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Thermocapillary migration of interfacial droplets
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10.1063/1.3112777
/content/aip/journal/pof2/21/4/10.1063/1.3112777
http://aip.metastore.ingenta.com/content/aip/journal/pof2/21/4/10.1063/1.3112777

Figures

Image of FIG. 1.
FIG. 1.

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).

Image of FIG. 2.
FIG. 2.

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

Image of FIG. 3.
FIG. 3.

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).

Image of FIG. 4.
FIG. 4.

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

Image of FIG. 5.
FIG. 5.

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).

Image of FIG. 6.
FIG. 6.

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.

Image of FIG. 7.
FIG. 7.

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) , , .

Image of FIG. 8.
FIG. 8.

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).

Image of FIG. 9.
FIG. 9.

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).

Image of FIG. 10.
FIG. 10.

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.

Image of FIG. 11.
FIG. 11.

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.

Image of FIG. 12.
FIG. 12.

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

Image of FIG. 13.
FIG. 13.

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.

Image of FIG. 14.
FIG. 14.

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 .

Image of FIG. 15.
FIG. 15.

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 .

Image of FIG. 16.
FIG. 16.

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.

Image of FIG. 17.
FIG. 17.

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.

Image of FIG. 18.
FIG. 18.

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 .

Image of FIG. 19.
FIG. 19.

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).

Image of FIG. 20.
FIG. 20.

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 .

Image of FIG. 21.
FIG. 21.

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 .

Image of FIG. 22.
FIG. 22.

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

Image of FIG. 23.
FIG. 23.

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

Generic image for table
Table I.

Dimensionless parameters describing thermocapillary migration of an interfacial droplet.

Generic image for table
Table II.

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.

Generic image for table
Table III.

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

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/content/aip/journal/pof2/21/4/10.1063/1.3112777
2009-04-16
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Thermocapillary migration of interfacial droplets
http://aip.metastore.ingenta.com/content/aip/journal/pof2/21/4/10.1063/1.3112777
10.1063/1.3112777
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