8618
Langmuir 1999, 15, 8618-8626
Thermocapillary Migration of a Fluid Sphere Parallel to an Insulated Plane Shih H. Chen Department of Chemical Engineering, Hwa Hsia College of Technology and Commerce, Taipei, Taiwan 235, Republic of China Received May 25, 1999. In Final Form: August 9, 1999 An analytical study is presented for thermocapillary migration of a fluid sphere in a constant prescribed temperature gradient parallel to an adiabatic plane. The Peclet and Reynolds numbers are assumed to be small, so that the temperature distributions and flow fields of the fluids inside and outside the droplet are governed by the Laplace and Stokes equations, respectively. A method of reflections is used to obtain the asymptotic formulas for temperature and velocity fields in the quasi-steady situation. The thermally insulated plane may be a solid wall (no-slip) and/or a free surface (perfect-slip). The boundary effect on the asymmetric thermocapillary motion of a droplet parallel to a plane is found to be weaker than that on the axisymmetric thermocapillary migration of a sphere normal to a plane with constant temperature. In comparison with the motion driven by gravitational force, the interaction between the particle and the boundary is less significant under thermocapillary migration. Even so, the interaction between a plane and a fluid sphere can be very strong when their gap thickness approaches zero. For thermocapillary migration of a droplet parallel to a solid plane, the plane surface reduces the translational velocity of the droplet. In the case of a droplet migrating parallel to a free surface due to thermocapillarity, the translating velocity of the particle can be either greater or smaller than that which would exist in the absence of the plane surface, depending on the relative thermal conductivity of the droplet and its relative distance from the plane. Both the droplet velocity and the region affected by the surface are evaluated by considering the droplet mobility. Generally speaking, a free surface exerts less influence on the particle movement than a solid wall.
Introduction Thermocapillary migration refers to the motion of fluid spheres, which are immersed in another immiscible liquid phase, in response to a temperature gradient. This phenomenon was first described by Young et al. in 1959,1 observing the movement of bubbles in a vertical liquid bridge in the gap between the anvils of a micrometer. The lower anvil was heated to produce a temperature gradient to arrest the buoyant rise of the bubbles or even to drive the bubbles downward. Since they are a mechanism for moving fluid particles from a cold surface to a hot surface, thermocapillary phenomena have become a subject of current interest following the development of orbiting spacecraft and opportunities for experimenting and manfacturing under near-weightless conditions. For example, the removal of unwanted gas bubbles or liquid drops in a continuous phase by thermocapillary forces is a possibility during the processing of materials in the reduced gravity environment provided by an orbiting laboratory. By assuming a small Peclet number and a small Reynolds number, Young et al.1 also obtained a theoretical prediction for the migration velocity of a spherical droplet of radius a, present in an immersion fluid of viscosity η and thermal conductivity k with an imposed linear temperature distribution T∞(x). The droplet velocity U(0) is related to the uniform prescribed temperature gradient ∇T∞ by
U(0) )
a ∂γ 2 ∇T∞ (2 + ki/k)(2 + 3ηi/η) η ∂T
(
)
(1)
Here, ηi and ki are the viscosity and thermal conductivity, respectively, of the fluid sphere and ∂γ/∂T is the gradient of the interfacial tension γ with the local temperature T. (1) Young, N. O.; Goldstein, J. S.; Brock, M. J. J. Fluid Mech. 1959, 6, 350.
In eq 1, all physical properties are assumed to be constant except for the interfacial tension, which is assumed to vary linearly with temperature. In practical applications of thermocapillary migration, fluid particles are not isolated and will move in the presence of neighboring particles2-15 and/or boundaries.16-22 During the past 2 decades, much progress has been made in the theoretical analysis concerning the applicability of eq 1 for a fluid droplet in a variety of bounded systems.4,7,9,23-25 Several important conclusions have resulted (2) Meyyappan, M.; Wilcox, W. R.; Subramanian, R. S. J. Colloid Interface Sci. 1983, 94, 243. (3) Meyyappan, M.; Subramanian, R. S. J. Colloid Interface Sci. 1984, 97, 291. (4) Anderson, J. L. Int. J. Multiphase Flow 1985, 11, 813. (5) Feuillebois, F. J. Colloid Interface Sci. 1989, 131, 267. (6) Keh, H. J.; Chen, S. H. Int. J. Multiphase Flow 1990, 16, 515. (7) Zhang, X.; Davis, R. H. J. Colloid Interface Sci. 1992, 152, 548. (8) Keh, H. J.; Chen, L. S. J. Colloid Interface Sci. 1992, 151, 1. (9) Satrape, J. V. Phys. Fluids 1992, A4, 1883. (10) Keh, H. J.; Chen, L. S. Chem. Eng. Sci. 1993, 48, 3565. (11) Wei, H.; Subramanian, R. S. Phys. Fluids 1993, A5, 1583. (12) Wang, Y.; Mauri, R.; Acrivos, A. J. Fluid Mech. 1994, 261, 47. (13) Golovin, A. A. Int. J. Multiphase Flow 1995, 21, 715. (14) Wei, H.; Subramanian, R. S. J. Colloid Interface Sci. 1995, 172, 395. (15) Leshansky, A. M.; Golovin, A. A.; Nir, A. Phys. Fluids 1997, 9, 2818. (16) Meyyappan, M.; Wilcox, W. R.; Subramanian, R. S. J. Colloid Interface Sci. 1981, 83, 199. (17) Meyyappan, M.; Subramanian, R. S. J. Colloid Interface Sci. 1987, 115, 206. (18) Chen, S. H.; Keh, H. J. J. Colloid Interface Sci. 1990, 137, 550. (19) Barton, K. D.; Subramanian, R. S. J. Colloid Interface Sci. 1990, 137, 170. (20) Ascoli, E. P.; Leal, L. G. J. Colloid Interface Sci. 1990, 138, 220. (21) Loewenberg, M.; Davis, R. H. J. Colloid Interface Sci. 1993, 160, 265. (22) Chen, S. H. Langmuir 1999, 15, 2674. (23) Hetsroni, G.; Haber, S. Rheol. Acta 1970, 9, 488. (24) Acrivos, A.; Jeffrey, D. J.; Saville, D. A. J. Fluid Mech. 1990, 212, 95. (25) Felderhof, B. U. Phys. Fluids 1996, 8, 1705.
10.1021/la990641b CCC: $18.00 © 1999 American Chemical Society Published on Web 10/07/1999
Thermocapillary Migration of a Fluid Sphere
Figure 1. Geometric sketch for a fluid sphere in the proximity of a plane.
from these investigations of particle interactions in thermocapillary migration. First, the particle interaction effects on thermocapillary migration are in general much weaker than those on sedimentation. This is because the disturbance of the fluid velocity field caused by a thermocapillary migrating droplet decays faster (as r-3, where r is the distance from the particle center) than that caused by a settling droplet (as r-1). In sedimentation, there is a net gravitational force exerted on the droplet, which is balanced by a nonzero hydrodynamic force. However, there is no hydrodynamic force exerted on the droplet in thermocapillary migration. As a consequence, the disturbance velocity fields in the surrounding fluid for the two situations decay at different rates with r. Second, for the two identical liquid droplets aligned parallel to the prescribed temperature gradient, the interaction effects make each droplet move faster than it would if isolated, while, for the two identical droplets undergoing the thermocapillary motion normal to their center line, each droplet migrates slower than when isolated. Third, the migrating velocity of each bubble in a collection of identical bubbles, which can be arbitrarily oriented, is unaffected by the presence of the others. Fourth, the influence of interactions between droplets is in general far greater on the smaller droplet than on the larger one. The objective of this work is to obtain an analytical solution to the problem of thermocapillary motion of a fluid sphere moving parallel to a nearby surface, a problem which has not been previously formulated. The infinite plane boundary which is thermally insulated can be a solid wall and/or a free surface. The quasi-steady energy and momentum equations applicable to the system are solved by using a method of reflections. The analytical results to correct eq 1 for various cases of flow are presented. The asymptotic solution obtained in the present article is not exact; however, the solution, which is in an analytical form, is convenient for application. It is found that the effect of the presence of the plane surface on the droplet velocity can be significant when the distance between the droplet and the plane becomes small. In addition, the region affected by the surface for thermocapillary motion of a droplet parallel to an insulated plane is also investigated. Description of the Problem Figure 1 shows the thermocapillary motion of a spherical droplet with radius a in the direction parallel to an infinite surface, which is thermally insulated at a distance h from the center of the sphere. A linear temperature field T∞(x) with a uniform thermal gradient E∞ez (equal to ∇T∞) is imposed on the surrounding fluid medium far away from
Langmuir, Vol. 15, No. 25, 1999 8619
the sphere; and ez is the axial unit vector in the righthanded Cartesian coordinate system (x, y, z). The droplet center is chosen to be at the origin of the coordinate frame for convenience. In addition to the rectangular coordinates, the spherical coordinate system (r, θ, φ) is also employed. The fluids, inside and outside the droplet, are assumed to be Newtonian and incompressible. Gravitational effects are ignored. Our purpose here is to determine the correction to eq 1 for the droplet velocity in the presence of a solid plane and/or a free surface. The transport of momentum and energy is inherently unsteady in this problem concerning thermocapillary migration of a spherical drop parallel to an adiabatic plane. However, the problem can be considered quasi-steady if the Peclet and Reynolds numbers are small (the convective effects are neglected). The energy equation governing the temperature distribution T(x) for the outside fluid phase with constant conductivity k and viscosity η is the Laplace equation
∇2T ) 0
(2a)
For the fluid inside the droplet, one has
∇2Ti ) 0
(2b)
where Ti(x) is the temperature distribution inside the droplet. The boundary condition at the droplet surface requires the temperature and the normal heat fluxes to be continuous. Also, the temperature of the surrounding fluid must approach the linear prescribed field far away from the droplet and the temperature inside the droplet is finite everywhere. In addition, the boundary condition at the plane surface requires that the normal heat flux be zero because of the thermally insulated property of the plane. Thus one has
r)a
T ) Ti k
∂Ti ∂T ) ki ∂r ∂r
(3a) (3b)
r
