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Dynamics of a harmonically driven fluid interface in a capillary
E. Charlaix, H. Gayvallet
To cite this version:
E. Charlaix, H. Gayvallet. Dynamics of a harmonically driven fluid interface in a capillary. Journal de Physique II, EDP Sciences, 1992, 2 (11), pp.2025-2038. �10.1051/jp2:1992249�. �jpa-00247786�
Classification Physics Abstracts
47.55K 03.40G
Dynamics of a harmonically driven fluid interface in a capillary
E. Charlaix and H. Gayvallet
Laboratoire de Physique, Ecole Normale Supdrieure de Lyon, 46Allde d'ltalie, 69364 Lyon Cedex 7, France
(Received ii May 1992, accepted in final for 2 August 1992)
RdsiJmd. Nous utilisons une technique harrnonique pour dtudier les propridtds dynamiques d'une interface fluide dons un capillaire. On mesure en fonction de l'amplitude et de la frdquence
d'excitation la chute de pression capillaire causde par le ddplacement des fluides. Pour des
ddplacements de basse frdquence, il existe un seuil de l'amplitude au-delh duquel la ligne de
contact se ddsancre et glisse ; ce seuil est lid h l'hystdrdsis statique de l'angle de contact. Lorsqu'on
augmente la frdquence h amplitude constante, on observe que la ligne se rdancre progressivement.
L'amortissement mesurd suggdre que l'angle de contact dynamique depend non seulement de la vitesse de la ligne de contact mais aussi de la frdquence d'excitation.
Abstract. We use an harmonic technique to probe the dynamics of a liquid-liquid interface in a
capillary. The capillary pressure drop induced by a harmonic displacement of the fluids is recorded
as a function of the excitation amplitude and frequency. At low frequency there is an amplitude
threshold above which the contact line depins and slides, and this threshold is related to the static
hysteresis of the contact angle. When the contact line depins, the damping due to its motion is measured from the component of the capillary pressure in phase with the fluids velocity. At
frequencies higher than one hertz we observe a phenomenon of dynamic pinning. The behaviour of the damping suggests that the variation of the contact angle with the three-phase line velocity is
frequency-dependent-
l~ Introduction~
Flows involving a three-phase line formed by a fluid interface and a solid wall have an
important role in many engineering applications, such as coating of solids, enhanced oil recovery, multiphase flows, etc. The main problem is that the conventional no-slip boundary condition on the solid surface leads to a divergence of the stresses and energy dissipation on the
three-phase line when it moves at non-zero velocity. A number of theoretical [1-4] and
experimental works [5-8] have focussed on the characterisation of the dynamic contact angle as
a function of the contact line Velocity in steady-state flows. In this paper we investigate the motion of the contact line at non-zero frequency. This aspect is of particular interest in the case of heterogeneous solid surfaces associated with static contact angle hysteresis : in this case the
motion of the contact line is made of successive « jumps » from one defect to another [3, 4], and a steady-state characterization does not provide a complete description of its dynamical properties.
Our experiments are performed on fluid-fluid interfaces held in glass capillaries of millimetric diameters. They consist in driving the interface with a harmonic fluid displacement
of frequency lying between 0.I Hz and 10 Hz, and measuring the component of the capillary
pressure drop at the excitating frequency. Harmonic excitation has already been used by Dimond et al. [9 to study the resonance of an interface which occurs when the contact line is
pinned at the solid sudace. Here we use this technique to study the dynamics of the contact line.
Section 2 is devoted to the description of the experimental set up, and to the study of small
amplitude excitations in two systems: a water/decane underdamped intedace and a
glycerol/silicone-oil overdamped interface. We recall the linear theory for small amplitude
oscillations [91 and find good agreement with our experimental results. In section 3 we study
the capillary pressure drop ind,Jced by the motion of the contact line as a function of the
amplitude and frequency of the drive. Special emphasis is given to the pressure component lying in phase with the fluid velocity, which measures the damping due to the contact line
motion. In sectinn 4 we develop a simple model for-the pressure response based on the steady-
state variation of the dynamic contact angle versus the contact line velocity, and show that it
cannot fully account for fine experimental results. We conclude that the dependence of the contact angle with the three-phase line Velocity is frequency-dependent.
2~ Experimental setup and dynamics of the pinned interface~
The experiments are done in the cell drawn in figure I. The interface is located in a horizontal
glass capillary of radius R
= 0.75 mm and length L
=
5 cm. The cell preparation is the
following : it is boiled in distilled water, acetone washed, rinsed and dried. It is first filled with
one of the fluids the interface is then introduced by pushing the other fluid in the cell and located at about 5 mm from one extremity of the capillary tube. The contact angle hysteresis is usually important.
An oscillatory fluid displaceinent of pulsation w is induced in the system by means of a vibration excitator driving a soft latex membrane. The pressures P
j and >°~ at the ends of the
capillary containing the interface are measured with Omega PX170 pressure transducers of high sensitivity. Their component at pulsation w, P
j(~o) and P
~(w ), are recorded in amplitude
and phase as a function of frequency by a Hewlett-Packard 35660 signal analyser with a source output which is used to drive the system. The pressure response is studied for frequencies ranging from 0. I Hz to 50 Hz. For th~ measure of the fluid displacement, a second capillary
referred to as « velocity capillary » is placed in series between the transducer measuring
Pi and a reservoir at atmospheric pressure. At the frequencies used, the fluids can be
considered as uncompressible and the flow rate is uniform through the cell. The velocity
vihration
excitator velocity
fluid ~~~~~~~
~~~~~~~
fl~~jd ~
~2 'tran~ucers~
~~ ~~~~~~°~~
signal anayser
Fig. j Experimental setup.
vjo~ R
o dx
Fig. 2. Spherical interface in a capillary. The volume of the spherical cap formed by the meniscus is V(o) and dx is the fluid displacement.
U(w in the capillary holding the interface is then obtained from the pressure Pi (w ) and the
dynamic permeability «'(w) of the velocity capillary (see hereafter), and the amplitude
xo of the fluid displacement is xo
=
iU (w )/w. The meniscus shape and motion are recorded by
the mean of a Leica macroscope hooked to a video camera. The resolution obtained is about 50 ~Lm. A strobe light is used to slow down or freeze the interface motion at high frequency.
The linear theory for small amplitude oscillations can be summarized as follows. It is assumed that the interface keeps a spherical shape and forms with the wall an uniform contact
angle 0 (Fig. 2). When the contact line is pinned, a displacement dx it )
= xo e~ ~~~ of the fluids induces a change in the contact angle, and the surface tension y creates a restoring force. Let Vi 0) be the volume limited by the spherical cap and the plane containing the contact line, and AP~~~ the capillary pressure drop across the interface.
Then
Vi 0)
=
arR~(2 + sin ) cos 0/3 ii + sin 0)~
and AP~~~ =
2 y cos 0/R. When the starting contact angle is 00, the change in AP~~~ induced by a small displacement dx is :
d AP~~~ d AP~~~ dv jo i dv 2 y sin 00 ii + sin 00)~
= ~~ ~~ ~ = ~ = C loo) II)
On the other hand, the pressure drop AP
~~~ due to the flow in the capillary is related to the a-c-
permeability « (w ) characterising the harmonic flow of a single homogeneous fluid of density
p and viscosity ~ through a pipe of radius R [101 uiw )
= « iw AP~iw )/~L
~ (w ) = # ~ wpR~
pm ~
k~y)
=
2 Jj(fi)
~~~
~iJo(fi)
The single fluid flow has the following properties : at low frequency (w ~ w
~ =
8 ~/pR~) the
motion is govemed by viscous damping and AP~(w)/U(w)m 8 ~L/R~, whereas at high
frequency inertial forces dominate and AP~(w )/U(w ) m ipLw. In our case there are two fluids in the capillary. However if the meniscus is located very close to one extremity of the
capillary, and if the viscosities and densities of the fluids are not very different, then (2) can be used with a good approximation. The linear response of the system is thus :
AP (W) 6P cap + 6P flow lC 1°0) ~L
UIW ) ~ UIW) ~ W ~ K iW ) ~~~
single fluid
interface
(a) 2
o
log f (Hz)
_~ single fluid
j
c-$
8 -2
5
to
interface
o 2
logf (Hz)
Fig. 3. Modulus of the frequency response U(w)Ii° (w obtained at low displacement amplitude (X~ =
3x~/2 R
~ 0. I in the overdamped system (a) and in the underdarnped system (b). (D) data obtained with a single fluid in the experimental cell. U(w )Ii° (w is the dynamic permeability of the capillary. (x) data obtained with the interface located as in figure I. The continuous line is the best fit of the data points
with the linear theory (Eq. (3)).
The first term of the right-hand-side reflects the action of the interface as a spring of stiffness C loo). One can define the frequency wo of the undamped oscillator : wo
= [C loo)/pL]'~~
Two types of oscillations occur for the meniscus :
if wow w~ the oscillator is overdamped the pressure response has a smooth transition from a capillary response at low frequency to a viscous response at w ~ w(/w~ ;
if wow w~ the oscillator is underdamped and the transition between the capillary
response at low frequency and the inertial response at high frequency occurs through a
resonance.
We study the underdamped oscillations with distilled water and n-decane, and the
overdamped one with silicone oil 47V20 and a solution of 70 fb glycerol and 30 fb water in volume. In both case, the oils are the fluids filling most of the capillary. The kinematic
viscosities of the fluids are measured at room temperature with a Ubbelhode viscosimeter, and the surface tensions with the pendant drop method. Their values are listed in table I as well as w~ and the typical value for wo in each system. The two couple of fluids are held in contact for at least 24 h before any measurement or experiments, in order for their surface tension to
stabilize.
Table I. Properties of the fluids used in the experiment at room temperature (20 °C), and
characteristic JFequency of the overdamped and underdamped system : w~
=
8 ~/pR~. A
typical value of wo is (8 y/pLR~)'~~ obtained with 00 = ar/2.
~ p Y
~~
Wc/2 ~ ~°o/2
~~~~~~
~~2/~ g cm~~
Hz ~~
mineral oil 0.2 0.97
overdamped 39 45 18
system glycerol 70 fb 0.209 1.18
+water 30 fb
underdamped decane 1.2 X 10~~ 0. 73
system 41 2.7 20
water
The typical small amplitude responses obtained in the overdamped and underdamped system
are shown in figure 3. At low frequency, capillary forces dominate and U(w )/AP (w grows linearly with
w. The slope is the inverse of the meniscus stiffness Cl 00). At higher frequency
we observe the cross-over to a flat viscous response in the overdamped system, and the
resonance peak in the underdamped one. In both cases the response is very well fitted by
equation (3) with C loo as an ajustable parameter.
In the overdamped system, we study the dependence of the meniscus stiffness C loo on the initial contact angle 00. Various values of 00 in the hysteresis range are obtained by tuning the offset position of the vibration excitator.
@o is determined optically from the width h of the meniscus on the axis of the capillary measured at rest before the experiment. (The ratio between axial and radial magnification, different from unity because of the cylindrical shape of the capillary, is estimated from the ratio between the extemal and internal diameter of the pipe
measured on the image). The main incertitude is due to distortion of the interface under gravity
forces and is estimated from the angle between the capillary axis and the normal to the contact line plane: 30m3°. Values of
@o and C loo) are listed in table II. Figure 4 shows
Table II. The values of 00 and C loo) plotted in figure 4. The incertitude in 00 is ± 3°, and the incertitude in Cl 00) is ± 200 dyne/cm3.
o c loo)
° dynes cm -3
79~ 5.34 X 104
79° 5.37 X 104
76° 5.31 X 104
67° 4.99 X 104
67° 5.10 X 104
63° 4.96 X 104
58° 3.69 X 104
55° 4.13 X 104
36° 2.45 X 104
6
o
~4
E
. u
m
~ C
~
~
~ .
~~2
U
0 1 2 3 4
sin8~(1+sin8 )2
Fig. 4. -Meniscus stiffness C(fo) obtained for various values of the initial contact angle oo as a function of sin oo(I + sin oo)~. The data are listed in table II. The linear regression gives a value of the surface tension y
=
40.2 dynes/cm.
C(@~) as a function of sin @~(l + sin @~)~. a linear regression leads to a value of y =
40.2 dynes/cm, which is in good agreement with the static value y
=
39 dynes/cm.
3~ Large amplitude oscillations.
In the following we use the non-dimensional spatial variable X
=
3 x/2 R. The factor 3/2 is
chosen for the adimensional displacement X be unity when the amplitude of the volume
displaced is 2 arR~/3. Thus, X~
= I is the highest possible excitation for which the contact line
can stay pinned at the wall and the interface keep a spherical shape. The pressure divided by velocity responses obtained in section 2 are independent of the adimensional amplitude
X~ for X~ smaller than about 0.I. At higher amplitudes however, U/AP becomes amplitude
dependent : in the overdamped system the low frequency part becomes distorted, whereas in the underdamped system, the resonance peak shifts and becomes wider. This latter feature
suggest that excess damping appears in the system. In order to study the motion of the contact line, we record the meniscus motion at the same time as the pressure signal, and perform two
types of experiments : experiments at constant frequency in which the amplitude X~ is varied and experiments at constant amplitude in which the frequency is varied. We define the
adimensional capillary pressure response p(w, X~) as : P(W, X~) = ~cap(W )R
~ ~'~° (~)
With APcap(W) "P21W)~PliW)- ~[)jj~
and calculate it from the experimental data with the values given in table I. For convenience the
velocity U is chosen as phase reference; the real part p'(w,X~) of the response thus characterizes extra-damping, I-e- the damping due to the presence of the interface in the
capillary, and the imaginary part p"(w, X~) is the stiffness of the meniscus. The frequency
domain in which we study p spans from 0. I Hz to 10 Hz ; at higher frequency the pressure drop
across the capillary is dominated by the flow and AP
~~~
is not obtained with enough precision.
3.I DEPENDENCE IN AMPLrrUDE. At low frequencies, typically f ~ l Hz, one observes
successively as the amplitude X~ is increased :
pinned oscillations without macroscopic motion of the contact line (X~ ~ 0.3) ;
a depinning transition in which the motion of the contact line is irregular : some portions only may move while other stay pinned, motion may not appear at each period, etc. During this
depinning transition rearrangements of the meniscus may occur, I-e- the contact line jumps
from its initial position to a more stable one, with a change in the average contact angle ;
a well established sliding motion of the contact line (typically for X~ ~ 0.6). The periodic
motion of the interface is formed of four sequences : I) the interface slides forward, it) the contact line pins and the contact angle changes, iii) the interface slides backward, and iv) the contact line pins and the contact angle recovers its advancing value. During that motion the
shape of the meniscus remains spherical. In the glycerol/silicon oil system, we observe at high amplitude the formation of a macroscopic film at the wall. We do not observe such
macroscopic film in the water/decane system, possibly because we do not go to a high enough Capillary number. A detailed discussion of the problem of macroscopic liquid film left behind
an interface can be found in [I Il. In the following, we restrict our attention to the range of
amplitude in which the film does not form.
The variation of the pressure response is shown in figure 5. The main feature is the important damping appearing when the contact line depins. At the lowest amplitudes (X~ ~ 0.2 p' stays
in the noise level of the experiment : when the contact line is pinned, the presence of the interface does not introduce measurable damping. However, p' starts to increase before the first visual signs of depinning. We believe this is due to small scale motion of portions of the
contact line. When p' reaches its maximum value, the dissipation of the moving contact line
reaches a significant level compared to the one of the flow in the capillary. The transition between pinned and sliding motion is also reflected by the stiffness : p" has its maximum value at low amplitude when all the displacement is stored into meniscus deformation, and decreases
when sliding starts (Fig. 5b).
JOURNAL DE PHYS'OUE'I T 2, N' il. NOVEMBER 1992 76
o.5
(a)
0A
0.3
0.2
o-i
o
2 Xo
2
'CL
(b)
o
o 2
Xo
Fig. 5. Real part la) and imaginary part (b) of the capillary pressure response p(Xo in the overdamped system as a function of the displacement amplitude X~. At the lowest frequencies, the rust visual signs of
depinning appear about X~=0.3, and sliding becomes well established for X~ above 0.6. At f
=
2 Hz, there is no clear macroscopic depinning. With (-z-) f
=
0.2 Hz, (-A-) f
= 0.5 Hz,
(-~) f Hz, (-O-) f = 2 Hz.
