Experiments of periodic forcing of Saffman-Taylor fingers

We report on an experimental study of long normal Saffman-Taylor fingers subject to periodic forcing. The sides of the finger develop a low amplitude, long wavelength instability. We discuss the finger response in stationary and nonstationary situations, as well as the dynamics towards the stationary states. The response frequency of the instability increases with forcing frequency at low forcing frequencies, while, remarkably, it becomes independent of forcing frequency at large forcing frequencies. This implies a process of wavelength selection. These observations are in good agreement with previous numerical results reported in LedesmaAguilar et al., Phys. Rev. E 71, 016312 2005 . We also study the average value of the finger width, and its fluctuations, as a function of forcing frequency. The average finger width is always smaller than the width of the steady-state finger. Fluctuations have a nonmonotonic behavior with a maximum at a particular frequency.


I. INTRODUCTION
Frequency dependent flows of fluids in confined media can be found in several fields.Cleaning of filters ͓1͔ and modern techniques of secondary oil recovery ͓2͔ are two examples of it.
Many of these situations imply the presence of more than one fluid.In particular, the displacement of a viscous fluid by a less viscous fluid is susceptible of undergoing a Saffman-Taylor instability ͓3͔, in which fingers of the less viscous fluid penetrate the more viscous fluid.In this paper we report on an experimental study of the destabilization of Saffman-Taylor ͑ST͒ fingers in frequency-dependent situations.
ST fingers can be generated in a controlled way in a Hele-Shaw ͑HS͒ cell ͓4͔, a pair of glass plates parallel to each other that form an almost two-dimensional channel in which the flow takes place.When a viscous fluid in the cell is displaced by a less viscous one, the interface between the two fluids becomes unstable to the formation of fingers.After a dynamic process of finger competition, a steady state is reached in which a single finger ͑the ST finger͒ occupies the channel.Due to its relative simplicity, the ST instability in a HS cell has become an archetype of interfacial instabilities in two-phase flow ͓5-8͔.
In the absence of anisotropy ST fingers are always wider than half the channel width.These fingers are called normal, in order to differentiate them from narrow fingers obtained in the presence of anisotropy, called anomalous ͓9-11͔.While lateral instabilities in anomalous fingers have been known for two decades ͓10,11͔, normal ST fingers were considered for a long time to have stable flat sides.Recently, however, there have been several observations of lateral fluctuations of normal ST fingers.Moore et al. have reported experiments of lateral fluctuations of the ST finger in wide channels ͓12͔.Also Torralba et al. have shown experimentally that quenched noise induces a small-amplitude long-wavelength instability on the sides of normal ST fingers ͓13͔.Fluctuations on the finger sides have a dominant wavelength, indicating that the system acts as a selective amplifier of static noise.A numerical integration of a phase field model ͓14͔ has made evident two different ways of inducing lateral instabilities on normal ST fingers.First, by forcing the finger with a time-dependent pressure gradient, superimposed upon the constant pressure gradient necessary to produce the steady-state finger ͓15͔.Second, through the presence of spatial perturbations ͓16͔.Numerically, it has been found that both spatial and temporal perturbations are able to destabilize the finger flat sides with a small amplitude and long wavelength instability.
Here we present an experimental study of the destabilization of a normal air-oil ST finger in a HS channel, subject to periodic forcing.The steady-state finger is generated by withdrawing the oil at constant average flow rate.The finger is then periodically forced by imposing a periodic pressure signal on the side of the air.We find that the finger responds by generating a wave on its tip that propagates towards the finger sides.There is a dynamic process and, in most cases, the instability reaches a stationary state with a wavelength that no longer changes in the course of the experiment.
The paper is organized as follows.The experimental setup and the protocol used to analyze the experimental data are described in Sec.II, together with a qualitative description of our observations.The experimental results are presented in Sec.III.As reference, we have introduced two quantities that have units of frequency in the unperturbed problem.We discuss the finger response in stationary and nonstationary situations as well as the dynamics towards the stationary states.We also study how the average value of the finger width and its fluctuations depend on forcing frequency, and the role of finger velocity.Finally in Sec.IV we discuss the results and draw the conclusions.tally one above the other, separated by a brass frame carefully machined to provide a gap spacing of 1.00 mm, and firmly clamped to each other.A mechanical strain gauge with 20 m resolution, placed in contact with the top plate, does not reveal any measurable flexing of the plate in the course of the experiments.A sketch of the experimental setup is presented in Fig. 1.The bottom plate has a wide opening at each end.The distance between both openings is L = 1.15 m.The channel provided by the brass frame has a width W = 25.00 mm.This gives an aspect ratio L / W Ͼ 40, large enough to reach the wavelength selection regime reported in ͓15͔.
We have used a silicone oil ͑Rhodorsil 47V500͒ of dynamic viscosity = 0.518Ϯ 0.002 Pa s at 20 °C, density = 975Ϯ 10 kg/ m 3 , and surface tension = 20.7 mN/ m, as viscous fluid.The oil has been colored with Oil Blue N ͑Sigma-Aldrich͒ without a measurable change in viscosity and surface tension, as verified by rheological and tensiometric tests.
The oil fills initially the cell.It is withdrawn from one end of the cell at constant average flow rate by means of a syringe pump.We selected flow rates of 36.0 and 71.0 mm 3 / s.The corresponding withdrawal velocities ͑far away from the finger tip͒ are V ϱ = 1.42 and 2.83 mm/ s.At the opposite end of the cell the air inlet is connected to an air pump designed to produce a periodic oscillation of the air pressure around ambient pressure.The air pump consists of a mobile piston of 4400 mm 2 of cross-sectional area, placed inside a plastic cylinder.The piston is able to oscillate at a frequency that can be set in the range 0 -1 Hz.The pump admits air from outside when the position of the piston is at one extreme.Initially the pump is off, and the piston is placed such that air may enter freely into the cell at ambient pressure.Quickly after fluid withdrawal starts, a finger of incoming air is formed.When this newly formed finger reaches a steady state ͑at a distance of approximately 6W͒ the air pump is switched on.
Figure 2 shows the pressure oscillations in the two fluids, recorded with pressure transducers placed in parallel to the fluid flow at the two ends of the cell.At the air inlet ͑bottom plot͒ the pressure perturbation is a periodic signal with a period formed of two parts: One period of a sinusoidal, to a very good approximation, and a constant signal.Both are produced by the oscillatory motion of the piston.The sinu-  soidal part is produced when the piston moves ahead of the small hole that allows air to fill the pump.The constant part is produced when the piston is behind the small hole and therefore the pressure in the cell is essentially the atmospheric pressure.We have checked that the frequency of the pressure signal in the air follows exactly the frequency of forcing.At the oil outlet ͑middle plot͒ the pressure signal is still periodic.A frequency analysis shows that it contains the periodicity of the forcing and several other frequency components.The average pressure signal ͑obtained from smoothing out this oscillating pressure signal͒ takes its most negative value at the beginning, reflecting the suction of the syringe pump.As the finger tip advances in the cell at a constant average velocity, and approaches the oil outlet, this average pressure rises up to the atmospheric pressure linearly in time ͑top plot͒, because the pressure difference between the finger tip and the oil outlet becomes progressively smaller ͑the average pressure gradient remains constant and keeps the average tip velocity constant͒.
While experiments with ST fingers are usually controlled by a constant liquid flow rate or by a constant pressure difference, our forcing is a combination of the two.We control the average flow rate with the syringe pump, while the average input pressure is left free.But at the same time we control the pressure fluctuations with the air pump and let the flow rate fluctuate freely.This arrangement may seem complicated, but it is optimal for two reasons: ͑i͒ The controlled pressure fluctuations in the gas are transmitted nearly instantaneously to the finger tip, and ͑ii͒ the viscous pressure dissipation on the oil is so large that the flow rate fluctuations at the oil outlet are small compared to the average flow rate.
The cell is illuminated from below with a 35 kHz fluorescent lamp, 1200 mm long, covered with white plastic to diffuse light.Images of the air finger are recorded by a CCD camera placed above the cell.The camera provides images of 640ϫ 480 pixels with a resolution of 0.16 mm/ pixel.Since our cell is 25 mm wide, the region of interest captured in each image spans the whole cell width and a length of 102 mm along the finger corresponding approximately to 4W.

B. Qualitative observations
Once the periodic forcing is switched on, the finger tip responds with a periodic oscillation ͑a periodic backward and forward motion in the reference frame moving with the average velocity of the finger tip͒.This generates a wave that is born close to the tip and propagates towards the finger sides.The finger tip looks like the one of a normal ST finger, and the finger sides present a low-amplitude longwavelength instability reminiscent of incipient dendrites.
As the wave propagates away from the tip, the wavelength of the lateral instability undergoes a dynamic process.For most forcing frequencies the lateral modulations of the finger reach a stationary state, with a wavelength and amplitude that no longer change in time.When this happens, the shape of this part of the finger remains static in the laboratory frame of reference.However, there is an intermediate range of forcing frequencies ͑as discussed in the next sec-tion͒ for which a stationary state is never reached.
For forcing frequencies larger than in = 0.08, we have checked that the shape of the wave in stationary states is nearly perfectly sinusoidal.An example of this wave is shown in Fig. 3.For in Ͻ 0.08 Hz the forcing is so slow that the wave follows the details of the forcing ͑which is periodic but not perfectly sinusoidal, as discussed above͒.The result is that the wave is not sinusoidal anymore.This is shown in Fig. 4.

Regular data analysis
Except for the lowest two forcing frequencies, discussed later, the recorded images have been analyzed in the following way.
The lateral modulations have been recorded as a function of time.To this purpose the camera has been fixed in the laboratory frame of reference and, as the fingertip advances towards the oil outlet, pictures in a fixed window of the HS cell have been taken.This window has been chosen following two criteria.First, the CCD camera must be placed far enough from the oil outlet to allow for the lateral instability of the finger to reach a stationary state.Second, to discard possible transient effects after switching on the periodic forcing, the window cannot be placed too close to the air inlet.With these two criteria in mind, the window has been set at 12.5W of the air inlet.Image acquisition begins when the finger tip enters the window, and ends when the tip of the finger has travelled a distance of approximately 22W beyond the window.A typical view of the recorded images ͑cropped to 640ϫ 154 pixels͒ is shown in Fig. 3.We have acquired about 300 images per experiment for the low flow rate and about 150 images for the high flow rate.Out of these we have analyzed one image every T / 10 s, T being the total duration of an experiment.
The dynamic process undergone by the lateral modulations of the finger is visible in the time sequence of recorded images.When the wavelength of the lateral modulationmeasured as the distance between adjacent maxima or minima-no longer changes, we observe a saturation of the shape.Then, we compute the average wavelength of the modulations in one image, ⌳, and take this value as the wavelength of the lateral instability for one realization of the experiment.In the same image we measure also the average finger width, .From these two measurements, a response frequency out ͑the frequency that would be measured in the reference frame moving with the average velocity of the finger tip͒, for a given realization of the experiment, can be determined as where V ϱ is the velocity of the oil far away from the finger tip ͓17͔.We have performed five independent realizations of the experiment for each in studied.The final response frequency of the lateral instability-at a given in -is the weighted average of out in the five realizations.

Data analysis at low forcing frequency
For the two lower values of in , the modulation of the finger sides is not sinusoidal.Instead, it presents a sawtooth profile, as shown in Fig. 4, which contains several frequencies.In addition, the largest wavelength of the profile is longer than the lengthwise span of our window of observation.The window cannot be made larger, because we would lose the resolution required to measure the amplitude of the instability.In general, since the wavelength evolves as it propagates far from the tip, an image reconstruction of the finger made of pictures taken at different times is not possible.However, for very low in the structure that is born close to the fingertip remains unaltered in the laboratory frame of reference as the fingertip moves forward.We have thus decided to measure the lateral modulations in this kind of fingers by ͑i͒ letting the camera travel along the cell while it takes pictures of the finger, and ͑ii͒ making an image reconstruction of the whole finger by superposition of the recorded images.This technique was already used and described in ͓13͔.
In this case we have carried out three realizations of the experiment for each value of in .Each realization provides a reconstructed image of the whole finger.Each reconstructed image is then digitized, and a fast Fourier transform ͑FFT͒ is applied to repeated structures on one of the finger's sides.The position of the largest peak of the FFT is associated with the wavelength of the instability, ⌳, for each realization.This, together with the mean finger width, , allows for the calculation of out .The final value of out is the weighted average over the three realizations.
In order to check that this method gives reasonable results, we have applied both types of data analysis to the experiments at the lowest four in for which the regular data analysis is possible.We have confirmed that the FFT analysis of the reconstructed image leads to the same values of out than the regular analysis described at the beginning of this subsection.

III. RESULTS
In order to describe our experimental results in terms of characteristic magnitudes of the unperturbed ST problem, we introduce two quantities that have dimensions of frequency.The first one is determined by the flow very far from the finger tip, and can be defined as ϱ = V ϱ / W. The second one is defined as fing = U / ͑W͒ = V ϱ / ͑ 2 W͒, where U is the finger velocity and W is the finger width.The second equality comes from mass conservation in two dimensions ͓17͔.
In our experiments the forcing causes the width to oscillate.Hence, in the expressions above W must be understood as the average finger width.Moreover, the average finger width is a function of in , as shown below.This means that, for given W and V ϱ , there is no longer a single fing , but a band of frequencies characteristic of the finger, ⌬ fing .
Next, we present our experimental results in terms of response frequencies, out , in order to make contact with the phase-field simulations reported in ͓15͔ and with our own work on lateral instabilities induced by quenched disorder ͓13͔.It is also interesting to present the results in terms of measured wavelengths, and this is done at the end of this section.

A. Stationary states
Figure 5 presents the results for out , for a withdrawal velocity V ϱ = 1.42 mm/ s.Open symbols correspond to stationary states.
Increasing zone.At low forcing frequency ͑ in in the range 0 -0.22 Hz͒ the instability is already born with a wavelength very similar to the one exhibited by the finger when its shape saturates.This stationary state is reached at a distance d sat Ӎ 2.5W from the fingertip.A typical time sequence of the window of observation for a finger in this range of in is shown in Fig. 6.As mentioned for in Ͼ 0.08 Hz the lateral modulation is very close to a sinusoidal.Note that the triangular-wave shape is an effect of the compression along one axis of the figure.
Our results for out ͑Fig.5͒ show that out increases monotonically as a function of in , up to in Ӎ fing .We will refer to this range of frequencies as increasing zone.Response frequencies in this zone seem to be bounded by in and in / 2. At very low in the response is linear.As in increases, the response deviates from being linear, and the slope of out versus in decreases systematically.
Selection zone.For in Ն 0.30 Hz the instability is born with a wavelength much smaller than the final one reached in the stationary state.In this case the finger shape saturates at d sat Ӎ 14W.A typical time sequence of the window of observation in this range of in is shown in Fig. 7.The amplitude of the instability is much smaller here than in the increasing zone, but the wavelength is still measurable following the procedure described in Sec.II C.
In Fig. 5 we observe that out is almost independent of forcing frequency for in Ͼ fing ͑i.e., at the right of the band ⌬ fing ͒.Consequently, we refer to this range of forcing frequencies as selection zone.The weighted average of out in the selection zone, for in in the range 0.30-0.54Hz gives a selected frequency equal to sel = 0.096Ϯ 0.002 Hz.It is worth noting that sel lies between the characteristic frequencies of the system, ϱ Ͻ sel Ͻ fing .See Table I.
Finally it is interesting to observe in Fig. 5 that, in the whole range of in explored in the present experiments, out always lies below the band ⌬ fing , characteristic of the finger.

B. Nonstationary states
In a narrow frequency range between the increasing zone and the selection zone the instability does not reach a sta-tionary state.At early and intermediate times the finger presents a sinusoidal response like the one observed in the increasing zone.Calculating out , by the usual procedure, for the last image that presents a sinusoidal profile in the time sequence, gives values of out ͑the inverted black triangles in Fig. 5͒ that follow the trend of our previous results in the increasing zone.For some experimental realizations, however, we find that the sinusoidal profile of the fingers evolves into a slightly different pattern with a longer wavelength.This behavior is illustrated in Fig. 8.Although a stationary state is not reached, we have measured the wavelength of the last recorded frame and computed the corresponding value of out .results, shown as black dots in Fig. 5, have a similar value than sel .
Hence the solid symbols in Fig. 5 indicate that, in this intermediate frequency range, a stationary state is not reached in the course of our experiment.For a number of experimental realizations, the response frequency out of the lateral modulations seems to evolve from the value that would correspond to the increasing zone to the value of the selection zone.

C. Average width and fluctuations of fingers
We have computed the average finger width, , and the root mean square fluctuations of the finger width, ␦ , within the window of observation.The top plot in Fig. 9 shows that depends on in in a nonmonotonic way.We have also plotted the maximum and minimum finger width, max and min .They are indicative of the amplitude of the lateral instability In Fig. 9 we have also indicated the values of and ␦ measured in the absence of forcing ͑ in =0͒.Our results show that the average finger width of the unperturbed ST finger is always larger than the average finger width of the forced ST finger.Conversely, the rms fluctuations of the width of the unperturbed ST finger are smaller than those of the forced fingers.This last result is indicative, first, of the experimental uncertainty of the steady-state finger width.And, second, it gives an estimate of the spatial resolution of our measurements.In other words, if for a given forcing frequency fluctuations were smaller than the ones measured for the steady state, we would not be able to detect them.Figure 9 shows that this is not the case, and thus the fluctuations reported can always be attributed to forcing.

D. Dependence on finger velocity
We have studied another withdrawal V ϱ = 2.83 mm/ s, in order to characterize the lateral instability of faster fingers.V ϱ = 2.83 mm/ s is a compromise value between two requirements: to have V ϱ large enough for the dimensionless finger width to approach 0.5, and to have enough time in the course of the experiment to reach the steady state.At this higher velocity, both ϱ and the band ⌬ fing move towards higher frequencies, and the frequency range between ϱ and fing widens.The characteristic frequencies for V ϱ = 2.83 mm/ s in Fig. 10 have been written with an apostrophe.
Figure 10 shows that the increasing zone at this higher velocity also extends up to frequencies of the order of fing , as it was observed for the lower velocity.Interestingly, as long as in is below the band ⌬ fing , the response frequency out is independent of V ϱ .To study the selection regime at this higher velocity, one would need to apply frequencies much higher than the ones accessible in our experiment.ν in (Hz) 10.Frequency of the lateral modulations, out , as a function of forcing frequency, in .᭺: Stationary states for a withdrawal velocity V ϱ = 2.83 mm/ s.The measurements have been obtained directly from the recorded images, as described in the text.The other symbols correspond to V ϱ = 1.42 mm/ s, and are the same as in Fig. 5.The frequencies ϱ , ϱ Ј , and the frequency bands ⌬ fing and ⌬ fing Ј ͑the primed variables corresponding to the higher V ϱ ͒ are also represented for reference.
Results for the average finger width and the rms fluctuations ␦ at V ϱ = 2.83 mm/ s are shown in Fig. 11.The qualitative behavior of both magnitudes is very similar to the case V ϱ = 1.42 mm/ s-in the frequency range from 0 to fing .At this higher velocity the average finger width is closer to the asymptotic value 1 / 2, and remains nearly constant with in .The maximum of ␦ , on the other hand, is slightly shifted to in Ͼ ϱ Ј .

E. Dimensionless wavelength of the lateral instability
Figure 12 presents our results for the wavelength of the lateral instability ⌳, rescaled by the finger width W, as a function of forcing frequency in , for the two withdrawal velocities V ϱ = 1.42 and 2.83 mm/ s.
The dimensionless wavelength decreases with in but remains always larger than 1, i.e., the measured wavelength is always larger than the finger width.Actually, the end of the increasing zone is characterized by a dimensionless wavelength that approaches 1.
At higher in the lateral instability readjusts itself to a larger wavelength ͑for V ϱ = 1.42 mm/ s͒.Notice that the finger width decreases slightly with in in the selection zone.Thus, upon rescaling the measured wavelength with W, we find that the resulting dimensionless wavelength is independent of in in the selection zone.
Finally, we notice that the results in terms of wavelengths do depend on the withdrawal velocity V ϱ .

IV. DISCUSSION AND CONCLUSIONS
Normal Saffman-Taylor fingers respond to a periodic pressure forcing by generating a wave on its tip.As the fin-ger advances, this wave propagates away from the tip towards the finger sides at constant velocity ͑in the finger frame of reference͒.For most forcing frequencies the amplitude and the shape of the wave saturate and give rise to a strictly periodic pattern on the sides of the finger.
These observations are in good qualitative agreement with the results of phase-field simulations reported in ͓15͔ in several frequency intervals.Thus, at very low forcing frequency the response is such that out = in , and the final frequency of the stationary state is reached near the finger tip.Also, at large forcing frequencies, a selection zone is reached where out is almost independent of in ͑although in the experiments it increases slightly with the forcing frequency͒, and falls between the two characteristic frequencies of the system, ϱ Ͻ out Ͻ fing .In this zone the stationary state is reached rather far from the finger tip.However, at intermediate forcing frequencies ͑ ϱ Ͻ in Ͻ fing ͒ the experimental observations differ from the numerical predictions.In the numerical integration of the phase-field model ͓15͔ out = in / 2 in this frequency range, the steady state has a frequency lower than the initial state, and the transition from the increasing zone to the transition zone is very abrupt.In experiments we find that out increases with in , with a progressively decreasing slope between 1 and 1 / 2, up to fing ͑increasing zone͒.Furthermore, the lateral modulations reach a steady state at a short distance of the finger tip.
There are few forcing frequencies between the increasing zone and the selection zone for which a stationary state is in .The symbols ᭝, ᮀ, ᭢, and b correspond to V ϱ = 1.42 mm/ s and the same measurements as in Fig. 5.The symbol ᭺ corresponds to V ϱ = 2.83 mm/ s and the same measurements as in Fig. 10.The frequencies ϱ , ϱ Ј , and the frequency bands ⌬ fing and ⌬ fing Ј ͑the primed variables corresponding to the higher V ϱ ͒ are also represented for reference.Top: Magnification of the selection zone for V ϱ = 1.42 mm/ s. never reached.For these forcing frequencies, the structures observed on the finger sides are roughly evolving from shapes characteristic of the increasing zone to shapes characteristic of the selection zone.It is possible that with a longer cell, a stationary state on the selection zone would have been reached.It is also possible that for these forcing frequencies two different response frequencies are equally likely, since we do not observe this shape evolution for all of the experimental realizations.Nevertheless, a thorough study of the non-stationary states falls beyond the scope of the present investigation.
The analysis of our results in terms of a dimensionless wavelength reveals that the average width of the finger, W, is a relevant length scale in the system.First, in the stationary state we do not observe wavelengths smaller than W. Thus, when the wavelength generated at the finger tip is shorter than the finger width, the system selects a much higher wavelength.It is also worth pointing out that the slight increase of out observed in the selection zone is compensated by a slight decrease of the average finger width, resulting in a constant dimensionless wavelength ⌳ / W in this zone.
We find that fingers subject to periodic forcing have an average finger width that depends on the forcing frequency in a nonmonotonic way.Moreover, we find that the average finger width is always smaller than the steady-state finger width.Also, we find that rms fluctuations of the finger width have a nonmonotonic behavior as a function of forcing frequency.At a certain forcing frequency, fluctuations are maximal.This maximum moves towards higher frequencies as the finger velocity increases.
ST fingers are known to become unstable when the modified capillary number 1 / B =12U / ͑W / b͒ 2 becomes large.The tip-splitting instability has been studied by Park and Homsy ͓18͔ and by Tabeling et al. ͓19͔.According to these authors tip-splitting takes place above 1 / B Ӎ 3000.In our experiments the finger is driven at moderate velocities, such that 1 / B Ӎ 500 or 1000 and, consequently, we do not have any observational evidence of tip-splitting.
It is interesting to compare the lateral modulations of normal ST fingers in our experiments with lateral modulations on anomalous ST fingers subject to forcing ͓11͔.The anomalous fingers ͑narrower than half the channel width͒ are produced by some source of anisotropy in the system, for instance by a thread or two grooves.For anomalous fingers when periodic forcing is applied with an amplitude larger than the natural noise, there is side branching with strictly periodic undulations.In our experiments, where periodic forcing is applied, the wave of the stationary state is also strictly periodic.Interestingly, for anomalous fingers an optimal forcing frequency makes the amplitude of the lateral modulations maximal.In normal fingers we have been unable to observe the instability for low forcing amplitudes.For large frequencies, there is no instability observed in anomalous fingers.In our case, not only there is an instability at large frequencies but it presents the interesting regime of selection.
The present experimental study for forced fingers, our own experimental work on fingers in the presence of quenched disorder ͓13͔ and the numerical integration of the phase-field model for both, dynamic ͓15͔ and static ͓16͔ perturbations, lead to the conclusion that in normal Saffman-Taylor fingers lateral instabilities seem to be due to selective amplification of perturbations at the finger tip, just as it happens in the case of anomalous fingers ͓11͔ and in sidebranching in solidification ͓20,21͔.
From an applied point of view the presence of a lateral instability on normal ST fingers, driven with a timedependent pressure gradient, is relevant in designing secondary oil recovery procedures and medical applications that exploit the possibilities of periodic forcing.It has been argued that the amplitude of the instability might be very large at some particular ͑resonance͒ frequencies if the fluid displaced is viscoelastic ͓22͔.A method for optimizing the displacement of a viscoelastic fluid has been reported on ͓23͔.The importance of forcing has also been emphasized to enhance viscoelastic flow during occlusions ͓24͔.The present study on Newtonian fluids provides a basis for future studies on the periodically forced displacement of complex fluids.

FIG. 2 .
FIG.2.Pressure modulations measured at the air inlet ͑bottom͒ and the oil outlet ͑middle and top͒, for an average flow rate of 36.0 mm 3 / s and a forcing frequency in = 0.11 Hz.The ascending line on the top plot, drawn to guide the eye, is the average pressure signal obtained by smoothing out the actual oscillating signal.
FIG.3.Typical image recorded in our experiments.In this case the picture corresponds to a late stage of the dynamic process.Here the withdrawal velocity is V ϱ = 1.42 mm/ s and the forcing frequency in = 0.19 Hz.
FIG. 5. Frequency of the lateral modulations, out , as a function of forcing frequency, in , for a withdrawal velocity V ϱ = 1.42 mm/ s.Measurements represented by the symbols ᮀ ͑stationary states͒ and ᭢, b ͑nonstationary states͒ have been obtained directly from the recorded images, as described in the text.The two measurements at low in represented by ᭝ have been obtained from a FFT analysis of the image reconstruction.The error bars include data dispersion and instrumental uncertainty.For stationary states data dispersion is negligible for in Ͼ 0.1 Hz.For nonstationary states, data dispersion is dominant.The dashed straight lines are guides to the eye, with slopes 1 and 1 / 2. The frequency ϱ and the frequency band ⌬ fing are also represented, for reference.See text for details.

FIG. 9 .
FIG. 8. Time sequence of images of the window of observation for a finger in a nonstationary state ͑ in = 0.29 Hz and V ϱ = 1.42 mm/ s͒.Frames are separated 25 s ͑approximately 1 / 10 of the total duration of the experiment͒.The last frame displays lateral modulations of longer wavelength than the first frame.
FIG. 11.Finger width, , and rms fluctuations of the finger width, ␦ , as a function of forcing frequency in , for V ϱ = 2.83 mm/ s.Top: ᭞, max ; , ; ᭺, min .Bottom: , ␦ .The value ϱ Ј and the frequency band ⌬ fing Ј are also shown for reference.FIG.12. Bottom: Wavelength of the lateral instability, ⌳, rescaled by the finger width W, as a function of the forcing frequency