# Elastic fingering in a rotating Hele-Shaw cell

Benjamin Foster<EMAIL_ADDRESS>Physics Department, University of
California at Berkeley, Berkeley, California 94720, USA  Edgar Knobloch
<EMAIL_ADDRESS>Physics Department, University of California at
Berkeley, Berkeley, California 94720, USA

###### Abstract

We consider the steady-state fingering instability of an elastic membrane
separating two fluids of different density under external pressure in a
rotating Hele-Shaw cell. Both inextensible and highly extensible membranes are
considered, and the role of membrane tension is detailed in each case. Both
systems exhibit a centrifugally-driven Rayleigh-Taylor–like instability when
the density of the inner fluid exceeds that of the outer one, and this
instability competes with the restoring forces arising from curvature and
tension, thereby setting the finger scale. Numerical continuation is used to
compute not only strongly nonlinear primary finger states up to the point of
self-contact but also secondary branches of mixed modes and circumferentially
localized folds as a function of the rotation rate and the externally imposed
pressure. Both reflection-symmetric and symmetry-broken chiral states are
computed. The results are presented in the form of bifurcation diagrams. The
ratio of system scale to the natural length scale is found to determine the
ordering of the primary bifurcations from the unperturbed circle state as well
as the solution profiles and onset of secondary bifurcations.

Keywords: Rayleigh-Taylor instability; nonlinear elastica; wrinkling;
bifurcation

##  I Introduction

The Saffman-Taylor instability, also known as the viscous fingering
instability, occurs at the interface of two fluids in a Hele-Shaw cell when a
lower viscosity fluid is injected into a more viscous fluid, leading to a
dynamic process of finger-like pattern formation at the fluid-fluid interface
[1, 2, 3, 4, 5]. In contrast, when a higher (or equal) viscosity fluid invades
a lower viscosity fluid, the interface is stable and forms a uniformly
spreading front. Many variations of this instability have been constructed
such as those which destabilize conventionally stable Hele-Shaw flows through
the implementation of variable geometry of the Hele-Shaw cell [6, 7, 8], the
introduction of surfactants [9, 10, 11, 12, 13, 14, 15], or the presence of
$A+B\to C$–type chemical reactions at the fluid interface [16, 17, 18]. One
variation of interest is the introduction of a global rotation of the Hele-
Shaw cell about a perpendicular axis through the center at a prescribed
frequency [19, 20, 21, 22, 23]. If the density of the inner fluid exceeds that
of the outer, the system is susceptible to a centrifugal instability
resembling the well-known Rayleigh-Taylor instablity of superposed fluids as
the denser fluid in the center is now unstable to outward displacement. In
general the two fluids may have different viscosities such that the
instability is mediated by competing centrifugal and viscous effects, or the
viscosities may be taken to be the same so that only inertial effects drive
the instability. More recently, this instability has been considered in
systems where the interface has additional properties such as a curvature-
dependent bending modulus due to a chemical reaction between the two fluids or
constrained length [24, 25, 20, 26, 27]. If the interface is taken to have
nonzero bending modulus, the length scale of the emerging fingering patterns
is set by the competition between the bending of the elastica and the
centrifugal driving [28]. The fingering structures which emerge can take the
form of a periodic finger pattern commonly referred to as wrinkles, or
isolated single or multiple localized structure(s) called folds. These states
represent steady states of the system characterized by force and torque
balance. Similar pattern formation occurs in many other systems beyond the
Hele-Shaw geometry, for example, during the dynamic buckling of a membrane
bounding a popped soap film [29], the dynamic wrinkling of a sheet due to drop
impact [30], or the dynamic buckling of pressurized circular rings [31].
Similar pattern formation may also occur quasistatically, for example, in the
buckling of a ring due to a geometrically simple confinement [32], compression
or contact-induced wrinkling and folding of a floating elastic sheet [33, 34,
35, 36, 37], three-dimensional deformations of water droplets under rotation
[38], the wrinkling and puckering of supported growing elastic struts [39,
40], or the folding in biological systems such as airways [41, 42], ocular
surfaces [43, 44] or arteries [28, 27].

In this article, we examine the deformation of an elastic interface between
two fluids in a rotating Hele-Shaw cell. This system has been studied in a
series of papers by Carvalho et al [25, 26, 20] on the assumption that the
pressures in the interior and exterior fluids are identical and that the
interface cannot support tension. The second assumption is consequential: it
implies there is no force that resists changes in the interface length. As a
result, Carvalho et al report a series of steady state profiles with different
interface lengths without organizing these states into a bifurcation scenario
that describes how the steady states of the system vary with system
parameters.

We adopt here a different approach. We include an interface tension comprised
of a curvature-dependent term and a Lagrange multiplier $T$ required by the
assumed inextensibility of the interface, and refer to $T$, for simplicity, as
the tension. This quantity is determined by solving a nonlinear eigenvalue
problem and quantifies the response of the system to changes in the system
parameters, at fixed interface length. It is therefore possible to plot the
solutions in a $(P,T)$ plane, measuring the response of the system (the
tension $T$) to changes in the pressure difference $P$ between the inner and
outer fluids. If $P$ is held fixed and the rotation rate $\Omega$ is varied,
one may instead show the $(\Omega,T)$ plane. Such diagrams are examples of
bifurcation diagrams and they allow one to track changes in the solution
profile as a parameter is varied since the profile is determined in the
process of solving the eigenvalue problem for $T$. Numerical continuation
techniques are ideally suited to this purpose and we use them here. In the
case where $T\equiv 0$, i.e. the length is unconstrained, we use the length
$L$ of the interface as a new parameter and track how $L$ changes as a
function of $P$ or, equivalently, of the rotation rate $\Omega$.

The steady states of the rotating two-fluid Hele-Shaw system are described by
a model equation derived from force and torque balance using the simplest
nonlinear bending energy possible for an interface with constant bending
modulus. In previous work, we studied an equivalent equation describing the
wrinkling of an elastic lining of an artery under compression and provided an
explanation via weakly nonlinear analysis and numerical continuation for how
primary and secondary solutions emerge and the forms they take [27]. We also
noted that the finger profiles of this system map onto the buckled states of
an elastic ring under pure compression, albeit at different locations in
parameter space [45]. Since the latter problem is integrable in terms of
elliptic functions, this is also the case for the present problem [46, 47, 48,
45]. It is not, however, the case for the secondary states comprising mixed
modes and spatially localized folds whose properties we also investigate. The
net result is a rather complete picture of the steady states of this
interesting system.

This paper is organized as follows. In Sec. II we formulate the problem.
Section III introduces the length-constrained problem and summarizes the
linear stability properties of a circular interface. Section IV summarizes
parallel results for the case $T=0$. Section V describes the formulation of
our numerical continuation approach, and the numerical tests carried out to
validate it. This is followed in Secs. VI–VIII by the results tracking the
properties of fingers, mixed modes and folds as a function of the parameters
in the length-constrained case, followed by a summary of our results in the
unconstrained case in Secs. IX–XI. Finally, in Sec. XII we consider asymmetric
or chiral states and investigate their origin in parameter space. The paper
concludes with a brief summary in Sec. XIII.

##  II The system

We consider a Hele-Shaw cell with a fluid of density $\rho_{i}$ surrounded by
a fluid of density $\rho_{o}$ and separated from it by a closed, elastic
membrane of length $L=2\pi R$ and bending modulus $\mathcal{B}$, assumed to be
independent of the curvature (Figure 1). The system rotates with constant
angular velocity $\Omega$ about an axis perpendicular to the cell at
$\textbf{r}=0$. In equilibrium, the inner liquid occupies a circular region
with the axis of rotation at its center. We are interested in understanding
the properties of this equilibrium state as either the rotation rate increases
for a given pressure difference $P$ between the fluids, or as $P$ varies for a
given rotation rate $\Omega$.

We parametrize the perturbed interface with the arclength $s$ such that the
curvature of the interface at location $s$ or equivalently at
$\textbf{r}(s)=(x(s),y(s))$ relative to the origin is given by
$\kappa=\partial_{s}\phi$, where $\phi(s)$ is the angle between the tangent to
the interface at point $s$ and the $x$ axis (Fig. 1). We define the density
difference $\Delta\rho\equiv\rho_{i}-\rho_{o}$ between the interior and
exterior fluids and impose a pressure difference $P\equiv P_{i}-P_{o}$ between
them. A tension $T$ in the elastica is required to maintain its
inextensibility. We call the resulting problem the constrained length problem.
In contrast, when $T$ is set to zero the length of the interface is
unconstrained, and we call the resulting problem the unconstrained length
problem. Both forms of the problem exhibit steady-state solutions, and it is
solely these solutions we consider in this paper.

Figure 1: Top view of a rotating Hele-Shaw cell containing two fluids of
densities $\rho_{i}$ and $\rho_{o}$ separated by an elastic membrane wrinkled
with wave number $m=5$. The interface is parametrized by arclength $s$, with
$(x(s),y(s))$ providing a parametric representation of the interface profile.

##  III Constrained Length

With the system thus defined, we can derive an equation for the interface by
balancing the normal forces and the torque on an element of length $ds$ of the
interface. The normal force includes contributions from the elastica bending
modulus, tension and centrifugal force together with the force from the
imposed pressure difference. The resulting Kirchhoff equations for the
interfacial elastica are given in the Supplementary Material [49], and can be
manipulated to yield the governing equation

$\mathcal{B}\left(\partial_{s}^{2}\kappa+\frac{1}{2}\kappa^{3}\right)-T\kappa-P-\frac{1}{2}\Delta\rho\,\Omega^{2}r^{2}=0\,.$
(1)

Here $r(s)$ is defined implicitly via the geometrical constraints
$\partial_{s}x=\cos\phi$ and $\partial_{s}y=\sin\phi$. In the unconstrained
case the resulting equation is identical to that used in previous work on
centrifugally driven instabilities in a Hele-Shaw cell [50, 20, 26, 21].

We expect a centrifugal instability to set in when $\Delta\rho>0$, i.e., when
the density of the inner fluid exceeds that of the outer one, and use this
fact to introduce the natural length scale of the instability

$\lambda\equiv\left(\frac{\mathcal{B}}{\Delta\rho\,\Omega^{2}}\right)^{\frac{1}{5}}\,.$
(2)

We use this scale to construct a dimensionless parameter $\ell\equiv
R/\lambda$ that measures the radius $R$ of the elastica in units of the
natural length $\lambda$.

Rescaling Eq. (1) according to $s\sim R$, $\kappa\sim R^{-1}$, $r\sim R$,
$T\sim\mathcal{B}/R^{2}$, $P\sim\mathcal{B}/R^{3}$, we obtain

$\partial_{s}^{3}\phi+\frac{1}{2}\left(\partial_{s}\phi\right)^{3}-T\partial_{s}\phi-P-\frac{1}{2}\ell^{5}r^{2}=0\,.$
(3)

Instability of the circular membrane arises when the denser interior fluid is
displaced outwards, thereby increasing the outward force upon it. When this
force exceeds the restraint arising from the curvature of the interface, the
tension $T$, and the imposed pressure difference $P$, instability sets in and
it is this balance which gives rise to the wavelength selection.

In the unperturbed problem, the interface is circular and of length $L=2\pi$,
with $T=\frac{1}{2}(1-\ell^{5})-P$. A linear stability analysis of this state
yields the dispersion relation for the mode number $m$ of the fingering
instability [27]

$m^{4}-\left(2+P+\frac{\ell^{5}}{2}\right)m^{2}+\left(1+P+\frac{3}{2}\ell^{5}\right)=0.$
(4)

From this relation, we can determine the wave number $m^{*}$ of the first
unstable mode that sets in as $P$ increases and the critical value $P=P^{*}$
at which it does so:

$\displaystyle m^{*}$ $\displaystyle=(1+\ell^{5/2})^{1/2}\,,\qquad
P^{*}\equiv(-\ell^{5}+4\ell^{5/2})/2\,,$ (5)

all for a fixed rotation rate $\Omega$. Alternatively, we may fix $P$ and
increase the rotation rate leading to the critical rotation rate
$\Omega(m,P)$,

$\displaystyle\Omega$ $\displaystyle\equiv\left[\frac{B}{\Delta\rho
R^{5}}\left(\frac{2(m^{2}-1)(P-m^{2}+1)}{3-m^{2}}\right)\right]^{1/2}\,,$ (6)

for the appearance of a mode with wave number $m$; minimizing this expression
over $m$ for fixed $P$ recovers the results in Eq. (5). Figure 2 shows several
examples of the marginally stable wave number $m$ as $P$ and $\ell^{5}$ vary
and shows that, as $\ell^{5}$ increases, instability sets in at lower and
lower values of $P$ and with larger and larger values of $m^{*}$, i.e., larger
finger wave numbers. This is the fingering instability whose nonlinear
development is key to understanding the constrained system. When $m^{*}$ is
not an integer, the figure shows that the primary instability corresponds to
the integer $m$ nearest to the $m^{*}$ given by Eq. (5).

Figure 2: The perturbation wave number $m$ as a function of $P$ across a range
of $\ell^{5}$ values corresponding to onset wave numbers $m^{*}=2,3,4$ (red
dots). As $\ell^{5}$ increases, instability sets in at lower values of $P$ and
with larger values of $m^{*}$, i.e., larger finger wave numbers. Since
$L=2\pi$ the $\ell^{5}$ values have been chosen to yield integer wave numbers
at onset.

##  IV Unconstrained Length

When the interface cannot support any tension, the instability may lead to
interface growth. In this case $T=0$ and the length $L$ of the interface
becomes a free parameter. We are interested in this case because earlier work
by Carvalho et al. [26] presented a number of solutions to Eq. (3) with $T=0$
with interfaces of different (and unspecified) lengths. With $L$ as a free
parameter, nondimensionalization requires rescaling Eq. (1) using the natural
length scale $\lambda$. We take
$s\sim\lambda,\kappa\sim\lambda^{-1},r\sim\lambda,P\sim\mathcal{B}/\lambda^{3}$
such that the rescaled governing equation is now given by

$\partial_{s}^{2}\kappa+\frac{1}{2}\kappa^{3}-P-\frac{1}{2}r^{2}=0.$ (7)

In the case of a circular solution, this yields the (monotonic) relation
between the pressure and interface length: $P=((L/2\pi)^{-3}-(L/2\pi)^{2})/2$,
so in equilibrium there is only one $L$ for a given $P$.

A linear stability analysis of this state, similar to that leading to (4),
yields

$m^{4}-\frac{5}{2}m^{2}+\left[\frac{3}{2}+\left(\frac{L}{2\pi}\right)^{4}\right]=0\,,$
(8)

after dividing out the marginally stable mode $m=0$. This relation has no real
roots for $L\geq\pi$, indicating that solutions of wave number $m$ do not
emerge as primary bifurcations from the circle state as $P$ varies.

##  V Method

Both versions of the problem, Eqs. (3) and (7), are implemented in AUTO [51]
as a 5-dimensional boundary value problem encompassing the third order ODE for
$\phi$ and the two first order ODEs for $(x,y)$. For the constrained length
sections of this paper, we construct the problem on the domain $s\in[0,\pi]$,
representing half of a closed elastic interface of length $2\pi$, subject to
the boundary conditions

$\displaystyle\phi(0)=\pi/2\,,\quad\phi(\pi)=3\pi/2\,,$ (9a) $\displaystyle
x(0)=x_{0}\,,\quad x(\pi)=x_{1}\,,$ (9b) $\displaystyle y(0)=y(\pi)=0\,,$ (9c)
$\displaystyle\partial_{s}^{2}\phi(0)=\partial_{s}^{2}\phi(\pi)=0\,.$ (9d)

The final two conditions represent force-free conditions following previous
work [52, 27] and allow us to generate the full circle solution via reflection
in the $x$ axis. For the unconstrained length sections, $s\in[0,L/2]$ where
$L$ is free to vary. Since the system is 5-dimensional with 8 boundary
conditions, numerical continuation is performed in the four parameters
$(P,T,x_{0},x_{1})$ for the constrained length problem and $(P,L,x_{0},x_{1})$
for the unconstrained length problem [53], i.e., for a given change in $P$,
the new tension $T$ is found as a nonlinear eigenvalue of the problem, while
$x_{0},x_{1}$ are adjusted to satisfy the force-free boundary conditions.

The results of this procedure have been carefully compared to those from a
high order weakly nonlinear analysis and are found to be in excellent
agreement [27]. The nonlinear results have also been confirmed to match exact
analytical solutions to within numerical tolerance [45]. Self-contact forces
can be included in order to continue solutions beyond our current range [54,
55], but we do not do so in this paper. Self-intersecting solutions are not
shown.

The boundary conditions (9) impose a reflection symmetry on all the solutions
generated by the above procedure and so prevent the computation of states that
break this symmetry. Since such states are also expected to be present [26],
we discuss in the penultimate section, Sec. XII, an alternative formulation of
the above problem that permits the computation of such solutions.

##  VI Constrained length: fingers

The order with which primary finger states bifurcate from the circle state is
determined by the parameter $\ell^{5}$ as it sets the critical wave number
$m^{*}$ via Eq. (5). Subsequent bifurcations as $P$ increases lead to modes
with wave numbers $m$ alternately above and below $m^{*}$. Once $m=2$ is
reached only modes with large wave numbers $m$ remain and these continue to be
destabilized as $P$ increases as seen in Fig. 3. The figure shows the response
of the system, as indicated by the tension $T$, to changes in the imposed
pressure difference $P$ for three values of $\ell^{5}$ obtained using
numerical continuation starting from the neutral modes of the circle state
(black line). The corresponding solution profiles at the point of first self-
contact are shown alongside. We see that for large enough $\ell^{5}$ the
primary finger states bifurcate to secondary branches of mixed-mode states
connecting a primary branch with $m<m^{*}$ to a primary branch with $m>m^{*}$
(Fig. 3(b), shown in green), as well as to circumferentially localized states
we call folds (Fig. 3(a,b), shown in yellow) which do not connect to another
branch. The mixed modes do not set in prior to self-contact when $\ell^{5}$ is
small but begin to proliferate with increasing $\ell^{5}$; for this reason
they are omitted from panel (c). The primary mode with $m=2$ is called here a
buckling mode (labeled $B$) because of its radically different behavior at
large $\ell^{5}$, with negative modulus $dP/dT$, behavior typical of buckling
processes [27]. In the absence of a natural wavelength $\lambda$ or as
$\lambda$ becomes large relative to the domain, the $m=2$ mode is the first to
become unstable [54].

Figure 3: Bifurcation diagrams in the $(T,P)$ plane for (a) $\ell^{5}=64$
($m^{*}=3$), (b) $\ell^{5}=576$ ($m^{*}=5$) with some states beyond self-
contact shown with dashed lines, and (c) $\ell^{5}=14400$ ($m^{*}=11$),
showing the succession of color-coded primary branches of finger states as $P$
increases: primary fingers with $m=m^{*}$ (purple), $m>m^{*}$ (blue), and
$m<m^{*}$ (red). Mixed-mode states are shown in (a) and (b) (green), together
with branches of folds (yellow). In (c) only the primary finger modes are
shown. Primary bifurcation points are labeled by the corresponding wave
number, while points of self-contact are indicated with crosses (for $m\leq 7$
in panel (b)); in panel (c) points of self-contact fall outside the parameter
range shown. Selected solution profiles at the point of self-contact are shown
on the right.

Single folds bifurcate from the first primary branch ($m=m^{*}$) and do so
prior to self-contact whenever $m^{*}\geq 3$. Figure 3(a) shows the case
$m^{*}=3$ ($\ell^{5}=64$) and shows that these states emerge in pairs, here a
single protrusion fold $F_{s^{+}}$ and a single intrusion fold $F_{s^{-}}$
(for profiles see Figs. 7 and 8). The resulting branches track closely but do
not self-contact at the same point; mixed-mode branches are absent. In panel
(b), for $\ell^{5}=576$, fingering, buckling and mixed modes are all present,
as well as additional folds. In (c), for $\ell^{5}=14400$, the number of mixed
modes and folds becomes large, and these states are omitted.

The finger profiles are $\ell^{5}$-independent [56, 47, 48, 45], so the
displayed self-contact solutions hold for any of the bifurcation diagrams,
although their location in the $(T,P)$ plane does vary with $\ell^{5}$ [45].

###  VI.1 Constrained length: $P=0$ fingers

Finger solutions when the applied pressure $P=0$ are of physical interest as
they dictate solutions which may be observed in ambient conditions, emerging
solely from the competition between inertial and elastic energies. Of course,
these solutions have the same shape as before, but the length and tension at
which they arise are not known a priori. Figure 4 shows their location in the
$(T,L)$ plane. Solutions bifurcate with $m$ increasing monotonically from
$m^{*}=2$ as $L$ increases. We note that the size of the marginally stable
circle state may be smaller than in Fig. 3 for finger solutions with $m$
values close to $m^{*}$ since these typically bifurcate from the circle state
already at more negative $P$, i.e. for these states $L<2\pi$. Moreover, for
the value of $\ell^{5}$ used in Fig. 4, $\ell^{5}=576$, the states with
$m=2,3$ reach self-contact at $L<2\pi$ and at negative values of $T$, i.e.,
under compression. In contrast, for larger values of $m$, the circle at
threshold is generally larger than $L=2\pi$ and self-contact generally occurs
at small or slightly positive values of the tension $T$.

Figure 4: The $P=0$ steady state solutions in the $(T,L)$ plane for
$\ell^{5}=576$ ($m^{*}=5$ in purple), where $L$ is the interface perimeter.
All profiles are plotted on the same scale. The horizontal dashed line
indicates $L=2\pi$, the length used in all other constrained problem
computations.

##  VII Constrained length: Mixed modes

In Figure 5, we show an example of the richness of the mixed-mode connections
at higher values of $\ell^{5}$, here $\ell^{5}=6400$ ($m^{*}=9$), where more
states with $m<m^{*}$ become closely interspersed with states with $m>m^{*}$.
The profiles above the diagram provide examples of mixed-mode states with
comparable contributions from both wave numbers (black dots). In contrast, the
profiles along the side correspond to profiles along a single mixed-mode
branch, here the branch M3,19 connecting the $m=3$ and $m=19$ primary finger
branches.

Figure 5: A partial bifurcation diagram for $\ell^{5}=6400$ ($m^{*}=9$) with
all complete mixed-mode connections between the $m=3$ finger state and other
primary branches. The profiles above the bifurcation diagram correspond to the
black points proceeding from left to right, while the profiles on the right
correspond to the blue points on the mixed-mode branch M3,19, proceeding
downwards.  Figure 6: A partial bifurcation diagram for $\ell^{5}=14400$
($m^{*}=11$) with four mixed-mode connections between $m=23$ and the primary
branches $m=2\text{ (B)},3,4,5$. Solutions beyond the point of self-contact
are shown with broken lines. The lower panels show profiles at the locations
indicated by black dots (two on the M23,2 branch).

In Fig. 6, we show another example of mixed-mode connections. The figure shows
the connections along a primary branch with $m=23$, i.e., $m>m^{*}$, as
opposed to Figure 5 where we showed all the secondary connections for a $m=3$
primary branch, $m<m^{*}$. The solutions in Figs. 5 and 6 match closely those
previously identified via a different computational procedure (see Figure 4 of
Ref. [26]).

##  VIII Constrained length: Folds

Additional secondary branches which do not connect to any other branches
bifurcate from the finger branches at or near the critical branch with
$m=m^{*}$ (Fig. 3(a,b), yellow curves). These correspond to localized
solutions and typically come in pairs. The first pair always has a single
intrusion or protrusion ($\mathrm{F}{}_{s^{\pm}}$) and bifurcates from the
critical finger branch (Fig. 3, yellow curves). Intruding and protruding
states follow the same path in the bifurcation diagram and reach self-contact
at almost the same point in the $(T,P)$ plane (Fig. 3(a)).

Figure 7 explores the $\ell^{5}$-dependence of the simplest fold states
F${}_{s^{+}}$ at self-contact and shows that the width of the localized
structure decreases in the expected way relative to the perimeter length as
$\ell\equiv R/\lambda$ or equivalently the rotation rate increases.

Figure 7: $F_{s^{+}}$ solutions at self-contact for increasing values of
$\ell^{5}$ corresponding to $m^{*}=3,4,5,6,9,11$.

Changes in $\ell^{5}$ also have an effect on the localized state branches in
parameter space. Figure 8(a) shows that for moderate values of $\ell^{5}$ the
branches of $\mathrm{F}{}_{s^{\pm}}$ fold on themselves, each exhibiting a
saddle-node bifurcation. These bifurcation points can be followed numerically
and Fig. 8(b) shows the result of such a computation for
$\mathrm{F}{}_{s^{+}}$ starting from the saddle-node bifurcation at
$\ell^{5}=225$ and both increasing and decreasing $\ell^{5}$. Evidently, as
$\ell^{5}$ varies, so does the wave number $m^{*}$ of the branch from which
the F${}_{s^{+}}$ bifurcate. This is in fact a continuous process: as
$\ell^{5}$ increases, for example, the secondary bifurcation to F${}_{s^{+}}$
moves down along the $m^{*}=4$ branch to the primary bifurcation point. As
this happens, the $m=5$ primary bifurcation passes through the $m^{*}=4$
primary bifurcation point, so that for larger $\ell^{5}$ the circle state
loses stability first to $m^{*}=5$, followed by $m=4$, i.e., the $m=4$ and
$m=5$ branches exchange positions. Beyond this point the secondary bifurcation
to F${}_{s^{+}}$ moves up the new $m^{*}$ branch, and the whole process
repeats [37, 57]. However, despite the jumps in $m^{*}$ the movement of the
saddle-node of the fold states F${}_{s^{+}}$ as $\ell^{5}$ varies is
continuous. With increasing $\ell^{5}$, the saddle-node moves past the point
of self-contact, and we terminate the continuation when this first happens
($\ell^{5}\approx 570$).

Figure 8: Numerical continuation in the $(T,P)$ plane of the maxima along the
branch of F${}_{s^{+}}$ localized fold states starting at $\ell^{5}=64$
($m^{*}=3$) as $\ell^{5}$ increases as indicated in the labels. Since $T$
measures the response of the system to changes in the pressure difference $P$,
these maxima represent saddle-node bifurcations. The continuation is
terminated when the saddle-node reaches the point of self-contact (crosses) or
the bifurcation point from the primary branch (squares). The corresponding
solution profiles at the labeled saddle-nodes are shown alongside.  Figure 9:
Bifurcation diagram for $\ell^{5}=6400$ depicting the first few finger modes
($m^{*}=9$, and $m=10,8,11$, color-coded) and the multitude of remarkable fold
states that bifurcate from them (yellow). Solutions at the point of self-
contact, arranged in ascending order and labeled with dots of the
corresponding color, are shown at the top. Branches of intrusions and
protrusions are almost degenerate leading to overlapping dots.

As $\ell^{5}$ increases, the variety of localized fold states increases
dramatically since fold states now bifurcate not only from the critical finger
state with $m=m^{*}$, but also from the subsequent primary finger branches. In
Fig. 9, we show all localized fold states for a given $\ell^{5}$
($\ell^{5}=6400$). The critical primary branch supports the most secondary
bifurcations to fold solutions. Although subsequent branches have a similar
overall number of secondary bifurcations, the number of bifurcations to fold
states decreases while the number of bifurcations to mixed-mode states
increases. The details depend on the number of factors of each integer $m$ and
hence the symmetry of the branch. For example, Fig. 9 shows several localized
states with a $\frac{2\pi}{3}$ rotational symmetry emerging from the $m=9$
finger branch which shares this symmetry. Similarly, we see a $\frac{2\pi}{5}$
rotational symmetry in one of the localized states emerging from $m=10$ and a
$\frac{\pi}{2}$ rotational symmetry in one that emerges from $m=8$.

##  IX Unconstrained length: Fingers

We now consider the same system but this time take $T$ to be zero and the
interface length $L$ to be a free parameter, resulting in Eq. (7). Numerical
continuation for steady-state symmetric finger states may be performed in the
same way as before. Figure 10 shows the result of continuing several finger
states in the $(L,P)$ plane. We observe that as the interior pressure drops,
the area and perimeter of the steady-state finger profile grow dramatically.
With a smaller contribution from internal pressure, a larger contribution from
the centrifugal force is needed to support a steady state, requiring larger
$L$. In contrast to the results with nonzero tension, in this case the finger
states no longer bifurcate from the circle state. Moreover, although secondary
bifurcations do set in as $L$ increases, for these small values of $m$ they do
so far beyond the point of first self-contact. Figure 11 shows that this is no
longer so for larger values of $m$.

Figure 10: Numerical continuation of $m=2,3,4,5$ finger states (orange, pink,
dark and light blue, respectively) in the $(L,P)$ plane. The branch of circle
states is shown in black, and self-contact is indicated by crosses. The finger
states no longer bifurcate from the circle state.

##  X Unconstrained length: Mixed modes

Mixed-mode states connecting two primary branches with different numbers of
fingers can also be found. Figure 11 shows three mixed-mode branches (light
green lines) originating on the $m=2$ finger branch (grey line), shown in the
$(P,L)$ plane. The branches terminate on the $m=12$ (red), $m=13$ (dark green)
and $m=14$ (blue) finger states, respectively. In each case the solutions are
realizable near the high $m$ end of the branch (solid light green lines) but
as $P$ becomes more and more negative they make self-contact beyond which the
solutions are no longer realizable. The top panel in the figure shows the pure
finger state at the right end of each branch (lowest profile), a physical
solution before self-contact (second profile from the bottom), the solution at
first self-contact (third profile from bottom) and finally an unphysical,
self-intersecting profile very close to the termination of the mixed-mode
branch on the $m=2$ branch (top profile).

Figure 11: Bottom: mixed-mode connections (light green) between $m=2$ (grey)
and $m=12$ (red), $m=13$ (dark green) and $m=14$ (blue) finger states shown in
the $(P,L)$ plane. Dashed lines indicate solutions beyond first self-contact.
Top: sample solutions at the color-coded locations indicated in the bottom
panel proceeding from the $m=2$ start of the mixed-mode branch to its endpoint
at the other end. In each case the top profile corresponds to an unphysical,
self-intersecting solution on the $m=2$ branch while the profile below
corresponds to last self-contact (lower panel zoom). Beyond this point the
solutions are realizable; the final profile corresponds to a pure finger state
at the right end of the branch.

##  XI Unconstrained length: Folds

Fold states arise through bifurcations from the primary finger states,
although this time almost exclusively from states far beyond self-contact.
Figure 12 shows a number of examples in the $(P,L)$ plane.

Figure 12: Bifurcation diagram depicting the $m=2,3,4,5,6$ finger states in
thick lines (red, orange, green, blue, and purple, respectively), and the many
fold states bifurcating from them (thin lines, corresponding color). Self-
contact for primary states is marked with a cross, and states beyond this
point are dashed. Sample solutions (all rescaled to equal size for clarity)
are arranged in ascending order according to the number of localized
intrusions and protrusions on the $2\pi$ domain (profile labels correspond to
branch labels in the bifurcation diagram). The number of distinct states
created in each bifurcation grows in successive bifurcations as $P$ decreases
($L$ increases) and depends on the wave number $m$ of the underlying finger
state. At large negative $P$ (large $L$) branches with the same number of
intrusions and protrusions are almost degenerate since both contribute only
slightly to the length, despite originating in general from different primary
finger states as indicated by the color-coded profiles at the top panels. For
example, there are four branches labeled 2, two of which originate from the
$m=2$ finger branch (red) and two of which originate from the $m=3$ finger
branch (orange). The branches originating in the same branch are strictly
degenerate in this projection, although they reach self-contact at slighly
different locations (not shown). Profiles originating in subsequent primary
bifurcations are arranged vertically and correspond to $P\approx-20$.

We see that each primary finger branch (thick solid lines) yields a multitude
of secondary branches of fold states with varying numbers of intrusions and
protrusions, collectively referred to as folds. The first fold state to emerge
from a primary finger state with wave number $m$ generates a state with $m-1$
folds, and each subsequent secondary bifurcation adds one additional fold.
These folds can be intruding, protruding, or come in antisymmetric pairs (due
to the symmetry constraint imposed by our boundary conditions). There is
always a special pair of intruding and protruding fold states which respect
the symmetry of the finger state they bifurcate from.

As $P$ becomes more negative and the length of the domain increases, the
profiles gradually deform from the self-intersecting, nonphysical states they
bifurcate from into non-self-intersecting, physically realizable states.
Beyond the point of last self-contact, the profiles remain physical even as
their length continues to increase. Simultaneously, the details of the shape
of the folds become less important, and the branches collapse to evenly spaced
curves, each corresponding to a particular number of folds, as indicated on
the left of the bifurcation diagram. Each of these asymptotic lines consists
of multiple branches originating from different primary states. For example,
there are four distinct 2-fold states, two of which are symmetric states with
either two intrusions or two protrusions originating from the second primary
bifurcation on the $m=2$ finger branch, and two additional states, one with
one intrusion and one protrusion, and one with a pair of antisymmetric folds,
originating from the first primary bifurcation on the $m=3$ finger branch. In
the present projection, these pairs of branches lie on top of one another and
one therefore sees only two branches, labeled 2. Likewise there are three
branches labeled 3, originating in the third bifurcation on the $m=2$ branch,
the second bifurcation on the $m=3$ branch and the first bifurcation on the
$m=4$ branch, each with appropriate multiplicity (see profiles in row labeled
3). As in the case of the fold states for the constrained length (Fig. 9),
many fold states respect the symmetry of the primary branch they bifurcate
from. For example, the second pair of states bifurcating from $m=3$ features a
solution with three intruding folds and a solution with three protruding
folds, while $m=6$ bifurcates to a state with three pairs of antisymmetric
folds and a solution with six alternating intrusions and protrusions with
overall symmetry under rotations through $2\pi/3$. The different states of
alternating symmetric intrusions and protrusions as well as antisymmetric
folds resemble the folded states of a floating planar elastic sheet under
compression [37], particularly when $L$ is large 111The profiles in Fig. 4 of
[37] are labeled in the opposite order to the branches in Fig. 3..

##  XII Asymmetric states

In the unconstrained case, many interesting steady-state finger profiles have
been reported in the literature [26], but the continuation scheme summarized
in Eqs. (9) rules out many of them since it can only generate solutions that
are symmetric about the $x$ axis. However, using an explicit Runge-Kutta
method and numerical shooting, we are able to construct a variety of full-
domain solutions which do not obey the previous symmetry restriction imposed
by our continuation procedure. Figure 13 shows examples that are similar in
nature to those discovered in previous work [26]. In the top row, we show
mixed-mode solutions with no reflection symmetry, while the bottom row shows
chiral versions of the localized states shown in Fig. 9. Owing to the symmetry
of Eq. (3) under reflection $(\phi,s)\to-(\phi,s)$, each left-handed solution
is accompanied by an identical but right-handed solution.

Figure 13: Asymmetric states computed on the full domain. Top left:
$L=15.714$, $\ell^{5}=8232.56$; top right: $L=15.714$, $\ell^{5}=6257.5$;
bottom left: $L=6.283$, $\ell^{5}=650$; bottom center: $L=25.649$,
$\ell^{5}=0.916$; bottom right: $L=15.707$, $\ell^{5}=295$, all computed for
$P=0$.

To determine the origin of these states in parameter space we implemented a
full-circle extension of our numerical continuation in AUTO using periodic
boundary conditions for
$(\partial_{s}\phi,\partial_{s}^{2}\phi,\partial_{s}^{3}\phi,x,y)$ with
Dirichlet boundary conditions for $\phi$ at $s=0,2\pi$ to pin the phase on the
full domain. The continuation of the first few primary and secondary branches
of symmetric and asymmetric states is shown in Fig. 14.

Figure 14: Full-domain numerical continuation for $\ell^{5}=576$ in the
$(T,P)$ plane showing the first three symmetric and asymmetric fold states
emerging simultaneously from secondary bifurcations of the $m=5$ and $m=6$
symmetric finger states (blue lines) which in turn bifurcate from the circle
states (grey line). The branches of symmetric and asymmetric states coincide
in this projection. Color-coded solution profiles corresponding to the
locations indicated in the top panel are shown along the bottom.

Figure 14 shows that asymmetric states appear via secondary bifurcations from
the symmmetric finger states, a result consistent with the fact that all
primary states in O(2)-symmetric steady state bifurcation problems are
necessarily reflection-symmetric [59]. It has previously been observed that
the branches of symmetric intruding and protruding folds emerging from a
common bifurcation point coincide in the $(T,P)$ plane [27]. Figure 14 shows
that in fact all secondary branches (intruding, protruding, symmetric, or
chiral) emerging from a common bifurcation point coincide in the $(T,P)$ plane
although they do not reach self-contact at the same location. We believe, but
have been unable to check, that the multifold chiral states shown in Fig. 13
likewise originate in secondary bifurcations from symmetric finger states.

##  XIII Conclusion

We have studied the bifurcation properties of equilibrium states arising from
a Rayleigh-Taylor-like instability of a higher density fluid confined within a
lower density fluid in a rotating Hele-Shaw cell. The interface separating the
two fluids is modeled as a thin elastic membrane, with the centrifugal force
playing the role of effective outward gravity. The bending modulus of the
interface introduces an intrinsic length scale into the problem that defines
the scale of the resulting fingering instability. We examined two cases, one
in which the interface was taken to be inextensible (the constrained case) and
one in which the interface was permitted to proliferate freely (the
unconstrained case). In both cases, we used numerical continuation to follow
strongly nonlinear equilibrium finger states in parameter space through to the
point of self-contact. We showed that, depending on parameters, these states
may undergo secondary bifurcations leading to two types of secondary states,
mixed modes and folds. The former form secondary connections between fingers
with distinct wave numbers, while the latter form progressively more localized
intrusions or protrusions as one follows each fold branch away from the
secondary bifurcation that generates it.

The unconstrained or tension-free case is of particular interest. Here the
primary finger states were found to be disconnected from the circle states, in
contrast to the constrained case. However, as in the constrained case, the
finger states exhibit instabilities to both mixed-mode states and to fold
states. The latter take the form of symmetric intrusions or protrusions or
antisymmetric folds that come in pairs to maintain the overall reflection
symmetry imposed by our numerical continuation scheme. In particular, we
demonstrated that the first bifurcation of a finger state with wave number $m$
generates a fold state with $m-1$ folds of various types. States with the same
number of folds but different orientation or shape are created in bifurcations
from other finger states with different but smaller wave number but these
bifurcations are necessarily subsequent bifurcations and not the first. For
example, the first bifurcation of the $m=3$ finger states creates a state with
two folds, and so does the second bifurcation of the $m=2$ finger state (Fig.
12). The folds localize away from these bifurcations as $P$ becomes more and
more negative and the interface length grows, resulting in asymptotic
degeneracy of all branches with the same number of folds, regardless of type
and origin. This is a consequence of the fact that in this regime the folds
take up an increasingly small fraction of the overall length $L$. The
unconstrained system thus recapitulates similar behavior found earlier in a
floating elastic sheet under compression [34, 37], a system described by an
equation similar to Eq. (3).

Moreover, the finger states can be either symmetric $m$-finger states, or
break this symmetry, forming an $m$-finger chiral state of definite
handedness. However, these chiral states appear via secondary bifurcations
from an already existing finger state, at the same bifurcation as the
symmetric folds, and cannot form in a primary bifurcation of the circle state.
Together these results shed light on the origin and organization of the states
reported previously in the unconstrained case with zero pressure difference
across the interface [26].

Similar progressive localization of folds takes place in the constrained case
as well, as the parameter $\ell$ or equivalently the rotation rate increases
(Fig. 7). In both cases this is a consequence of the fact that the folds
bifurcate subcritically, much as in the Swift-Hohenberg equation with a
bistable nonlinearity [60].

This work was supported in part by the National Science Foundation under grant
DMS-$1908891$. We thank L. Gordillo and N. Verschueren for valuable
discussions.

## References

## References

  * Hill [1952] S. Hill, Channeling in packed columns, Chemical Engineering Science 1, 247 (1952).
  * Taylor and Saffman [1958] G. I. Taylor and P. G. Saffman, The penetration of a fluid into a porous medium or hele-shaw cell containing a more viscous liquid, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences (1958).
  * Paterson [1981] L. Paterson, Radial fingering in a Hele Shaw cell, Journal of Fluid Mechanics 113, 513 (1981).
  * Nye  _et al._ [1984] J. F. Nye, H. W. Lean, and A. N. Wright, Interfaces and falling drops in a Hele-Shaw cell, European Journal of Physics 5, 73 (1984).
  * Chen [1987] J. D. Chen, Radial viscous fingering patterns in Hele-Shaw cells, Experiments in Fluids 5, 363 (1987).
  * Zhao  _et al._ [1992] H. Zhao, J. Casademunt, C. Yeung, and J. V. Maher, Perturbing Hele-Shaw flow with a small gap gradient, Physical Review A 45, 2455 (1992).
  * Al-Housseiny  _et al._ [2012] T. T. Al-Housseiny, P. A. Tsai, and H. A. Stone, Control of interfacial instabilities using flow geometry, Nature Physics 8, 747 (2012).
  * Al-Housseiny and Stone [2013] T. T. Al-Housseiny and H. A. Stone, Controlling viscous fingering in tapered Hele-Shaw cells, Physics of Fluids 25, 092102 (2013).
  * Guo  _et al._ [1992] H. Guo, D. C. Hong, and D. A. Kurtze, Surface-tension-driven nonlinear instability in viscous fingers, Physical Review Letters 69, 1520 (1992).
  * Guo  _et al._ [1995] H. Guo, D. C. Hong, and D. A. Kurtze, Dynamics of viscous fingers and threshold instability, Physical Review E 51, 4469 (1995).
  * Chan and Liang [1997] C. K. Chan and N. Y. Liang, Observations of Surfactant Driven Instability in a Hele-Shaw Cell, Physical Review Letters 79, 4381 (1997).
  * Chan [2000] C. K. Chan, Surfactant wetting layer driven instability in a Hele-Shaw cell, Physica A 288, 315 (2000).
  * Krechetnikov and Homsy [2004] R. Krechetnikov and G. M. Homsy, On a new surfactant-driven fingering phenomenon in a Hele-Shaw cell, Journal of Fluid Mechanics 509, 103 (2004).
  * Fernandez  _et al._ [2005] J. Fernandez, R. Krechetnikov, and G. M. Homsy, Experimental study of a surfactant-driven fingering phenomenon in a Hele-Shaw cell, Journal of Fluid Mechanics 527, 197 (2005).
  * Rocha and Miranda [2013] F. M. Rocha and J. A. Miranda, Manipulation of the Saffman-Taylor instability: A curvature-dependent surface tension approach, Physical Review E 87, 013017 (2013).
  * Almarcha  _et al._ [2010] C. Almarcha, P. M. J. Trevelyan, P. Grosfils, and A. De Wit, Chemically Driven Hydrodynamic Instabilities, Physical Review Letters 104, 044501 (2010).
  * Mishra  _et al._ [2010] M. Mishra, P. M. J. Trevelyan, C. Almarcha, and A. De Wit, Influence of Double Diffusive Effects on Miscible Viscous Fingering, Physical Review Letters 105, 204501 (2010).
  * Riolfo  _et al._ [2012] L. A. Riolfo, Y. Nagatsu, S. Iwata, R. Maes, P. M. J. Trevelyan, and A. De Wit, Experimental evidence of reaction-driven miscible viscous fingering, Physical Review E 85, 015304 (2012).
  * Carrillo  _et al._ [1996a] L. Carrillo, F. X. Magdaleno, J. Casademunt, and J. Ortín, Experiments in a rotating Hele-Shaw cell, Physical Review E 54, 6260 (1996a).
  * Carvalho  _et al._ [2014a] G. D. Carvalho, H. Gadêlha, and J. A. Miranda, Elastic fingering in rotating Hele-Shaw flows, Physical Review E 89, 053019 (2014a).
  * Álvarez Lacalle  _et al._ [2004] E. Álvarez Lacalle, J. Ortín, and J. Casademunt, Nonlinear Saffman-Taylor Instability, Physical Review Letters 92, 054501 (2004).
  * Schwartz [1989] L. W. Schwartz, Instability and fingering in a rotating Hele-Shaw cell or porous medium, Physics of Fluids A: Fluid Dynamics 1, 167 (1989).
  * Alvarez-Lacalle  _et al._ [2004] E. Alvarez-Lacalle, J. Ortín, and J. Casademunt, Low viscosity contrast fingering in a rotating Hele-Shaw cell, Physics of Fluids 16, 908 (2004).
  * He  _et al._ [2012] A. He, J. Lowengrub, and A. Belmonte, Modeling an Elastic Fingering Instability in a Reactive Hele-Shaw Flow, SIAM Journal on Applied Mathematics 72, 842 (2012).
  * Carvalho  _et al._ [2013] G. D. Carvalho, J. A. Miranda, and H. Gadêlha, Interfacial elastic fingering in Hele-Shaw cells: A weakly nonlinear study, Physical Review E 88, 053006 (2013).
  * Carvalho  _et al._ [2014b] G. D. Carvalho, H. Gadêlha, and J. A. Miranda, Stationary patterns in centrifugally driven interfacial elastic fingering, Physical Review E 90, 063009 (2014b).
  * Foster  _et al._ [2022a] B. Foster, N. Verschueren, E. Knobloch, and L. Gordillo, Pressure-driven wrinkling of soft inner-lined tubes, New Journal of Physics 24, 013026 (2022a).
  * Pocivavsek  _et al._ [2008] L. Pocivavsek, R. Dellsy, A. Kern, S. Johnson, B. Lin, K. Y. C. Lee, and E. Cerda, Stress and fold localization in thin elastic membranes, Science 320, 912 (2008).
  * Box  _et al._ [2020] F. Box, O. Kodio, D. O’Kiely, V. Cantelli, A. Goriely, and D. Vella, Dynamic Buckling of an Elastic Ring in a Soap Film, Physical Review Letters 124, 198003 (2020).
  * Box  _et al._ [2019] F. Box, D. O’Kiely, O. Kodio, M. Inizan, A. A. Castrejón-Pita, and D. Vella, Dynamics of wrinkling in ultrathin elastic sheets, Proceedings of the National Academy of Sciences 116, 20875 (2019).
  * Katifori  _et al._ [2009] E. Katifori, S. Alben, and D. R. Nelson, Collapse and folding of pressurized rings in two dimensions, Physical Review E 79, 056604 (2009).
  * Hazel and Mullin [2017] A. L. Hazel and T. Mullin, On the buckling of elastic rings by external confinement, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160227 (2017).
  * Roman and Bico [2010] B. Roman and J. Bico, Elasto-capillarity: deforming an elastic structure with a liquid droplet, J. Phys.: Condens. Matter 22, 493101 (2010).
  * Diamant and Witten [2011] H. Diamant and T. A. Witten, Compression Induced Folding of a Sheet: An Integrable System, Physical Review Letters 107, 164302 (2011).
  * King  _et al._ [2012] H. King, R. D. Schroll, B. Davidovitch, and N. Menon, Elastic sheet on a liquid drop reveals wrinkling and crumpling as distinct symmetry-breaking instabilities, Proceedings of the National Academy of Sciences 109, 9716 (2012).
  * Rivetti [2013] M. Rivetti, Non-symmetric localized fold of a floating sheet, Comptes Rendus Mécanique 341, 333 (2013).
  * Gordillo and Knobloch [2019] L. Gordillo and E. Knobloch, Fluid-supported elastic sheet under compression: Multifold solutions, Physical Review E 99, 043001 (2019).
  * Brown and Scriven [1980] R. A. Brown and L. E. Scriven, The shape and stability of rotating liquid drops, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, 27 (1980).
  * Michaels  _et al._ [2019] T. C. T. Michaels, R. Kusters, A. J. Dear, C. Storm, J. C. Weaver, and L. Mahadevan, Geometric localization in supported elastic struts, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, 20190370 (2019).
  * Michaels  _et al._ [2021] T. C. T. Michaels, R. Kusters, and L. Mahadevan, Puckering and wrinkling in a growing composite ring, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, 20200999 (2021).
  * Heil and White [2002] M. Heil and J. P. White, Airway closure: surface-tension-driven non-axisymmetric instabilities of liquid-lined elastic rings, Journal of Fluid Mechanics 462, 79 (2002).
  * Moulton and Goriely [2011] D. E. Moulton and A. Goriely, Circumferential buckling instability of a growing cylindrical tube, Journal of the Mechanics and Physics of Solids 59, 525 (2011).
  * Pournaras  _et al._ [2000] C. J. Pournaras, G. Donati, P. D. Brazitikos, A. D. Kapetanios, D. L. Dereklis, and N. T. Stangos, Macular epiretinal membranes, Seminars in Ophthalmology 15, 100 (2000).
  * Neff [2013] K. D. Neff, 20 - Conjunctivochalasis, in  _Ocular Surface Disease: Cornea, Conjunctiva and Tear Film_, edited by E. J. Holland, M. J. Mannis, and W. B. Lee (W.B. Saunders, London, 2013) pp. 161–166.
  * Foster  _et al._ [2022b] B. Foster, N. Verschueren, E. Knobloch, and L. Gordillo, Universal Wrinkling of Supported Elastic Rings, Phys. Rev. Lett. 129, 164301 (2022b).
  * Arreaga  _et al._ [2002] G. Arreaga, R. Capovilla, C. Chryssomalakos, and J. Guven, Area-constrained planar elastica, Phys. Rev. E 65, 031801 (2002).
  * Djondjorov  _et al._ [2011] P. A. Djondjorov, V. M. Vassilev, and I. M. Mladenov, Analytic description and explicit parametrisation of the equilibrium shapes of elastic rings and tubes under uniform hydrostatic pressure, International Journal of Mechanical Sciences 53, 355 (2011).
  * Djondjorov  _et al._ [2012] P. Djondjorov, V. Vassilev, M. Hadzhilazova, and I. Mladenov, Analytic description of the viscous fingering interface in a rotating Hele-Shaw cell (Avangard Prima, Sofia, 2012) pp. 107–114.
  * [49] See supplementary material at …, .
  * Carrillo  _et al._ [1996b] L. Carrillo, F. X. Magdaleno, J. Casademunt, and J. Ortín, Experiments in a rotating Hele-Shaw cell, Physical Review E 54, 6260 (1996b).
  * Doedel  _et al._ [2008] E. J. Doedel, A. R. Champneys, F. Dercole, T. Fairgrieve, Y. Kuznetsov, B. Oldeman, R. Paffenroth, B. Sandstede, X. Wang, and C. Zhang,  _AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations_ (2008).
  * Kodio  _et al._ [2020] O. Kodio, A. Goriely, and D. Vella, Dynamic buckling of an inextensible elastic ring: Linear and nonlinear analyses, Physical Review E 101, 053002 (2020).
  * Doedel  _et al._ [1991] E. J. Doedel, H. B. Keller, and J. P. Kernevez, Numerical analysis and control of bifurcation problems (II) Bifurcations in infinite dimensions, International Journal of Bifurcation and Chaos 1, 745 (1991).
  * Flaherty  _et al._ [1972] J. E. Flaherty, J. B. Keller, and S. I. Rubinow, Post buckling behavior of elastic tubes and rings with opposite sides in contact, SIAM Journal on Applied Mathematics 23, 446 (1972).
  * Flaherty and Keller [1973] J. E. Flaherty and J. B. Keller, Contact problems involving a buckled elastica, SIAM Journal on Applied Mathematics 24, 215 (1973).
  * Vassilev  _et al._ [2008] V. M. Vassilev, P. A. Djondjorov, and I. M. Mladenov, Cylindrical equilibrium shapes of fluid membranes, J. Phys. A: Mathematical and Theoretical 41, 435201 (2008).
  * Dangelmayr [1986] G. Dangelmayr, Steady-state mode interactions in the presence of o(2) symmetry, Dynamics and Stability of Systems 1, 159 (1986).
  * Note [1] The profiles in Fig. 4 of [37] are labeled in the opposite order to the branches in Fig. 3.
  * Crawford and Knobloch [1991] J. D. Crawford and E. Knobloch, Symmetry and symmetry-breaking bifurcations in fluid dynamics, Annual Review of Fluid Mechanics 23, 341 (1991).
  * Knobloch [2015] E. Knobloch, Spatial localization in dissipative systems, Annual Review of Condensed Matter Physics 6, 325 (2015).


# The novel XYU-GEM to resolve ambiguities

K.J. Flöthner 11footnotetext: Corresponding author. F. Brunbauer  S. Ferry  F.
Garcia  D. Janssens  B. Ketzer  M. Lisowska  H. Muller  R. de Oliveira  E.
Oliveri  G. Orlandini  D. Pfeiffer  L. Ropelewski  J. Samarati  F. Sauli  L.
Scharenberg  M. van Stenis  A. Utrobicic  R. Veenhof

###### Abstract

Removing ambiguities within a single stage becomes crucial when one can not
use multiple detectors behind each other to resolve them which naturally is
the case for neutral radiation. An example would be RICH detectors. Commonly
pixilated readout is choosen for this purpose. However, this causes a
remarkable increase in quantity of channels and does not scale up well.
Therefore, the XYU-GEM was proposed as a three coordinate strip-readout which
is combined with a triple GEM detector. The readout complements a common XY
readout with an additional projection which is tilted by 45°. The
overdetermination due to three projections can be used to resovle ambiguities.
Following the detector design will be explained, first measurements discussed
to understand the response of the detector and a way to change the charge
sharing without changing the manufacturing parameters of the readout.

##  1 Introduction

Within limits, ambiguities can be resolved by using electronics which give
access to better time resolution or additional information about signal
amplitude. However, solutions can be found as well at detector level by
specific designs [1][2]. The proposed XYU readout would be such a solution
which does not depend on the used electronics. A cross-section and microscope
picture of the proposed XYU readout can be seen in fig. 1.

Figure 1: Left: Schematic cross section. Right: Microscope picture of the
three strip layers.

How the three coordinates can be used to resolve ambiguities is shown in fig.
2. As depicted on the left it is not possible to resolve ambiguities with a
binary readout with only using two projections. But obvious with the third
coordinate. On the right resolving most of the ambiguities by simple
correlation is already possible. Still present ghosts might be resolved by
more complex combinatory algorithm, e.g. looping over the multiple
possibilities and requiring to use up all signals. A multi-hit event like
illustrated in 2 would be impossible to reconstruct with two projections,
while one would be able resolve this using the third projection. This has to
be verified with actual multi-hit events, caused by showers, from test beam
data.

Figure 2: Left: Schematic of ambiguities. Right: Illustrated Multi-hit
example.

##  2 Detector Design and Setup

From simulations a smaller charge collection for U is predicted for the
current parameters which are limited by manufacturing constrains [3]. In
principle it would be desired to have an equal sharing to avoid the need of
operating the detector at higher gains to achieve full efficiency with all
projections. These simulations were based on the Ramo-Shockley theorem [4][5]
and can provide the total induced charge on the different strip layers. The
picture in fig. 1 shows all three conductive layers of the readout which are
exposed to the gas volume. On top visible the X strips, by 90° tilted the Y
strips and by another 45° the U strips. The manufacturing process was
developed by the CERN Micro Pattern Technology (MPT) group [6]. The process
involves the glueing of two copper-cladded polyimide foils, hosting the
different strip-layers. To reveal projections, three dielectric etchings need
to be performed. The result can be seen in fig. 1 as it was stated above. The
detector itself is based on the standard COMPASS-like triple-GEM detector [7].
The gas mixture Ar/CO2 70/30 was used and an passive voltage divider (R in
M$\Omega$: 1/0.55/1/0.5/1/0.44/1). For the readout electronics the VMM3a ASIC
with the RD51 Scalable Readout System (SRS) was used [8].

##  3 General response of the Detector

The response of the XYU detector was investigated with 55Fe in the laboratory
and with Minimum Ionizing Particles (MIPs) during the RD51 October test-beam
at the SPS H4 beam-line. Due to the lower energy deposition from MIPs,
compared to 55Fe, the gain for the test-beam is increased by a factor of
three222The gain during laboratory measurements was ~10k.. As shown by the ADC
counts, fig. 3, the distribution looks similar for all three coordinates.
Altough one can observe a shift to lower adc channel for the U coordinate,
which indicates the lower charge collection. Extracting the position of the
photopeak and the MPV, the charge sharing can be determined. This leads to
41/38/21 (X/Y/U)333A minor change of sharing with radiation can be observed as
well caused by charging up of the insulator [3]. for the 55Fe and 39/36/25 for
the beam measurements, which matches with the simulations [3].

Figure 3: Left: ADC spectra for 55Fe. Right: ADC spectra for 150 GeV/c muons

##  4 Biasing Strips and Charge Sharing Variation

Applying a voltage on the strips can be used to compensate the unequal charge
collection. As shown in fig. 4 one can see a clear shift of the spectrum
collected from the X strips while applying different voltages on the U strips.
Therefore one could achieve equal sharing without changing the manufacturing
parameters.

Figure 4: Spectra from the X strips for different bias voltages on U strips.

##  5 Summary and Outlook

The first prototype of the XYU-GEM could operated and it is possible to
collect reasonable data using an 55Fe source and 150 GeV/c muons. The observed
characteristics are as expected. It is shown, that one can influence the
sharing of the signal by applying a bias voltage on the strips. However,
further investigations are needed to see the effect simultaneously on all
projections. Using the beam data the detector response with actual multi-hit
events, caused by showers, is under investigation. As well as the possible
influence of the third coordinate on the resolution. Another aspect worth to
investigate would be a different pitch of the U projection. This can be done
merging channels and therefore changing the signal-to-noise-ratio on
individual strips which could be compared to simulations.

## Acknowledgments

This work has been sponsored by the Wolfgang Gentner Programme of the German
Federal Ministry of Education and Research (grant no. 13E18CHA). The work has
been performed in the context of the CERN Strategic Programme on Technologies
for Future Experiments. https://ep-rnd.web.cern.ch/

## References

  * [1] F. Sauli, _Novel cherenkov photon detectors_ , _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 553 (2005) 18.
  * [2] I. Laktineh, _A new readout scheme for rpc and other gaseous detectors_ , in _ILCX2021_ , 2021.
  * [3] K.J. Floethner, D. Janssens, F. Sauli, F. Brunbauer, S. Ferry, B. Ketzer et al., _The xyu-gem: Ambiguity-free coordinate readout of the gas electron multiplier_ , _Submitted to NIM-A, RD51-NOTE-2022-09_ .
  * [4] W. Shockley, _Currents to conductors induced by a moving point charge_ , _Journal of Applied Physics_ 9 (1938) 635.
  * [5] S. Ramo, _Currents induced by electron motion_ , _Proceedings of the IRE_ 27 (1939) 584.
  * [6] R.D. Oliveira, _Xyv read out board for gem detectors_ , 2022, https://indico.cern.ch/event/1110129/contributions/4720923/.
  * [7] C. Altunbas, M. Capéans, K. Dehmelt, J. Ehlers, J. Friedrich, I. Konorov et al., _Construction, test and commissioning of the triple-gem tracking detector for compass_ , _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 490 (2002) 177.
  * [8] M. Lupberger, L. Bartels, F. Brunbauer, M. Guth, S. Martoiu, H. Müller et al., _Implementation of the vmm asic in the scalable readout system_ , _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 903 (2018) 91.