10. The time evolution of the fingering pattern in a Newtonian oil
10.8. Conclusion
Chapter 10. The fingering pattern in a Newtonian oil
can be linked to the initial conditions. In the simulations, higher amplitudes of the initial perturbations lead to higher amplitudes during the whole debonding process.
Greater insight into the finger growth could be gained pursuing experiments with a sinusoidal perturbation of controlled wavelength and amplitude at the outline of the probe.
A second effect we studied in this Chapter is the influence of the fingering insta- bility on the lifting force. This effect was unclear so far, as different working groups have found contradictory results [113, 92, 40, 75]. We showed that the force is at high control parameters well described by an equation for a retracting oil circle. We observed a visible influence of the fingering pattern only for small enough control pa- rameters and strong finger amplitudes. Higher confinement, corresponding to higher amplitudes, leads to lower debonding forces.
10.8. Conclusion
Appendix to Chapter 10
In this Appendix, we discuss some possible additional effects on the number of fingers. First, we consider 3D effects and apply a theory that accounts for the different initial shape of the contact line. Second, we discuss a possible influence of volume loss during a test.
A 3D effect?
In the derivation of Darcy’s law, the Hele–Shaw cell is approximated by a two dimensional system. Naturally, the lubrification approximation breaks down at some point with increasing cell thickness. Thus, we will discuss here some possible three dimensional effects.
Solving the full Navier-Stokes equations with free boundaries analytically in three dimensions is impossible [10]. Instead, Ben Amaret al. included three dimensional effects into their calculations by investigating the influence of the meniscus shape at the cell edge [10]. Comparing to experiments with different control parameters, the authors found that the influence of the meniscus shape is becoming more important for higherb0. Reference [10] is to our knowledge the only study including 3Deffects into the stability analysis. Thus we investigate here whether their theory can predict the influence of the confinement we observed.
The crucial parameter in the theoretical analysis isρ, a parameter that measures the deviation of the initial viscous blob from a perfect cylindrical shape. Att= 0, it is defined as
ρ= 4|R(b=b0/2)−R(b=b0)|
b0 . (10.24)
R is the blob radius. ρ evolves with time during the experiment.
On the one hand, the authors derived limiting equations that are only valid in the case of very small or largeρ. On the other hand, they calculate an equation for arbitraryρ that is valid for the boundary condition of an oscillating contact line. ρ discriminates in which regime the system falls. In our experiments, we estimateρ as follows. We assume the silicone oil to be perfectly wetting and estimate therefore the initial curvature to beb0/2. Thus, for very small times, equation10.24simplifies and reads
ρ= 4|R0−b0/2−R0|
b0 = 2. (10.25)
ρis of the order of 1 and we cannot use the limiting equations. Instead, we use the general equation for allρ:
Chapter 10. The fingering pattern in a Newtonian oil
N2
·1
ρarctanh ρ (1 +ρ2)1/2
¸ + 2
3 ρ (1 +ρ2)3/2
R20b0
b(t)3 +L−1
3ρ arctanh ρ (1 +ρ2)1/2
−L 3
1
(1 +ρ2)1/2 = 1−ηrel 6T
v(t) v(0)
Ã
b(0)5/2R20 b(t)9/2
! .
(10.26) The notations are the following: R0 is the radius of the Hele–Shaw cell, b0 and b(t) the initial and changing gap width, respectively, v(0) and v(t) the initial (and possibly changing) lifting speed,T the control parameter (b0γ)/(12R0ηv(0)), andγ the surface tension. L is the smallest radius of the viscous blob, that is, the radius at half the cell height. It is non-dimensionalized by R0. ηrel is the ratio of the viscosities of surrounding and inner fluid, so that 1−ηrel≈1. In the following, we discuss small timest and estimate that
L= L(t)
R0 = L0−b0/2
R0 = 1− b0
2R0 ≈1. (10.27)
Thus equation10.26 simplifies to
N2
"
1
ρarctanh ρ (1 +ρ2)1/2
# +2
3 ρ (1 +ρ2)3/2
R02b0 b(t)3 −1
3 1 (1 +ρ2)1/2
= 12R0ηv0 6b0γ
Ã
b05/2R02 b(t)9/2
! .
(10.28)
At t= 0, it reads N2
"
1
ρarctanh ρ (1 +ρ2)1/2
# +2
3 ρ (1 +ρ2)3/2
R02 b02 −1
3 1
(1 +ρ2)1/2 = 2R03ηv0
b03γ . (10.29) The 2Dlimit is recovered forρ→0. Equation 10.28can be solved numerically. We investigated the experimental parameters γ = 20mN/m, R0 = 3 mm, and ρ = 2. b0, v0, and η were varied according to each experiment.
Figure10.27compares the experimental results to linear 2Dtheory and numerical 3D predictions. Special care has to be taken considering the time dependence.
Previously, we have shown that the number of fingers changes strongly, especially at the beginning of the experiment.
Graph 10.27(a) displays the number of fingers at the first moment we were able to count them, tfirst count. We compare experimental results to predictions from linear 2D theory and to calculations from equation10.28at the corresponding time tfirst count for each experiment. The results all fall in the same range, but no clear tendencies are observed. The approximations concerning the shape of the viscous blob, that is,L= 1 andρ= 2 might not be true anymore even at small timest >0.
10.8. Conclusion
100 90 80 70 60 50 40
N
120x10-6 100 80 60 40 20 0
b0 [m]
Experiment 2D 3D
@ time of first count
(a)t0=t0first count
100 90 80 70 60 50 40 Nfirst count
0.50 0.45 0.40 0.35 0.30 0.25 0.20
t'first count
2D 3D
(b)
Figure 10.27.: The comparison of the number of fingers in the experiment (squares) to results from 2D and 3Dtheory (triangles and circles, respectively).
But the finger growth is not very developed at that point so that we consider these approximations still as reasonable.
The number of fingers predicted from both theories fluctuates strongly. Figure 10.27(b)displays the number of fingers versus the timetfirst count they are calculated at. The linear relation between number of fingers and time shows that these fluctu- ations can be attributed to the time dependence of the number of fingers. Even if the time of first count differs only a little ( 0.23< t0 <0.48) and is small compared to the experimental duration (t0 ≈ 15), the number of fingers changes strongly at the initial stages of the experiment. We state that the number of fingers for different realizations of one control parameter is slightly increased when including the menis- cus shape. However, the general tendency of increasing total finger number with increasing confinement is not reproduced by the 3D theory including the curvature effect.
Volume loss
In this section, we discuss the possible influence of volume loss on the control pa- rameter.
A thin oil layer is left on the glass and the steel plate behind the advancing air fingers. The thickness of this draining film has first been studied by Bretherton [14]
for air bubbles rising in capillary tubes. He found that its thickness varies likeCa2/3. Tabelinget al. investigated the influence of this draining film on the finger selection in the linear Hele–Shaw cell [111,112]. They showed that Bretherton’s law breaks down for capillary numbers higher than 0.01. The layer thickness saturates at a value 0.1b0 that depends on the cell geometry. In our experiments, the capillary number is always higher than 0.01. Presuming a layer with thickness 0.1b0 on each wall of the cell and leads to an effective thicknessbeff = 0.8b0 at early times. However, the control parameter is changed about the same percentage for different confinements
Chapter 10. The fingering pattern in a Newtonian oil
6
1
2 4 6
10
2 4 6
100
N
12 8
4 0
t'
Figure 10.28.: Comparison between the number of fingers calculated from linear theory with volume conservation (solid line) and calculated from a mean radius determined experimen- tally from the pictures (markers). Filled symbols = oil 100, open symbols = oil 10. C= 30 (lack circles), C= 40 (red squares), C= 120 (green diamonds).
C. The draining film can hence not explain the confinement dependence.
A second origin for volume loss is material flow due to gravity. We checked whether the loss of oil volume affects the control parameter strongly enough to be reflected in the number of fingers. Therefore, we measured the contact areaAduring a test as a function of time. We calculated then an effective mean radiusReff by approximating the measured area Awith a circle. Reff is given by
Reff = µA
π
¶1/2
. (10.30)
In equation10.8, we replace R(t) withReff(t). It then reads τ(t) = γb(t)3
12ηv0Reff(t) = γb03(1 +t0)3
12ηv0Reff(t) . (10.31) Figure 10.28 displays the calculated number of fingers as a function of time. The solid line represents N(t) as predicted from linear stability analysis. Markers cor- respond to calculations from equation10.28. Filled symbols represent oil 100, open symbols oil 10. Black circles correspond to C = 30, red squares to C = 40, and green diamonds toC = 120.
Obviously, the deviation is only small. Note thatN is plotted here on a logarithmic scale. Especially for small times, no influence is observed. In conclusion, the volume loss has no important influence on the control parameter at early times.