Residual Film Dynamics in Glass Capillaries

JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.

182, 483–491 (1996)

0492

Residual Film Dynamics in Glass Capillaries M. CACHILE,* R. CHERTCOFF,* A. CALVO,* ,1 M. ROSEN,* J. P. HULIN,†

AND

A. M. CAZABAT‡

*Grupo de Medios Porosos, Facultad de IngenierıB a, Universidad de Buenos Aires, Paseo Colo´n 850, 1063, Buenos Aires, Argentina; †Laboratoire FAST (URA CNRS No. 871), Campus de l’Universite´ Paris–Sud, Baˆtiment 502, 91405 Orsay, France; and ‡Laboratoire de Physique de la Matie`re Condense´e (URA CNRS No. 542), Colle`ge de France, 11 Place Marcelin Berthelot, 75005 Paris, France Received October 10, 1995; accepted April 26, 1996

We report measurements of the residual film thickness profiles resulting from the displacement of a fluid by an immiscible fluid in a 50-cm-long and 1.1-mm i.d. capillary tube. A radioactive tracer technique is used to measure the variations of the local thickness along the tube. Spatial variations of the film thickness are observed experimentally increasing with the viscosity ratio M Å m2 / m1 between the displacing and displaced fluids (10 03 õ M õ 30), with the capillary number Ca (5 1 10 06 õ Ca õ 3) and with the distance from the meniscus tip (up to 70 cm). The thickness along the tube is constant only for low M values and/or low Ca values. The film thickness averaged over the tube length increases with the capillary number as Ca 0.6 : this variation is independent of the relative viscosity of the fluids and very similar to the observations of other authors using different sets of fluids. Differential equations verified by the film thickness variations with time and distance far from the meniscus tip are established and their relations with the observed results are discussed. q 1996 Academic Press, Inc.

Key Words: residual film thickness; liquid–liquid interfaces; film profiles; glass capillaries; viscous displacement.

INTRODUCTION

Residual films appearing during the displacement of a liquid by another immiscible liquid are very important for practical applications and for the fundamental understanding of diphasic flows in confined geometries. In particular, such effects are a limiting factor in the percentage of oil that can be recovered from porous rocks during water flooding operations and in many cleaning or chemical processes. A particularly important problem is the dependence of the amount of residual liquid on the solid walls on such factors as the flow velocity or the ratio of the fluids viscosities. We report in the present paper experimental measurements of the residual phase distribution in a capillary tube initially filled with a liquid and flooded with another. We used a radioactive tracer technique that allowed us to determine the residual film thickness profile along the length of the tube. 1

To whom correspondence should be addressed.

Numerous previous works have dealt with similar problems experimentally and numerically. Most experiments have been performed with low-viscosity fluids displacing more viscous fluids: some authors measure the variation of the length of a liquid plug filling the whole capillary section as it has left behind a given length of residual film. Others (1, 2) use air bubbles moving inside a liquid saturated tube and estimate the average film thickness around the bubble by comparing the bubble velocity to the mean flow velocity through the capillary; Goldsmith and Mason (3) use the same geometry and determine the mean film thickness with an optical technique, while Marchessault and Mason (4) and Chen (5) use a resistivity measurement for that purpose. Several works deal with fingers of air or of low-viscosity fluids: the average film thickness can be determined by comparing the injected flow rate and the velocity of the displacing finger (6) or by direct photographic visualization (7, 8). These experiments allowed them to determine the variation of the mean film thickness with the flow rate of the displacing fluid. The latter is generally characterized by the capillary number Ca Å m1Vt / g ( m1 is the displaced fluid viscosity, g is the interfacial tension between the two fluids, and Vt is the mean fluid velocity across the tube section). The variation of the average thickness » h … with Ca follows a power law » h … } Ca a , for 10 05 õ Ca õ 10 01 , but the exponent a varies between 21 and 32 depending on the authors. At higher capillary numbers, the thickness increases slower as it becomes on the order of 0.3 a (a being the capillary radius). A general limitation of these experiments is the fact that the thickness variation with distance parallel to the flow is not measured: this does not allow one to understand thoroughly the displacement mechanism. Furthermore, using a low-viscosity injected fluid does not reproduce realistically oil–polymer–water solutions displacement processes in which the fluid viscosities are much more comparable and the viscous stresses at the interface may play a significant part. A number of theoretical predictions have also been obtained by using the lubrication approximation and by assuming Ca values below 10 02 . These models will be discussed

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0021-9797/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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TABLE 1

Pairs of fluids

Viscosity of displaced fluid (cp)

Viscosity of displacing fluid (cp)

Water–cyclohexane Glycerol solution–silicon oil Glycerol solution–silicon oil Glycerol solution–cyclohexane Glycerol solution–silicon oil

1 50 400 900 12

1 50 400 1 400

in more detail below but all of them predict a variation of the film thickness proportional to Ca 2 / 3 : however, a major limitation of these theories is the fact that the film thickness is always assumed to be constant at large enough distances from the interface meniscus. In the transition zone between the meniscus and the constant thickness region, an exponential decrease of the thickness is predicted and observed experimentally (7, 9, 10). In the present work we analyze, in contrast, the variations of the local film thickness with distance from the meniscus at various capillary numbers and for various ratios of the injected and displaced fluid viscosities. We shall first present our experimental technique based on radioactivity measurements using a tagged liquid to saturate initially a glass capillary tube. Then we report film profile measurements at capillary numbers between 10 05 and 3 for viscosity ratios between the injected and the displaced fluid ranging between 10 03 and 33. We shall demonstrate that the constant thickness hypothesis is not always verified: this is particularly the case when the viscosity of the outer film is lower or equal to that of the displacing fluid and at high capillary number values. Finally we discuss possible origins of the dependence of the profile shape on Ca. EXPERIMENTAL PROCEDURE

We use glass capillary tubes manufactured by heating and stretching larger tubes. The internal diameter is 2a Å 1.1 mm and the length is 500 mm; we checked that the internal radius of the tube is constant to within {3 1 10 02 mm (11, 12). The tubes received from the factory are longer but 10 cm are cut off from each end in order to reduce the effect of pollution by the surrounding atmosphere. The tubes are used only for one experiment and then discarded. We verified previously that the reproducibility of the results from one experiment to another (and therefore from one tube to another) is excellent. We used cyclohexane and silicon oils of various viscosities as the displacing fluid and water or water–glycerol solutions of various concentrations as the more wetting, dis-

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Range of capillary numbers

Range of Reynolds numbers

5 1 1006 –1003 1003 –2 1 1001 1002 –1 1002 –3 1004 –2 1 1002

0.23–46 1.3 1 1002 –2.6 1.8 1 1003 –0.18 3.9 1 1004 –0.12 2.4 1 1002 –4.9

Surface tension (dynes/cm) 42 30 27 29 32

placed fluid. The viscosities of the fluids and the corresponding capillary and Reynolds numbers and surface tensions are listed in Table 1; the Reynolds number Re is taken equal to Re Å 2Vta/ m1 . In three series of experiments, the viscosities of the two fluids are identical; in the fourth series, the invading fluid is much less viscous than the displaced one and in the last series, the displaced fluid is of lower viscosity than the invading one. Varying the displaced fluid viscosity extends the range of capillary numbers investigated in our experiments. In all cases, well-purified invading and displaced fluids have been used in our experiments so that the residual film surface will be considered as surfactant free and mobile. The duration of most experiments is also short enough so that migration of large amounts of surfactant toward the surface is not likely to take place. Furthermore, as shown by other authors (13), the effect of surfactants would be highest at the front part of the interface: they would then modify the mean film thickness rather than influence its dynamics downstream from the meniscus tip. The experimental technique is based on the use of a tagged liquid initially filling the capillary tube: this enables us to determine the final amount of this liquid after the displacement process and its spatial distribution along the tube. We use as tracer the radioactive isotope 131I. The experimental setup is shown in Fig. 1. The capillary tube assembly can be translated horizontally along its full length of 50 cm in front of a g-ray scintillation detector with a measurement window of 4 cm. At the beginning of the experiment the capillary tube is completely saturated with the tagged fluid (water or water– glycerol solution) and we measure the radioactivity values Ai at ten points located every 5 cm along the tube. Then, this fluid is displaced at constant velocity by a nonradioactive fluid (cyclohexane or silicone oil): one lets the interface meniscus move up to the end of the glass tube and stops the flow thereafter. The first measurement point, taken in the following as the origin of distances, is placed at 6 cm behind the final location of the meniscus: this eliminates the influence of the highly radioactive liquid regions ahead of the meniscus.

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and on the capillary number values. Let us first discuss the case of pairs of fluids of identical viscosities. In the case of low capillary numbers (obtained in particular for lowviscosity fluids), the residual film thickness is constant along the tube length. Figure 2 displays this result in a range of capillary numbers 10 05 õ Ca õ 5 1 10 03 and for two different pairs of fluids: —water and cyclohexane ( m1 Å m2 Å 1 cp) —silicon oil and water–glycerol solution ( m1 Å m2 Å 50 cp).

FIG. 1. Experimental setup.

The whole displacement process is observed optically and recorded on videotape in order to detect the possible appearance of interfacial instabilities during the experiment: it has been indeed known both theoretically (14) and experimentally (15) that Rayleigh-type waves may develop on liquid films in such geometries. We verified particularly that neither the liquid film nor the core fluid broke during the displacement: the few experiments in which the film or the core fluid broke before the end of the displacement were discarded. The film thickness measurements themselves are taken only after the displacement and the measurements are not perturbed by instabilities occurring at this stage. The film is indeed observed to break into patches smaller than the measurement windows: then, the radioactive fluid mass detected inside a given window is left unchanged and the measured thickness is the same. The radioactivities A *i are measured after the injection at the same ten points as mentioned previously. Then the effective local thickness hi in the measurement window is computed from the formula hi Å10 a

r

10

A *i . Ai

In contrast, at capillary numbers 10 02 õ Ca õ 1, but still for two fluids with the same viscosity (silicon oil displacing glycerol with m1 Å m2 Å 50 cp and 400 cp), the film thickness varies with distance along the capillary tube. Figure 3 displays this variation. For capillary numbers of a few 10 02 , the film thickness is about constant near the tip of the meniscus and decreases farther from it. The spatial extent of the plateau region decreases from 35 to 10 cm as the capillary number increases from Ca Å 10 02 to 5 1 10 02 . At higher capillary numbers (Ca § 9 1 10 02 ), the thickness decreases steadily away from the meniscus in the whole 50-cm-long measurement domain. In order to detect whether the thickness still decreased at larger distances from the tip of the meniscus, we performed two different experiments at the same capillary number Ca Å 0.26 ( m1 Å m2 Å 400 cp). The first test used a 50-cm tube as in the other cases, while the second test used an 80-cm tube. In both cases, the interface meniscus is allowed to move up to the end of the tube and the flow is stopped thereafter; the measurements are

[1]

In this way, small errors due to variations of the tube geometry and position are compensated: the effective thickness is measured with a precision on the order of 3% and films as thin as a few micrometers are detected. Equation [1] allows us to determine from the measurements the relative fraction of the tube volume occupied by the residual film in each 5cm segment: hi represents the actual thickness of the film only if the film thickness on the tube walls is constant in the corresponding 5-cm length of each measurement. EXPERIMENTAL RESULTS

Effective Film Thickness Profiles along the Capillary Tubes Two markedly different behaviors have been observed, depending on the relative viscosities of the fluids being used

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FIG. 2. Effective thickness profiles for pairs of liquids of equal viscosity (M Å 1): l, cyclohexane displacing water (1 cp); ¨, silicon oil displacing a water–glycerol solution (50 cp). Respective capillary number values are, from top to bottom: 5 1 10 03 , 2 1 10 03 , 10 03 , 10 03 , 8 1 10 04 , 5 1 10 04 , 2 1 10 04 .

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FIG. 3. Effective thickness profiles for pairs of liquids of equal viscosity (M Å 1): silicon oil displacing a water–glycerol solution, j and l (400 cp); ¨ (50 cp). Respective capillary number values are, from top to bottom: 6.5 1 10 01 , 2.5 1 10 01 , 8.7 1 10 02 , 4.5 1 10 02 , 2.2 1 10 02 , 1.2 1 10 02 , 2.2 1 10 02 , 9 1 10 03 . All data points correspond to a 50-cm-long tube except for the l points which were obtained with a 70-cm-long tube.

performed in its half length farther from final position of the meniscus. This procedure allows one to overcome the limiting length of 50 cm of the measurement domain in our experimental setup. Both sets of experimental points are plotted together in Fig. 3 and overlap precisely in their common section: we observe that the thickness decreases by a factor of 10 in the 70-cm range of distances investigated. Let us also emphasize that we verified visually in this experiment that no rupture of the film is detectable during the displacement of the meniscus: this ensures that the film fluid does not evolve into discontinuous patches changing the final measured distribution. We then investigated how these results were modified when two fluids of different viscosities are used. For a lowviscosity fluid (cyclohexane 0 m2 Å 1 cp) displacing a highviscosity fluid (water–glycerol solution 0 m1 Å 900 cp) the film thickness remains constant along the tube length, even for capillary numbers as high as 3 for which a thickness variation is observed in the case of fluids of equal viscosities. In contrast, for a high-viscosity fluid (silicon oil 0 m2 Å 400 cp) displacing a low-viscosity fluid (water–glycerol solution 0 m1 Å 12 cp), the thickness varies along the tube length down to Ca Å 10 04 : this is far below the limit value observed for fluids of same viscosities (Ca Å 10 02 ). These results are shown in Figs. 4 and 5, respectively. Hence, the extent of the domain of capillary number values over which the thickness is constant with distance clearly increases as the viscosity ratio M Å m2 / m1 becomes lower. This explains why experiments performed by other authors in the case of invading fluids of very low viscosities did not indicate any spatial variations of the residual film thickness.

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FIG. 4. Effective thickness profiles for low-viscosity cyclohexane (1 cp) displacing a water–glycerol solution (900 cp): low-viscosity ratio M Å 1.1 1 10 03 . Respective capillary number values are, from top to bottom: 1.5, 3.6 1 10 01 , 1.2 1 10 01 , 3.6 1 10 02 , 1.2 1 10 02 .

Mean Film Thickness Variations with the Capillary Number From the above results, we have determined the dependence on the capillary number Ca of the effective residual film thickness » h … over the whole measurement zone. We first take the average » A *i /Ai … of all ratios A *i /Ai whether they depend on distance or not. » A *i /Ai … represents the fraction of the total tube volume occupied by the residual film: » h … is then obtained by applying Eq. [1] to » A *i /Ai … . Figure 6 displays the variation of » h … /a with Ca for all five fluid pairs used in our experiments. Curves corresponding to

FIG. 5. Effective thickness profiles for a high-viscosity silicon oil (400 cp) displacing a low-viscosity water–glycerol solution (12 cp): high-viscosity ratio M Å 33. Respective capillary number values are, from top to bottom: 1.75 1 10 02 , 5.85 1 10 03 , 5.85 1 10 04 .

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over the capillary tube length has a nearly identical dependence on Ca for values of the viscosity ratio M Å m2 / m1 ranging between 10 03 and 33. Furthermore, all variations of » h … /a with Ca which we measure experimentally are in fair agreement with the results of other authors using different sets of fluids. In particular, for Ca £ 0.1 the ratio » h … /a is proportional to Ca 0.6 which is higher than the exponent 0.5 found experimentally by many authors. In the next section, first, we compare the above results to the theoretical predictions of other authors assuming a thickness h(x) independent of the distance x; then, we discuss the differential equations verified by the dependence of h(x) on x and on time. THEORETICAL ANALYSIS FIG. 6. Variation of the average thickness » h … of the residual film with the capillary number Ca: m, cyclohexane (1 cp) displacing water (1 cp); l, cyclohexane (1 cp) displacing a water–glycerol solution (900 cp); , silicon oil (400 cp) displacing a water–glycerol solution (12 cp); silicon oil displacing a water–glycerol solution of same viscosity: ¨ (50 cp), j (400 cp). ---, Empirical model of Fairbrother (1); — , theoretical predictions of Bretherton (2); – – – , experimental results of Taylor (6).

the theoretical and experimental results of other authors (1, 2, 6) have also been drawn for comparison (continuous and dotted lines): let us note that the experimental techniques used by these authors only provided the average film thickness and not the full profile. It can be seen that all data points follow rather precisely the same mean curve of variation with the capillary number, independent of the variation of the viscosity ratio and of the displaced fluid viscosity (only mean thickness values corresponding to M Å 33 seem to be consistently lower than those corresponding to the mean trend). At low capillary numbers Ca õ 0.1, » h … /a can be approximated by » h … /a Å 1.1 Ca a , with a Å 0.6 { 0.05. This result is in good agreement with the empirical relation (continuous line) developed by Fairbrother and Stubbs (1) from their experiments: our experiments indicate therefore that their relation can be used for fluids of equal viscosity as well as for an invading fluid of much lower viscosity than the other fluid as in their work. At high capillary numbers, the increase of » h … /a with Ca levels off for Ca ú 10 01 : in this case too, data points corresponding to M õ 1, M Å 1, and M ú 1 almost coincide, while viscosity contrasts between fluids for each couple are very different. Our experimental » h … /a values are also similar to the previous experimental results of Taylor (6) (although a little lower for Ca Å 10 02 ). In conclusion, the viscosity ratio influences only the variation of the local thickness with the distance x from the tip of the interface and not its average value over the tube length. The normalized average » h … /a of the residual film thickness

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Comparison of Experimental Results and Theoretical Predictions for the Average Film Thickness Let us first compare our experimental results with previous theoretical predictions of other authors. Except for the empirical analysis of Fairbrother (1), most of these deal with the case of low capillary numbers (typically Ca £ 10 02 ). Landau and Levich (16) studied the case of a vertical plane surface pulled out, at a constant velocity U, from the free surface of a liquid of density r and viscosity m. The residual film thickness h is assumed to be independent of height far enough from the horizontal free surface and the contact angle on the solid plane is taken to be equal to zero. One also assumes that the flow velocity in the film is vertical (lubrication approximation) and that the Reynolds number Re Å rUh/ m is ! 1. Under these conditions, these authors predict a variation h } Ca 2 / 3 . Derjaguin (17) obtained the same 23 power law with a different prefactor in the case of a circular geometry. Bretherton (2) used these assumptions in the geometry of a long bubble of negligible viscosity moving at constant velocity U in a cylindrical tube: the fluid film thickness h around the bubble (also assumed to be constant) is also proportional to U 2 / 3 . All these models assume the viscosity of the invading fluid to be negligible compared to that of the displaced fluid. Teletzke et al. (18) extended these models to two fluids of finite viscosity ratios flowing in a horizontal tube or between plane surfaces: they obtained the same power law h } Ca 2 / 3 for Ca õ 10 02 , still assuming a film thickness h constant with distance. These theories (and particularly that of Teletzke et al.) agree therefore fairly with our experimental results at Ca õ 10 02 : however, as for most previous works—except for Que´re´ (9) —the experimental exponent a à 0.6 in the relation » h … } Ca a is lower than the predicted value a Å 23. Film Thickness Variations with Time and Distance The previous models are no longer usable when the film thickness decreases with distance, as in our experiments,

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£x 1 Å

1 Ìp1 m 1 Ìx

F

r r2 0 a2 / D ln 4 a

G

[3a]

and £x 2 Å

FIG. 7. Schematic view of the interface geometry used for the theoretical analysis of the interface distortion.

with fluids of equal viscosities at capillary numbers Ca § 10 02 or for m2 Å 33 m1 and Ca § 10 04 . Let us now write the differential equation verified by the film thickness variations with time and with the distance from the meniscus tip. The purpose of this computation is not to determine quantitatively the complete film profile: rather, we analyze the respective magnitudes of the viscous and capillary effects for various values of Ca and of the ratio M Å m2 / m1 of the fluid viscosities, and we estimate the film thickness variations during the transit time of the meniscus along the experimental zone. We use the following assumptions: —The film thickness h(x) is much smaller than the tube radius a and varies slowly with distance ( Ìh/ Ìx ! 1) which allows us to use the lubrication approximation. This simplifying assumption will be valid in the intermediate regions far enough from both the rear end and the front part of the interface at the tip of the meniscus. In particular, we do not take into account periodic wavelike thickness variations which may appear in some cases: this assumption is justified since all measurements of h(x) profiles correspond to an average over a distance along the tube (5 cm) which is very large compared to the wavelength (a few mm at most). We also do not take into account capillary pressure terms due to the curvature Ì 2h/ Ìx 2 of the interface in the diametral plane: while these play a key role close to the front part of the meniscus, they are negligible in the regions of the film that we are dealing with. —The geometry of the interface has a circular symmetry around the capillary tube axis and the pressure difference p2 (x) 0 p1 (x) at the interface between the invading fluid (2) and the displaced fluid (1) verifies (Fig. 7)

p2 (x) 0 p1 (x) Å

g . a 0 h(x)

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F

r2 /C 4

G

,

[3b]

in which the coefficients C and D verify Ìp2 Ìx

Y

Ìp1 2D ÅRÅ1/ Ìx (a 0 h) 2

or R01 (a 0 h) 2 2

DÅ

[3c]

and CÅ0

F

a2 0 4

h 2 ah 0 4 2

G F F /

M R

/M 0

0

h2 h3 / 2 6a

G

ah 3h 2 h 3 / 0 2 4 6a

G

.

[3d]

The values of C and D are obtained by writing the continuities of the velocity component £x and of the stress at the interface and using third-order developments with respect to h/a. Integrating Eqs. [3a] and [3b] over the part of the section of the tube occupied by each fluid, one obtains the respective volume flow rates Q1 (x) and Q2 (x) at a distance x (C and D have been replaced by their values from Eqs. [3c] – [3d]): Q1 (x) Å 0

p Ìp1 4 3 p Ìp2 ah 0 2m1 Ìx 3 2m1 Ìx

F

a 2h 2 0

7 3 ah 3

G

[4a] Q2 (x) Å

[2]

Under the previous assumptions, one obtains the classical laminar velocity profiles

1 Ì p2 m2 Ìx

p Ìp2 2m2 Ìx

F

0

a4 / a 3h(1 0 M) 4

/

a 2h 2 ah 3 (7M 0 3) / (3 0 13M) 2 3

0

p Ìp1 2m1 Ìx

F

a 2h 2 0

G

7 3 ah . 3

G [4b]

The developments have been limited to the third order in h/a. The total volume flow rate Qt Å Q1 / Q2 verifies then

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Qt Å 0

pa 4 Ìp2 pgM Ìh 2 2 / 2 (a h / ah 3 ) 8m2 Ìx 2a m2 Ìx /

F

G

p Ìp2 3 (1 0 M) a 3h 0 a 2h 2 / ah 3 2m2 Ìx 2

[5]

after summing Eqs. [4a] – [4b] and approximating the derivative of Eq. [2] as Ìp2 Ìp1 g Ìh 0 Å 2 Ìx Ìx a Ìx

F

h 1/2 a

G

.

1 m1

F S Ìp2 Ìx

0

a 2h 2 ah 3 / 2 2

D

/

2 g 3 Ìh h 3a Ìx

G

.

[7]

For obtaining the differential equation verified by h(x, t), we use the mass conservation relation corresponding to fluid 1 written as 02p(a 0 h)

Ìh ÌQ1 Å . Ìt Ìx

[8]

In order to compute ÌQ1 / Ìx from Eq. [7], one first determines Ìp2 / Ìx and Ì 2 p2 / Ìx 2 from [5]. Replacing ÌQ1 / Ìx in Eq. [8] by the value obtained and multiplying the result by 0a 2 /2Qt , one has finally a Ìh 1 Å0 V t Ìt 3Ca 1

F

( Ì / Ìx)((h 3 /a)( Ìh/ Ìx)) / M(h 4 /a 2 )( Ì 2h/ Ìx 2 ) 1 0 4(h/a)(1 0 M) 0 4M

Ìh Ìx

F

1 0 2(h/a)(1 0 M) (1 0 4(h/a)(1 0 M)) 2

G

,

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SD h a

3

.

Assuming that h/a is on the order of the mean value » h … /a Å 1.1 Ca 0.6 (Ca õ 0.1), one obtains dh 1a É0 (Ca) 0.8 õ 10 04 . h 3L

For Ca ú 1, but Mh/a ! 1, » h … /a Å 0.3 and dh/h É 10 05 / Ca which is still smaller. Finally, if Mh/a @ 1 one finds in the same way that the variations are of the same order of magnitude. We conclude that in all cases the capillary terms in Eq. [9] are negligible in the range of x values for which the lubrication approximation used to establish this equation is valid. This corresponds to distances from the tip large enough so that the curvature of the interface in the diametral plane, which is used in most models predicting film thickness variations (14), can be neglected. By suppressing the corresponding terms, Eq. [9] becomes

a

Ìh Ìh Å 04MVth Ìt Ìx

F

1 0 2(h/a)(1 0 M) (1 0 4(h/a)(1 0 M)) 2

G

.

[10]

Let us now estimate in the same way as above relative film thickness variations during the experiment due to the viscous term in Eq. [10]. We first investigate the case of M values close to 1 or lower (lower or matched viscosity invading fluid). Neglecting all (1 0 M)h/a terms, Eq. [10] reduces to

[9]

with, as above, Ca Å m1Vt / g and Vt Å Qt / pa 2 . The three terms of Eq. [9] represent, respectively, the time dependence of the film thickness, the influence of capillarity, and the convection of the interface at the local velocity. The development of the prefactors in all these three terms has been limited to the lowest orders in h/a: expression [9] is valid

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dh 1 a É0 h 3Ca L

[6]

Qt must be a constant with time and distance since the fluids are incompressible and the flow is stationary. Let us point out that the last term of Eq. [5] is equal to zero when both fluids have the same viscosity and that it reduces to the classical Poiseuille equation when capillary effects are in addition negligible. Ìp1 / Ìx can also be eliminated from Eq. [4a] using [6] so that Q1 (x) Å

far enough from the meniscus tip so that the film thickness is small compared to the tube radius. First, we estimate the relative variation dh/h of the thickness associated with the capillary forces (first term on the right-hand side of Eq. [9]) during the characteristic duration texp Å L/Vt of the displacement (L is the film length). When Mh/a is much smaller than 1, only the ( Ì / Ìx)((h 3 /a)( Ìh/ Ìx)) term must be taken into account and one has

a

Ìh Ìh Å 04MVth . Ìt Ìx

[11]

Let us integrate this equation with respect to x between the rear end of the film where its thickness cancels out ( x Å xc , h Å 0) and a point (x Å x0 , h Å h(x0 )). Calling » h(xc , x0 ) … the average value of h between x Å xc and x Å x0 , one obtains

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a(x0 0 xc )

Ì» h(xc , x0 ) … Å 02MVth 2 (x0 ). Ìt

[12]

This implies that the velocity of the rear end of the film is zero or low compared to Vt so that xc can be considered as constant. Equation [12] describes the balance between the variations of the fluid mass in the film and the flow of liquid through the section x Å x0 . We estimate now the variation d» h(xc , x0 ) … of the thickness of the film at x Å x0 during a characteristic duration of the experiment taken to be on the order of L/Vt (this is the time for the interface to move by one tube length L). From [12], one has L Mh(x0 ) d» h(xc , x0 ) … É 02 . h(x0 ) x0 0 xc a

[13]

We note that d» h(xc , x0 ) … represents a mean value of the thickness variation averaged over the distance x0 0 xc . The prefactor L/(x0 0 xc ) increases as one moves away from the interface meniscus toward the rear end so that thickness should decrease accordingly as is indeed observed experimentally. For Ca £ 0.1, h(x0 )/a may be approximated by h(x0 )/a É » h … /a Å 1.1 Ca 0.6 and, in this range of Ca values Eq. [13] becomes MLCa 0.6 d» h(xc , x0 ) … É 02 . h(x0 ) x0 0 xc

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d» h(xc , x0 ) … L É0 . h(x0 ) x0 0 xc

[16]

and

From Eq. [16] a strong thickness decrease with distance should occur at all Ca values in this limit: in experiments corresponding to M @ 1 (Fig. 5), Mh/a is, respectively, equal to 4.6, 2.1, and 0.66 in the thickest parts of the film. Thus, the condition Mh/a @ 1 is only verified locally and for the two largest capillary numbers: we observe that a very sharp drop of the thickness with distance toward very low values is indeed observed in the corresponding curves. In low film thickness regions and for the curve Ca Å 6 1 10 04 , the equations of the previous part will remain valid: applying Eq. [13] to the full length of the profile for Ca Å 6 1 10 04 (x0 0 xc Å 50 cm) gives d» h(xc , x0 ) … /h(x0 ) É 1 in reasonable agreement with the experimental curve. We observe in particular that, for M Å 33, variations of the thickness profile with distance occur, as expected, at capillary numbers much lower (Ca É 10 04 ) than those for matched viscosity fluids (M Å 1, Ca § 10 02 ). CONCLUSION

[14]

From Eq. [14], it is clear that the film thickness variations increase with the viscosity ratio between the two fluids, with the capillary number Ca, and with the distance from the meniscus tip. Let us estimate a threshold value Cat for observing film thickness variations with distance. Assuming that a 50% relative variation of the mean thickness occurs over the last 20% of the film length farthest from the front meniscus, Eq. [14] gives Cat Å 6 1 10 03 for M Å 1: this is in reasonable agreement with the curves of Figs. 2 and 3 in which such thickness variations are observed roughly for Ca § 10 02 . For M É 10 03 , Eq. [14] gives Cat Å 750 with the same threshold criterion: however, since the power law dependence » h … /a É 1.1 Ca 0.6 is only valid up to Ca Å 0.1, an upper limit d» h(xc , x0 ) … /h(x0 ) É 3 1 10 03 is obtained by taking » h … /a Å 0.3 (6). This is far below the sensitivity of our measurements: thus Eq. [14] also explains why we did not observe thickness variations experimentally in the case of cyclohexane displacing a water–glycerol solution (M É 10 03 , Fig. 4). Let us now analyze the case of a large viscosity ratio M @ 1 for which the condition Mh/a ! 1 may not be verified. In the limit Mh/a @ 1 Eqs. [10] and [13] become

AID

Ìh Vt Ìh Å0 Ìt 2 Ìx

09-10-96 10:09:01

In conclusion, we have demonstrated experimentally that the assumption of a constant local residual film thickness is only valid rigorously for low enough viscosity ratios M and/ or at low enough capillary numbers: in other cases, it decreases as one moves away from the front interface meniscus. On the other hand, the average value of the film thickness over its length at a given capillary number is, within a factor of two, independent of the viscosity ratio M (10 03 £ M £ 33). The driving mechanism of these thickness variations is the viscous drag of the residual film by the displacing fluid: capillary forces due to variations of the film thickness and the interface curvature play a negligible role at distances from the meniscus larger than a few tube radii. The drag effect increases strongly with M and Ca so that, for M Å 33, large thickness gradients appear along the film and little fluid is left near the rear end of the residual film. In contrast, for small M values (M Å 10 03 ) corresponding to a low viscosity displacing fluid, the entrainment effect is reduced and no thickness variation is observed in the range of capillary numbers which we used. Overall, the threshold capillary number Cat below which the thickness is constant should vary roughly as M 01.7 (at least for Ca õ 1): near Cat , thickness variations are localized in the rear part of the film and spread thereafter over its full length when Ca increases.

coidas

AP: Colloid

RESIDUAL FILM DYNAMICS IN GLASS CAPILLARIES

Another point is the fact that previous models (2, 9, 14, 16–18) are only valid close to the interface meniscus: however, they predict reasonably adequately the mean film thickness values, although they assume a limiting constant value at large distances. This is probably due to the fact that, while significant thickness gradients occur, the local slope of the interface remains always small. An important problem for future work will be to analyze the dynamics of the residual film in zones closer to the meniscus inside which the influence of capillary pressure gradients is not more negligible. APPENDIX: NOMENCLATURE

a Ca h L M p1 p2 Q1 Q2 Qt R Vt x g m1 m2

AID

Tube radius. Capillary number Ca Å m1Vt / g. Displaced film thickness. Capillary tube length. Ratio m2 / m1 of the viscosities of the invading and displaced liquids. Pressure of displaced fluid. Pressure of invading fluid. Flow rate of displaced liquid 1. Flow rate of invading liquid 2. Total flow rate: Qt Å Q1 / Q2 . Ratio of the pressure gradients in the invading and the displaced fluids. Mean velocity Qt / pa 2 in tube section. Coordinate parallel to tube axis. Superficial tension between the two liquids. Viscosity of the displaced liquid. Viscosity of the invading liquid.

JCIS 4368

/

6g15$$$$24

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491

ACKNOWLEDGMENTS We have benefited in the theoretical part from helpful discussions with L. Limat. This work has been realized in part with the financial support of the PICS-CNRS No. 145 of the EEC Program CHRX-CT94-0448 ‘‘mouillage et re´activite´ aux interfaces’’ and of the IN 010, SECyT grant from the University of Buenos Aires.

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coidas

AP: Colloid