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行政院國家科學委員會專題研究計劃成果報告 Coagulation of Thermocapillary Migration between Two Liquid Droplets 執行期限：90 年 8 月 1 日至 91 年 7 月 31 日 主持人：陳時欣 私立華夏工商專科學校化工科 教授 Abstract- This paper presents an analytical study of the thermocapillary motion of an adiabatic gas bubble
and a liquid drop with constant temperature by using a method of reflections. The particles are allowed to
differ in radius, and the droplet viscosity is arbitrary, which can be limited to a solid sphere. The Peclet and
Reynolds numbers are assumed small, so that temperature and flow fields are governed respectively by the
Laplace and Stokes equations. The method of reflections is based on an analysis of the thermal and
hydrodynamic disturbances produced by an insulated gas bubble and by a single drop with constant
temperature, placed in an arbitrarily varying temperature field. The effect of interactions on thermocapillary
migrations of an insulated gas bubble and a thermally uniform droplet are generally stronger than that on the
motions influenced by the interaction of two gas bubbles or droplets. Meanwhile, the particle behaviors are
quite different from those we known.
neighboring particles2-15 or boundaries,16-24 and this Introduction
causes the particle motion to deviate from the When a liquid drop or a gas bubble is submerged prediction of eq 1 significantly. Some important in an immiscible liquid phase with a non-uniform conclusions resulted from these investigations were temperature distribution, the fluid particle migrates addressed by Chen.24 The objective of the present from the cold region to the hot region of the study is to discuss the thermocapillary interactions surrounding. This temperature-induced interfacial between a gas bubble and a fluid droplet. The tension gradient driving the particle motion is known particles are allowed to differ in radius. The gas as thermocapillary migration, and it has been the bubble is thermally insulated. While, the droplet is subject of considerable investigations for many years, with a constant internal temperature and its viscosity is arbitrary. The quasi-steady energy and momentum Assuming a small Peclet and Reynolds number equations applicable to the system, which there may Young et al.1 pioneered the formulation for the exist a prescribed temperature gradient or not, are migrating velocity of a spherical droplet of radius a, solved by using a method of reflections. which is suspended in an immersion fluid of Thermocapillary Migration of a Single Sphere
viscosity η and thermal conductivity k. The droplet By applying of the method of reflections to obtain the interactions between two spherical particles, it is essential to realize that thermal and hydrodynamic effects result when a single particle exists in an arbitrary temperature field T (x) . We consider that
a spherical particle of radius a exists in a In eq 1, η and k are the respective viscosity and surrounding fluid. The particle is a liquid drop with thermal conductivity of the droplet; and ∂γ / T constant temperature T . The particle could also be the gradient of interfacial tension γ with the local a gas bubble with zero heat flux at the interface; i.e., temperature T. All physical properties are assumed a thermally insulated bubble. The instantaneous to be constant except the interfacial tension, which is center of the particle is positioned at x , and the
assumed to vary linearly with temperature on the relative position vector is defined as r = x − x .
Although x changes with time, the problem can
In practical situations of thermophoresis, fluid be dealt with as a quasi-steady state process if both droplets are not isolated and move in the presence of the Peclet and Reynolds numbers are small. Because hydrodynamics by setting the internal-to-external the boundary conditions of the fluid velocity field viscosity ratio equal to ∞ and 0 respectively, the are coupled with the temperature gradient at the following derivations are based on the droplet particle surface, it is necessary to determine the concept. Based on the low Reynolds numbers encountered in thermocapillary migration, the fluid The energy equation governing the temperature flows inside and outside the droplet satisfy the T (x , for the external fluid is Stokes equations
v − ∇p = 0 ∇ ⋅ v = 0 , (7a,b)
η v − ∇p = 0 ∇ ⋅ v = 0 , (8c,d)
The boundary conditions at the particle surface where v and v are fluid velocities for the internal
require that temperature continuity for liquid drops and zero heat flux for gas bubbles must occur. The and external flows of the droplet, respectively; p fluid temperature must approach the prescribed field and p are the corresponding dynamic pressures. far away from the particle. Thus, we have Due to the continuity of the fluid velocities and the r = a T = T liquid drop thermocapillarity along the droplet-fluid interface as e ⋅ ∇T = 0 gas bubble
well as the surrounding fluid at rest far from the sphere, the boundary conditions for the flow fields r → ∞ T → T , where er is the radial unit vector in the spherical
r = a v = v e ⋅ (v − U) = 0 , (9a,b)
(I − e e )e : (τ − τ ) = −
T = T +   T − T −   r ⋅ ∇T
r → ∞ v → 0 . (9d)
Here, τ and τ are the viscous stress tensors for
the external flow and for the flow inside the droplet; −   rr (∇∇T
γ is the local interfacial tension for the droplet; U is
the instantaneous migrating velocity of the droplet to be determined. The surface temperature gradient, T = T +   r ⋅ ∇T
T , has been obtained in eqs 5 and 6. The thermocapillary migrating velocity is found +   rr (∇∇T
U = 0 , (for liquid drop)(10a)
(∇T ) . (for gas bubble)(10b) The tangential component of the temperature gradient at the particle surface, ∇ T , is needed to The corresponding flow fields inside and outside evaluate the thermal creep velocity in a later the particle can be obtained by expressions found in derivation. It can be obtained by differentiating eq 7 Happel & Brenner.26 The external flow field is at r = a, substituting ∇T by its Taylor expansion v = 0 , (droplet)
about x = x , and eliminating the normal
 (3e e − I ⋅ ∇T
∇ T = 0 + O(∇∇∇T ) (5) for the particle with constant temperature, and +   2Ir − 5
I − e e ⋅ ∇T
: (∇∇T
For the motion of a freely suspended droplet + a(I − e e e : ∇∇T
under an arbitrarily applied temperature gradient ∇T and flow field v in an unbounded fluid, the
Since the solid particle and gas bubble can both translational velocities of the particle can be be limited by the droplet from the viewpoint of obtained by combining eqs 11 and the modified a ∇ v
(I + 3ee) (0)
where U denotes the thermocapillary migration
As expected, both particles will move with the velocity predicted by eq 18; U means the
velocity that would exist in the absence of the other hydrodynamic velocity evaluated by Faxen’s law. for any arbitrary orientation of the particles as Interactions of a bubble-droplet pair
r → ∞ ( U = U(0) U
= 0 ). It should be noted
We now consider the quasi-steady from eq 31 that the direction of thermocapillary low-Peclet-number interactions of two spherical migration of particle 1 (gas bubble) or of particle 2 particles of radii a1 and a2 in thermocapillary (liquid droplet) is deflected by the other, unless the migration. The particle 1 is specified as a gas bubble, temperature gradient prescribed is either parallel or and the particle 2 is a liquid droplet with constant perpendicular to the line of particle centers. internal temperature T and viscosity η . They Conclusions
are oriented in an arbitrary direction to the The analytical formulation of the thermocapillary
prescribed temperature gradient E . The particles
interactions between a thermally insulated gas are assumed to be sufficiently close to interacting bubble and a thermally uniform droplet is studied in thermally and hydrodynamically with each other, but this article. The prescribed temperature sufficiently far from boundary walls for the distribution can be linear or uniform. A method of
surrounding fluid to be regarded as unbounded. Let e
reflections, which is correct to O( 7 be the unit vector pointing from the center of particle applied to obtain the temperature distributions and 1 to the center of particle 2 and r be the flow fields inside and outside the particles. Both center-to-center distance between the particles, as the mobility functions (or dimensionless velocities) illustrated in Figure 1. E is assumed to be
of the gas bubble and the liquid drop (or solid constant over distances comparable to r , and the particle) in thermocapillary migration are examined. The thermocapillary behavior of the gas bubble is generally the same to the corresponding situation For conciseness, detailed derivations of the influenced by the interaction between two gas reflected temperature and velocity fields and their bubbles in a prescribed temperature gradient. corresponding particle velocities are omitted. The However, the hydrodynamic behavior of the liquid translational velocity of thermocapillary migration drop (or the solid particle) with constant temperature of bubble 1 and the induced moving velocity of is quite different from the results we imaged, whatever the prescribed temperature gradient existing or not. When the interacting pair is far (I − 3ee) (0)
from each other and there is no applied temperature gradient, the two particles attract each other as the drop temperature (T ) higher than the surrounding temperature (T ); while the couple repulses each other as T < T . However, the pair migrates one after one if the gap between the particle surfaces is small; the bubble is heading in front for T < T , and quite the reverse for T > T . This phenomenon is remarkable if the particle size of the I − 3ee ⋅ U
droplet (or solid particle) is much smaller than that Acknowledgment
This research was supported by the National Science Council of the Republic of China under 27. Batchelor, G. K. An Introduction to Fluid REFERENCES
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