10. The time evolution of the fingering pattern in a Newtonian oil
10.2. Materials and methods
In this section, we characterize first the materials and describe then the experimental protocol.
10.2. Materials and methods
Nominal viscosity y0 A B
oil 100 100 Pa s 0 142.17±1.29 0.0152±0.0003 oil 10 10 Pa s 0 17.202±0.246 0.0164±0.0005
Table 10.1.: Fit parameters to an exponential law describing the temperature dependence of the viscosity for two Newtonian oil, see figure10.2.
120 100 80 60 40 20 0
h [Pa.s]
1 2 3 4 5 6 7 8 9
10 2
dg/dt [1/s]
@ 25.5°C
Figure 10.1.: The viscosity versus the shear rate for oil 100 (η ≈100 Pa s) and oil 10 (η ≈ 12 Pa s) at 25.5◦C.
10.2.1. Materials
We used two silicone oils (PDMS) with different viscosities. The oil called “oil 100”
in the following had a nominal viscosityη= 100 Pa s, the oil called “oil 10” had a a nominal viscosityη = 10 Pa s. They were purchased atAldrich.
Figure 10.1 shows the shear viscosity as a function of shear rate for both oils.
The data was obtained in aHaake RS 100 Rheostress rheometer using a cone-plate geometry with diameter d = 35 mm and an angle θ = 2◦ at T = 25.5◦C. The viscosity does not vary with the shear rate and is considered as constant in the experimentally relevant range of shear rates. A oscillatory frequency sweep revealed thatG00ÀG0 in this range of frequencies; the oils behave as Newtonian liquids.
However, the viscosity depends on the temperature. As we cannot control the temperature in the experimental setup, we determinedη as a function ofT between T = 24◦C and T = 40◦C at ˙γ ≈51/s. The data could be fitted with an exponential law
η(T) =y0+Ae−B T , (10.5)
see figure10.2. The resulting values of the fit parameters are given in table10.1.
Chapter 10. The fingering pattern in a Newtonian oil
100 95 90 85 80 75
h [Pa.s]
40 36 32
28 24
T [°C]
(a) Oil 100 at ˙γ= 6 s−1.
12.0 11.5 11.0 10.5 10.0 9.5 9.0
h [Pa.s]
40 36 32 28 24
T [°C]
(b) Oil 10 at ˙γ= 4 s−1.
Figure 10.2.: Viscosity as a function of temperature. The solid line is a fit to an exponential law.
10.2.2. Experimental protocol
Experiments were performed in the µ-tack machine described earlier, yet with a slightly different experimental protocol. It was not possible to prepare samples of a given thickness on microscope glass slides as in the previous experiments, as the oil was flowing too fast. Instead, we proceeded in the following way.
The lifted Hele–Shaw cell consists in an upper microscope glass slide and a lower circular steel probe with a given radius R0. The probe radius determines the size of the cell. The upper plate can be lifted at a defined speed. We used a thin PSA film on a microscope glass slide for a coarse alignment of probe and upper glass slide. Then we precleaned a microscope glass side with acetone and ethanol and put it into the apparatus. We approached the metal probe by hand slowly to the glass slide until contact was established and the force became non-zero. At the same time, interference fringes were observed upon contact through the camera mounted on the microscope. The final alignment was done at this moment by rendering the fringes as symmetric as possible. Once the alignment was satisfying, we moved the probe away from the glass slide in very small steps. We determined the zero position b = 0µm as the force became zero again and the interference fringes disappeared.
Lowering the probe by several millimeters, a drop of silicone oil was then placed on the probe with a syringe. Finally, we brought the probe slowly into the desired position. A typical value was b0 = 100µm. The experiment itself started when we lifted the upper glass slide at a given speed v0. During the experiment, images were taken by the digital camera, allowing for a good visualization of the emerging pattern and its evolution in time.
A crucial point in the course of an experiment was to have reproducible boundary conditions. It was impossible to put the exact volume of oil that corresponds to the initial cell volume onto the probe in the first place. Thus, we always had to remove
10.2. Materials and methods an excess of oil at the borders of the probe when the probe arrived at its starting position. The borders had to be cleaned very properly with small instruments made for this purpose. Every small dust particle represents a perturbation of the boundary and disturbs the fingering pattern, and every extra amount of oil surrounding the probe introduces time shifts in the evolution of the finger pattern.
Figure 10.3.: The Saffman-Taylor instability at different times in the lifted Hele–Shaw cell.
R0= 3 mm,b0= 50µm,v0= 8µm/s,η= 96 Pa s.
Figure 10.3 shows a typical experiment in top view at different times. At the beginning, the system is at rest and the 2D projection of the interface between oil and air is ideally a perfect circle. The contact line is not visible as it lies on the border of the probe. The light area represents the metal probe covered with oil, the black line the interface between air and oil. At t = 0, the oil starts to flow inwards and the circular interface retracts as the upper glass plate moves upwards.
The interface is soon destabilized and starts to undulate. The amplitude of these undulations increases and air fingers grow inwards. Figure10.4shows an overlay of the contact line for different times. Lighter colors correspond to later times. The finger amplitude can be strong or weak depending on the experimental parameters.
The finger growth is very pronounced in figure10.3. The number of fingers decreases with time. Finally all the fingers disappear, the interface becomes circular again, and one final fibril remains between glass plate and probe. Its stretching is dominated by material flow. It finally breaks up triggered by the Rayleigh–Plateau instability [90, 95], the test is stopped, and the glass plate and probe return to their initial position.
We investigate different realizations of the same control parameter τ0. To do so, we change the initial gap width and the oil viscosity and adapt at the same time the lifting speed to keepτ0 constant. As the temperature cannot be controlled in
Chapter 10. The fingering pattern in a Newtonian oil
Figure 10.4.: Superposition of the contact lines from figure10.3. Lighter colors correspond to later times.
our setup, we noted the temperature for each experiment and accounted for the change in the viscosity in the calculation of τ0. We performed several experiments in one series, that is, with the same oil and differentb0. When an experiment at a certain initial gap width was completed, we returned to the starting positionb=b0. Then we moved the steel probe a little closer to the glass plate and proceeded to a second experiment. It is important to make sure that all traces from the previous experiments are deleted to exclude memory effects. The interface was again cleaned from excess oil and the next test was started. In this way, we performed three to four experiments in one series.
We checked that using several times the same oil is a valid protocol by performing experiments where fresh oil was used for every single experiment, including a new setting of the zero point. The results were in perfect agreement with the ones obtained from the protocol that uses several times the same oil.
Determination of t0
A crucial point was the determination of the starting time t0 = 0, that is the moment where the motors start to lift the upper glass plate and the oil starts to move. Normally, the experimental setup is equipped with a trigger that activates a timer at the moment the motors start to move. In this way, each image can be linked to a time stamp from the timer. However, technical problems due to an incompatibility of camera and trigger made it impossible to use the trigger. We decided instead to determine t0 visually on the images. We started the camera before the actual experiment. The camera software automatically generates a time stamp Ti for each image i. The image on which the oil started to move could be determined from very small dust particles, which were always present in the oil and were visible on a zoom. We determined the time ti of an image i relative to the moment where the oil first moved as
10.3. The number of fingers in comparison to linear 2D theory
@T = 25◦C η[Pas] b0[µm] v0[µm/s] η[Pas] b0[µm] v0[µm/s]
τ0 '9.6×10−6 97.2 100 66 11.4 75 238
97.2 75 28 11.4 50 71
97.2 50 8 11.4 35 24
97.2 25 1 11.4 25 9
Table 10.2.: Different realizations of the control parameterτ= 9.6×10−6.
ti=Ti−Tfirst move+Tfirst move−Tlast still
2 . (10.6)
Ti is the time stamp of image i. Tfirst move and Tlast still denote the time stamps of the two images between which the oil started to move first. In this way, t0 lies between the last image on which the oil did not move and the first image on which the dust particles changed their position. This method yields an uncertainty in the time of the order of the inverse frame acquisition rate of the camera.