact line, , are not shown.

Stream plots for a submerged [(a), (c), and (e)] and interfacial [(b), (d), and (f)] drop in the plane. Panels (a) and (b) correspond to , (c) to and (d) to , and (e) and (f) correspond to .

Stream plots for a submerged [(a), (c), and (e)] and interfacial [(b), (d), and (f)] drop in the plane. Panels (a) and (b) correspond to , (c) to and (d) to , and (e) and (f) correspond to .

The absolute value of the velocity in the plane for the submerged droplet [(a), (c), and (e)] and interfacial droplet [(b), (d), and (f)]. (a) and , (b) and , (c) and , (d) and , (e) and , and (f) and .

The absolute value of the velocity in the plane for the submerged droplet [(a), (c), and (e)] and interfacial droplet [(b), (d), and (f)]. (a) and , (b) and , (c) and , (d) and , (e) and , and (f) and .

The interior flow field for the light glycerin/water mixture. Shown are the streamlines for the submerged (a) and interfacial (b) droplet in the plane and the magnitude of for the submerged (c) and interfacial (d) droplet in the plane. and 0.33 in (c) and (d), respectively. The parameters are as in Table III.

The interior flow field for the light glycerin/water mixture. Shown are the streamlines for the submerged (a) and interfacial (b) droplet in the plane and the magnitude of for the submerged (c) and interfacial (d) droplet in the plane. and 0.33 in (c) and (d), respectively. The parameters are as in Table III.

Invariant sets of the flow inside (a) the interfacial droplet and (b) the fully submerged droplet. The open squares represent the set of elliptic fixed points in the plane. The solid lines are sample 3D steamlines and the dashed line is a heteroclinic orbit connecting the stable and unstable spiral fixed points. The surface of the droplet (light gray), the plane (dark gray), and the contact line are also invariant sets.

Invariant sets of the flow inside (a) the interfacial droplet and (b) the fully submerged droplet. The open squares represent the set of elliptic fixed points in the plane. The solid lines are sample 3D steamlines and the dashed line is a heteroclinic orbit connecting the stable and unstable spiral fixed points. The surface of the droplet (light gray), the plane (dark gray), and the contact line are also invariant sets.

Streamlines of the flow in the plane. (a) The blowup of Fig. 16(b) near a spiral fixed point. (b) A spiral fixed point of the flow produced by the model for and .

Streamlines of the flow in the plane. (a) The blowup of Fig. 16(b) near a spiral fixed point. (b) A spiral fixed point of the flow produced by the model for and .

The exterior flow field for the interfacial droplet with the parameters taken from Table III. Shown are 2D streamlines in the plane (a) and sample 3D streamlines near the droplet surface (b).

The exterior flow field for the interfacial droplet with the parameters taken from Table III. Shown are 2D streamlines in the plane (a) and sample 3D streamlines near the droplet surface (b).

The interior flow field for a pure water droplet. Shown are the streamlines for the submerged (a) and interfacial (b) droplet in the plane and the magnitude of for the submerged (c) and interfacial (d) droplet in the plane. and 2.55 in (c) and (d), respectively. The parameters are as in Table III, except for .

The interior flow field for a pure water droplet. Shown are the streamlines for the submerged (a) and interfacial (b) droplet in the plane and the magnitude of for the submerged (c) and interfacial (d) droplet in the plane. and 2.55 in (c) and (d), respectively. The parameters are as in Table III, except for .

The interior flow field for the heavy glycerine/water mixture. Shown are the streamlines for the submerged (a) and interfacial (b) droplet in the plane and the magnitude of for the submerged (c) and interfacial (d) droplet in the plane. and 0.15 in (c) and (d), respectively. The parameters are as in Table III, except for .

The interior flow field for the heavy glycerine/water mixture. Shown are the streamlines for the submerged (a) and interfacial (b) droplet in the plane and the magnitude of for the submerged (c) and interfacial (d) droplet in the plane. and 0.15 in (c) and (d), respectively. The parameters are as in Table III, except for .

Interior flow with complex topology. Shown are streamlines in the plane. The parameters are as in Table III, except for .

Interior flow with complex topology. Shown are streamlines in the plane. The parameters are as in Table III, except for .

Invariant structures of the flow are shown for the interfacial (a) and the submerged droplet (b). The open squares and crosses represent the sets of elliptic and saddle fixed points, respectively, in the plane. The solid lines are sample 3D streamlines and the dashed line is a heteroclinic orbit connecting the stable and unstable spiral points. The invariant sets contained inside the plane are not shown. The parameters are as in Table. III, except for .

Invariant structures of the flow are shown for the interfacial (a) and the submerged droplet (b). The open squares and crosses represent the sets of elliptic and saddle fixed points, respectively, in the plane. The solid lines are sample 3D streamlines and the dashed line is a heteroclinic orbit connecting the stable and unstable spiral points. The invariant sets contained inside the plane are not shown. The parameters are as in Table. III, except for .

## Tables

Dimensionless parameters describing thermocapillary migration of an interfacial droplet.

Dimensionless parameters describing thermocapillary migration of an interfacial droplet.

The coefficients of Lamb’s expansions (34) and (35) of the interior and exterior flow field for thermocapillary migration of a completely submerged droplet for , (all nonzero coefficients are shown). The column labeled “theory” corresponds to the analytical solution (41) and the column labeled “error” shows the difference between the analytical and the numerical solution.

The coefficients of Lamb’s expansions (34) and (35) of the interior and exterior flow field for thermocapillary migration of a completely submerged droplet for , (all nonzero coefficients are shown). The column labeled “theory” corresponds to the analytical solution (41) and the column labeled “error” shows the difference between the analytical and the numerical solution.

Dimensionless parameters computed from Grigoriev *et al.* (Ref. 19).

Dimensionless parameters computed from Grigoriev *et al.* (Ref. 19).

Article metrics loading...

Full text loading...

Commenting has been disabled for this content