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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

**e***r* 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

(3

**e e **−

**I **⋅ ∇

*T*
∇

*T *= 0 +

*O*(∇∇∇

*T *) (5)
for the particle with constant temperature, and
+ 2

**Ir **− 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 **+ 3

**ee**) (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

*a*1 and

*a*2 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 **− 3

**ee**) (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 **− 3

**ee **⋅

**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.

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Source: http://hwhir.hwh.edu.tw/bitstream/987654321/57/1/1041302005.pdf

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