and $\psi_{i}(U_{i})$ is either an open subset of $\mathbb{C}^{n}$ or
$\psi_{i}(U_{i})$ is of the form $\\{z\in U:f(z)\leq 0\\},\leavevmode\nobreak\
U\subset\mathbb{C}^{n}$ for $f$ a $C^{\infty}$-submersion.

######  Remark 7.3 (Complex manifolds with an abstract boundary, Section 2 in
[57]).

One says that a given smooth manifold with boundary ${\overline{M}}^{2n}$
carries an almost complex structure $J$ which is _integrable up to the
boundary_ , if there exists vector bundle $\mathcal{H}\subset TM^{\mathbb{C}}$
(the complexified tangent bundle of $M$) over $M$ with $n$-dimension complex
fibers such that $\mathcal{H}_{x}\cap\overline{\mathcal{H}_{x}}=\\{0\\}$ for
each $x\in M$ and for each two-sections $P,\leavevmode\nobreak\
Q\in\Gamma(\mathcal{H})$ it holds that $[P,Q]\in\Gamma(\mathcal{H})$. A
complex manifold with such a structure is termed _complex manifold with an
abstract boundary_.

Inherited by the notion of complex manifolds with boundary (Definition 7.2),
there exist $n$ complex-valued vector fields $\\{P_{1},\ldots,P_{n}\\}$ in
$M$, given in holomorphic coordinate charts by

$P_{k}=\frac{1}{2}\left(\frac{\partial}{\partial
x_{k}}+\sqrt{-1}\frac{\partial}{\partial y_{k}}\right),\leavevmode\nobreak\
\quad k\in\\{1,\ldots,n\\}$

such that

  1. (a)

$\\{P_{k}\\}\cup\\{\overline{P_{k}}\\}$ are linearly independent over
$\mathbb{C}$

  2. (b)

for each $i,j\in\\{1,\ldots,n\\}$ there exist smooth complex-valued functions
$\\{c_{i}j^{k}\\}$ such that $[P_{i},P_{j}]=\sum_{i,j=1}^{n}c_{ij}^{k}P_{k}$.

Thus, complex manifolds with a boundary can always be seen as complex
manifolds with an abstract boundary. We could ask whether the structure of a
complex manifold with an abstract boundary extends to that of a complex
manifold with a boundary. This holds whenever $\overline{M}^{2n}$ is regarded
with an integrable, almost complex structure that is real analytic up to the
boundary; see Theorem 5 in [57]. ∎

######  Definition 7.4 ((Almost) CR structure, Section 1 in [96]).

Let $\overline{M}^{2n}$ be a smooth manifold with smooth boundary
$\partial\overline{M}$. An _almost CR structure_ on $\partial\overline{M}$ is
a $2n-2$-dimension subbundle $E$ of the tangent bundle $T\partial\overline{M}$
with an automorphism $J\in\Gamma\mathrm{Hom}(E)$555i.e., it is a section of
the bundle of automorphisms $\mathrm{Hom}(E)$ of $E$ such that
$J^{2}=-\mathrm{Id}$.

Consider the complexified bundle
$E\otimes_{\mathbb{R}}\underline{\mathbb{C}},\leavevmode\nobreak\
\underline{\mathbb{C}}:=\overline{M}\times\mathbb{C}$, and extend $J$ to be
$\mathbb{C}$-linear in the usual manner. Let
$E^{\prime}:=\mathrm{ker}\leavevmode\nobreak\
(J-\sqrt{-1}\mathrm{Id}),\leavevmode\nobreak\
E^{\prime\prime}:=\mathrm{ker}\leavevmode\nobreak\ (J+\sqrt{-1}\mathrm{Id})$.

######  Definition 7.5.

An almost CR structure $J$ on $\partial\overline{M}$ is said to be a _CR
structure_ if $[E^{\prime\prime},E^{\prime\prime}]\subset E^{\prime\prime}$.

######  Definition 7.6 (CR isomorphism).

Let $\partial{\overline{M}}_{1},\leavevmode\nobreak\
\partial{\overline{M}}_{2}$ to closed manifolds bounding $2n$-complex
manifolds with boundary ${\overline{M}}_{1},\leavevmode\nobreak\
{\overline{M}}_{2}$. We say that there exists a CR-isomorphism between
$\partial{\overline{M}}_{1},\leavevmode\nobreak\ \partial{\overline{M}}_{2}$
if there exists a smooth map between the representing CR bundles
$E_{1},\leavevmode\nobreak\ E_{2}$,
$f:\partial{\overline{M}}_{1}\rightarrow\partial{\overline{M}}_{2}$ such

  1. (a)

$\mathrm{d}fE_{1}\subset E_{2}$

  2. (b)

$\mathrm{d}f\circ J_{1}=J_{2}\circ\mathrm{d}f$.

######  Lemma 7.7 (p. 228 in [96]).

Let $\overline{M}^{2n}$ be a compact complex manifold with a boundary. Then,
the boundary $\partial\overline{M}$ admits a CR structure induced from
$\overline{M}$.

###### Sketch of the proof.

In a local $C^{\infty}$-boundary coordinate chart, one can take a suitable
linear combination of $P_{1},\ldots,P_{n}$ (such as that given in Remark 7.3)
given by $\\{\widetilde{P}_{k}\\}_{k=1,\ldots,n}$ with
$\\{\widetilde{P}_{k}\\}_{\\{k=1,\ldots,n-1\\}}$ tanget to $\partial M$ at
points $x\in\partial M$. Taking the brackets
$\left[\widetilde{P}_{i}\Big{|}_{\partial
M},\widetilde{P}_{j}\Big{|}_{\partial M}\right]$, the standard Frobenius
theorem ensures the existence of the required $E$. The distinguished section
$J_{E}$ is obtained by restricting $J$ to $E$ and can be determined computing
$J$ in the elements $\left\\{\widetilde{P}_{k}\Big{|}_{\partial M}\right\\}$.
∎

Let $X_{0}:=\mathrm{D}^{4}\times\mathrm{S}^{3}\times\mathrm{S}^{1}$. We claim
to be possible to regard $X_{0}$ as a complex manifold, thus constituting a
complex manifold with a boundary with an induced CR structure in
$\mathrm{S}^{3}\times\mathrm{S}^{1}$. The first step is to observe that there
exists a unique (up to diffeomorphism) smooth structure of manifold with
boundary in $X_{0}$ whose boundary is a smooth realization of $\partial
X_{0}:=\mathrm{S}^{3}\times\mathrm{S}^{3}\times\mathrm{S}^{1}$.

######  Lemma 7.8.

$\mathrm{D}^{4}$ is a complex manifold with boundary.

###### Proof.

The disc $\mathrm{D}^{4}$ can be described as $\\{q\in\mathbb{H}:|q|^{2}\leq
1\\}$, where $\mathbb{H}$ stands to the quaternions. As a complex manifold
description of it, we can identify $\mathbb{H}\cong\mathbb{C}\oplus\mathbb{C}$
(as $\mathbb{R}$-vector spaces). Thus,
$\mathrm{D}^{4}\cong\\{(z,w)\in\mathbb{C}^{2}:|z|^{2}+|w|^{2}\leq 1\\}.$ Take
$f(z,w):=|z|^{2}+|w|^{2}-1$. Then
$\mathrm{D}^{4}=\\{(z,w)\in\mathbb{C}^{2}:f(z,w)\leq 0\\}$. Since $f$ is a
$C^{\infty}$-submersion, one gets desired. ∎

Lemmas 7.7 and 7.8 imply

######  Corollary 7.9.

The boundary $\mathrm{S}^{3}=\partial\mathrm{D}^{4}$ admits a CR structure.

Lemma 7.10 below ensures that the holomorphic charts making $\mathrm{D}^{4}$ a
complex manifold with boundary is essentially constrained by CR structures in
$\partial\mathrm{D}^{4}=\mathrm{S}^{3}$.

######  Lemma 7.10 (Theorem 2.1 in [96]).

The moduli space of complex structures (for manifolds with boundary) in
$\mathrm{D}^{4}$, up to biholomorphism, is in one-to-one correspondence with
that of CR structures, up to CR isomorphisms, in $\mathrm{S}^{3}$.

Once Lemma 7.10 in hand, we prove:

######  Proposition 7.11.

There is an injective map $\mathcal{G}$

$\mathcal{G}:\mathcal{CR}(\mathrm{S}^{3})\times\mathcal{H}\rightarrow\mathcal{M}_{\mathrm{D}^{4}\times\mathrm{S}^{3}\times\mathrm{S}^{1}}$

where $\mathcal{CR}(\mathrm{S}^{3})$ is the moduli space of CR structures in
$\mathrm{S}^{3}$ and $\mathcal{H}$ is the moduli space of holomorphic
structures in $H=\mathrm{S}^{3}\times\mathrm{S}^{1}$, to to the moduli space
$\mathcal{M}_{\mathrm{D}^{4}\times\mathrm{S}^{3}\times\mathrm{S}^{1}}$ of
complex structure (as a complex manifold with boundary) of
$\mathrm{D}^{4}\times\mathrm{S}^{3}\times\mathrm{S}^{1}$.

###### Proof.

As the first step, we show how to induce a complex manifold with boundary
structure in $\mathrm{D}^{4}\times\mathrm{S}^{3}\times\mathrm{S}^{1}$ once
given complex structures (for manifold with boundary and not) in
$\mathrm{D}^{4}$ and $\mathrm{S}^{3}\times\mathrm{S}^{1}$.

For simplicity, we initially assume given the holomorphic structure (single
chart) in $\mathrm{D}^{4}$ presented in Lemma 7.8 and a holomorphic structure
$\\{(V_{i},\phi_{i})\\}$ in $\mathrm{S}^{3}\times\mathrm{S}^{1}$. To build a
complex structure in the manifold with boundary
$\mathrm{D}^{4}\times\mathrm{S}^{3}\times\mathrm{S}^{1}$ we consider the
coordinate open sets $\\{\mathrm{D}^{4}\times V_{i}\\}$ and coordinate charts
$\Psi_{i}(x,y):=((z_{1},z_{2}),\phi_{i})$ where
$x\in\\{(z_{1},z_{2})\in\mathbb{C}^{2}:|z|^{2}\leq 1\\},\leavevmode\nobreak\
y\in\mathrm{S}^{3}\times\mathrm{S}^{1}$. Once the images $\phi_{i}(V_{i})$ are
open in $\mathbb{C}^{2}$, it is only left to verify that the transition
functions are biholomorphisms. But this is immediate since they are given as
$\Psi_{i}\circ\Psi_{j}^{-1}((z_{1},z_{2}),w)=((z_{1},z_{2}),\phi_{i}\circ\phi_{j}^{-1}(w)),\leavevmode\nobreak\
w\in\phi_{i}(U_{i}\cap U_{j})$. To the general case, pick an altas
$\mathcal{A}_{\mathrm{D}^{4}}$ in $\mathrm{D}^{4}$ making this a complex
manifold with a boundary. Separe the coordinate sets in
$\mathcal{A}_{\mathrm{D}^{4}}^{\mathrm{in}}\cup\mathcal{A}_{\mathrm{D}^{4}}^{\mathrm{out}}$
where the superscripts $\mathrm{in},\leavevmode\nobreak\ \mathrm{out}$
designate open subsets in $\mathbb{C}^{2}$ or sets of the form
$\\{z\in\mathbb{C}:f(z)\leq 0\\}=:[f\leq 0]$ for $f$ a smooth submersion,
respectively. Given the atlas $\\{(V_{i},\phi_{i})\\}$ in
$\mathrm{S}^{3}\times\mathrm{S}^{1}$, we take as new coordinate sets to
$\mathrm{D}^{4}\times\mathrm{S}^{3}\times\mathrm{S}^{1}$ the collection
$\\{V_{i}\times U_{i},\leavevmode\nobreak\
U_{i}\in\mathcal{A}_{\mathrm{D}^{4}}^{\mathrm{in}}\\}\cup\\{V_{i}\times[f\leq
0]\\},\leavevmode\nobreak\ [f\leq
0]\in\mathcal{A}_{\mathrm{D}^{4}}^{\mathrm{out}}$. Choosing diffeomorphisms
between each $[f\leq 0]$ with $\mathrm{D}^{4}$ as described in the first case,
a simple adaptation ensures the result.

Next, we construct $\mathcal{G}$. To it, we show that the resulting
holomorphic structure in
$\mathrm{D}^{4}\times\mathrm{S}^{3}\times\mathrm{S}^{1}$ is independent of
choosing representatives of a certain ((CR)isomorphism) class of CR structures
in $\mathrm{S}^{3}$ and representatives of a fixed class in the moduli space
of complex structures in $\mathrm{S}^{3}\times\mathrm{S}^{1}$.

Once an atlas is chosen, it induces a CR structure in
$\mathrm{S}^{3}\times\mathrm{S}^{3}\times\mathrm{S}^{1}$ represented by the
bundle

$\widetilde{E}:=E^{\prime}\times\left(T(\mathrm{S}^{3}\times\mathrm{S}^{1})\right)_{1,0},$

where $E^{\prime}=\ker(J+\sqrt{-1}\leavevmode\nobreak\ \mathrm{Id})\subset
E\otimes_{\mathbb{R}}{\underline{\mathbb{C}}}$ and
$(T\mathrm{S}^{3}\times\mathrm{S}^{1})_{1,0}$ is the holomorphic tangent
bundle of $\mathrm{S}^{3}\times\mathrm{S}^{1}$, with a distinguished section

$J_{\widetilde{E}}:=\begin{pmatrix}J_{E}&0\\\
0&J\end{pmatrix}\in\mathrm{Hom}(\widetilde{E},\widetilde{E}).$

$J_{E}$ specifies the chosen holomorphic structure in $\mathrm{D}^{4}$
uniquely (this is Lemma 7.10). Elements in the (CR)isomorphic class as $J_{E}$
and a holomorphic class of $J$ designate the complex structure in
$\mathrm{D}^{4}\times\mathrm{S}^{3}\times\mathrm{S}^{1}$ uniquely. Hence,
$\mathcal{G}$ is well-defined. It is left to show the injectivity claim.

Suppose two atlases $\mathcal{A},\leavevmode\nobreak\ \mathcal{A}^{\prime}$
are given in $\mathrm{D}^{4}\times\mathrm{S}^{3}\times\mathrm{S}^{1}$ and
obtained as above. If
$(\mathrm{D}^{4}\times\mathrm{S}^{3}\times\mathrm{S}^{1},\mathcal{A}),\leavevmode\nobreak\
(\mathrm{D}^{4}\times\mathrm{S}^{3}\times\mathrm{S}^{1},\mathcal{A}^{\prime})$
are biholomorphic, then the induced CR structures are (CR)isomorphic. In
particular, the CR structures in $\mathrm{S}^{3}$ are isomorphic, and the
complex structures in $\mathrm{S}^{3}\times\mathrm{S}^{1}$ are biholomorphic.∎

######  Lemma 7.12.

Given two copies $X_{0},\leavevmode\nobreak\ X_{1}$ of
$\mathrm{D}^{4}\times\mathrm{S}^{3}\times\mathrm{S}^{1}$ regarded with
analytic atlases, for any homotopy hopf manifold
$\Sigma^{7}\times\mathrm{S}^{1}$ where $\Sigma^{7}$ is a Milnor sphere, one
can find holomorphic structures on $X_{0},\leavevmode\nobreak\ X_{1}$ with
isomorphic CR structures in $\partial X_{0},\leavevmode\nobreak\ \partial
X_{1}$ which globally extend to a complex structure in
$\Sigma^{7}\times\mathrm{S}^{1}$.

###### Proof.

Pick a holomorphic structure (for complex manifolds with boundary) in
$\mathrm{D}^{4}$ and denote by $(E,J_{E})$ the corresponding induced CR
structure. Now, pick any complex structure in
$\mathrm{S}^{3}\times\mathrm{S}^{1}$ and consider the corresponding
holomorphic atlas (as given in Proposition 7.11) $\mathcal{A}$ in
$\mathrm{D}^{4}\times\mathrm{S}^{3}\times\mathrm{S}^{1}=:X_{1}$ whose
underlined real atlas is analytic. One gets a CR structure in
$\mathrm{S}^{3}\times\mathrm{S}^{3}\times\mathrm{S}^{1}$ represented by the
bundle
$\widetilde{E}:=E^{\prime}\times\left(T(\mathrm{S}^{3}\times\mathrm{S}^{1})\right)_{1,0}$,
where $E^{\prime}=\ker(J+\sqrt{-1}\leavevmode\nobreak\ \mathrm{Id})\subset
E\otimes_{\mathbb{R}}{\underline{\mathbb{C}}}$ and
$(T\mathrm{S}^{3}\times\mathrm{S}^{1})_{1,0}$ is the holomorphic tangent
bundle of $\mathrm{S}^{3}\times\mathrm{S}^{1}$. Such a bundle is regarded with
the distinguished section $J_{\widetilde{E}}:=\begin{pmatrix}J_{E}&0\\\
0&J\end{pmatrix}\in\mathrm{Hom}(\widetilde{E},\widetilde{E})$.

Since $\Sigma^{7}=M_{m,1-m}$ for some $m\in\mathbb{Z}$, pick
$f:=(f_{t_{m1-m}},\mathrm{1}):X_{0}\times\mathrm{S}^{1}\rightarrow
X_{1}\times\mathrm{S}^{1}$ and consider the pullback bundle
$f^{*}(\widetilde{E})$ with the pullback $f^{*}J_{\widetilde{E}}$. Since $f$
is a diffeomorphism one readily recovers that
$(f^{*}(E),f^{*}J_{\widetilde{E}})$ yields a CR structure in $X_{0}$. Theorem
A.2 concludes desired. ∎

We finally accomplish the proof of Theorem 7.1.

###### Proof of Theorem 7.1.

The part of the statement concerning the existence and the injection are the
contents of Proposition 7.11 and Lemma 7.12. We show now that
$\mathcal{M}_{\Sigma}$ is non-discrete for any Milnor sphere $\Sigma^{7}$. We
use the Kodaira-Spencer theorem [63, Theorem 6.2.12, p.272]. It suffices to
show for any complex structure in $\Sigma^{7}\times\mathrm{S}^{1}$, if
$T(\Sigma^{7}\times\mathrm{S}^{1})_{1,0}$ stands to the holomorphic tangent
bundle of $\Sigma^{7}\times\mathrm{S}^{1}$ then
$\mathrm{H}^{2}(\Sigma^{7}\times\mathrm{S}^{1};T(\Sigma^{7}\times\mathrm{S}^{1})_{1,0})=0$
(see [70, Theorems 6.2, 7.2 and 8.3] or [63, Chapter 6]). According to [63,
Corollary 4.1.14] we have

$\displaystyle\mathrm{H}^{2}(\Sigma^{7}\times\mathrm{S}^{1};T(\Sigma^{7}\times\mathrm{S}^{1})_{1,0})$
$\displaystyle\cong[\mathrm{H}^{2}(\Sigma^{7}\times\mathrm{S}^{1};\Omega^{2}(T^{*}\Sigma^{7}\times\mathrm{S}^{1}_{1,0}))]^{*}$
$\displaystyle\cong[\mathrm{H}^{2,2}(\Sigma^{7}\times\mathrm{S}^{1};\mathbb{C})]^{*}.$

Since $\mathrm{H}^{2}(\Sigma^{7}\times\mathrm{S}^{1};\mathbb{C})=0$ one
concludes desired. ∎

## Acknowledgments

The São Paulo Research Foundation FAPESP supports L. F. C grants 2022/09603-9
and 2023/14316-1. L. G’s research is partially supported by São Paulo Research
Foundation FAPESP grants 2018/13481-0 and 2023/13131-8. L. K. is supported by
The Basic Research Program of the National Research University Higher School
of Economics, the Simons Investigators Award of the Simons Collaboration on
Homological Mirror Symmetry, and partially supported by the National Science
Fund of Bulgaria, National Scientific Program “Excellent Research and People
for the Development of European Science” (VIHREN), Project No. KP-06-DV-7.

The authors acknowledge Denis Auroux, Maxim Kontsevich, Ernesto Lupercio,
Llohann Sperança, and Yuri Tschinkel for many fruitful conversations related
to this work. A large part of this work was conceived at the University of
Miami during several visits of L. G. and L. F. C, who are enormously thankful
for the hospitality and staff’s work.

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##  Appendix A Straightening angles: the smooth and the holomorphic cases

We first recall the following classical result in differential topology.
Later, we obtain its analogous in the holomorphic realm.

Let $M$ and $N$ be two manifolds with common boundaries $\partial M=\partial
N$. Let $f:\partial M\rightarrow\partial N$ be a diffeomorphism.

######  Theorem A.1.

There exists only one smooth structure in $X:=M\cup_{f}N$ satisfying

  1. (a)

the inclusions $i:M\rightarrow X,\leavevmode\nobreak\ j:N\rightarrow X$ are
smooth embeddings such that $\mathrm{d}i|_{\partial M},\leavevmode\nobreak\
\mathrm{d}j|_{\partial N}$ are injective

  2. (b)

if two maps $f,\leavevmode\nobreak\ f^{\prime}:\partial M\rightarrow\partial
N$ are isotopic, then $X:=M\cup_{f}N,\leavevmode\nobreak\
X^{\prime}:=M\cup_{f^{\prime}}N$ are diffeomorphic.

###### Proof.

We begin proving uniqueness. Assume $X$ is regarded with a smooth structure
satisfying the requirements in the hypotheses. Let $X^{\prime}$ be
homeomorphic to $X=M\cup_{f}N$ via $q:X\rightarrow X^{\prime}$ satisying
$q|_{M}:=i^{\prime}:M\rightarrow X^{\prime},\leavevmode\nobreak\
q|_{N}:=j^{\prime}:N\rightarrow X^{\prime}$. Item (a) ensures that
$i^{\prime},\leavevmode\nobreak\ j^{\prime}$ are smooth embeddings with
injective derivatives. We claim that $q$ is a diffeomorphism. Indeed, for any
$x\in M$ we have that
$\mathrm{d}q(x)=\mathrm{d}i^{\prime}(i(x))\mathrm{d}i(x)$, which item (a)
ensures to be an injection. Similarly, for $x\in N$, we have that
$\mathrm{d}q(x)$ is also injective. Thus, $q$ is a homeomorphism with
injective derivative (immersion), thus a diffeomorphism. We now construct a
smooth structure to $X$.

Let $U$ to be a collar neighborhood for $\partial M$, i.e., $U$ is
diffeomorphic to $]-2,0]\times\partial M$ via a diffeomorphism $\psi$. Let
$U^{\prime}$ be a collar neighborhood for $\partial N$ via a diffeomorphism
$\psi^{\prime}:[0,2[\times\partial N\rightarrow U^{\prime}$. For any
$\tau\in]0,2[$ let $U_{\tau}:=\psi(]-\tau,0]\times\partial
M),\leavevmode\nobreak\
U^{\prime}_{\tau}:=\psi^{\prime}([0,\tau[\times\partial N)$. Let

$\widetilde{f}:U_{\tfrac{1}{4}}\rightarrow U^{\prime}_{\tfrac{1}{4}}$
$\widetilde{f}(\psi(t,x)):=\psi^{\prime}(t+\tfrac{1}{4},x).$

Being $U_{\tfrac{1}{4}},\leavevmode\nobreak\ U^{\prime}_{\tfrac{1}{4}}$ open
sets, there exists a unique smooth structure in $M\cup_{\widetilde{f}}N$
making the inclusions in the statement smooth embeddings. These maps have
injective derivatives because they are inclusions.

Now let $f^{\prime}:\partial M\rightarrow\partial N$ be any other
diffeomorphism and $H:]-1,0]\times\partial M\rightarrow\partial N$ a smooth
isotopy such that

$H(t,x)=\begin{cases}f(x),\leavevmode\nobreak\ t\in]-2,-\tfrac{2}{3}[\\\
f^{\prime}(x),\leavevmode\nobreak\ t\in[-\tfrac{1}{3},0]\end{cases}$

Let $\widetilde{H}(t,x):U_{1}\rightarrow U^{\prime}_{1}$ be defined as

$\widetilde{H}(t,x):=H\circ\psi(t,x)=\psi^{\prime}(t+1,H(t,x))$

and let $X_{H}:=M\cup_{\widetilde{H}}N$. Assume that $\widetilde{f^{\prime}}$
is obtained from $f^{\prime}$ in the same manner $\widetilde{f}$ is obtained
from $f$. We outline now the proof that $M\cup_{\widetilde{f}}N$ and
$M\cup_{\widetilde{f^{\prime}}}N$ are diffeomorphic to
$M\cup_{\widetilde{H}}N$, finishing the result.

There exists a differentiable function
$\varphi:[0,2]\rightarrow[\tfrac{3}{4},2]$ such that

  1. (a’)

$\varphi^{\prime}(t)>0$ for all $t\in[0,2]$

  2. (b’)

$\varphi(t)=t+\tfrac{3}{4}$ for every $t\in[0,\tfrac{1}{4}]$

  3. (c’)

there exists $0<\delta\ll 1$ such that $\varphi(t)=t$ in $[2-\delta,2]$.

Let $\ell:M\rightarrow M,\leavevmode\nobreak\ \ell^{\prime}:N\rightarrow N$
defined as

$\displaystyle\ell(x)$ $\displaystyle:=\begin{cases}x,\leavevmode\nobreak\
x\in M\setminus U_{1}\\\ \psi(-\varphi(-t),\psi(t,x)),\leavevmode\nobreak\
\text{otherwise}\end{cases}$ $\displaystyle\ell^{\prime}(x^{\prime})$
$\displaystyle:=\begin{cases}x^{\prime},\leavevmode\nobreak\ x^{\prime}\in
N\setminus U^{\prime}_{1}\\\
\psi(\varphi(t),\psi(t,x^{\prime})),\leavevmode\nobreak\
\text{otherwise}\end{cases}$

It can be checked that both $\ell,\leavevmode\nobreak\ \ell^{\prime}$ are
injective with injective derivatives, thus defining embeddings. Moreover,
$\ell,\leavevmode\nobreak\ \ell^{\prime}$ give rise to bijective maps
$\widetilde{\ell}:X\rightarrow X_{H},\leavevmode\nobreak\
\widetilde{\ell^{\prime}}:X^{\prime}\rightarrow X_{H}$ via the commutative
diagram

${M}$${M}$${M}$${U_{1/4}}$${U_{1}}$${U_{1/4}}$${U^{\prime}_{1/4}}$${U^{\prime}_{1}}$${U^{\prime}_{1/4}}$${N}$${N}$${N}$$\scriptstyle{\ell}$$\scriptstyle{\widetilde{f}}$$\scriptstyle{\widetilde{H}}$$\scriptstyle{\widetilde{f^{\prime}}}$$\scriptstyle{\ell^{\prime}}$

Such induced $\widetilde{\ell},\leavevmode\nobreak\ \widetilde{\ell^{\prime}}$
are the desired diffeomorphisms. ∎

We now prove the holomorphic version of the former result. To now,

######  Theorem A.2 (Holomorphic Straightening angles).

Let $M,\leavevmode\nobreak\ N$ be two complex manifolds with boundaries
$\partial M,\leavevmode\nobreak\ \partial N$, respectively. Assume that

  1. (a)

$M,\leavevmode\nobreak\ N$ are real analytic manifolds with real analytic
isomorphic boundaries $\partial M\cong\partial N$

  2. (b)

the complex structures in $M,\leavevmode\nobreak\ N$ are real analytic up to
the boundary

  3. (c)

it exists a (CR)isomorphism $f:\partial M\rightarrow\partial N$ between the CR
structures induced by the holomorphic atlases in $M,\leavevmode\nobreak\ N$,
that is also a real analytic isomorphism.

Then, a unique (up to biholomorphism) complex manifold structure in
$X:=M\cup_{f}N$ exists.

The most critical aspect for the proof of Theorem A.2 is the existence of
“holomorphic collar neighborhoods”. Fortunately, it was already obtained in
[58, Theorem 1]. Following their main result, one observes that the hypothesis
requiring real analyticity can be relaxed, but since it suffices for our
purposes, one chooses to state it in this manner. Due to the former
observation, we proceed to the proof, following a prompt adaptation of Theorem
A.1.

###### Proof of Theorem A.2.

Consider holomorphic collar neighborhoods $U$ and $U^{\prime}$ of $\partial M$
and $\partial N$, respectively, defined as $2n$-open submanifolds $M_{0}$ and
$N_{0}$ of $M$ and $N$ containing $\partial M$ and $\partial N$. Their complex
structures are induced from $M$ and $N$ (i.e.,
$J_{M_{0}}=J_{M}\Big{|}_{M_{0}}$ and $J_{N_{0}}=J_{N}\Big{|}_{N_{0}}$).

Let $\mathcal{A}_{M}$ and $\mathcal{A}_{N}$ be compatible holomorphic atlases
in $M$ and $N$, respectively, extending the holomorphic atlases in $M_{0}$ and
$N_{0}$. Extend the CR-isomorphism $f$ to biholomorphisms between $M_{0}$ and
$N_{0}$.

Theorem A.1 yields a real analytic manifold $M\cup_{f}N$. Simultaneously, as
$f$ is a biholomorphism between two open holomorphic submanifolds of $M$ and
$N$, the identity theorem guarantees a unique holomorphic structure in
$M_{0}\cup_{f}N_{0}$. The assumption of real analyticity for $M$ and $N$
combined with the identity theorem ensures its extension to $M\cup_{f}N$. ∎

##  Appendix B Chen-Ruan orbifold cohomology

Let $U$ be a connected topological space and $V$ be a connected $n$-dimension
smooth manifold with a smooth action by a finite group $G$. Therein, assume
that the fixed-point set of each group element is either the whole space or of
codimension at least two. Observe that this may make the $G$-action
ineffective. Consequently, the non-fixed-point set is locally connected if it
is not empty. The ineffective kernel of the $G$-action consists of elements
fixing the whole space $V$.

######  Definition B.1 (Uniformizing system).

A $n$-dimension uniformizing system of $U$ is a triple $(V,G,\pi)$, where
$\pi:V\rightarrow U$ is a continuous map inducing a homeomorphism between the
quotient space $V/G$ and $U$. Two uniformizing systems
$(V_{i},G_{i},\pi_{i})$, $i=1,2$, are isomorphic if there is a diffeomorphism
$\phi:V_{1}\rightarrow V_{2}$ and an isomorphism $\lambda:G_{1}\rightarrow
G_{2}$ such that $\phi$ is $\lambda$-equivariant, and
$\pi_{2}\circ\phi=\pi_{1}$. If $(\phi,\lambda)$ is an automorphism of
$(V,G,\pi)$, then there is $g\in G$ such that $\phi(x)=g\cdot x$ and
$\lambda(a)=gag^{-1}$ for any $x\in V$ and $a\in G$. The element $g$ is unique
if, and only if, the action of $G$ on $V$ is effective.

Assume that $U$ is a locally connected topological space. For any point $x\in
U$, we can define the germ of uniformizing systems at $x$. Let
$(V_{1},G_{1},\pi_{1})$ and $(V_{2},G_{2},\pi_{2})$ be uniformizing systems of
neighborhoods $U_{1}$ and $U_{2}$ of $x$. We say that $(V_{1},G_{1},\pi_{1})$
and $(V_{2},G_{2},\pi_{2})$ are equivalent at $x$ if they induce isomorphic
uniformizing systems for a neighborhood $U_{3}$ of $x$.

######  Definition B.2 (Topological spaces with orbifold structures).

Consider a Hausdorff second countable topological space $M$. An $n$-dimension
orbifold structure on $M$ entails that for every point $x\in M$, there exists
a neighborhood $U_{x}$ and an $n$-dimension uniformizing system
$(V_{x},G_{x},\pi_{x})$ over $U_{x}$. This system satisfies the condition that
for any point $y\in U_{x}$, the uniformizing systems $(V_{x},G_{x},\pi_{x})$
and $(V_{y},G_{y},\pi_{y})$ are equivalent at $y$, implying they yield the
same germ at $y$.

Given a germ of orbifold structures, $M$ is designated an _orbifold_.

######  Remark B.3.

  1. (i)

An open subset $U$ of $M$ is called a uniformized open subset if it is
uniformized by a $(V,G,\pi)$ such that for each $x\in U$, $(V,G,\pi)$ defines
the same germ with $(V_{x},G_{x},\pi_{x})$ at $x$.

  2. (ii)

If $G_{x}$ acts effectively for every $x$, then $M$ is said to be a _reduced
orbifold_.

  3. (iii)

The notion of orbifold with boundary can be similarly given. If $M$ is an
orbifold with a boundary, its boundary $\partial M$ inherits an orbifold
structure from $M$ and becomes an orbifold.

∎

Let $M$ be an orbifold. For any point $x\in M$, let $(V_{x},G_{x},\pi_{x})$ be
a local chart at $x$ and $(g)_{G_{x}}$ be the conjugacy class of $g$ in
$G_{x}$. Consider the set of pairs

$None$ $\widetilde{M}=\\{(x,(g)_{G_{x}})|x\in M,g\in G_{x}\\},$

If there is no risk of confusion, we will omit the subscript $G_{x}$ to
simplify the notation. There is a surjective map $\pi:\widetilde{M}\rightarrow
M$ defined by $(x,(g))\mapsto x$. It can be checked that the set
$\widetilde{M}$ is an orbifold with the orbifold structure given by

$\\{\pi_{x,g}:(V_{x}^{g},C(g))\rightarrow V_{x}^{g}/C(g):x\in M,g\in
G_{x}.\\},$

where $V_{x}^{g}$ is the fixed-point set of $g$ in $V_{x}$, $C(g)$ is the
centralizer of $g$ in $G_{x}$.

Every point $x$ has a local chart $(V_{x},G_{x},\pi_{x})$ which gives a local
uniformized neighborhood $U_{x}=\pi_{x}(V_{x})$. If $y\in U_{x}$, up to
conjugation, there is an injective homomorphism $G_{y}\rightarrow G_{x}$. For
$g\in G_{y}$, the conjugacy class $(g)_{G_{x}}$ is well-defined. We define an
equivalence relation $(g)_{G_{y}}\sim(g)_{G_{x}}$. Let $T$ be the set of
equivalence classes. Abusing notation we shall denote by $(g)$ the equivalence
class which $(g)_{G_{y}}$ belongs to $\widetilde{M}$ is decomposed as a
disjoint union of connected components

$\widetilde{M}=\bigsqcup_{(g)\in T}M_{(g)},$

where

$M_{(g)}=\\{(x,(g^{\prime})_{G_{x}})|g^{\prime}\in
G_{x},(g^{\prime})_{G_{x}}\in(g)\\}.$

######  Definition B.4.

$M_{(g)}$ for $g\neq 1$ is called a twisted sector. $M_{(1)}=M$ is termed the
non-twisted sector.

Constructing bundles over orbifolds and considering sections for these bundles
is possible – [4, Section 2.3]. If an orbifold $M$ has an almost complex
structure $J$, i.e., it exists a smooth section of the orbifold bundle
$\mathrm{End}(TM)$ such that $J^{2}=-\mathrm{Id}$ then $\widetilde{M}$
inherits an almost complex structure from the one on $M$, and the map
$\pi:\widetilde{M}\rightarrow M$ defined by $(x,(g)_{G_{x}})\rightarrow x$
becomes “pseudo-holomorphic”, i.e., its differential commutes with the almost
complex structures on $\widetilde{M}$ and $M$.

Let $x$ be any point of $M$. The almost complex structure on $M$ gives rise to
a representation $\rho_{x}:G_{x}\rightarrow\mathrm{GL}(n,\mathbb{C})$ (here
$n=\dim_{\mathbb{C}}X$). For any $g\in G_{x}$, we write $\rho_{x}(g)$ as a
diagonal matrix

$\mathrm{diag}(e^{2\pi\mathbf{i}m_{1,g}/m_{g}},\cdots,e^{2\pi\mathbf{i}m_{n,g}/m_{g}}),$

where $m_{g}$ is the order of $\rho_{x}(g)$, and $0\leq m_{i,g}<m_{g}$. This
matrix depends only on the conjugacy class $(g)_{G_{x}}$ of $g$ in $G_{x}$. We
define a function $\iota:\widetilde{X}\rightarrow{\mathbb{Q}}$ by

$\iota(p,(g)_{G_{x}})=\sum_{i=1}^{n}\frac{m_{i,g}}{m_{g}}.$

It is straightforward to check that $M_{(g)}\rightarrow{\mathbb{Q}}$ is the
constant which we denote as $\iota_{(g)}$. Moreover,

######  Lemma B.5.

  1. (a)

$\iota_{(g)}$ is integral if and only if
$\rho_{x}(g)\in\mathrm{SL}(n,\mathbb{C})$.

  2. (b)

$\iota_{(g)}+\iota_{(g^{-1})}=\mathrm{rank}(\rho_{x}(g)-\mathrm{I}),$

which is the “complex codimension”
$\dim_{\mathbb{C}}M-\dim_{\mathbb{C}}M_{(g)}=n-\dim_{\mathbb{C}}M_{(g)}$ of
$M_{(g)}$ in $M$. As a consequence, $\iota_{(g)}+\dim_{\mathbb{C}}M_{(g)}<n$
when $\rho_{x}(g)\neq\mathrm{I}$.

Let $\mathrm{H}^{\ast}(M_{(g)})$ stands to the singular cohomology of
$M_{(g)}$ with real coefficients.

######  Definition B.6 (Chen-Ruan cohomology).

The _Chen-Ruan orbifold cohomology groups_ $\mathrm{H}^{d}_{\mathrm{orb}}(M)$
of $M$ are defined as

$\mathrm{H}^{d}_{\mathrm{orb}}(M)=\oplus_{(g)\in
T}\mathrm{H}^{d-2\iota_{(g)}}(M_{(g)})$

and _orbifold Betti numbers_ are
$b^{d}_{\mathrm{orb}}=\sum_{(g)}\dim\mathrm{H}^{d-2\iota_{(g)}}(M_{(g)})$.

###  B.1. Twisted Sectors and Twisted forms

Here, we recall a prolific way of computing Chen-Ruan’s cohomology. Therein,
we fix an orbifold $\mathcal{G}$ such that each group model $G_{x}$ for
choosing a uniformizing system (Definition B.1) is Abelian. We treat orbifolds
as their presentations as groupoids (see [4, Section 1.4]). We denote by
$\mathcal{G}_{0}$ the set of objects and by $\mathcal{G}_{1}$ the set of
arrows. As usual, we let $s$ and $t$ stand to the _source_ and _target_ maps,
respectively. The _inertia orbifold_ $\wedge\mathcal{G}$ is a sub orbifold of
$\mathcal{G}$ whose embedding defined from the object set
$(\wedge\mathcal{G})_{0}=\\{g\in\mathcal{G}_{1}|s(g)=t(g)\\}$ is just
$e(g)=s(g)=t(g)$.

For each $g\in(\wedge\mathcal{G})_{0}$ we have an action on the fiber
$e^{*}(T_{x}\mathcal{G}_{0})$ for $x=s(g)=t(g)$. Decompose
$e^{*}T_{x}\mathcal{G}_{0}=\oplus_{j}E_{j}$ where $E_{j}$ is the eigenspace
associated with the eigenvalue $e^{2\pi\mathbf{i}\tfrac{m_{j}}{m}},$ where
$m=\mathrm{oder}\leavevmode\nobreak\ g$. Regard it with the order-relation
$m_{i}\leq m_{j}$ if $i\leq j$. Then $\iota_{(g)}=\sum_{j}\tfrac{m_{j}}{m}$.

Let $v\in\mathcal{G}_{1}$ be an arrow with $s(v)=x$. As an element of
$(\wedge\mathcal{G})_{1}$, $v$ connects $g$ with $vgv^{-1}$. By taking
differentials for each $v$, one can map $(E_{j})_{g}$ in an eigenspace with
the same eigenvalue. In the case the multiplicity of the eigenvalue associated
with $(E_{j})_{g}$ is one, this map preserves the splitting of
$e^{*}T_{x}\mathcal{G}$, and $E_{j}$ constitutes a line bundle over
$\wedge\mathcal{G}$ for each $j$. Although this does not always hold, therein,
we assume this is the case, aiming to be as introductory as possible.

For each $j$, let $\theta_{j}$ stand to the Thom form of the line bundle
$E_{j}$. We convention that $\theta_{j}^{0}$ is just the identity.

######  Definition B.7 (Tiwsted factors).

Fix a twisted sector $M_{(g)}$. The _twisted factor_ $t(g)$ of
$\mathcal{G}_{(g)}$ is the formal product

$\prod_{j=1}^{m}\theta_{j}^{\frac{m_{j}}{m}}.$

The degree $t(g)$ is

$\mathrm{deg}\leavevmode\nobreak\ t(g)=2\iota_{(g)}.$

######  Definition B.8 (Twisted forms).

Let $\omega\in\Omega^{*}(\mathcal{G}_{(g)})$ be an invariant form. The formal
product $t(g)\omega$ is termed _a twisted form_ associated with
$\mathcal{G}_{(g)}$.

It can be constructed as a de Rham complex of twisted forms. Indeed, we have
the following definition.

######  Definition B.9 (de Rham complex of twisted forms).

$\Omega^{p}(\mathcal{G}):=\\{\omega_{1}t(g_{1})+\ldots+\omega_{k}t(g_{k}):\sum_{i}\mathrm{deg}\leavevmode\nobreak\
(\omega_{i})\mathrm{deg}\leavevmode\nobreak\ (t(g_{i}))=p\\}$

with co-boundary operator $d(\omega_{i}t(g_{i})):=(d\omega_{i})t(g_{i})$. The
co-boundary map defines a co-chain complex. The resulting cohomology is
denoted by $\mathrm{H}^{*}_{CR}(\mathcal{G};\mathbb{R})$ and termed _the de
Rham model for the Chen-Ruan cohomology_.

It is possible to introduce appropriate operations in
$\mathrm{H}^{*}_{CR}(\mathcal{G};\mathbb{R})$, making it a ring. The following
is true.

######  Theorem B.10 (Theorem 5.5 in [4]).

There exists an isomorphism of rings

$\mathrm{H}^{*}_{\mathrm{orb}}(\mathcal{G})\cong\mathrm{H}^{*}_{CR}(\mathcal{G};\mathbb{R}).$

Theorem B.10 is to accomplish with the proof of Theorem 5.12. Observe that the
latter provides the twisted-forms description of the Chen-Ruan cohomology we
use for the computation in Theorem 5.12.


††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Gas compression systematically delays the onset of viscous fingering

Liam C. Morrow  Callum Cuttle  Christopher W. MacMinn
<EMAIL_ADDRESS>Department of Engineering Science,
University of Oxford, Oxford, OX1 3PJ, UK

###### Abstract

Using gas to drive liquid from a Hele-Shaw cell leads to classical viscous
fingering. Strategies for suppressing fingering have received substantial
attention. For steady injection of an incompressible gas, the intensity of
fingering is controlled by the capillary number $\mathrm{Ca}$. Here, we show
that gas compression leads to an unsteady injection rate controlled primarily
by a dimensionless compressibility number $\mathcal{C}$. Increasing
$\mathcal{C}$ systematically delays the onset of fingering at high
$\mathrm{Ca}$, highlighting compressibility as an overlooked but fundamental
aspect of gas-driven fingering.

When a viscous liquid is displaced from a Hele-Shaw cell or porous medium by
the injection of a gas, the gas-liquid interface is hydrodynamically unstable
and tends to deform into branched, finger-like structures [1, 2, 3, 4]. This
viscous-fingering instability has been extensively studied as an archetype of
interfacial pattern formation [5, 6, 7, 8] and for its relevance to a variety
of practical applications, including the operation of fuel cells [9, 10], the
remediation of groundwater contamination [11, 12], and the subsurface
sequestration of CO2 [13, 14] or storage of hydrogen [15]. A concern in all
applications is understanding the key mechanisms that promote or suppress the
development of this instability, which is driven by viscous forces in the
defending phase and opposed by capillary forces at the interface. The
capillary number $\mathrm{Ca}$ compares the magnitudes of these two forces and
is thus the key parameter in determining the intensity of fingering. Recent
decades have seen an increased focus on exploring the impact of perturbations
to the flow or to the flow cell at a given $\mathrm{Ca}$, including active
modulation of the injection rate [16, 17] and use of a geometry that varies in
space [18, 19, 20, 21, 22, 17] and/or in time [23, 24, 25, 17, 26].

One fundamental aspect of this problem that has thus far been ignored is the
volumetric compression of the injected gas, which is practically unavoidable
under the typical pressures associated with displacing a viscous liquid from a
confined geometry. Previous theoretical studies of fingering have taken both
fluids to be incompressible and many previous experimental studies have
implicitly made the same assumption. Gas compression can lead to unsteady flow
or even stick-slip motion, thereby exerting a fundamental influence on
displacement processes and pattern formation [27, 28, 29, 30], but the
implications of compression on viscous fingering have been overlooked. Here,
we use mathematical modelling, simulations, and experiments to study viscous
fingering driven by the steady compression of a gas reservoir. We find that
steady compression leads to an unsteady injection rate that is dictated by a
dimensionless compressibility number $\mathcal{C}$, which can be interpreted
as the rate of viscous depressurisation relative to the rate of compressive
pressurisation [30]. We find that increasing $\mathcal{C}$ delays the onset of
the instability to a degree that is comparable to that of decreasing
$\mathrm{Ca}$ by a similar magnitude. Our results suggest that compression can
play a fundamental role in viscous fingering, with implications for many
previous studies of interfacial instabilities during gas-driven displacement.

Figure 1: (a) A body of gas with initial volume $V_{g}(0)$ is compressed at a
steady volumetric rate $Q$ in order to displace liquid from the gap $b$
between two plates, leading to viscous fingering. (b) Experiments performed
with $Q=2.5$ mL/min for (left to right) $V_{g}(0)=25$, $50$, $100$, and 200
mL. Other parameters are $b=420$ $\mu$m, $R_{c}$ = 10.5 cm, $\mu=0.97$ Pa
$\cdot$ s, and $\gamma=2.1\times 10^{-2}$ N/m, such that
$\mathrm{Ca}=5.21\times 10^{3}$ and $\alpha=250$ in all cases. (c)
Corresponding fingering simulations [i.e., numerical solutions to Eqs.
(3)-(9)]. Column one shows the incompressible case (black box); columns two to
six show the compressible case with $p_{\mathrm{atm}}=1.01\times 10^{5}$ Pa
and $V_{g}(0)=25$, $50$, $100$, $200$, and $400$ mL. For both experiments and
simulations, each snapshot shows interface contours at equal time intervals of
$\hat{t}_{0.9}/10$. The arrow indicates increasing compressibility number
$\mathcal{C}$, which is proportional to $V_{g}(0)$ and thus increases from
left to right.

We consider a radial Hele-Shaw cell comprising a narrow gap of thickness $b$
between two rigid, circular plates of radius $R_{c}$ [Fig. 1(a)]. The gap is
initially filled with an incompressible liquid of viscosity $\mu$ and surface
tension $\gamma$. This liquid is displaced by injecting gas through a hole in
the centre of one plate. Gas injection is driven by the steady compression of
a connected gas reservoir at volumetric rate $Q$, such that the volume of the
reservoir is $V_{\mathrm{res}}(t)=V_{\mathrm{res}}(0)-Qt$. Figure 1(b) shows
the time-evolution of the gas-liquid interface in experiments, where silicone
oil (1,000 cSt, Sigma) was displaced by air compressed with a syringe pump.
These images suggest that increasing the initial reservoir volume
$V_{\mathrm{res}}(0)$ (left to right) while holding all other parameters
constant systematically delays the onset of fingering, in the sense that the
interface remains near-circular up to larger radii. We investigate these
observations quantitatively below using full numerical simulations of viscous
fingering, complemented by experiments and by a simplified axisymmetric model.
We observe strong qualitative and often quantitative agreement between
experiments and simulations, as illustrated in Figure 1(c). We compare our
experiments and simulations in detail alongside further analysis of the
axisymmetric model in a companion study [31]. Here, we focus on the impact of
compressibility on the fingering pattern.

To simulate viscous fingering, we model flow of the liquid as Stokes flow
averaged over the gap $b$. We make the problem dimensionless via

$\begin{split}\hat{\boldsymbol{x}}=\frac{\boldsymbol{x}}{R_{c}},\quad&\hat{t}=\frac{Q}{\pi
R_{c}^{2}b}t,\quad\hat{p}=\frac{\pi b^{3}}{6\mu Q}p,\\\
&\hat{\boldsymbol{v}}=\frac{2\pi
R_{c}b}{Q}\boldsymbol{v},\quad\hat{V}=\frac{V}{\pi R_{c}^{2}b},\end{split}$
(1)

leading to the following dimensionless parameters:

$\begin{split}\mathrm{Ca}=\frac{12\mu QR_{c}}{2\pi
b^{3}\gamma},\quad\mathcal{C}=\frac{12\mu
QV_{g}(0)}{\pi^{2}R_{c}^{2}b^{4}p_{\mathrm{atm}}},\quad\mathcal{V}=\frac{V_{g}(0)}{\pi
R_{c}^{2}b},\\\
\alpha=\frac{R_{c}}{b},\quad\mathcal{R}=\frac{r_{0}}{R_{c}},\end{split}$ (2)

where the total volume of gas $V_{g}=V_{\mathrm{res}}+V_{b}$ combines the
volume $V_{\mathrm{res}}(t)$ of gas in the reservoir and the volume $V_{b}(t)$
of gas in the cell. The capillary number $\mathrm{Ca}$, commonly referred to
as the “modified” capillary number, incorporates the role of the aspect ratio
$\alpha$. The new compressibility number $\mathcal{C}$ compares viscous and
compressive (atmospheric) pressure scales or, equivalently, the rates at which
these pressures vary. We fix the geometry of the flow cell and the radius
$r_{0}$ of the initially circular bubble, such that $\alpha=250$ and
$\mathcal{R}=0.025$. The nondimensional flow equations are then

$\displaystyle\hat{\mbox{\boldmath$\nabla$}}^{2}\hat{p}$
$\displaystyle=0\quad$ in
$\displaystyle\hat{\boldsymbol{x}}\in\mathbb{R}^{2}\backslash\Omega,$ (3)
$\displaystyle\hat{v}_{n}$
$\displaystyle=-\frac{1}{1-f_{1}}(\hat{\mbox{\boldmath$\nabla$}}\hat{p})\cdot\boldsymbol{n}\quad$
on $\displaystyle\partial\Omega,$ (4) $\displaystyle\Delta\hat{p}$
$\displaystyle=\Delta\hat{p}_{g}(\hat{t})-\left(\frac{\pi\hat{\kappa}}{4}+2\alpha
f_{2}\right)\quad$ on $\displaystyle\partial\Omega,$ (5)
$\displaystyle\Delta\hat{p}$ $\displaystyle=0\quad$ on
$\displaystyle|\hat{\boldsymbol{x}}|\in 1,$ (6)

where $\partial{\Omega}$ denotes the gas-liquid interface, $\hat{v}_{n}$ is
the magnitude of the normal velocity of the interface, $\boldsymbol{n}$ is the
unit normal to the interface, and $\hat{\kappa}$ is the signed in-plane
curvature of the interface. Gauge pressures of the liquid and gas, $\Delta p$
and $\Delta p_{g}$, respectively, are measured relative to $p_{\mathrm{atm}}$.
To capture the effects of thin residual films of the wetting liquid in the gas
region [32], the functions $f_{1}$ and $f_{2}$ modify the kinematic and
dynamic boundary conditions, respectively, to account for volume conservation,
and enhanced curvature and viscous stresses at the interface. Following Peng
_et al._ [33], we adopt the empirical expressions

$\displaystyle f_{1}$ $\displaystyle=\frac{\left|\mu
v_{n}/\gamma\right|^{2/3}}{0.76+2.16\left|\mu v_{n}/\gamma\right|^{2/3}},$ (7)
$\displaystyle f_{2}$ $\displaystyle=1+1.59\left|\mu
v_{n}/\gamma\right|+\frac{\left|\mu
v_{n}/\gamma\right|^{2/3}}{0.26+1.48\left|\mu v_{n}/\gamma\right|^{2/3}},$ (8)

which were derived by fitting to simulations of viscous fingering in a rigid
Hele-Shaw channel [34].

We ignore pressure gradients in the gas, taking $p_{g}(t)$ to be spatially
uniform. We also assume isothermal compression, such that
$p_{g}(t)=p_{g}(0)V_{g}(0)/V_{g}(t)$, which is justified by our experimental
observations (see Appendix A of Ref. [31]). The dimensionless gauge gas
pressure may then be written

$\Delta\hat{p}_{g}=\frac{2\mathcal{C}^{-1}\left(\hat{t}+\mathcal{R}^{2}-\hat{V}_{b}\right)+\mathrm{Ca}^{-1}\left(\frac{\pi}{4\mathcal{R}}+2\alpha\right)}{1-\mathcal{V}^{-1}\left(\hat{t}+\mathcal{R}^{2}-\hat{V}_{b}\right)}.$
(9)

In addition to the above compressible model, we also consider an analogous
incompressible model, in which Eqs. (3)-(8) are unchanged but
$\Delta\hat{p}_{g}(\hat{t})$ is no longer governed by Eq. (9), instead
evolving in time such that
$\hat{Q}_{b}\equiv\mathrm{d}{\hat{V}_{b}}/\mathrm{d}\hat{t}=1$ (i.e.,
$Q_{b}\equiv\mathrm{d}{V}/\mathrm{d}{t}=Q$).

We solve both compressible and incompressible models numerically using a
scheme proposed by Morrow  _et al._ [35], which utilises the level-set method
[36]. Our simulations use the perturbed initial condition

$\displaystyle\hat{R}(\theta,0)=\mathcal{R}\left\\{1+\varepsilon\sum_{n=2}^{12}\cos\left[n(\theta-2\pi
r_{n})\right]\right\\},$ (10)

where $\hat{R}(\theta,\hat{t})$ is the radial position of the interface at
azimuthal coordinate $\theta$, and $\varepsilon\ll 1$ and $0\leq r_{n}\leq 1$
are random numbers sampled from uniform distributions. Simulations are
performed on the domain $0\leq\hat{r}\leq 1$ and $0\leq\theta<2\pi$ using
1,000 $\times$ 1,000 equally-spaced nodes, and are concluded when $\hat{R}\geq
0.99$ for any value of $\theta$.

To better understand the coupling between gas compression and fluid
displacement, we also consider an axisymmetric model in which the interface is
taken to be a circle of radius $\hat{R}_{0}\left(\hat{t}\right)$. By further
taking the limits that the air reservoir is much larger than the volume of the
Hele-Shaw cell ($\mathcal{V}\gg 1$) and that the capillary pressure is
negligible relative to atmospheric pressure
($2\mathcal{C}^{-1}\gg\mathrm{Ca}^{-1}[\pi/(4\mathcal{R})+2\alpha]$), and
ignoring thin-film effects ($f_{1}=0$, $f_{2}=1$), Eqs. (3)-(9) reduce to

$\displaystyle\frac{\textrm{d}\hat{R}_{0}}{\textrm{d}\hat{t}}=\frac{\Delta\hat{p}_{g}}{2\hat{R}_{0}\ln\left(1/\hat{R}_{0}\right)},\quad\Delta\hat{p}_{g}=\frac{2(\hat{t}+\mathcal{R}^{2}-\hat{R}_{0}^{2})}{\mathcal{C}}.$
(11)

The dynamics of this axisymmetric model [Eq. (11)] are then determined solely
by the compressibility number. For the incompressible case ($\mathcal{C}=0$),
the solution to the axisymmetric model is
$\hat{R}_{0}=\sqrt{\mathcal{R}^{2}+\hat{t}}$ and
$\Delta\hat{p}_{g}=\ln(1/\hat{R}_{0})$.

Figure 2: Nondimensional (left) injection rate $\hat{Q}_{b}$ and (right) gauge
gas pressure $\Delta\hat{p}_{g}$ for $\mathcal{C}=0.018$, $0.036$, $0.071$,
$0.14$, $0.28$, $0.57$, $1.13$, $2.27$, and $4.54$ (dark blue to dark red;
arrows). Blue and red curves are for $\mathcal{C}<1$ and $\mathcal{C}>1$,
respectively. The incompressible case is also shown (dashed black). Panels (a)
and (b) are from the axisymmetric model [Eq. (11)]. Panels (c) and (d) are
from fingering simulations (solid curves) and experiments (symbols), both for
$\mathrm{Ca}=2.08\times 10^{4}$ with
$\mathcal{V}=0.215,0.430,0.859,\textbf{1.72, 3.44, 6.87, 13.7},27.5,$ and 55
(experimental values in bold).

We solve Eq. (11) numerically to plot the actual, time-varying injection rate
$\hat{Q}_{b}\left(\hat{t}\right)=\textrm{d}\hat{V}_{b}/\textrm{d}\hat{t}=2\hat{R}_{0}(\textrm{d}\hat{R}_{0}/\textrm{d}\hat{t})$
and the gauge gas pressure in Figs. 2(a,b), respectively, for a range of
$\mathcal{C}$. Both quantities are initially zero for all $\mathcal{C}>0$.
Both then increase toward and then beyond the incompressible solution,
corresponding to an initial period of increasing injection rate due to gas
pressurisation. The pressure evolves non-monotonically in time for all
$\mathcal{C}$, increasing at early times when the compression rate exceeds the
injection rate ($\hat{Q}_{b}<1$) and then decreasing at late times, once the
injection rate exceeds the compression rate ($\hat{Q}_{b}>1$). The injection
rate $\hat{Q}_{b}$ is determined by a balance between the gauge gas pressure
and the viscous resistance in the draining liquid. This observation can be
used to demonstrate the existence of two dynamical regimes in the axisymmetric
model [see 31]. For $\mathcal{C}\lessapprox 1$ (blue curves), the breakout
time $\hat{t}_{f}\equiv\hat{t}(\hat{R}_{0}=1)$ is unity and the breakout
pressure $\Delta\hat{p}_{g}(\hat{R}_{0}=1)$ is zero, corresponding to a
nonmonotonic evolution of $\hat{Q}_{b}$, which tends to a finite value at
breakout. For $\mathcal{C}\gtrapprox 1$ (red curves), breakout is delayed
($\hat{t}_{f}>1$) and the breakout pressure is significantly greater than
zero. This overpressure leads to a monotonically increasing and ultimately
divergent injection rate as the viscous resistance vanishes at breakout. These
regimes arise in the axisymmetric model due to the coupling of time-evolving
compressive and viscous forces, independent of viscous fingering. The same
features were recently identified and analysed in detail in the context of a
capillary tube [30].

Figures 2(c, d) show $\hat{Q}_{b}$ and $\Delta\hat{p}_{g}$, respectively, from
fingering simulations at the same values of $\mathcal{C}$ as in Figs. 2(a, b)
and for $\mathrm{Ca}=2.08\times 10^{4}$, demonstrating that the qualitative
dynamics of the axisymmetric model and the fingering simulations are
strikingly similar. The presence of viscous fingering, and to a lesser extent
residual films, leads to a few key differences. Notably, the fingering
simulations achieve breakout much earlier (typically around
$\hat{t}_{f}\approx 0.5$ for low $\mathcal{C}\lessapprox 1$) because the
irregular, branched interface bypasses a substantial fraction of the defending
liquid. In addition, the transition to monotonically increasing $\hat{Q}_{b}$
occurs significantly below the threshold $\mathcal{C}\approx 1$ from the
axisymmetric model. (Note that the threshold $\mathcal{C}$ is approximate in
the Hele-Shaw geometry because of a secondary dependence on $\mathcal{R}$
[31].) Unlike in the axisymmetric model, the values of $\hat{Q}_{b}$ and
$\Delta\hat{p}_{g}$ from the fingering simulations do also depend on the value
of $\mathrm{Ca}$, but we find that this dependence is comparatively weak, as
discussed below. Nonetheless, this comparison suggests that the basic physics
of compression-driven displacement underpin the dynamics of this system, even
in the presence of viscous fingering. For comparison, we also plot
experimental measurements at four values of $\mathcal{C}$ in Fig. 2(c, d),
which show excellent qualitative agreement with the fingering simulations (see
Ref. [31] for a detailed comparison).

Figure 3: Nondimensional (a) injection rate $\hat{Q}_{b}$ and (b) gauge gas
pressure $\Delta\hat{p}_{g}$ from fingering simulations for $\mathcal{C}=0.14$
and $\mathrm{Ca}=2.61\times 10^{3}$, $5.21\times 10^{3}$, $1.04\times 10^{4}$,
and $2.08\times 10^{4}$ (solid blue curves; arrows show increasing
$\mathrm{Ca}$). The solution to the axisymmetric model [Eq. (11)] is shown for
comparison (dashed black). Panels (c) and (d) show the evolution of the
fingering pattern from corresponding experiments and simulations,
respectively, plotted at equal time intervals of $\hat{t}_{0.9}/10$.

The importance of $\mathcal{C}$ is further illustrated by considering
fingering simulations at fixed $\mathcal{C}$ and varying $\mathrm{Ca}$ as
plotted in Figs. 3(a, b). In practice, this can be achieved by varying $Q$
while holding $QV_{g}(0)$ and all other parameters constant. The results of
the axisymmetric model are included for reference (dashed black). The
injection rate $\hat{Q}_{b}$ [Fig. 3(a)] is particularly insensitive to
variations in $\mathrm{Ca}$ and, hence, to the evolution of the fingering
pattern. The gauge gas pressure $\Delta\hat{p}_{g}$ [Fig. 3(b)] appears to be
somewhat more sensitive to the pattern, featuring a more pronounced
$\mathrm{Ca}$-dependent deviation from the axisymmetric solution after the
peak pressure is reached. However, variations in $\mathrm{Ca}$ clearly have
much less of an impact on $\hat{Q}_{b}$ and $\Delta\hat{p}_{g}$ than
variations in $\mathcal{C}$ of a similar magnitude [cf. Figs. 2(c, d)]. This
insensitivity to $\mathrm{Ca}$ is surprising, given the strong impact of
$\mathrm{Ca}$ on the fingering pattern, as illustrated in Fig. 3(c, d) for the
corresponding experiments and simulations, respectively. The severity of the
fingering pattern increases with $\mathrm{Ca}$, with the instability beginning
at smaller radii, generating narrower and more-branched fingers, and leading
to earlier breakout—see the values of the near-breakout time $\hat{t}_{0.9}$
at which the interface first crosses $\hat{R}=0.9$, in Figs. 3(c, d). However,
the evolving balance between compressive and viscous forces is a bulk feature
that is relatively insensitive to the shape of the interface.

Figure 4: (a) The onset radius $\hat{R}^{*}$ at which fingers begin to
develop, defined as the maximum radius of the interface when
$\mathcal{I}=1.1$, from fingering simuations at varying $\mathcal{C}$ with
$\mathcal{V}=$ 0.215, 0.430, 0.859, 1.72, 3.44, 6.87, 13.7, 27.5, 55, and 110.
(b) The subsequent values of $\mathcal{I}$ at the near-breakout time
$\hat{t}_{0.9}$. Dashed lines denote the corresponding incompressible cases.
Arrows indicate the direction of increasing $\mathrm{Ca}=2.61\times
10^{3},5.21\times 10^{3},1.04\times 10^{4}$, and $2.08\times 10^{4}$.

To further examine the influence of $\mathcal{C}$ and $\mathrm{Ca}$ on viscous
fingering, we perform a linear stability analysis of the full model [Eqs.
(3)-(9)] by considering the growth of perturbations to a near-circular
interface
$\hat{R}(\theta,\hat{t})=\hat{R}_{0}(\hat{t})+\varepsilon\gamma_{n}(\hat{t})\cos(n\theta)+\mathcal{O}(\varepsilon^{2})$,
where $\varepsilon\ll 1$, and $\gamma_{n}$ is the amplitude of mode $n$. This
analysis adopts the same assumptions as the axisymmetric model:
$\mathcal{V}\gg 1$, capillary pressure is negligible compared with atmospheric
pressure (but not negligible overall), and we ignore thin films. As such, the
base state $\hat{R}_{0}(\hat{t})$ is determined by Eq. (11), but we retain
terms with $\mathrm{Ca}$ in the stability analysis. Following the methodology
of Paterson [5], it can be shown that the growth rate of each mode is given by

$\displaystyle\lambda_{n}=\frac{\dot{\gamma}_{n}}{\gamma_{n}}=\frac{n-1}{2\hat{R}_{0}^{2}}\left[\hat{Q}_{b}-\frac{n(n+1)}{\mathrm{Ca}\hat{R}_{0}}\right],$
(12)

where $\hat{Q}_{b}=2\hat{R}_{0}(\mathrm{d}\hat{R}_{0}/\mathrm{d}\hat{t})$ for
a circular interface and $\mathrm{d}\hat{R}_{0}/\mathrm{d}\hat{t}$ is
determined by Eq. (11). The most unstable mode is then

$\displaystyle n_{\max}=\sqrt{\frac{1+\mathrm{Ca}\hat{R}_{0}\hat{Q}_{b}}{3}},$
(13)

which comes about by solving $\partial\lambda_{n}/\partial n=0$ for $n$.
Increasing $\mathrm{Ca}$, $\hat{R}_{0}$, and $\hat{Q}_{b}$ all increase
$\lambda_{n}$ and $n_{\max}$, thus promoting the instability (i.e.,
perturbations grow faster and generate more fingers). In the incompressible
system, for which Eqs. (12) and (13) were originally derived and where
$\hat{Q}_{b}=1$ and $\hat{R}_{0}=(\mathcal{R}^{2}+\hat{t})^{1/2}$,
$\mathrm{Ca}$ is the key parameter in determining the onset of viscous
fingering. For the compressible case, however, both $\hat{Q}_{b}(\hat{t})$ and
$\hat{R}_{0}(\hat{t})$ depend primarily on $\mathcal{C}$. The very low initial
injection rates $\hat{Q}_{b}\ll 1$ observed the axisymmetric model, the
fingering simulations, and the experiments [Fig. 2(a, c)] tend to reduce
$\lambda_{n}$ and $n_{\max}$ considerably at early times relative to an
incompressible flow, suggesting that fingering is delayed by compressibility
[see 31].

To compare the predictions of this linear stability analysis with the
nonlinear evolution of the instability in the fingering simulations, we
quantify the severity of the fingering pattern using the isoperimetric ratio

$\displaystyle\mathcal{I}\left(\hat{t}\right)=\frac{\hat{L}^{2}\left(\hat{t}\right)}{4\pi\hat{A}\left(\hat{t}\right)},$
(14)

where $\hat{L}$ and $\hat{A}$ denote the perimeter and area of the gas region,
respectively. For a circular interface, $\mathcal{I}=1$; any deformation away
from a circle results in $\mathcal{I}>1$. Hence, the initial deviation of
$\mathcal{I}$ away from 1 is an indicator of the onset of instability. Figure
4(a) shows the onset radius $\hat{R}^{*}$ at which fingers begin to develop
(defined here as $\max(\hat{R})$ when $\mathcal{I}=1.1$) as a function of
$\mathcal{C}$ for different $\mathrm{Ca}$. As $\mathcal{C}\to 0$,
$\hat{R}^{*}$ asymptotes to the incompressible case. Increasing $\mathcal{C}$
or decreasing $\mathrm{Ca}$ systematically delays the onset of fingering
relative to the incompressible case. Varying either $\mathrm{Ca}$ or
$\mathcal{C}$ by a similar magnitude has a comparable effect, highlighting the
power of $\mathcal{C}$ as a control parameter. While compressibility delays
onset primarily by reducing the initial injection rates at higher
$\mathcal{C}$, the overshoot in $\hat{Q}_{b}$ at later times could drive
anomalously fast growth and exacerbate the final severity of fingering. As
shown in Fig. 4(b), however, the delayed onset with increasing $\mathcal{C}$
also correlates with lower $\mathcal{I}(\hat{t}_{0.9})$, meaning a less severe
fingering pattern near breakout. Hence, $\mathcal{C}$ not only delays the
onset of the instability, but also hinders the nonlinear growth of the
fingering pattern. We observe the same qualitative behaviour in experiments
over a comparable range of $\mathcal{C}\sim 10^{-2}$ – $10^{0}$ [see 31, for a
discussion of the distinct patterns formed in experiments and simulations].

Thus, compressibility naturally and passively leads to an unsteady injection
rate that can significantly delay the onset of fingering and reduce the
severity of the fingering pattern, with the compressibility number
$\mathcal{C}$ having a comparable effect to that of $\mathrm{Ca}$, the
traditional control parameter. Even for relatively small $\mathcal{C}$, which
would be unavoidable even in systems that are nominally incompressible, the
onset radius is consistently larger and the final isoperimetric ratio is
consistently smaller than for the incompressible case [Fig. 4], suggesting
that compressibility is rarely truly negligible. Our study is of particular
relevance to flows in porous media where the presence of compressibility is
already widely appreciated in sub-surface fluid injection [37], although its
full implications for fluid displacement and fingering were previously not.
There is broad scope for taking advantage of compressibility in gas-driven
flows in porous media and our study provides a strong foundation for doing so.
Further, our results imply that features in other physical systems that play a
role analogous to compressibility, such as elasticity, heat capacity, or
electrical capacitance, can play an important and even controlling role in the
evolution of those systems.

Data availability — The supporting data for this study are openly available on
Zenodo [38].

Published version — This article is published as L. C. Morrow, C. Cuttle, and
C. W. MacMinn. Gas compression systematically delays the onset of viscous
fingering. _Physical Review Letters_ , 131:224002, 2023.

###### Acknowledgements.

We are grateful to Mr. Clive Baker for technical support. This work was
supported by the European Research Council (ERC) under the European Union’s
Horizon 2020 Programme [Grant No. 805469], by the UK Engineering and Physical
Sciences Research Council (EPSRC) [Grant No. EP/S034587/1], and by the John
Fell Oxford University Press Research Fund [Grant No. 132/012].

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