Bubbles Creeping Inclined Surface

8/6/2019 Bubbles Creeping Inclined Surface

1/7

Europhys. Lett., 59 (3), pp. 370376 (2002)

EUROPHYSICS LETTERS 1 August 2002

Bubbles creeping in a viscous liquid

along a slightly inclined plane

P. Aussillous and D. Quere

Laboratoire de Physique de la Matiere Condensee, URA 792 du CNRS

College de France - 75231 Paris Cedex 05, France

(received 1 February 2002; accepted in final form 13 May 2002)

PACS. 47.15.Gf Low-Reynolds-number (creeping) flows.PACS. 68.15.+e Liquid thin films.PACS. 83.50.Lh Slip boundary effects (interfacial and free surface flows).

Abstract. We describe the upwards movement of an air bubble creeping along a slightlyinclined plane immersed in a viscous liquid which totally wets the solid. After characterizingthe shape of the static bubble under a horizontal plane, we tilt the plane and study the resultingmotion. The bubble reaches a steady velocity, which is described as a function of the bubblesize, the liquid viscosity and the tilting angle. We interpret our results by considering the

viscous dissipation in the so-called dynamic meniscus, which is the zone where a lubricatingfilm forms.

Introduction. We consider a tank filled with a viscous liquid, in which a bubble isintroduced. If the upper boundary of the tank is tilted, the bubble creeps along this boundaryonce it has reached it (fig. 1). In the (common) situation where the liquid totally wets thesolid, a thin lubricating film of thickness e forms between the bubble and the solid. We denoteby V the bubble velocity and by the tilting angle of the solid.

The bubble moves at a constant velocity and an interesting practical problem is to under-stand the source of dissipation. First, we study the shape of a bubble under a horizontal planeimmersed in a fluid. Then, we present measurements of creeping velocities, as a function ofthe bubble size, the nature of the liquid and the solid slope.

Our experiment consists in placing in a tank of oil an air bubble of controlled size (wenote R0 the initial radius) below a planar solid. This solid is a transparent polymer (PMMA)sheet, with millimetric graduations engraved on the top surface. Then, the solid is slightlytilted. We measure the position x of the front meniscus as a function of time, from which wededuce the bubble velocity (V = dx/dt). Each run is typically 10 cm long, and the velocityis found to be constant all along the run. The experimental error on the measurement of Vis of the order of 1%. The liquid is a viscous silicone oil (viscosity = 970 mPa s, density

= 970kg/m3) of low surface tension ( = 20mN/m). The oil completely wets the solid, sothat the bubble is in a situation of non-wetting. This is confirmed by a direct observation.Side views of the bubble show that it joins the solid tangentially: the contact angle of thebubble on the solid is 180 (see fig. 1). Furthermore, all the bubbles (even the smallest ones,

c EDP Sciences


2/7

P. Aussillous et al.: Bubbles creeping in a viscous liquid etc. 371

Fig. 1 An air bubble (of volume 200 l) creeping along an inclined plane in a viscous wetting liquid(silicone oil of viscosity 970 mPa s). The bar is 1.5 mm, which is the capillary length.

of the order of 100 m in size) were observed to move, which also proves a purely non-wettingbehaviour: the contact-angle hysteresis associated with the existence of a contact line wouldnecessarily stick such tiny bubbles.

Static shape. The shape of the bubble under the solid results from a balance betweenthe surface forces and Archimedes pressure. The relative importance of these forces can beunderstood by introducing the capillary length 1 =

/g (1.5 mm in our experiments).

Bubbles smaller than 1 adopt a spherical shape, and larger ones are flattened by the actionof gravity, forming what is sometimes called pancakes (see fig. 1).

In both cases, a contact can be defined as the portion of liquid/air interface parallel tothe solid. We were interested in the radius of this contact, in the spirit of previous studies,either theoretical [1] or experimental [2], about the contact between non-wetting liquid dropsand solid substrates.

The case of large bubbles (R0 1) is the simplest. As for liquids, gravity tends to

flatten such a bubble, which forms a pancake of thickness h. The thickness h results from abalance between gravity (which tends to make the pancake as thin as possible) and surfaceforces which have the opposite action [3]. In the particular case of non-wetting, the thickness isfound to be simply 21, twice the capillary length (3 mm in our experiments, which was easyto check). Conservation of the bubble volume (4R0

3/3 = 221, neglecting the bubble

0.01

0.1

1

10

0.1 1 10

RoFig. 2 Radius of the contact zone as a function of the bubble size. Both lengths are scaled by thecapillary length. The straight lines represent the slopes 2 and 1.5, as expected from eqs. (2) and (1).


3/7

372 EUROPHYSICS LETTERS

0.0001

0.001

0.01

0.1

1

10

100

0.01 0.1 sin

V (mm/s)

Fig. 3 Creeping velocity of the bubble as a function of the plate slope. The black squares correspondto a large bubble (R0 = 6 mm) and the open ones to a small one (R0 = 0.3 mm). The straight lines,of respective slope 1.4 and 1.3, indicate that the dependence is not linear.

ends) immediately yields the contact size

=

2

3

R03/2

1/2. (1)

On the other hand, a bubble smaller than the capillary length (R0 < 1) adopts a nearly

spherical shape, except close to the top, where it is slightly deformed by Archimedes pressure,as stressed by Mahadevan and Pomeau [1]. The centre of mass of the bubble goes up by asmall distance , which can be evaluated by balancing (dimensionally) the restoring capillaryforce with Archimedes force gR0

3. Then, using the geometric relation 2 R0 gives, asa size for the contact,

R0

2

1. (2)

Experimental results are found to be in good agreement with these predictions. Figure 2shows the variation of the contact zone vs. the bubble size. It is observed that the contactsuccessively scales as R0

2 and as R03/2, below and above the capillary length. The best fit

with the data provides numerical coefficients of 0.90 and 0.74, of the order of unity as expected.

Creeping motion.

E x p e r i m e n t a l o b s e r v a t i o n s. As soon as the plate is tilted by an angle , thebubble creeps upwards, driven by Archimedes force 4/3gR0

3 sin . If it were balanced bya viscous force of a Stokes type F1 V R, denoting by R the maximum size of the bubble(R R0 for small bubbles, and R R0

3/21/2 for large ones), this would provide aconstant rise velocity proportional to sin . This is not the case, as can be observed in fig. 3,where the creeping velocity is plotted vs. sin , for two different bubble sizes, and found toscale as sin1.4 for the pancake and as sin1.3 for the bubble.

Similarly, taking for R the largest characteristic length of the bubble, we would expectfrom the same Stokes balance a velocity scaling as R0

3/2 for a pancake and as R02 for a

spherical bubble. This is once again not observed, in particular for the pancakes, as reportedin fig. 4, where the creeping velocity is plotted vs. R0, for two different angles.

In the pancake regime (R0 1), the creeping velocity scales as R0

y, with y between2.22 and 2.34. For smaller bubbles, it seems that the curve bends, and the power law could be


4/7


0.0001

0.001

0.01

0.1

1

10

100

0.1 1 10Ro (mm)

V(mm/s)

Fig. 4 Creeping velocity of a bubble vs. its initial size R0 (black squares: = 5.7, open squares:

= 1

). For R0 > 1.5 mm (pancake regime), the data obey simple power laws (straight lines),yielding exponents of 2.22 and 2.34, much larger than 1.5, resulting from a simple Stokes law.

compatible with 2, the classical Stokes exponent. Note that this decrease of the characteristicexponent y is more sensible for the smallest slope (open squares).

These different behaviours, which reveal a non-trivial law for the viscous force, can beanalyzed by considering the lubricating film which intercalates between the solid and thebubble. At a small velocity, we can consider that the bubble keeps its static shape, except inthe region where the film forms.

S c a l i n g l a w s. We consider here the viscous dissipation associated with the formation

of a lubricating film between the pancake and the solid. As is usually done for characterizingsuch a film, we work in the reference frame where the bubble is static. Then, the film isdeposited because of the motion of a solid at a speed V, as sketched in fig. 5.

This problem is quite similar to the entrainment of a liquid film by a moving plate [4], orto the deposition of a film behind a drop running in a tube [5,6] or in a Hele-Shaw cell [7]. Ata small capillary number Ca (Ca = V/), the bubble is slightly deformed by the movementand the deposition process can be described by three zones: the lubrification film of constantthickness e, a static meniscus of curvature of about h1 (the other curvature, of the order of1, is negligible for large pancakes), both being connected through a dynamic meniscus oflength .

The flow in the dynamic meniscus can be described by balancing the capillary force op-posing the formation of the dynamic meniscus with the viscous force supplying the film. Thepressure difference between the film and the static meniscus is of the order of /h, i.e. , so

e

h

P~P0

P~P0 -/h

air

liquid

V

Fig. 5 Zoom on the region where the lubricating film forms, drawn in the reference frame of themoving bubble. is the length of the dynamic meniscus and the thin line indicates the position ofthe liquid/vapour interface for V = 0.


5/7


0.00001

0.0001

0.001

0.01

0.1

1

0.1 1 10Ro

V/

Fig. 6 Capillary number Ca = V/ vs. the size of the bubble normalised by the capillary length.

The open, grey and black squares correspond to = 0.7, 2 and 5.7, respectively. The straight linesindicate the power 9/4 for R0 > 1 (see eq. (6)) and eq. (7) for R0 < 1.

that this balance dimensionally writes

V

e2

. (3)

In addition, matching the Laplace pressures, and thus the curvatures, between the static andthe dynamic menisci yields a relation between the length and the thickness e:

e

2 . (4)

The usual laws for coating can be deduced from eqs. (3) and (4), together with a dimensionalexpression for the viscous force F2

Ve . Using eqs. (3) and (4), we deduce that

F2 Ca2/3. (5)

Supposing that this friction dominates the Stokes friction (F2 F1), and balancing it withthe driving force g sin R0

3, we find in the pancake case (using eq. (1))

V

(R0)

9/4 sin3/2 . (6)

The condition for establishing eq. (6) (F2 F1) finally appears to be a condition of smallcapillary number, also supposed for deriving eqs. (3) and (4), and satisfied in the experiments.The thickness deduced from eqs. (3) and (4) is typically between 10 m and 100 m, and thusshould be much larger than the roughness scale of the solid (typically 1 m). At very smallvelocity, the viscous force F2 vanishes together with the film thickness (and with the samescaling), and the viscous force associated with the liquid trapped in the solid roughness shouldbecome dominant. This would finally lead to a friction force linear in velocity (and thus varyingas the driving force, as sin , for example).

We displayed in fig. 6 different series of data, in the coordinates suggested by eq. (6): thedimensionless velocity V/= Ca is plotted as a function ofR0, the bubble size normalized

by the capillary length, for different slopes ( = 0.7, 2 and 5.7). In the regime of pancakes(R0 > 1), the data are very well fitted by a power law with the exponent 9/4 predicted byeq. (6) (Ca = (R0)

9/4). The coefficient is observed to increase with the slope, and thecorresponding dependence is reported in fig. 7.


6/7


0.0001

0.001

0.01

0.1

0.01 0.1 sin

Fig. 7 Coefficient vs. the sinus of the tilting angle of the solid. The straight line has a slope 3/2,as expected from eq. (6).

It is found that the data are indeed quite well fitted by the power 3/2, as expected fromeq. (6). A numerical coefficient can finally be deduced from the fit, and found to be of theorder of 0.34 (this value corresponds to the straight line drawn in fig. 7).

The case of small bubbles (R0 < 1) is more complicated. The lengths e and are scaledby the bubble radius R0 and the condition F2 F1 is now written Ca

1/3 R0, which is

not necessarily satisfied. Corrections due to the Stokes force F1 must be taken into account.Balancing the quantity F1 + F2 with the driving force g sin R0

3 (and using eq. (2) for thesize of the contact) leads to the following (scaling) equation for the velocity:

a Ca + b Ca2/3R0 sin R02, (7)

where a and b are unknown numerical coefficients. Looking for asymptotic cases, it is foundthat the solution should scale as a function of R0, with an exponent between 3/2 and 2,and as a function of sin , with an exponent between 1 and 3/2. This corresponds to theobservations displayed in figs. 3, 4 and 6 but it is quite hazardous to be more quantitative:the range on capillary numbers is too small to observe the asymptotic behaviours. However,a fit of the form 7 is found to fit the data (on this small range), as observed in fig. 6. Therespective numerical coefficients which can be deduced from such a fit are a = 5.0 (largerthan the Stokes coefficient, which is 3 for a bubble in an infinite medium) and b = 2.3. Notefinally that the velocity always increases with the bubble size, a situation (logically) differentfrom the case where the liquid and the gas are inverted (viscous liquid pearl running down aninclined plate): then, the velocity is found to increase when decreasing the pearl size [1,2].

Conclusions. We described in this paper the creeping motion of a bubble along aninclined solid which is not wetted by this bubble. We first discussed the characteristic sizeof the contact between a static bubble and the plate, i.e. the size on which the bubbleis deformed by the action of gravity. Then, we showed that the creeping motion cannot bedescribed by a simple Stokes law: the presence of a lubricating film implies non-trivial lawsfor the viscous dissipation, which were analyzed by scaling arguments. We focused moreparticularly on the case of large bubbles: they form pancakes because of the action of gravity,which were found to creep with well-characterized scaling laws.

It would be interesting to complete this study by comparing this situation with the one of asolid of same shape creeping similarly (or in the opposite direction, if denser than the liquid).In this case, the interface between the creeping object and the liquid is non-deformable, whichraises the question of the selection of the film thickness. Besides, the dissipation should be


7/7


dominated in this case by the lubrication film a case thus very different from the situationdescribed in this paper. For bubbles, it should be also worth studying the case of inviscid

liquids, for which both the question of the film selection and the deformations of the bubbleshape should lead to quite different behaviours.

REFERENCES

[1] Mahadevan L. and Pomeau Y., Phys. Fluids, 11 (1999) 2449.[2] Aussillous P. and Quere D., Nature (London), 411 (2001) 924.[3] Taylor G. I. and Michael D. H., J. Fluid Mech., 58 (1973) 625.[4] Landau L. D. and Levich B., Acta Physicochim. USSR, 17 (1942) 42.[5] Taylor G. I., J. Fluid Mech., 10 (1961) 161.[6] Bretherton F. P., J. Fluid Mech., 10 (1961) 166.[7] Park C. W. and Homsy G. M., J. Fluid Mech., 139 (1984) 291.

Documents

Bubbles Creeping Inclined Surface