# Measurement of Pressure Gradients near the Interface in the Viscous
Fingering Instability

Savannah D. Gowen<EMAIL_ADDRESS>Department of Physics and The James
Franck and Enrico Fermi Institutes, University of Chicago, Chicago, IL 60637,
USA  Thomas E. Videbæk  Department of Physics and The James Franck and Enrico
Fermi Institutes, University of Chicago, Chicago, IL 60637, USA  Martin A.
Fisher School of Physics, Brandeis University, Waltham, MA 02453, USA  Sidney
R. Nagel  Department of Physics and The James Franck and Enrico Fermi
Institutes, University of Chicago, Chicago, IL 60637, USA

###### Abstract

The viscous fingering instability, which forms when a less-viscous fluid
invades a more-viscous one within a confined geometry, is an iconic system for
studying pattern formation. For both miscible and immiscible fluid pairs the
growth dynamics change after the initial instability onset and the global
structures, typical of late-time growth, are governed by the viscosity ratio.
Here we introduce an experimental technique to measure flow throughout the
inner and outer fluids. This probes the existence of a new length scale
associated with the local pressure gradients around the interface and allows
us to compare our results to the predictions of a previously proposed model
for late-time finger growth.

††preprint: APS/123-QED Figure 1:  Viscous fingering patterns imaged with
alternate injections of dyed and undyed fluid rings. The low-viscosity inner
fluid is dyed in alternate rings of blue. The outer fluid is transparent. The
scale bar represents 1 cm, $b=420~{}\upmu\rm{m}$, and
$(\eta_{\rm{in}}/\eta_{\rm{out}})=0.067$.

##  I Introduction

Pattern formation is often seeded by an instability; The selection criteria
for the initial pattern shape and length scale have in many cases been well
established [1, 2, 3, 4]. Pattern development at later times has received less
attention. Here we consider the patterns formed by the viscous fingering
instability – a fluid instability caused by a less viscous fluid (of viscosity
$\eta_{\rm{in}}$) displacing a more viscous one (of viscosity
$\eta_{\rm{out}}$) within a confined geometry [5, 6, 7, 8, 9, 10]. Pattern
formation begins when the initially smooth fluid-fluid interface becomes
unstable to the formation of undulations. At onset, the patterns can be
characterized by a most unstable wavelength. However, as pattern growth
continues experiments have shown that the parameters setting the emergent
global structures may be different from those important at onset [11, 12].
This demonstrates that the growth dynamics change even after the instability
has formed. Here we concentrate on the late-time growth of patterns using a
visualization technique, demonstrated in Fig. 1, to observe flows throughout
the inner and outer fluids.

Viscous fingering can be observed in pairs of fluids, with viscosity
difference $\Delta\eta\equiv\eta_{\rm{out}}-\eta_{\rm{in}}$, confined in a
Hele-Shaw cell consisting of two parallel glass plates separated by a uniform
gap of width $b$, as illustrated in Fig. 2a. For pairs of immiscible fluids,
Saffman and Taylor [13] showed that perturbations at the interface between a
less-viscous invading fluid and a more-viscous one will grow with a critical
wavelength: $\lambda_{c}=\pi b(\sigma/\Delta\eta V)^{1/2}$ where $\sigma$ is
the interfacial tension and $V$ is the velocity of the interface. In the case
of miscible fluids, where $\sigma$ is negligible, the most unstable wavelength
varies as $\lambda_{c}\propto Cb$ [9, 14, 15, 16, 17], where $C$ is a constant
that has been shown to depend weakly on the viscosity ratio
$(\eta_{\rm{in}}/\eta_{\rm{out}})$ [18]. Despite these differences, for both
miscible and immiscible pairs of fluids, viscous fingering produces similar
global structures: the patterns have an inner stable radius, $R_{\rm{in}}$,
characterized by the inner circular region where the outer fluid has been
completely displaced and an outer radius, $R_{\rm{out}}$, which contains the
entire interface as shown in Fig. 2b.

A detailed investigation of this late-time growth [11] showed that for
patterns with the same $\lambda_{c}$, the inner stable radius grows faster
with increasing $(\eta_{\rm{in}}/\eta_{\rm{out}})$, independent of surface
tension and qualitatively independent of the geometry of the interface in the
third dimension between the plates. In an attempt to account for the emergence
of the viscosity ratio as a control parameter for these ubiquitous global
features, that paper also proposed a model to describe the late-time growth of
the fingers once they had formed. The key assumptions of that model are two
length scales related to the decay of local pressure gradients from the
interface into the inner and outer fluids. The experimental technique
introduced here allows direct access to the pressure gradients around the
interface, and can therefore be used to test these assumptions.

We conduct experiments in a radial Hele-Shaw cell as shown in Fig. 2a and
described in detail in the Methods section. Miscible fluids are injected at a
constant flow rate through a small hole in the top plate. The outer fluid is
injected first followed by the less viscous invading fluid. To visualize flow
throughout the cell we create dyed and clear versions of both the inner and
outer fluids, being careful that the addition of dye to either fluid preserves
both its density and viscosity. We then alternately inject small volumes of,
e.g., the inner fluid in contrasting colors such that dyed volumes of fluid
can be tracked as they propagate outward radially as rings. Whereas the
interface is commonly tracked by using inner and outer fluids with contrasting
colors as was shown in Fig. 2b, alternate injection of dyed and clear fluid
allows contrast for observing flow anywhere in the cell and does so without
perturbing the flow.

In a confined flow, the fluid velocity $\langle\vec{u}\rangle$ is proportional
to the pressure gradient, $\vec{\nabla}P$, as given by Darcy’s Law:

$\langle\vec{u}\rangle=-\frac{\kappa}{\eta}\vec{\nabla}P$ (1)

where $\eta$ is the fluid viscosity and $\kappa$ is the permeability of the
medium. Thus by tracking dyed fluid rings we can measure the local flow
velocity and obtain the local pressure gradients. The technique, the details
of which are described in the Methods section, can be used separately in
either the inner [Fig. 1] or outer fluid or in both fluids simultaneously
[Fig. 3a].

Figure 2:  (a) Schematic of a radial Hele-Shaw cell consisting of two circular
glass plates separated by a thin uniform gap spacing, $b$. The less-viscous
fluid moves radially, displacing the more-viscous one. (b) Finger patterns of
late-time growth viewed from above. The inner fluid is dyed and the outer
fluid is left transparent for contrast. Characteristic length scales
$R_{\rm{in}}$ and $R_{\rm{out}}$ are labeled.

Our experiments demonstrate a non-invasive method for measuring the pressure
gradients as a function of angle in the neighborhood of the interface between
the fluids. We show that these pressure gradients decay with a characteristic
distance behind and beyond the undulating interface and thus determine two new
length scales associated with the fingering instability. In the Results
section, we use this technique to measure the local pressure gradients in both
the inner and outer fluids as a function of distance relative to the moving
interface; we extract the decay lengths and analyze them in terms of the
experimental variables. In the Discussion section, we relate these lengths to
those that appeared as hypothesized free parameters in the model for late-time
interface evolution [11]. We compare our results with the model predictions
and propose modifications that agree well with experimental measurements.

##  II Results

###  II.1 Detection of Local Pressure Gradients

Using our experimental technique, we observe that as rings of dyed fluid
approach the unstable interface there is a finite distance at which
undulations form along the rings. Flow in this system is driven by an imposed
pressure gradient between the inlet pressure and the atmospheric pressure of
the air surrounding the open cell. In a stable system, alternately injected
rings of dyed fluid would remain circular. The variations in the local
velocity field that create the observed undulations reveal the presence of
local pressure gradients and a length scale over which they influence flow. In
Fig. 3a, we estimate the region surrounding the interface over which local
pressure gradients perturb the flow, bounded in the inner fluid by
$r_{\rm{dp_{\rm{in}}}}$, and in the outer fluid by $r_{\rm{dp_{\rm{out}}}}$.

Figure 3: Length scales of viscous fingering patterns. (a) One pattern
quadrant is shown that has been imaged using alternate dyed-ring injection in
both the inner and outer fluids. A region surrounding the interface over which
local pressure gradients perturb flow is estimated with the bounds
$r_{\rm{dp_{\rm{in}}}}$ and $r_{\rm{dp_{\rm{out}}}}$. (b) Dyed rings are
analyzed in local pattern segments pertaining to a finger and an adjacent
valley. Length scales $r_{\rm{finger}}$ and $r_{\rm{valley}}$ along a single
dyed ring are labeled in the inner and outer fluids. The scale bars represent
0.5 cm.

###  II.2 Local Pressure Gradient Decay in the Inner Fluid

To quantify the length scale associated with the local pressure gradient
behind the interface, we measure the magnitude of undulations along a single
dyed fluid ring. As shown in Fig. 3b, we identify the radial location of the
dyed ring behind a finger protrusion as $r_{\rm{finger}}$, and behind an
adjacent valley as $r_{\rm{valley}}$; we calculate the amplitude of an
undulation, $\Delta r_{\rm{in}}\equiv r_{\rm{finger}}-r_{\rm{valley}}$. These
measurements are taken instantaneously and are repeated as parameters evolve
with the introduction of more inner fluid volume into the cell. Measurements
along a ring terminate when the fluid arrives at the interface. We repeat this
procedure behind several fingers.

Figure 4:  Undulation magnitude versus the negative distance behind the
interface for a single dyed ring. The interface, $R_{\rm{in}}$, is marked by
the dashed vertical line. Deformations are measured behind four different
fingers. Experimental parameters are: $b=420~{}\mathrm{\upmu m}$ and
$(\eta_{\rm{in}}/\eta_{\rm{out}})$ = 0.104.  Figure 5:  Characterization and
scaling of perturbation growth in the inner fluid. (a) The magnitude of
undulations versus distance behind the interface for five dyed rings labeled
in order of their relative injection times. Curves are fit by an exponential
function to determine a characteristic decay length, $\ell_{\rm{in}}$. The
inset shows $\ell_{\rm{in}}$ increasing as a function of ring injection
sequence (time). (b) The x- and y-axes are re-scaled such that the deformation
curves from sequential rings collapse onto a single curve. Experimental
parameters are: $b=420~{}\mathrm{\upmu m}$ and
$(\eta_{\rm{in}}/\eta_{\rm{out}})$ = 0.104.

Despite local differences in the sizes and shapes of the interfacial fingers,
the resulting deformations develop similarly for fingers that onset at
approximately the same time. Figure 4 shows the magnitude of undulations,
$\Delta r_{\rm{in}}$, behind four fingers along a single dyed ring versus the
negative distance behind the interface $-(R_{\rm{in}}-r_{\rm{valley}})$. The
zero on the x-axis corresponds to the location of the stable interface radius,
$R_{\rm{in}}$. Dyed rings far from the interface begin with zero deformations
(long-wavelength perturbations appearing farther behind the interface can lead
to artificially negative deformation values, which we subtract away). $\Delta
r_{\rm{in}}$ grows with the fluid’s proximity to the interface in a similar
way for different fingers.

We repeat these measurements for the same four fingers but along rings
injected at different times throughout the pattern formation. Figure 5a shows
these deformation curves gathered from five dyed fluid rings. Fitting these
curves to an exponential form $\Delta
r_{\rm{in}}=A\exp[-(R_{\rm{in}}-r_{\rm{valley}})/\ell_{\rm{in}}$] shows good
agreement with the data. The associated length scale, $\ell_{\rm{in}}$, is
shown as a function of ring injection sequence in the inset. $\ell_{\rm{in}}$
increases with time, showing the growth of local pressure gradients from the
interface into the inner fluid.

To collapse data collected from different dyed rings, we examined many
combinations of system parameters to re-scale the x- and y-axes. These
included the radial locations associated with the tips and valleys of the dyed
rings in the bulk; $r_{\rm{finger}},\ r_{\rm{valley}}$; the radial locations
of the interface (measured locally) and its average radius: $R_{\rm{in}}$,
$R_{\rm{out}}$, $R_{\rm{int}}\equiv\frac{(R_{\rm{in}}+R_{\rm{out}})}{2}$; the
radius at which the interface initially became unstable for a finger,
$R_{\rm{onset}}$; as well as the length, width, and tip diameter of individual
fingers. The re-scaled axes must, in the end, be dimensionless. Our best re-
scaling of these curves is shown in Fig. 5b where the x-axis is scaled by
$R_{\rm{int}}-R_{\rm{onset}}$ and the y-axis by
$(R_{\rm{int}}-R_{\rm{onset}})R_{\rm{int}}/b$. This scaling yields a single
master curve in which each data point varies in time and space. We note that
$R_{\rm{int}}$ may have also been substituted by $R_{\rm{in}}$ with little
change to the result. The collapse of this data suggests a form of the
characteristic decay length given by the denominator of the re-scaled x-axis:
$\ell_{\rm{in}}\propto(R_{\rm{int}}-R_{\rm{onset}})$.

We repeat this scaling procedure for data collected from experiments with
varied gap widths to produce the collapsed curve shown in Fig. 6a (in this
analysis we have limited ourselves to a regime in which the finger length is
significantly larger than the finger width). These results are consistent with
the decay length of local pressure gradients, $\ell_{\rm{in}}$, being
independent of the gap spacing $b$.

We also examine the relation between the pressure-gradient decay and the
viscosity ratio, $(\eta_{\rm{in}}/\eta_{\rm{out}})$. Figure 6b shows that data
from a higher viscosity ratio experiment collapses onto that shown in Fig. 6a
when the y-axis is further scaled by $(\eta_{\rm{in}}/\eta_{\rm{out}})$. This
re-scaling does not affect the x-axis; we conclude that the decay length is
independent of the viscosity ratio, although the maximum deformation size is
sensitive to it.

Figure 6:  Scaled collapse of inner fluid experiments with varied control
parameters. (a) Scaled curves from six experiments for
$(\eta_{\rm{in}}/\eta_{\rm{out}})=0.06$ and varied gap widths,
$305~{}\mathrm{\upmu m}\leq b\leq 610~{}\mathrm{\upmu m}$. (b) The y-axis is
scaled by $(\eta_{\rm{in}}/\eta_{\rm{out}})$ to collapse data from experiments
with different viscosity ratios. Here, $b=420~{}\upmu\rm{m}$.

We fit the scaled data from all these experiments to extract a form for the
characteristic decay length of local pressure gradients in the inner fluid,
given by:

$\ell_{\rm{in}}\approx(0.09\pm 0.01)(R_{\rm{int}}-R_{\rm{onset}}).$ (2)

We interpret this length as a penetration depth for local pressure gradients
extending behind the interface. It depends on the radial distance traveled by
the interface since the initial onset of the instability.

###  II.3 Local Pressure Gradient Decay in the Outer Fluid

We perform an analysis for the outer fluid by measuring the undulations on
dyed fluid rings in front of the interface. Similar to our treatment of the
inner fluid, the magnitude of an undulation, $\Delta r_{\rm{out}}$, is
determined by taking the difference between the radius of a dyed ring in front
of a finger, $r_{\rm{finger}}$, and in front of the adjacent valley,
$r_{\rm{valley}}$, as demonstrated in Fig. 3b. Undulations decay with the
distance from the interface, $-(R_{\rm{out}}-r_{\rm{finger}})$, now defined
with respect to the outer radius, $R_{\rm{out}}$. It is possible for the
curves to plateau to positive or negative values rather than zero due to long-
wavelength effects. To account for this, we apply a small vertical shift so
that all curves plateau to zero far from the interface.

Similar to the inner fluid, deformation curves are well characterized by an
exponential function of the form, $\Delta
r_{\rm{out}}=A\exp[(-(R_{\rm{out}}-r_{\rm{finger}})/\ell_{\rm{out}}]$. Curves
associated with undulations along three dyed rings and ahead of four different
fingers are fit to extract the characteristic decay length,
$|\ell_{\rm{out}}|$, shown in the inset of Fig. 7a. $|\ell_{\rm{out}}|$
increases in time for each sequential ring but, compared to the inner fluid,
shows greater variability between fingers along a single ring. This
variability is reduced by re-scaling the x-axis of the deformation curves. Our
best collapse of the data is shown in Fig. 7a where we scale by the local
diameter of a protruding finger, $d_{\rm{t}}$, as characterized by fitting a
circle to the tip of the finger. All undulations appear to converge to a
similar maximum amplitude without scaling the y-axis. Collapsed data from the
experiment is fit to an exponential form to extract the normalized decay
length $\ell_{\rm{out}}/d_{\rm{t}}$.

Figure 7:  Perturbation decay in the outer fluid. (a) Deformation curves from
three sequential dyed rings (color-coded as in inset) ahead of four different
fingers (represented by square, diamond, circle, and triangle) collapse when
the x-axis is scaled by the local finger tip diameter, $d_{\rm{t}}$. The
scaled master curve is fit to an exponential form. The inset shows individual
fits for $|\ell_{\rm{out}}|$ as a function of ring injection sequence before
scaling. Experimental parameters are: $b=420~{}\mathrm{\upmu m}$ and
$(\eta_{\rm{in}}/\eta_{\rm{out}})$ = 0.107. (b) The normalized decay length
$|\ell_{\rm{out}}|/d_{\rm{t}}$ from different experiments is plotted versus
gap width, $b$. Different color-symbol combinations represent different
viscosity ratios. Individual experimental values agree with the overall fit
(represented by the horizontal line).

We repeat this analysis by collecting deformation curves along multiple rings
and behind several fingers for experiments with varied gap widths and
viscosity ratios. Data from all experiments collapse best when we re-scale the
y-axis by dividing by the gap width, $b$, and multiplying by the viscosity
ratio, $(\eta_{\rm{in}}/\eta_{\rm{out}})$. However, we note that other scaling
parameters could not be ruled out within the noise of this data; for example,
the y-axis could have been re-scaled by the finger tip diameter, $d_{\rm{t}}$,
but this introduced slightly more noise to the data collapse. Also, scaling
the x-axis by $R_{\rm{int}}-R_{\rm{onset}}$ yielded a slightly worse collapse,
but one still plausible within the noise.

Figure 7b shows the magnitude of the normalized decay length,
$|\ell_{\rm{out}}|/d_{\rm{t}}$, versus the gap width, $b$, and the viscosity
ratio, $(\eta_{\rm{in}}/\eta_{\rm{out}})$, for individual experiments. The fit
from the collapse of all the data is marked by the horizontal line. The decay
length $|\ell_{\rm{out}}|/d_{\rm{t}}$ appears constant, and thus independent
of control parameters $b$ and $(\eta_{\rm{in}}/\eta_{\rm{out}})$. From the fit
and x-axis scaling relation, we can extract a functional form for the outer
fluid pressure-gradient penetration depth:

$\lvert\ell_{\rm{out}}\rvert\approx(0.16\pm 0.02)d_{\rm{t}}.$ (3)

These experiments and analyses have allowed us to determine and characterize a
new pressure-decay length scale associated with the late time growth of
patterns. It should be noted that the dynamics in the inner and outer fluids
are different from each other; the same non-dimensional x- and y-axes used in
the inner fluid do a poor job of collapsing the data in the outer fluid.
Likewise, the outer-fluid scaling relation gives a poor collapse for data in
the inner fluid. This might be explained by differences in flow structure
within the gap for the inner and outer fluids, as we discuss in the following
section.

##  III Discussion

Our experiments show that local pressure gradients decay exponentially from
the interface. A number of earlier studies have examined the non-linear growth
of viscous fingering patterns [19, 20, 21, 22, 23, 24, 25, 26, 11, 12, 27, 28,
29, 30, 6], but only a few have endeavored to analyze pressure and local
velocity gradients more explicitly [31, 32, 33]. We were unable to find a
direct comparison between the observations of this literature and the pressure
decay length that we have reported here. Nevertheless, a comparison can be
made between our findings and the analytical form of pressure perturbations
proposed in the linear stability analysis. Exponential perturbation decay is
often assumed in the rectilinear geometry [7, 13]; however, in the radial
geometry, a power-law form of $r$ is more common [9]. In either case, the
decay of perturbations is related to the most-unstable wavelength. We see this
in our outer fluid where the decay length is proportional to the diameter of a
finger, which is related to the wavelength. Notably, the decay in the inner
fluid does not show the same dependence.

###  III.1 Comparison to model

A direct comparison can be made between our data and the model for the local
pressure drop between two adjacent uncoupled fluid channels proposed by
Bischofberger et. al. [11]. Figure 8 illustrates two neighboring radial flow
channels – one with a finger protrusion whose interface radius is marked as
$R_{\rm{out}}$, and the other an adjacent valley with the interface marked as
$R_{\rm{in}}$. While the net pressure drop across both channels is identical,
a local pressure difference near the interface leads to a faster interface
velocity for the finger, $V_{\rm{out}}$, than for the valley, $V_{\rm{in}}$.
The model distinguishes the range over which these local pressure gradients
influence velocity by defining radial positions in the inner and outer fluids,
$r_{\rm{dp_{\rm{in}}}}$ and $r_{\rm{dp_{\rm{out}}}}$ respectively, as bounds
where the local pressures become equal in both channels. Flow is assumed to be
entirely radial and to obey Darcy’s law [Eq. 1] such that the interface
velocity is governed by the pressure drop in each channel. This yields an
expression for the ratio of the outer to inner interface velocity:

Figure 8:  Two independent radial flow channels corresponding to a finger and
adjacent valley. The interface is denoted by $R_{\rm{out}}$ in the finger and
by $R_{\rm{in}}$ in the adjacent valley. The net pressure drop across each
channel is the same, however, the local pressure gradients are different
around the interface: flow is identical in both channels up to
$r_{\rm{dp_{\rm{in}}}}$ and beyond $r_{\rm{dp}_{out}}$.

$\frac{V_{\rm{out}}}{V_{\rm{in}}}=\frac{R_{\rm{in}}}{R_{\rm{out}}}\bigg{[}\frac{(\eta_{\rm{in}}/\eta_{\rm{out}})\ln(\frac{R_{\rm{in}}}{r_{\rm{dp_{\rm{in}}}}})+\ln(\frac{r_{\rm{dp_{\rm{out}}}}}{R_{\rm{in}}})}{(\eta_{\rm{in}}/\eta_{\rm{out}})\ln(\frac{R_{\rm{out}}}{r_{\rm{dp_{\rm{in}}}}})+\ln(\frac{r_{\rm{dp_{\rm{out}}}}}{R_{\rm{out}}})}\bigg{]}.$
(4)

This model predicts the flow after the initial instability onset and has the
viscosity ratio, $(\eta_{\rm{in}}/\eta_{\rm{out}})$, appearing as a control
parameter. The parameters $r_{\rm{dp_{\rm{in}}}}$ and $r_{\rm{dp_{\rm{out}}}}$
were introduced as free parameters. Our experiments validate their existence
and provide a functional characterization for each.

We compare the model’s predictions with our experiments for a single finger
protrusion by analyzing our data in a manner analogous to the model. We
measure the interface velocities of a finger tip and its adjacent valley;
these are the gradients of $R_{\rm{out}}$ and $R_{\rm{in}}$ as a function of
time. Figure 9a shows the ratio of the velocities,
$(V_{\rm{out}}/V_{\rm{in}})$, versus time once the finger has reached a steady
growth rate, thus excluding a region at the beginning of pattern growth where
the velocity ratio rises rapidly.

Figure 9:  Comparison of the measured and predicted velocity ratio. (a) The
velocity ratio, $V_{\rm{in}}/V_{\rm{out}}$, for a single finger-valley pair is
shown over the course of an experiment and is compared to model predictions.
Model data (connected plus symbols) is shown to differ significantly from
experimental data (circles) unless modified to account for flow structure in
the gap (line with the shaded region). Values for the $\beta$ model assume
that the equal-pressure point behind and ahead of the interface occurs where
ring undulations decay to 90% of their maximum value; the bounds of the shaded
region correspond to 75% (upper) and 95% (lower) thresholds. Data is shown
for: $b=420~{}\upmu\rm{m}$, $(\eta_{\rm{in}}/\eta_{\rm{out}})$ = 0.067. (b)
The average plateau value for $V_{\rm{out}}/V_{\rm{in}}$ (such as that seen in
(a)) is plotted versus $(\eta_{\rm{in}}/\eta_{\rm{out}})$. The experiment and
model outputs are marked with the same symbols as (a). For the model, we plot
an average of outputs for 3-4 fingers with the bar denoting the range from
smallest to largest values. The averaged $\beta$ adjusted model output is
marked by a cross. Uncertainty due to finger variation and parameter
definitions are smaller than the symbol size for the two larger viscosity
ratios and are thus omitted. The error bars shown correspond to the bounds of
the shaded region in (a). All experiments were done with
$b=420~{}\mathrm{\upmu m}$.

To compare our experiments to Eq. 4, we determine $r_{\rm{dp_{\rm{out}}}}$ and
$r_{\rm{dp_{\rm{in}}}}$ with respect to $\ell_{\rm{out}}$ and $\ell_{\rm{in}}$
($i.e.$, the parameters at which local pressure gradients from the interface
have ceased to affect the flow) determined experimentally and given by Eqs. 2
and 3. Here we choose the length at which undulations have decayed to 90% of
their value at the interface, $2.3\ell$, to characterize the model parameters:

$r_{\rm{dp_{\rm{in}}}}\equiv R_{\rm{in}}-2.3\ell_{\rm{in}}$ (5)

and

$r_{\rm{dp_{\rm{out}}}}\equiv R_{\rm{out}}+2.3\lvert\ell_{\rm{out}}\rvert.$
(6)

Using these, and the other parameters determined experimentally, in Eq. 4
yields a predicted value for the velocity ratio shown by the connected plus
signs in Fig. 9a. The velocity ratio predicted by the model is almost double
the experimental value (circles).

To address the discrepancy between the model and experiment we apply a
correction that takes into account the fact that there are three stratified
layers of miscible fluid in the gap: a thin layer of outer fluid adjacent to
both the top and bottom plates, and a tongue of inner fluid in the center [34,
14]. Ahead of the interface, there is only a single layer; behind the
interface, the inner fluid tongue occupies only a fraction, $\beta$, of the
gap. We average across the gap, incorporating the stratified nature of the
fluids, to find an effective viscosity ratio:

$\left(\frac{\eta_{\rm{in}}}{\eta_{\rm{out}}}\right)^{*}=\frac{\eta_{\rm{in}}}{\eta_{\rm{out}}}\bigg{(}\frac{1}{(\eta_{\rm{in}}/\eta_{\rm{out}})+\beta^{3}(1-(\eta_{\rm{in}}/\eta_{\rm{out}}))}\bigg{)}.$
(7)

We measure $\beta$ experimentally as a function of viscosity ratio and use
this averaged viscosity ratio, $(\eta_{\rm{in}}/\eta_{\rm{out}})^{*}$, in Eq.
4. This yields the curve shown by the dark blue line in Fig. 9a, which we will
refer to as the $\beta$-adjusted model. The $\beta$-adjusted model corresponds
well with the average magnitude and trend of the data.

We note that the greatest source of uncertainty in the model prediction comes
from the factor used in the definition of model parameters
$r_{\rm{dp_{\rm{out}}}}$ and $r_{\rm{dp_{\rm{in}}}}$. The model defines a
finite length behind and in front of the interface over which local pressure
gradients decay, while in the experiment we observe a smooth exponential
decay. We chose to compare the model length scales with respect to the
experiment as the length at which undulations have decayed to 90% of their
maximum value. In Fig. 9a the shaded region bounded by dashed lines represents
the range of model predictions had we chosen this factor to correspond to a
decay range between 75% and 95%. The inclusion of this error does not change
the conclusions of our results. Including other sources of error (such as the
uncertainty in the measured value of $\beta$, $\ell_{in}$, or $\ell_{out}$)
does not significantly increase this spread.

This comparison is repeated for multiple fingers and for experiments at varied
viscosity ratios. At late times the velocity ratio plateaus to a nearly
constant value, as demonstrated in Fig. 9a. In Fig. 9b, we show the average
value of the plateau region in experiments for different control parameters
and find that the experimental velocity ratio (circles) decreases with
increasing viscosity ratio, $(\eta_{\rm{in}}/\eta_{\rm{out}})$. This trend is
also captured by the model.

The correction for the stratification of the flow is necessary to produce a
good quantitative fit to the data for all of the experiments. This
quantitative comparison also works for larger gap widths. However, at smaller
gap widths, the comparison deteriorates as azimuthal flows become more
prominent in the pattern formation, as evidenced by tip-splitting events and
channel merging. These effects will also emerge for the gap widths shown, but
at larger radii than observed here.

##  IV Conclusions

The experiments reported here have shown that the fluid velocities, and
therefore the pressure gradients, throughout a Hele-Shaw cell can be
determined by a technique in which the fluid is differentiated by alternate
injection of dyed fluid. Using this technique, we have identified a radial
decay length for the pressure, extending both in front of and behind the
propagating interface. Our characterizations of the local pressure-gradient
decay lengths in the inner and outer fluid, $\ell_{\rm{in}}$ and
$\ell_{\rm{out}}$, represent the best fits to our data. As indicated in the
Results section, other scaling choices, within the noise and fluctuations of
the data, could have been made but would not have significantly changed our
conclusions.

The existence of a length scale related to the decay of local pressure
gradients was proposed to relate late-time pattern formation to the control
parameter, $(\eta_{\rm{in}}/\eta_{\rm{out}})$ [11]. Despite the
simplifications made in constructing that model, our experiments show that it
captures the local dynamics of pattern formation and the dependence of the
velocity ratio on the viscosity ratio as a control parameter surprisingly
well. It is predictive of the magnitude of the velocity ratio when a
correction is made for the stratified flows behind the interface. The need for
such a correction shows that fluid-structure within the third dimension of the
gap is important to pattern development.

While the results presented here were for viscous fingering with miscible
fluids, the technique of using alternately dyed rings should be adaptable to
immiscible fluids (as well other systems such as colloidal or non-Newtonian
fluids [35, 36]). The stratified flow behind the interface in miscible fluids
led to important differences between how the inner and outer fluids were
treated. For example, we saw that the predictions of the model for the
velocity ratio did not correspond quantitatively with the data unless the
effect of stratification was included. Therefore, it is perhaps not surprising
that the decay lengths of the inner and outer fluids scale differently. In the
case of immiscible fluids, this difference in scaling might disappear so that
the correspondence between the model and experiment would be sufficient
without any corrections.

## Methods

###  IV.1 Apparatus

Experiments are performed in a radial Hele-Shaw cell apparatus consisting of
two flat, circular glass plates of either 1.9 cm or 1.7 cm thickness and a
diameter of 28 cm. The two plates are separated by a uniform gap thickness
using spacers which range in size from $305~{}\mathrm{\upmu
m}-610~{}\mathrm{\upmu m}$. Fluid is injected through a 1.6 mm hole in the
center of the top plate and propagates outward radially at a fixed volumetric
flow rate Q=2 ml/min. Pattern formation is recorded at a typical frame rate of
8 f.p.s. using a Prosilica GX3300 camera. A blue light filter is used to
enhance contrast.

The fluids used in these experiments are miscible mixtures of glycerol and
water. Tuning the concentration of glycerol to water allows us to vary the
fluid viscosity between 1-1100 cP. We measure fluid viscosity using an SVM
3001 viscometer. Each fluid exists in an optically transparent version and a
dyed version. Dyed fluid is colored using Brilliant Blue G dye from Alfa Aesar
in a concentration of 0.08 mg/100 ml.

###  IV.2 Alternate Fluid Injection Technique

In these experiments, we exploit the nature of the Hele-Shaw flow geometry to
isolate fluid velocity at the cell’s mid-plane without adding any tracer other
than molecular scale dye. Our technique allows us to track flow velocity
throughout the cell and thus indirectly determine local pressure gradients by
observing variations in the velocity field. The confinement of the Hele-Shaw
cell renders traditional flow-tracking techniques like particle image
velocimetry (PIV) ineffective.

We alternately inject a dyed and clear version of the same fluid continuously
into the cell using two syringe pumps (NE-1000 from New Era Pump Systems
Inc.). Each syringe pump is connected via a TTL input to an SFG-205 function
generator (Global Specialties) which powers the pumps on and off using a
square wave signal. A T-junction joins separate feed tubing for dyed and
undyed fluid at the inlet of the cell. We set a duty cycle of 70% so that
there is slightly more undyed than dyed fluid and vary the frequency of
injection from 100 mHz to 500 mHz in order to better resolve the dyed fluid
volumes. Small volumes of dyed fluid will propagate outward from the inlet as
rings of fixed volume but with an increasing circumference. Injecting many of
such dyed volumes allows us to track dyed fluid both spatially and temporally.

###  IV.3 Alternate Injection Concept: Measuring Flow Velocity at the Mid-
plane of the Cell’s Gap

The nature of Poiseuille flow in the cell’s gap allows us to relate the
velocity of the dyed fluid ring as measured from above the cell with the
velocity along the mid-plane of the cell’s gap.

Figure 10:  An illustration of a small volume of dyed fluid viewed between two
glass plates stretched by the steady-state Poiseuille flow profile. The black
arrows in the gap denote the velocity as a function of height. Viewed from the
side, the $r$ location of the highest gap averaged concentration corresponds
to the boundary between the back of the dyed ring and the tip of the incoming
undyed fluid, which lies in the middle of the gap. Here the aspect ratio is
highly exaggerated, as $b\ll r$. Viewed from above, the highest concentration
of dye appears around the back of a dyed ring.

Figure 10 shows a schematic illustration of the morphology of a small volume
of dyed fluid surrounded by clear fluid propagating through the gap between
two glass plates. The fluid is stretched into a parabola by the Poiseuille
flow due to no-slip boundary conditions with the plates. As viewed from above
the plates, the highest concentration of dye appears at the back of the dyed
fluid ring. As viewed within the gap, this region of highest concentrated dye
corresponds to where the dyed fluid is displaced by the tip of the trailing
undyed fluid, which is located at the mid-plane of the cell’s gap. Thus by
tracking where dye appears most intensely concentrated at the back of the
fluid ring, we can isolate flow velocity at the mid-plane of the cell’s gap.
We have verified qualitatively the assumption of Poiseuille flow within the
gap using confocal microscopy to image a fluorescent-dyed ring of fluid. We
also compare the squared radial trajectory of a single dyed fluid ring with
that theoretically predicted at the gaps mid-plane and find an agreement to
within 5%. We note that the assumption of the highest dye concentration
corresponding to fluid flow at the mid-plane of the gap is also true in the
case of stratified flow.

## Acknowledgements

We would like to thank Samar Alqatari, Severine Atis, Irmgard Bischofberger,
Rudro Biswas, Justin Burton, and Zhaoning Liu for insightful discussions. This
work was primarily supported by the University of Chicago Materials Research
Science and Engineering Center, NSF-MRSEC program under award NSF-DMR 2011854.
The US Department of Energy, Office of Science, Basic Energy Sciences, under
grant DE-SC0020972 provided support for TEV.

## References

  * Gollub and Langer [1999] J. P. Gollub and J. S. Langer, Rev. Mod. Phys. 71, S396 (1999).
  * Cross and Hohenberg [1993] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
  * Walgraef [2012] D. Walgraef,  _Spatio-temporal pattern formation: with examples from physics, chemistry, and materials science_ (Springer Science & Business Media, 2012).
  * Langer [1980] J. S. Langer, Rev. Mod. Phys. 52, 1 (1980).
  * Homsy [1987] G. M. Homsy, Annual Review of Fluid Mechanics 19, 271 (1987).
  * Bensimon  _et al._ [1986] D. Bensimon, L. P. Kadanoff, S. Liang, B. I. Shraiman, and C. Tang, Rev. Mod. Phys. 58, 977 (1986).
  * Couder [2000] Y. Couder, in  _Perspectives in Fluid Mechanics - A Collective Introduction to Current Research_ , edited by G. Batchelor, H. Moffat, and M. Worster (Cambridge University Press, Cambridge, 2000) pp. 53–104.
  * Paterson [1981] L. Paterson, Journal of Fluid Mechanics 113, 513 (1981).
  * Paterson [1985] L. Paterson, The Physics of Fluids 28, 26 (1985).
  * Chuoke  _et al._ [1959] R. Chuoke, P. Van Meurs, and C. van der Poel, Transactions of the AIME 216, 188 (1959).
  * Bischofberger  _et al._ [2014] I. Bischofberger, R. Ramachandran, and S. R. Nagel, Nat. Commun. 5, 5265 (2014).
  * Bischofberger  _et al._ [2015] I. Bischofberger, R. Ramachandran, and S. R. Nagel, Soft Matter 11, 7428 (2015).
  * Saffman and Taylor [1958] P. G. Saffman and G. I. Taylor, Proc. R. Soc. Lond. , 312–329 (1958).
  * Lajeunesse  _et al._ [1997] E. Lajeunesse, J. Martin, N. Rakotomalala, and D. Salin, Phys. Rev. Lett. 79, 5254 (1997).
  * Nagel and Gallaire [2013] M. Nagel and F. Gallaire, Physics of Fluids 25 (2013).
  * Kim  _et al._ [2009] H. Kim, T. Funada, D. D. Joseph, and G. Homsy, Physics of Fluids 21 (2009).
  * Dias and Miranda [2013] E. O. Dias and J. A. Miranda, Phys.Rev. E 88, 013016 (2013).
  * Videbæk and Nagel [2019] T. E. Videbæk and S. R. Nagel, Phys. Rev. Fluids 4, 033902 (2019).
  * Praud and Swinney [2005] O. Praud and H. L. Swinney, Phys. Rev. E 72, 011406 (2005).
  * Tan and Homsy [1988] C. T. Tan and G. M. Homsy, The Physics of Fluids 31, 1330 (1988).
  * Chen [1989] J.-D. Chen, Journal of Fluid Mechanics 201, 223–242 (1989).
  * Booth [2010] R. J. S. Booth, Journal of Fluid Mechanics 655, 527–539 (2010).
  * Malhotra  _et al._ [2015] S. Malhotra, M. M. Sharma, and E. R. Lehman, Physics of Fluids 27, 014105 (2015).
  * Zimmerman and Homsy [1992a] W. B. Zimmerman and G. M. Homsy, Physics of Fluids A: Fluid Dynamics 4, 1901 (1992a).
  * Jha  _et al._ [2011] B. Jha, L. Cueto-Felgueroso, and R. Juanes, Phys. Rev. Lett. 106, 194502 (2011).
  * Zimmerman and Homsy [1992b] W. B. Zimmerman and G. M. Homsy, Physics of Fluids A: Fluid Dynamics 4, 2348 (1992b).
  * Cheng  _et al._ [2008] X. Cheng, L. Xu, A. Patterson, H. M. Jaeger, and S. R. Nagel, Nature Physics 4, 234 (2008).
  * Zheng  _et al._ [2015] Z. Zheng, H. Kim, and H. A. Stone, Phys. Rev. Lett. 115, 174501 (2015).
  * Vaquero-Stainer  _et al._ [2019] C. Vaquero-Stainer, M. Heil, A. Juel, and D. Pihler-Puzović, Physical Review Fluids 4, 064002 (2019).
  * Oliveira and Meiburg [2011] R. M. Oliveira and E. Meiburg, Journal of Fluid Mechanics 687, 431–460 (2011).
  * Gadêlha and Miranda [2009] H. Gadêlha and J. A. Miranda, Phys. Rev. E 79, 066312 (2009).
  * Holzbecher [2009] E. Holzbecher, in  _COMSOL conference, Milan_ (2009).
  * Weitz  _et al._ [1987] D. A. Weitz, J. P. Stokes, R. C. Ball, and A. P. Kushnick, Phys. Rev. Lett. 59, 2967 (1987).
  * Videbæk [2020] T. E. Videbæk, Phys. Rev. Fluids 5, 123901 (2020).
  * Xu  _et al._ [2016] F. Xu, J. Kim, and S. Lee, Journal of Non-Newtonian Fluid Mechanics 238, 92 (2016), viscoplastic Fluids From Theory to Application 2015 (VPF6).
  * Lindner  _et al._ [2002] A. Lindner, D. Bonn, E. C. Poiré, M. B. Amar, and J. Meunier, Journal of Fluid Mechanics 469, 237–256 (2002).