Fluctuations in Saffman-Taylor fingers with quenched disorder

We make an experimental characterization of the effect that static disorder has on the shape of a normal Saffman-Taylor finger. We find that static noise induces a small amplitude and long wavelength instability on the sides of the finger. Fluctuations on the finger sides have a dominant wavelength, indicating that the system acts as a selective amplifier of static noise. The dominant wavelength does not seem to be very sensitive to the intensity of static noise present in the system. On the other hand, at a given flow rate, rms fluctuations of the finger width, decrease with decreasing intensity of static noise. This might explain why the sides of the fingers are flat for typical Saffman-Taylor experiments. Comparison with previous numerical studies of the effect that temporal noise has on the Saffman-Taylor finger, leads to conclude that the effect of temporal noise and static noise are similar. The behavior of fluctuations of the finger width found in our experiments, is qualitatively similar to one recently reported, in the sense that, the magnitude of the width fluctuations decays as a power law of the capillary number, at low flow rates, and increases with capillary number for larger flow rates.

Figure 1

Finger in a cell with 10% coverage and gap spacing $b$ equal to $b=0.8\mathrm{mm}$. The velocity of the finger is $U=2\mathrm{mm}∕\mathrm{s}$. The bright region is defined as the finger body, that is, the portion of the finger for which the amplitude of the instability has saturated. The dark regions are the tail and tip of the finger. The tail of the finger experiences an instability whose amplitude changes in the course of the whole experiment.

Figure 2

Distributions of frequencies of the lateral instability corresponding to the three experimental configurations of Fig. 3. Vertical lines are the characteristic frequencies of the steady state [${\nu }_f$ at the right and ${\nu }_{\infty }$ at the left, as defined in Eqs. (2, 3)]. From top to bottom, $b=0.8\mathrm{mm}$ and 10% coverage, $b=1.3\mathrm{mm}$ and 10% coverage, $b=1.3\mathrm{mm}$ and 35% coverage.

Figure 3

Fingers in the presence of different configurations of quenched disorder. For all of them, the finger velocity is $U=2\mathrm{mm}∕\mathrm{s}$. From left to right, $b=0.8\mathrm{mm}$ and 10% coverage, $b=1.3\mathrm{mm}$ and 10% coverage, $b=1.3\mathrm{mm}$ and 35% coverage. Only finger tips and bodies are shown.

Figure 4

Mean finger width as a function of $1∕B\equiv 12(W∕b{)}^2\mathrm{Ca}$. The continuous line is the theoretical McLean and Saffman curve [17]. Solid squares are the experimental values of the mean width averaged over four different disorder configurations. As can be seen, our experiments give, in all cases, normal Saffman-Taylor fingers, that is, fingers whose width is larger than half of the channel width.

Figure 5

Fluctuations in the finger width vs $1∕B$. Solid squares are the average fluctuations over four different disorder configurations. Dotted line is the best fit for the low velocity regime ($1∕B$ from 0 to 250): ${\delta }_{\lambda }=(0.034\pm 0.008)(1∕B{)}^{(-0.30\pm 0.13)}$. Letters correspond to capillary numbers for which a finger is illustrated in Fig. 6.

Figure 6

Fingers in a cell with a 10% coverage and gap spacing $b=1.3\mathrm{mm}$ at several velocities. Corresponding capillary numbers are labeled with letters in Fig. 5. (A) $1∕B=18$, (B) $1∕B=88$, (C) $1∕B=198$, (D) $1∕B=339$, and (E) $1∕B=1091$, (F) $1300$.

Figure 7

Distributions of frequencies of the lateral fluctuations at different velocities. The symbols are $∎$, $1∕B=23$; $\circ$, $1∕B=39$; $▴$, $1∕B=63$; $▿$, $1∕B=82$; $\star$, $1∕B=118$; $\times$, $1∕B=198$; $◻$, $1∕B=254$; $•$, $1∕B=339$; $▵$, $1∕B=435$; $▾$, $1∕B=558$; $+$, $1∕B=925$; $◇$, $1∕B=1268$.

Figure 8

Top, mean of the distribution of frequencies of the lateral instability vs $1∕B$. Squares are experimental values. Dotted line is the best fit to the same ones, ${\nu }_{\mathrm{mean}}=(4.9\pm 0.4)\times {10}^{-4}(1∕B{)}^{(1.02\pm 0.04)}\mathrm{Hz}$. Bottom, standard deviation of the frequencies as a function of $1∕B$.