# Exact solutions for viscous Marangoni spreading

Thomas Bickel  Univ. Bordeaux, CNRS, Laboratoire Ondes et Matière d’Aquitaine,
F-33400 Talence, France  François Detcheverry  University of Lyon, Université
Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, F-69622 Villeurbanne,
France

###### Abstract

When surface-active molecules are released at a liquid interface, their
spreading dynamics is controlled by Marangoni flows. Though such Marangoni
spreading was investigated in different limits, exact solutions remain very
few. Here we consider the spreading of an insoluble surfactant along the
interface of a deep fluid layer. For two-dimensional Stokes flows, it was
recently shown that the non-linear transport problem can be exactly mapped to
a complex Burgers equation [Crowdy, SIAM J. Appl. Math., 81, 2526 (2021)]. We
first present a very simple derivation of this equation. We then provide fully
explicit solutions and find that varying the initial surfactant distribution –
pulse, hole, or periodic – results in distinct spreading behaviors. By
obtaining the fundamental solution, we also discuss the influence of surface
diffusion. We identify situations where spreading can be described as an
effective diffusion process but observe that this approximation is not
generally valid. Finally, the case of a three-dimensional flow with axial
symmetry is briefly considered. Our findings should provide reference
solutions for Marangoni spreading, that may be tested experimentally with
fluorescent or photoswitchable surfactants.

##  I Introduction

The spreading of surface-active species at aqueous interfaces is a long-
standing issue in interfacial science Matar_sm-2009 . Surfactants, of which
amphiphilic molecules are the prime example, accumulate at interfaces where
they lower the surface tension book_dGbwq-CapWet . As a consequence, a
gradient in surface concentration induces a Marangoni stress at the interface
Scriven_nat-1960 ; Manikantan_jfm-2020 . Inhomogeneities in the distribution
of surfactants thus drive a liquid flow, that in turn couples with the
surfactant distribution. This complex feedback mechanism may eventually lead
to the rigidification of the interface that can alter the rising motion of gas
bubbles BelFdhila_pf-1996 ; Takagi_arfm-2011 ; Palaparthi_jfm-2006 ;
Crowdy_jem-2021 . Another striking effect observed in microfluidic experiments
is that traces of surfactants can severely limit the drag reduction of
superhydrophobic surfaces Peaudecerf_pnas-2017 ; Landel_jfm-2020 ;
Baier_jfm-2021 . More generally, Marangoni stresses due to the presence of
surfactants, even at a very low concentration, are ubiquitous through
fundamental processes Bickel_epje-2019 ; Bickel_prf-2019 ; Venerus_sr-2015 ,
the natural world Trinschek_sm-2018 ; Stetten_cocis-2018 ; Botte_jms-2005  and
industrial applications Morciano_ees-2020 . The consequences are essential for
a variety of phenomena, including film thickness in coating Quere_arfm-1999 ,
dispersion relation for capillary waves Sauleda_jcis-2022  and stability of
foams book_Cantat-Foams ; Breward_jfm-2002 .

The dynamics of surfactant spreading is also relevant in the field of active
matter through the propulsion mechanism of Marangoni swimmers Fei_cocis-2017 ;
Grosjean_epje-2018 ; book_nplks-SelfOrganizedMotion . When a particle filled
with surface-active molecules such as camphor is placed at the air-water
interface, the resulting surface-tension gradient drives the spontaneous
motion of the particle, even if the latter is perfectly symmetric
Rednikov_jcis-1994 ; Boniface_pre-2019 ; Ender_epje-2021 . Although camphor
swimmers are known for centuries Boniface_pre-2019 , their rich individual and
collective dynamics is still a matter of investigation Suematsu_cej-2018 ;
Nakata_pccp-2015 ; Dietrich_prl-2020 ; Boniface_prf-2021 . To elucidate their
propulsion mechanism, a fine understanding of the transport of surfactants is
required. In particular, it has been suggested both experimentally
Suematsu_langmuir-2014  and theoretically Kitahata_jcp-2018 ; Bickel_sm-2019
that the transient spreading of camphor molecules could be described by an
effective diffusion process that would account for the advection by the
Marangoni flow. This point is actually quite subtle and requires an in-depth
investigation.

When a surfactant is released at the air-liquid interface, the surfactant-
covered domain grows in time and its size $L(t)$ is governed by a balance
between viscous stresses and surface tension gradients. At large time, the
size generally follows the scaling law $L(t)\sim t^{\alpha}$. The exponent
$\alpha$ depends on the dominant features of the system such as inertia,
gravity, capillarity or viscous dissipation, among others. Although most of
previous studies focused on thin films Jensen_jfm-1992 ;
AfsarSiddiqui_acis-2003 ; Dussaud_jfm-2005 ; Craster_rmp-2009 ;
Swanson_jem-2015 , where the lubrication approximation applies, we consider
here the deep layer limit. Regarding the dynamics of surfactant spreading,
asymptotic self-similar solutions have been thoroughly discussed in the wake
of the seminal work of Jensen Jensen_jfm-1995 . More recently, the
hydrodynamic signature of surfactant transport in a semi-infinite liquid has
also been investigated at steady state Roche_prl-2014 ; LeRoux_pre-2016 ;
Bandi_prl-2017 ; Mandre_jfm-2017 . This line of works focuses mainly on the
large Reynolds limit, where the surfactant-driven flow develops in a boundary
layer.

In contrast, much less is known regarding the properties of Marangoni
spreading with Stokes flow. In this case, the vorticity created by the
Marangoni shear stress at the interface penetrates deep into the liquid, so
that the transport equations are generally both nonlocal and nonlinear
Matar_sm-2009 . An important step forward was achieved, however, when it was
recognized that the interfacial velocity can be expressed as a convolution of
the surface concentration Thess_prl-1995 . Another essential finding, achieved
recently by Crowdy Crowdy_siamjam-2021 , has revealed that the mathematical
problem for insoluble surfactant can be mapped to a complex Burgers equation,
opening new perspectives for its resolution.

The aim of the present work is to exploit those advances to build a complete
set of exact solutions for viscous Marangoni spreading. In contrast to self-
similar solutions that apply asymptotically, our solutions are valid at all
time. We find that the non-linear character of the spreading problem leads to
a rich variety of possible behaviors and that the initial surfactant
distribution has a key influence on the subsequent evolution. Besides, our
exact solutions provide reference cases and a range of physical insights. In
particular, the question of Marangoni spreading as an effective diffusion
process can now be settled.

The study is organized as follows. In Sec. II, we first present the
hydrodynamic model with its underlying assumptions and propose a particularly
simple derivation of the complex Burgers equation that governs the dynamics of
insoluble surfactants in one dimension. Section III focuses on transient
Marangoni spreading in the absence of diffusion. Exact solutions are obtained
for a variety of initial surfactant distributions. The influence of surface
diffusion is investigated in Sec. IV, where we provide a fundamental solution
and discuss the resulting effective diffusion coefficient. Finally, we examine
in Sec. V the spreading dynamics in two dimensions with axial symmetry. We
conclude in Sec. VI with a physical discussion of our findings.

##  II From Marangoni spreading to Burgers equation

###  II.1 Dimensionless numbers and simplifying assumptions

We consider the spreading of surfactants at the air-water interface, as
illustrated in Fig. 1. Initial inhomogeneities in the surfactant distribution
induce Marangoni constraints and a fluid flow that drives the system toward a
homogeneous state. Our goal is to provide an exact description of this
relaxation process. We assume a two-dimensional flow and an infinitely deep
liquid layer. The liquid is Newtonian and incompressible, with dynamic
viscosity $\eta$, mass density $\rho$ and kinematic viscosity $\nu=\eta/\rho$.
The surface-active molecules are insoluble, i.e. they are irreversibly
adsorbed at the interface. We note $D_{\mathrm{s}}$ their surface diffusion
coefficient.

To introduce further modeling assumptions, we first discuss the relevant
dimensionless numbers. The contribution of advection to momentum and mass
transport can be rationalized by the Reynolds and Péclet numbers

$\displaystyle\text{Re}\equiv\frac{LU}{\nu}\quad\text{and}\quad\text{Pe}\equiv\frac{LU}{D_{\mathrm{s}}},$
(1)

with $L$ the length scale of the initial concentration perturbation and $U$
the velocity associated with the corresponding Marangoni flow. Both quantities
may span a large range in experiments, namely $L=10^{-3}-10^{-1}\,$m and
$U=10^{-3}-1\,$m s-1, resulting in Reynolds number $\text{Re}=1-10^{5}$ much
above or around unity. Because the surface diffusion coefficient
$D_{\mathrm{s}}\simeq 10^{-9}\,$ m${}^{2}\,$s-1 Shmyrov_langmuir-2019  is
several orders of magnitude smaller than the momentum transport coefficient
$\nu\simeq 10^{-6}\,$ m${}^{2}\,$s-1, the hierarchy $\text{Re}\ll\text{Pe}$
always applies Thess_jfm-1997 . The thickness of the hydrodynamic boundary
layer is therefore much larger than that of the mass boundary layer
LeRoux_pre-2016 . In the following, we neglect fluid inertia and focus on the
Stokes regime. This assumption is reasonable when $L$ lies below or within the
millimeter range and $U$ in the mm s-1 range at most. While this choice of the
Stokes limit is certainly restrictive, we will show in the following that it
is interesting in its own right. Emphasis is placed on nonlinearities that
occur in the mass transport equation and on the methods to handle them
exactly.

Figure 1:  Spreading of an insoluble surfactant above a semi-infinite liquid
layer with two-dimensional flow. The inhomogeneous distribution of surfactant
$\Gamma(x,t)$ induces a Marangoni flow with bulk velocity $\mathbf{u}(x,z,t)$
and interfacial velocity $u_{\mathrm{s}}(x,t)$.

The spreading of surfactant can induce an unsteady displacement of the free
interface. The competition between surface deformation and viscous stress is
quantified by the capillary number

$\displaystyle\text{Ca}\equiv\frac{\eta U}{\gamma},$ (2)

with $\gamma$ the surface tension of the interface. With $\eta\simeq
10^{-3}\,$Pa s, $\gamma=10^{-2}-10^{-1}\,$N m-1 and the range of velocity
scale given above, one gets $\text{Ca}=10^{-5}-10^{-1}$. We can therefore
neglect capillary effects and assume that the interface remains flat in the
formulation of the hydrodynamic problem. Still, interfacial deformations can
be determined afterward by invoking the normal stress continuity condition
Jensen_jfm-1995 .

Finally, some surface-active species such as camphor or alcohol molecules may
evaporate from the liquid to the gas phase. To account for this phenomenon, we
include a first-order kinetic evaporation with rate $k$. The ratio of
evaporative to convective mass transport is set by the Damköhler number

$\displaystyle\text{Da}\equiv\frac{kL}{U},$ (3)

that can take values in a very wide range, spanning from negligible
($\text{Da}\ll 1$) to prevalent ($\text{Da}\gg 1$) evaporation.

###  II.2 Marangoni flow and surfactant transport

In the viscous regime, the velocity field $\mathbf{u}=(u,v)$ and pressure $p$
satisfy the incompressible Stokes equations

$\displaystyle\bm{\nabla}\cdot\mathbf{u}$
$\displaystyle=0,\qquad\eta\Delta\mathbf{u}=\bm{\nabla}p,$ (4)

with $\bm{\nabla}$ the nabla operator and $\Delta$ the Laplacian. The velocity
is assumed to vanish far away from the initial disturbance. Along the free
interface, variations in surface tension $\gamma$ induce a tangential stress
given by the Marangoni boundary condition

$\displaystyle\eta\,\partial_{z}u\big{|}_{z=0}=-\partial_{x}\gamma,$ (5)

where the interface remains flat and corresponds to $z=0$. The surface tension
is related to the local surfactant concentration $\Gamma$ through an equation
of state $\gamma=\gamma_{\mathrm{o}}-\kappa\Gamma$ that is linear book_birdi-
HandSurfaceColloidChem_chapter4 , a convenient but approximate assumption
Pawar_pf-1996 ; Swanson_jem-2015 . Surface activity is then measured by the
positive constant $\kappa\equiv-\partial\gamma/\partial\Gamma$. Finally, the
concentration $\Gamma(x,t)$ of surfactant evolves according to the advection-
diffusion equation

$\displaystyle\partial_{t}\Gamma+\partial_{x}\left(\Gamma
u_{\mathrm{s}}\right)=D_{\mathrm{s}}\,\partial^{2}_{xx}\Gamma-k\Gamma,$ (6)

where $u_{\mathrm{s}}(x,t)\equiv u(x,z=0,t)$ is the velocity at the interface.
The last term in Eq. (6) accounts for evaporation.

###  II.3 Closure relation and Thess’ equation

Since the Stokes equation is linear, it should be possible to relate the flow
anywhere in the bulk to the distribution of Marangoni stress at the interface,
and thus ultimately to the distribution of surfactant. This step has indeed
been achieved by Thess and collaborators Thess_prl-1995 ; Thess_ps-1996 ;
Thess_jfm-1997  (see also Ref. Pismen_prl-1997 ). The end result is a closure
relation giving the surface velocity as a convolution of surfactant
concentration with a dimension-dependent kernel. When the surface velocity is
one-dimensional, the closure relation reads

$u_{\mathrm{s}}(x,t)=\frac{\kappa}{2\eta}\,\mathcal{H}[\Gamma(x,t)].$ (7)

Here, we define the Hilbert transform $\mathcal{H}$ of a function $f$

$\mathcal{H}[f](x)\equiv\breve{f}(x)\equiv\frac{1}{\pi}\mbox{p.v.}\int_{-\infty}^{\infty}\frac{f(x^{\prime})}{x-x^{\prime}}\text{d}x^{\prime},$
(8)

where the integral is understood in the sense of Cauchy principal value
(p.v.). The closure relation of Eq. (7) is non-local: the velocity at position
$x$ depends on the surfactant distribution everywhere on the surface. This is
in clear contrast with the thin layer limit, where the surface velocity is
simply proportional to the concentration gradient $\partial_{x}\Gamma$
Thess_jfm-1997 .

It is convenient at this point to switch to dimensionless variables. From now
on, we focus on the reduced surfactant concentration
$f=\Gamma/\Gamma_{\mathrm{o}}$ and the reduced velocity $u=u_{\mathrm{s}}/U$,
with $\Gamma_{\mathrm{o}}$ the concentration scale and
$U=\kappa\Gamma_{\mathrm{o}}/2\eta$ the characteristic velocity associated
with the Marangoni flow. The length and time scales are also expressed in
units of $L$ and $\tau=L/U$, with $L$ the relevant size pertaining to the
initial perturbation. Expecting no confusion by the reader, we keep the same
notations $x$ and $t$ for the dimensionless variables. According to Eq. (7),
the closure relation in dimensionless form simply reads
$u(x,t)=\breve{f}(x,t)$.

Putting all pieces together, the flow and surfactant dynamics described by
Eqs. (4)–(6) can finally be replaced by a unique equation originally
introduced by Thess Thess_ps-1996

$\partial_{t}f+\partial_{x}(f\breve{f})=-\alpha
f+\epsilon\,\partial^{2}_{xx}f\ ,$ (9)

where $\alpha$ and $\epsilon$ stand respectively for the Damköhler number
$\alpha\equiv\text{Da}$ and the inverse Péclet numer
$\epsilon\equiv\text{Pe}^{-1}$. By “integrating out” the features of the flow,
the coupled problem of momentum and mass transport is reduced to a single
equation, which is, however, nonlinear and nonlocal.

###  II.4 Complex Burgers equation

Although some specific solutions of Thess’ equation have been discussed
previously in the literature Thess_ps-1996 ; Thess_jfm-1997 ; Pismen_prl-1997
; Bickel_sm-2019 , a systematic method for solving the problem is still
missing. A major breakthrough has been achieved very recently by Crowdy
Crowdy_siamjam-2021 , recognizing that the problem defined by Eqs. (4)-(6) can
be mapped to a complex Burgers equation. The derivation is based on the
complex variable representation of the stream function and therefore relies on
advanced properties of analytic functions. Here we propose a different route,
inspired by an approach originally developed in the context of vorticity
transport Constantin_cpam-1985 .

By exploiting properties of Hilbert transforms, let us show that Crowdy’s
result can be derived in a straightforward manner. The first step is to apply
the Hilbert operator to Eq. (9) to obtain

$\displaystyle\partial_{t}\breve{f}+\breve{f}\partial_{x}\breve{f}-f\partial_{x}f=-\alpha\breve{f}+\epsilon\,\partial^{2}_{xx}\breve{f}.$
(10)

Here, we made use of the relations
$\mathcal{H}[\partial_{x}f]=\partial_{x}\breve{f}$,
$\mathcal{H}[fg]=\breve{f}g+f\breve{g}+\mathcal{H}[\breve{f}\breve{g}]$ and
$\mathcal{H}^{2}[f]=-f$, from which one also gets
$\mathcal{H}[f\breve{f}]=(\breve{f}^{2}-f^{2})/2$ book_King-HilbertTransformII
; Constantin_cpam-1985 . The second step is to introduce the complex function
$\Psi(x,t)$ as

$\displaystyle\Psi(x,t)\equiv\breve{f}(x,t)-if(x,t).$ (11)

Combining Eqs. (9) and (10), it is then a simple calculation to recover
Crowdy’s equation Crowdy_siamjam-2021

$\displaystyle\partial_{t}\Psi+\Psi\partial_{x}\Psi=-\alpha\Psi+\epsilon\,\partial^{2}_{xx}\Psi,$
(12)

with an additional term that accounts for evaporation. This equation expresses
that in the present setting, Marangoni spreading is formally governed by a
viscous Burgers equation for the complex function $\Psi(x,t)$. If a solution
can be found, the interfacial velocity and the concentration are deduced from

$\displaystyle u(x,t)=\mathrm{Re}[\Psi(x,t)],\qquad
f(x,t)=-\mathrm{Im}[\Psi(x,t)].$ (13)

Note that, since $f$ is a concentration, it should always satisfy $f(x,t)\geq
0$.

The inverse Péclet number $\epsilon$ appearing in Eq. (12) plays the role of
viscosity in the more common form of the Burgers equation, which was initially
introduced in the context of fluid turbulence book_Burgers-NonlinDifEq ;
Bonkile_pramanajp-2018 . The effect of evaporation is embodied through an
extra linear term monitored by the Damköhler number $\alpha$. The main
accomplishment of shifting from Thess’ to Crowdy’s formulation is that
locality is now restored in the Burgers Eq. (12). Moreover, a rigorous
mathematical transformation allows to further transform the nonlinear Burgers
equation into a linear partial differential equation Hopf_cpam-1950 . As a
consequence, analytical solutions for the transient spreading of insoluble
surfactants can be derived and discussed in a systematic way. We now proceed
to do so.

Figure 2:  Spreading of a surfactant pulse: Concentration and velocity
profiles for a Cauchy, circular and Gaussian pulse, from left to right. The
typical width is $a=1$ and the total amount of surfactant is the same in all
cases. Time is $t=0,0.5,1,2,5,10$ from top to bottom. Note that the vertical
scale changes from one graph to the other.

##  III Transient spreading at infinite Péclet number

The situation most amenable to analytical treatment is when diffusion and
evaporation are both absent ($\alpha=\epsilon=0$ or
$\text{Da}=\text{Pe}^{-1}=0$). The equation to solve in this case is the
inviscid Burgers equation book_Burgers-NonlinDifEq ; book_Whitham-
LinearNonlinearWaves ; Bonkile_pramanajp-2018

$\displaystyle\partial_{t}\Psi+\Psi\partial_{x}\Psi=0.$ (14)

If $\Psi_{\mathrm{o}}(x)\equiv\Psi(x,0)$ denotes the initial value, the
solution given by the method of characteristics book_aw-MathMethodsPhys  is
simply

$\displaystyle\Psi(x,t)=\Psi_{\mathrm{o}}(x^{*})\ ,\quad\text{with}\quad
x^{*}+t\Psi_{\mathrm{o}}(x^{*})=x.$ (15)

It remains only to solve the equation on $x^{*}$. We have identified five
relevant cases where this is possible: two types of surfactant “pulses”, two
types of surfactant “holes” and one periodic distribution.

###  III.1 Spreading dynamics of surfactant pulses

##### Cauchy pulse.

We first assume that the initial profile is a Cauchy (or Lorentzian)
distribution with amplitude $\mathcal{A}$ and width $a$. The initial values
are then

$\displaystyle
f_{\mathrm{o}}(x)=\frac{\mathcal{A}a^{2}}{a^{2}+x^{2}},\quad\text{and}\quad\breve{f}_{\mathrm{o}}(x)=\frac{x}{a}f_{\mathrm{o}}(x),$
(16)

so that $\Psi_{\mathrm{o}}(x)=a\mathcal{A}/(x+ia)$. Given the simplicity of
the expressions, Eq. (15) reduces to a second order equation for $x^{*}$

$\displaystyle{x^{*}}^{2}-(x-ia)x^{*}+\mathcal{A}at-iax=0,$ (17)

which leads to the solution

$\displaystyle\Psi(x,t)=\frac{1}{2t}\left(x+ia-i\left[(a-ix)^{2}+4a\mathcal{A}t\right]^{1/2}\right).$
(18)

Separating the real and imaginary parts 111To separate the real and imaginary
part, one can use the relation
$\sqrt{2}\sqrt{x+iy}=\sqrt{\rho+x}+i\operatorname{sign}(y)\sqrt{\rho-x}$,
where $x$ and $y$ are reals and $\rho=\sqrt{x^{2}+y^{2}}$., the concentration
and velocity profiles are obtained explicitly as

$\displaystyle f(x,t)$
$\displaystyle=\frac{1}{2t}\left(\Upsilon^{+}(\xi)-a\right),$ (19a)
$\displaystyle u(x,t)$
$\displaystyle=\frac{1}{2t}\left(x-\Upsilon^{-}(\xi)\right),$ (19b)

where we define $\xi\equiv a^{2}+4a\mathcal{A}t-x^{2}$ and

$\displaystyle\Upsilon^{\pm}(\xi)$
$\displaystyle\equiv\sqrt{\left(\sqrt{\xi^{2}+(2ax)^{2}}\pm\xi\right)/2}.$
(20)

To keep compact expressions, we have assumed $x>0$. The negative part can be
obtained by symmetry since $f(x,t)$ and $u(x,t)$ are respectively even and odd
with respect to $x$. The resulting concentration and velocity profiles are
shown in Fig. 2(a).

##### Circular pulse.

As a second instance, we consider an initial surfactant profile that is
elliptical or “circular” after rescaling:

$\displaystyle f_{\mathrm{o}}(x)$
$\displaystyle=\mathcal{A}\left[1-x^{2}/a^{2}\right]_{+}^{1/2},$ (21a)
$\displaystyle\breve{f}_{\mathrm{o}}(x)$
$\displaystyle=\frac{\mathcal{A}}{a}\left(x-\left[x^{2}-a^{2}\right]_{+}^{1/2}\right),$
(21b)

with radius $a$ and amplitude $\mathcal{A}$. The brackets $\left[.\right]_{+}$
indicate the positive part of the argument. Here we also assume $x>0$ for the
sake of brevity. The initial values lead to
$\Psi_{\mathrm{o}}(x)=\left(\operatorname{sign}(a-x)\sqrt{a^{2}-x^{2}}+ix\right)\mathcal{A}/a$.
Equation (15) is again second order for $x^{*}$ and proceeding as above, the
concentration and velocity profiles then read

$\displaystyle f(x,t)$
$\displaystyle=\frac{a\mathcal{A}}{a(t)}\left[1-\frac{x^{2}}{a^{2}(t)}\right]_{+}^{1/2},$
(22a) $\displaystyle u(x,t)$
$\displaystyle=\frac{a\mathcal{A}}{a^{2}(t)}\left(x-\left[x^{2}-a^{2}(t)\right]_{+}^{1/2}\right),$
(22b)

where $a(t)=\sqrt{a^{2}+2a\mathcal{A}t}$. The circular solution, plotted in
Fig. 2(b), is self-similar. Indeed, Eq. (22) shows that the profile at any
time can be obtained by replacing the initial radius $a$ with a time-dependent
radius $a(t)$, while keeping a fixed amount of surfactant. In the context of
Marangoni spreading, the circular solution was initially found by Thess
Thess_ps-1996  with an adhoc method. In the mathematical literature
Biler_crasp-2011 , it is known as a fundamental solution which arises when
assuming self-similarity. Note that like other self-similar solutions, such as
the Barenblatt-Pattle solution for the porous medium equation book_Barenblatt-
InterAsymptotics ; Pattle_qjmam-1959 , the circular pulse solution has a
finite support.

##### More general pulse shapes.

The Cauchy and circular pulses are special because they both lead to a second
order equation for Eq. (15). Other pulse shapes yield an equation on $x^{*}$
that in general cannot be solved explicitly 222 The derivative of a Cauchy
pulse leads to a third-order equation that is also solvable.. Still, it is
always possible to resort to numerical resolution. The example of a Gaussian
pulse treated this way is shown in Fig. 2(c). In the limit
$t\rightarrow\infty$, all pulses eventually approach the fundamental circular
solution 333 In the sense that the norm of the difference between the solution
and the circular solution tends toward zero at large time, that is
$||f(x,t)-f_{\mathrm{c}}(x,t)||_{p}\to 0$ for $t\to\infty$, with
$||\cdot||_{p}$ a $p$-norm.. However, it is apparent that such loss of memory
of the initial profile happens only at long time, for $t\simeq 10$ in
dimensionless units. This indicates that the shape of the initial pulse has a
significant influence on transient Marangoni spreading.

##### Effective diffusion coefficient.

We now examine whether the spreading process can be described as an effective
diffusion process, as suggested in several studies Suematsu_langmuir-2014 ;
Kitahata_jcp-2018 ; Xu_csa-2022 . For the circular pulse, the spatial
extension squared $a^{2}(t)=a^{2}+2a\mathcal{A}t$ increases linearly with
time, a feature typical of diffusion. But for the Cauchy pulse, the second
moment of the concentration profile is infinite so that the width of the
surfactant distribution is not properly defined. One can use instead the
position $x_{\mathrm{vmax}}$ where the velocity is maximal. In the large time
limit, one finds $x_{\mathrm{vmax}}\simeq 2\sqrt{a\mathcal{A}t}$, a behavior
again reminiscent of diffusion. An effective diffusion coefficient can thus be
defined as $D_{\mathrm{eff}}\sim a\mathcal{A}$. Interestingly,
$D_{\mathrm{eff}}$ is proportional to the total amount of surfactant in the
pulse. Thus, the more surfactant in the pulse, the faster the spreading. Such
a feature, which would not be permitted with plain diffusion since the
corresponding equation is linear, is a consequence of nonlinearity. Surfactant
spreading may thus be described as a diffusive process, but only under
specific circumstances and with regard to the time evolution of the
concentration profile.

Figure 3:  (a-b) Closing of a surfactant hole: Concentration and velocity
profiles for Cauchy and circular holes. Here $\bar{c}=a=\mathcal{A}=1$. Time
is $t=0,0.5,1,1.5,2,3,5$ from bottom to top. A singularity occurs at
${t^{\star}}=1$ and ${x^{\star}}=0$. (c) Relaxation of a periodic distribution
of surfactant: Concentration and velocity profiles for a sinusoidal initial
distribution, with $\bar{c}=a=\mathcal{A}=1$. Time is $t=0,0.2,0.5,1,1.5,2,5$.
A singularity occurs at ${t^{\star}}=1$ and ${x^{\star}}=\pm\pi$.

###  III.2 Closure dynamics of surfactant holes

We now discuss the reverse situation where surfactant is already present at
the interface but with a concentration near the origin that is lower than the
value $\bar{c}$ far away from it. Such surfactant “dimple” or for brevity
“hole” in the following is expected to close as a result of Marangoni flow. As
for the spreading case, we derive two exact solutions for this process: the
Cauchy hole and the circular hole.

##### Cauchy hole.

Consider the initial surfactant concentration given by

$\displaystyle
f_{\mathrm{o}}(x)=\bar{c}-\frac{a^{2}\mathcal{A}}{a^{2}+x^{2}},$ (23)

with $0\leqslant\mathcal{A}\leqslant\bar{c}$. Retracing the steps detailed
above, the $\Psi$ function, concentration and velocity can all be written
explicitly in a straightforward manner. For the sake of clarity, the
mathematical expressions are reported in App. C. An interesting feature occurs
when the initial concentration vanishes at the origin, i.e. when
$\mathcal{A}=\bar{c}$. In this case, the concentration profile exhibits a
singularity at a finite time ${t^{\star}}=a/\bar{c}$. Indeed, a small-$x$
expansion gives at lowest order

$\displaystyle x\rightarrow 0,\quad
f(x,{t^{\star}})=-\frac{u(x,{t^{\star}})}{\operatorname{sign}(x)}=\frac{\bar{c}}{\sqrt{2a}}\sqrt{|x|}.$
(24)

This indicates that the concentration profile has a cusp near the origin, as
illustrated in Fig. 3(a). Regarding the velocity profile, it has an infinite
slope at the origin when $t={t^{\star}}$. For larger time $t>{t^{\star}}$, the
profile becomes regular again. Note that this singularity has already been
identified in previous investigations Crowdy_siamjam-2021 , but focusing on
the dynamics of poles in the complex plane. For $\bar{c}=\mathcal{A}$, the
concentration at the origin has a remarkably simple expression

$\displaystyle t\leqslant{t^{\star}},\quad f(0,t)$ $\displaystyle=0,$ (25a)
$\displaystyle t\geqslant{t^{\star}},\quad f(0,t)$
$\displaystyle=\bar{c}\left(1-\frac{{t^{\star}}}{t}\right).$ (25b)

The concentration remains zero until the cusp forms at time ${t^{\star}}$,
after which it relaxes as $t^{-1}$.

##### Circular hole.

The case of an elliptic initial dimple can also be treated analytically. The
initial density reads

$\displaystyle
f_{\mathrm{o}}(x)=\bar{c}-\frac{\mathcal{A}}{a}\left[a^{2}-x^{2}\right]_{+}^{1/2},$
(26)

with the condition $0\leqslant\mathcal{A}\leqslant\bar{c}$. The exact solution
is provided in App. C. The concentration and velocity profiles are displayed
in Fig. 3(b). When $\mathcal{A}=\bar{c}$, the concentration exhibits a finite-
time singularity as well, with features similar to those discussed above. In
particular, Eq. (24) still holds, with the factor of 2 removed. Regarding the
concentration at the origin, one now gets

$\displaystyle t\leqslant{t^{\star}},\quad f(0,t)$ $\displaystyle=0,$ (27a)
$\displaystyle t\geqslant{t^{\star}},\quad f(0,t)$
$\displaystyle=\bar{c}\left(1-\frac{{t^{\star}}}{2t-{t^{\star}}}\right).$
(27b)

We therefore find that the concentration relaxes asymptotically with the same
$t^{-1}$ law.

###  III.3 Periodic distribution of surfactant

We now turn to a periodic initial distribution of surfactant with sinusoidal
variations of period $2\pi a$ and amplitude $\mathcal{A}$ around the mean
value $\bar{c}$ 444For the concentration to remain positive, one should have
$0\leqslant\mathcal{A}\leqslant\bar{c}$.. From
$f_{\mathrm{o}}(x)=\bar{c}+\mathcal{A}\cos(x/a)$ and
$i\Psi_{\mathrm{o}}(x)=\bar{c}+\mathcal{A}\exp(ix/a)$, the solution can be
found as

$\displaystyle
i\Psi(x,t)=\bar{c}+\frac{a}{t}W\left(\frac{t\mathcal{A}}{a}\exp\left[(ix-\bar{c}t)/a\right]\right),$
(28)

where $W$ is the principal value of the Lambert function that satisfies
$W(x)\exp\left[W(x)\right]=x$ Corless_acm-1996 . The corresponding profiles
are plotted in Fig. 3(c). Once again, if the initial concentration includes a
point of vanishing concentration (i.e. $\mathcal{A}=\bar{c}$), the profile
exhibits a singularity at $({t^{\star}},{x^{\star}})=(1,\pm\pi a)$ with the
same features as discussed previously. As a side remark, we note that the
solution of Eq. (28) derived for an infinite system also applies to a finite
domain $x\in[-\pi a,\pi a]$ with no-flux boundary conditions.

The asymptotic behavior can also be extracted from Eq. (28). In the large-time
limit, the relaxation is found to be exponential

$\displaystyle t\rightarrow\infty,\quad
f(x,t)-\bar{c}=\mathcal{A}\cos(x/a)e^{-t/\tau_{\mathrm{r}}}.$ (29)

The characteristic time $\tau_{\mathrm{r}}=a/\bar{c}$ is independent of the
modulation amplitude $\mathcal{A}$ but decreases with the mean concentration
$\bar{c}$. In other words, a surfactant-laden interface can erase initial
inhomogeneities more rapidly when richer in surfactant. At this point, it is
instructive to compare with a purely diffusive process. For diffusion, the
relaxation of a sinusoidal profile is exponential at all time with a
characteristic time $\tau_{D}=a^{2}/D$. Here, Marangoni spreading exhibits a
similar behavior: setting $\tau_{\mathrm{r}}=a^{2}/D_{\mathrm{eff}}$, one can
define an effective diffusion coefficient $D_{\mathrm{eff}}\sim a\bar{c}$ that
is again proportional to the total amount of surfactant involved, as already
discussed in Sec. III.1. The analogy with diffusion is, however, incomplete.
As clearly seen in Fig. 3(c), the concentration profile in surfactant-poor and
surfactant-rich regions evolves in a very asymmetric manner. Such a feature,
that arises from the nonlinearity of Eq. (14), would be proscribed in a
diffusion process.

##  IV Spreading dynamics at finite Damköhler or Péclet numbers

When either evaporation or diffusion are relevant, additional terms have to be
accounted for and the inviscid Burgers Eq. (14) has to be replaced by its more
general version Eq. (12). We discuss in this section the new solutions that
arise when considering finite Damköhler or Péclet numbers.

###  IV.1 Effect of evaporation

For finite Damköhler number, i.e. when evaporation is considered, the equation
to solve is $\partial_{t}\Psi+\Psi\partial_{x}\Psi=-\alpha\Psi$. One can
easily check that the solution $\Psi$ with evaporation ($\alpha>0$) is
directly related to the solution $\Psi^{\dagger}$ without evaporation
($\alpha=0$) by

$\displaystyle\Psi(x,t)=e^{-\alpha
t}\Psi^{\dagger}(x,\tau_{\mathrm{eff}}),\quad\tau_{\mathrm{eff}}=\frac{1-e^{-\alpha
t}}{\alpha}.$ (30)

Note in particular that combining Eq. (30) and the circular pulse solution Eq.
(22) yields back the solution obtained previously in Ref. Bickel_sm-2019 .

Evaporation has two effects. First, both the concentration and the velocity
include an exponential decay with rate $\alpha$. Second, the effective time
$\tau_{\mathrm{eff}}$, which is equal to $t$ at short times
$t\ll\tau_{\mathrm{eff}}$, subsequently reaches a plateau at longer times
$t\gg\tau_{\mathrm{eff}}$. The plateau value $\tau=\alpha^{-1}$ corresponds to
the characteristic time associated with evaporation. This implies that the
long-time behavior of the relaxation is always exponential, whatever the
initial distribution. Asymptotically, evaporation is thus the dominant
transport mechanism.

###  IV.2 Effect of surface diffusion

####  IV.2.1 Fundamental solution

We come back to the situation where there is no evaporation and focus on the
effect of surfactant diffusion along the interface. We assume that the
interface is initially clean, with $f(x,t=0)\to 0$ for $x\to\infty$. The
equation to solve is now the viscous Burgers equation

$\displaystyle\partial_{t}\Psi+\Psi\,\partial_{x}\Psi=\epsilon\,\partial^{2}_{xx}\Psi,$
(31)

with $\epsilon$ the inverse Péclet number which is proportional to the surface
diffusion coefficient $D_{\mathrm{s}}$. This nonlinear equation can actually
be converted to a linear partial differential equation thanks to the Hopf
transformation Hopf_cpam-1950 . Denoting as $\Psi_{\mathrm{o}}(x)$ the initial
value, the general solution can be expressed as

$\displaystyle\Psi(x,t)$ $\displaystyle=-2\epsilon\,\partial_{x}\log I(x),$
(32)

where we define

$\displaystyle I(x)$
$\displaystyle\equiv\int_{-\infty}^{\infty}\exp\left(-g(u)/\epsilon\right)\mathrm{d}u,$
(33) $\displaystyle g(u)$
$\displaystyle\equiv\frac{(u-x)^{2}}{4t}+\frac{1}{2}\int_{0}^{u}\Psi_{\mathrm{o}}(u^{\prime})\,\mathrm{d}u^{\prime}.$
(34)

This general solution is now specified for a Cauchy pulse because among the
initial profiles we consider, this case is the only one we found to be fully
tractable analytically. Considering the initial value
$\Psi_{\mathrm{o}}(x)=a\mathcal{A}/(x+ia)$, one gets

$\displaystyle
I(x)=\int_{-\infty}^{\infty}\left(1-iu/a\right)^{-\frac{a\mathcal{A}}{2\epsilon}}\exp\left[-\frac{(u-x)^{2}}{4\epsilon
t}\right]\,\mathrm{d}u.$ (35)

While the result of the latter integral is available for any width $a$ and
amplitude $\mathcal{A}$ 555 A Hubbard-Stratonovitch transform is first
introduced for the exponential term. After switching the integration order,
the double integral can also be obtained., we focus for simplicity on the
limit $a\rightarrow 0$ while keeping $a\mathcal{A}=1/\pi$, so that the total
amount of surfactant is unity. In this limit, the initial Cauchy pulse
approaches a Dirac distribution. The explicit expression for $I(x)$ is then

$\displaystyle I(x)=$
$\displaystyle\Gamma\left(\zeta\right)\,_{1}F_{1}\left(\zeta,\frac{1}{2},-\frac{x^{2}}{4\epsilon
t}\right)+$ $\displaystyle i\frac{2x}{\sqrt{4\epsilon
t}}\,\Gamma\left(\frac{1}{2}+\zeta\right)\,_{1}F_{1}\left(\frac{1}{2}+\zeta,\frac{3}{2},-\frac{x^{2}}{4\epsilon
t}\right),$ (36)

where $\Gamma$ denotes here the gamma function, ${}_{1}F_{1}$ is the confluent
hypergeometric function book_GradshteynRyzhik-TableInt  and $\zeta\equiv
1/4\pi\epsilon$. Using Eq. (32), fully explicit expressions can be written for
the velocity and concentration profiles. The resulting formulas then
constitute the fundamental solution for Marangoni spreading with surface
diffusion.

Figure 4:  Influence of surface diffusion on the transient spreading of a
Dirac pulse. Here, time is fixed to $t=0.5$ and the concentration profile is
shown for $\epsilon=0.1,0.3,1,3,10$ from top to bottom. The dashed lines
correspond to Gaussian distributions with variance
$2tD_{\mathrm{eff}}(\epsilon)$. Inset: Effective diffusion coefficient
$D_{\mathrm{eff}}(\epsilon)$ as given by Eq. (38). Approximations from Eqs.
(39) and (40) are also shown with dashed lines.

####  IV.2.2 Effective diffusion coefficient

The primary effect of diffusion is to smooth out the distribution. This is
illustrated in Fig. 4 at fixed time: the stronger the diffusion coefficient,
the wider the surfactant distribution. Because the spreading of a Cauchy pulse
is already diffusive-like when $\epsilon=0$, one expects that the whole
process — now including genuine surface diffusion — can be described with an
effective diffusion coefficient. To do so, we focus on the concentration at
the origin, which is given by the remarkably simple expression

$\displaystyle f(x=0,t)=\frac{1}{\sqrt{4\pi D_{\mathrm{eff}}t}},$ (37)

with an effective coefficient $D_{\mathrm{eff}}$ defined as

$\displaystyle
D_{\mathrm{eff}}(\epsilon)=\pi\epsilon\,\left[\frac{\Gamma\left(1+\zeta\right)}{\Gamma\left(\frac{1}{2}+\zeta\right)}\right]^{2},\quad\zeta\equiv
1/4\pi\epsilon.$ (38)

Equation (37) is valid at all time and matches rigorously what is expected for
a purely diffusive process. A similar dependence also holds asymptotically at
long time for any finite position $x$. The effective diffusion coefficient
$D_{\mathrm{eff}}(\epsilon)$ is plotted as a function of $\epsilon$ in the
inset of Fig. 4. We can identify the limiting behaviors

$\displaystyle\epsilon\rightarrow 0,$ $\displaystyle\quad
D_{\mathrm{eff}}(\epsilon)=\frac{1}{4}\left(1+\pi\epsilon+\frac{\pi^{2}}{2}\epsilon^{2}+...\right),$
(39) $\displaystyle\epsilon\rightarrow\infty,$ $\displaystyle\quad
D_{\mathrm{eff}}(\epsilon)=\epsilon+C_{1},$ (40)

where $C_{1}\approx 0.22$ is a numerical constant 666
$C_{1}=-(\gamma_{\mathrm{E}}+\psi^{(0)}(1/2))/2\pi$, with
$\gamma_{\mathrm{E}}$ Euler’s constant and $\psi$ a polygamma function.. One
thus recovers the expected values in two limits. For very weak diffusion
$\epsilon\ll 1$, the value of the effective coefficient approaches
$D_{\mathrm{eff}}=1/4$, in agreement with Eq. (19a) for the zero-diffusion
case 777Remember that we set $a\mathcal{A}=1/\pi$ and take the limit $a\to
0$.. On the other hand, one obtains $D_{\mathrm{eff}}\simeq\epsilon$ when
intrinsic diffusion dominates. Finally, since a Gaussian behavior is expected
for pure diffusion, the concentration profiles can be compared to Gaussian
distributions with variance $2tD_{\mathrm{eff}}(\epsilon)$, as done in Fig. 4.
Though small discrepancies are visible in the tails of the distributions, the
agreement is fairly good. The fundamental solution of Marangoni spreading may
thus be reasonably approximated as a diffusive process with effective
coefficient $D_{\mathrm{eff}}(\epsilon)$ given by Eq. (38).

We have focused so far on surfactant spreading in the transient regime. Yet
another situation of interest is when the surfactant is released continuously
Roche_prl-2014 ; LeRoux_pre-2016 ; Mandre_jfm-2017 ; Kitahata_jcp-2018 ;
Arangalage_sm-2018 ; Benouaguef_jfm-2021 . We describe in App. A the solutions
that are available for this steady state source in the one-dimensional case.

##  V Axially symmetric spreading in two dimensions

###  V.1 Closure relation and Riesz transform

In this section, we briefly discuss transient surfactant spreading in higher
dimensionality. Evaporation and diffusion are discarded everywhere. Space
dimensionality is noted $D$ and the dimension of the interface is $d=D-1$.
Keeping the assumptions made in Sec. II, the closure relation of Eq. (7) can
be generalized in any space dimension as Thess_jfm-1997

$\displaystyle\mathbf{u}_{\mathrm{s}}(\mathbf{x})$
$\displaystyle=-\bm{\nabla}(-\Delta)^{-1/2}f(\mathbf{x}),$ (41)

or, equivalently, in Fourier representation

$\displaystyle\hat{\mathbf{u}}_{\mathrm{s}}(\mathbf{q})$
$\displaystyle=-\frac{i\mathbf{q}}{q}\hat{f}(\mathbf{q}).$ (42)

Here, $\mathbf{u}_{\mathrm{s}}$ denotes the in-plane velocity at the
interface, $\mathbf{x}$ is the position along the interface and
$\hat{f}\equiv\mathcal{F}[f]$ is the Fourier transform of a function $f$
888Equation (42) depends on the convention chosen for the definition of
Fourier transform.. For the dimension $d=1$ discussed until now, the velocity
is given by the Hilbert transform of the concentration. For the dimension
$d=2$ considered in this section, the velocity can be expressed as the Riesz
transform of the concentration, $\mathbf{u}_{s}=\mathcal{R}[f]$, where we
define

$\displaystyle\mathcal{R}[f](\mathbf{x})\equiv\frac{1}{2\pi}\mbox{p.v.}\int\frac{\mathbf{x}-\mathbf{x}^{\prime}}{|\mathbf{x}-\mathbf{x}^{\prime}|^{3}}f(\mathbf{x}^{\prime})\mathrm{d}\mathbf{x}^{\prime}.$
(43)

Consequently, the closure relation is again non-local.

Whereas an extensive literature can be found on Hilbert transforms, the use of
Riesz transforms appears to be less common. Moreover, it does not seem
possible to recast the problem as a Burgers equation. From now on, we restrict
the discussion to radially symmetric pulses, such that the concentration $f$
depends only on the distance $r$ to the origin. Denoting as $\mathbf{e}_{r}$
the radial unit vector, the velocity is also radial with component
$u_{r}=\mathcal{R}[f]\cdot\mathbf{e}_{r}\equiv\check{f}$. The equation
governing the evolution of surfactant distribution $f(r,t)$ is then

$\displaystyle\partial_{t}f+\frac{1}{r}\partial_{r}\left(rf\check{f}\right)=0.$
(44)

Even within these assumptions, computing the Riesz transform is in general not
straightforward. Several expressions that are useful for this purpose are
given in App. B.

###  V.2 Transient spreading

By analogy with the one-dimensional case, we consider the spreading of a
circular pulse

$\displaystyle
f(r,t)=\frac{a^{2}\mathcal{A}}{a^{2}(t)}\left[1-\left(\frac{r}{a(t)}\right)^{2}\right]^{1/2}_{+},$
(45)

where $a$ and $\mathcal{A}$ are respectively the initial radius and amplitude,
and the time-dependent radius $a(t)$ is to be determined. Computing the Riesz
transform, we get

$\displaystyle\check{f}(r,t)$ $\displaystyle=\frac{\pi
a^{2}\mathcal{A}}{4a^{3}(t)}r,$ (46) $\displaystyle\check{f}(r,t)$
$\displaystyle=\frac{a^{2}\mathcal{A}}{2a^{3}(t)}\left[-\frac{a(t)}{r}\sqrt{r^{2}-a^{2}(t)}+r\arcsin\left(\frac{a(t)}{r}\right)\right].$

for $r\leqslant a(t)$ and $r\geqslant a(t)$ respectively. Equation (44) is
then satisfied provided $a^{\prime}(t)a^{2}(t)=-\pi a^{2}\mathcal{A}/4$, which
finally leads to

$\displaystyle a(t)=\left(a^{3}+\frac{3\pi
a^{2}\mathcal{A}}{4}t\right)^{1/3}.$ (47)

We recover the self-similar solution that was identified in the mathematical
literature Biler_crasp-2011  but using a completely different method. The
transient spreading in the two-dimensional geometry is therefore subdiffusive,
since the spatial extent increases as $a(t)\sim t^{1/3}$.

Figure 5: Concentration and velocity profiles for the circular solution to
transient spreading. Shown are the rescaled concentration
$f_{\mathrm{rs}}\equiv\lambda(t)f(r,t)$ and the rescaled velocity
$u_{\mathrm{rs}}\equiv\lambda(t)u(r,t)$, with a scaling factor
$\lambda(t)=(a(t)/a)^{d}/\mathcal{A}$, $d$ the dimension, $a(t)$ the radius at
time $t$ and $a$ the initial radius. The case $d=2$ (solid lines) and $d=1$
(dashed line) correspond respectively to Eqs. (45)-(46) and Eq. (22). The
dotted line is the large distance approximation of velocity from Eqs. (48) and
(49) taken at lowest order.

It is interesting to compare the characteristics of the circular pulse for
$d=1$ and $d=2$. As visible in Fig. 5, both solutions have a finite support
within which the velocity is linear and also feature an angular point with
infinite slope at the boundary $r=a(t)$. At distances much larger than $a(t)$,
the velocity behaves as

$\displaystyle d=1,\quad\frac{u(x)}{a\mathcal{A}}$
$\displaystyle=\frac{1}{2x}+\frac{a^{2}+2a\mathcal{A}t}{8x^{3}}+...,$ (48)
$\displaystyle d=2,\quad\frac{u(r)}{a^{2}\mathcal{A}}$
$\displaystyle=\frac{1}{3r^{2}}+\frac{a^{2}(t)}{10r^{4}}+...$ (49)

The velocity is thus proportional to the “volume” of the pulse $\sim
a^{d}\mathcal{A}$ and decays as a power law $u(r)\sim r^{-d}$.

###  V.3 Steady spreading from a source

As the last solvable case, let us consider a punctual source releasing a
steady flux of surfactant $Q$ that spreads in two dimensions, without
evaporation. Assuming steady state, Eq. (44) or surfactant conservation gives

$\displaystyle 2\pi r\,f(r)\,\check{f}(r)=Q.$ (50)

Using the Riesz transform expressions of App. B, one can show that a solution
is

$\displaystyle f(r)$
$\displaystyle=\sqrt{\frac{2}{\pi}}\frac{\Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{3}{4}\right)}\,\sqrt{\frac{Q}{r}},$
(51a) $\displaystyle u(r)$
$\displaystyle=\sqrt{\frac{2}{\pi}}\frac{\Gamma\left(\frac{3}{4}\right)}{\Gamma\left(\frac{1}{4}\right)}\,\sqrt{\frac{Q}{r}}.$
(51b)

The concentration $f$ and the velocity $u=\check{f}$ display the same power-
law dependence, a behavior clearly distinct from diffusive behavior where no
steady state exists when $d=2$. The $r^{-1/2}$ dependence might be understood
from a simple argument. Since there is no characteristic length scale other
than the distance to the source, one can expect a typical viscous stress
$u(r)/r$. On the other hand, the typical Marangoni stress is $f^{\prime}(r)$.
Equating viscous and Marangoni stresses and using Eq. (50), one finds
$f^{\prime}(r)f(r)\sim r^{-2}$ and $f(r)\sim u(r)\sim r^{-1/2}$.

For a steady source of insoluble surfactant Mandre_jfm-2017 ; Bandi_prl-2017 ,
Mandre found that the velocity field decays as $r^{-\nu}$ with $\nu=3/5$,
whereas we find an exponent $\nu=1/2$. The former applies when there is a
boundary layer for the flow, the latter holds for Stokes flow. An other point
of comparison is the exact solution from Bratukhin and Maurin for a steady
punctual source of heat or soluble surfactant in three dimensions
Bratukhin_jamm-1967 ; Shtern_jfm-1993 , in which the far-field velocity decays
as $r^{-1}$. Spreading of an insoluble surfactant in the Stokes regime has
thus a distinct hydrodynamic signature with the slowest velocity decay.

##  VI Discussion

To summarize, we developed a unified analytical approach to describe the
transient and steady Marangoni spreading of an insoluble surfactant in the
Stokes flow limit 999Whether in a specific experiment the surfactant may
actually be considered as insoluble depends on a number of factors, including
the characteristic time for desorption from the interface, the time over which
spreading is observed and the initial concentration of surfactant [44].. In
the one-dimensional case, the mapping between an intricate set of transport
equations and a complex Burgers equation is derived in a straightforward
manner using the properties of Hilbert transforms. The solutions previously
uncovered in the various fields of the literature – from applied mathematics
to physical chemistry through fluid mechanics – are gathered within a single
framework. Importantly, our approach allows to identify all cases where a
fully explicit solution is possible. By investigating a number of them, we
show that the nonlinearity of the Burgers equation may lead to a variety of
behaviors in spreading. Finally, for an initial surfactant distribution with
arbitrary shape, the solution will not be generally available in analytical
form but may be obtained numerically by solving a simple equation. With a set
of exact solutions in hands, we can now discuss more thoroughly the physical
insights they provide. To do so, we consider several points in turn, focusing
on the one-dimensional case if not mentioned otherwise.

##### Time scale for Marangoni spreading.

If we switch back to dimensional variables, the time scale involved with
transient surfactant spreading reads $\tau=2\eta L/\kappa\Gamma_{\mathrm{o}}$,
with $L$ the characteristic length scale of the perturbation. This time scale
is thus inversely proportional to the surfactant concentration. Taking
$\Gamma_{\mathrm{o}}\simeq 10^{3}$ molecules/µm2, which corresponds to a low
surface fraction $\varphi\simeq 10^{-3}$, $\kappa=k_{\mathrm{B}}T\simeq
4\times 10^{-21}\,$J at room temperature and the viscosity of water
$\eta\simeq 10^{-3}\,$Pa s, one gets a macroscopic time $\tau\simeq 1\,$s for
$L=1\,$mm. Our results are thus relevant with regards to experimental time
scales.

##### (Dis)similarities between pulses and holes.

To compare the complementary situations of pulses and holes, we examine the
long-time behavior of the concentration $f(x,t)$, which is normalized by the
“volume” $a\mathcal{A}$ of the initial perturbation so that whatever the
amount of surfactant, the spatial integral is unity. For any finite position
$x$, the concentration evolves asymptotically as

$\displaystyle
t\rightarrow\infty,\quad\frac{f(x,t)}{a\mathcal{A}}\simeq\frac{1}{\sqrt{\chi
a\mathcal{A}t}},$ (52)

for a pulse whereas for a hole

$\displaystyle
t\rightarrow\infty,\quad\frac{\bar{c}-f(x,t)}{a\mathcal{A}}\simeq\frac{1}{\chi\bar{c}t},$
(53)

with $\chi$ a numerical constant 101010$\chi=1$ and $2$ for Cauchy and
circular cases respectively.. Two differences can be emphasized. (i) The time
relaxation of pulses is essentially diffusive-like, with an effective
diffusion coefficient $D_{\mathrm{eff}}\sim a\mathcal{A}$ proportional to the
total amount of surfactant in the pulse. In contrast, hole closing cannot be
described as a diffusive process. (ii) The dynamics of hole closing is much
faster than pulse spreading. This difference arises from the non-linear nature
of the transport equation. Indeed, with a linear equation, holes and pulses
would evolve in the same way. This clearly indicates that any analogy with
effective diffusion has to be handled carefully.

##### Surfactant spreading on a contaminated surface.

Our results also indicate that Marangoni spreading is very sensitive to the
initial state of the surface: There is an essential difference in the behavior
expected with a perfectly clean surface and a surface already covered with
endogenous surfactants Grotberg_japphysio-1995 ; Sauleda_langmuir-2021 .
Indeed, the solution for the holes still holds when $\mathcal{A}<0$, that is
when a surfactant pulse is released on a surface with an initial homogeneous
concentration $\bar{c}$. In the long-time limit, the spreading dynamics thus
depends on whether the interface is clean or contaminated with surface-active
molecules. Such a change in behavior can be traced back to the additional term
$\bar{c}\,\partial_{x}\breve{f}$ appearing in Eq. (9), which describes the
transport of endogenous surfactants by the Marangoni flow due to the added
surfactant pulse. Because the distribution of endogenous surfactants may
become inhomogeneous, a contaminated interface with concentration $\bar{c}$ is
not equivalent to a clean interface with surface tension
$\gamma_{\mathrm{o}}-\kappa\bar{c}$. In particular, the asymptotic dynamics of
Eq. (52) for a clean interface cannot be recovered by taking the limit
$\bar{c}\rightarrow 0$ in Eq. (53). This point is especially relevant for
experiments with aqueous solvents. Indeed, it may provide a clear signature of
interfacial contamination that could be tested experimentally.

##### Mode decomposition.

Whether the decay of an initial surfactant pulse can be decomposed in
sinusoidal modes is a natural question. This issue has been addressed recently
in a linearized version of the transport equations Mcnair_jfm-2022 . Here, we
solved the full nonlinear equations. We note in particular that, when the
amplitude of the perturbation is small with respect to mean level
($\mathcal{A}\ll\bar{c}$), the surfactant profile indeed remains sinusoidal
while decaying exponentially. This is expected since this assumption makes the
problem linear Shardt_jfm-2016 . On the other hand, nonlinear effects become
relevant as soon as $\mathcal{A}/\bar{c}$ is not well below unity and the mode
decomposition does not apply anymore.

##### Effective diffusion and space dimension.

In the one-dimensional geometry, we have identified several cases – including
surfactant pulse with and without surface diffusion – where the Marangoni
spreading can be mapped to a diffusion process, though with specific features.
For a two-dimensional interface, the algebra is much more involved and only
two analytical solutions are so far available: the spreading dynamics for a
circular pulse and the steady state source. For the former, the spreading in
the long-time limit is subdiffusive with exponent $1/3$. For the latter, the
velocity decays as a power law, at odds with a diffusive behavior. Taken
together, these findings point to the impossibility of defining an effective
diffusion coefficient in the two-dimensional case. Since this situation is the
most relevant experimentally, this calls for caution. Even though the idea of
effective diffusion is sometimes invoked, it appears difficult, in general, to
map a transport process dominated by Marangoni convection onto a simple
diffusive process.

To conclude, we briefly discuss how our findings may be tested experimentally.
Predictions for Marangoni spreading have long been confronted to experiments
Matar_sm-2009 , using for instance fluorescent surfactants Fallest_njp-2010 ;
Usma_langmuir-2022 . To reproduce the situations considered in this work, two
peculiar features are needed. First, the set-up must allow to tailor the
initial profile of surfactant to a prescribed shape. Second, the one-
dimensional geometry is preferable because it is best understood and amenable
to complete predictions. The controlled deposition of surfactant at the
interface might be a possible option. However, the use of photoswitchable
surfactant molecules Eastoe_sm-2005 ; Liu_jcis-2009 ; Chevallier_sm-2013 ;
Kavokine_acie-2016  could be an ideal method to induce the pulse, hole and
periodic patterns that we have considered theoretically. Indeed, because the
activated surfactants appear right at the interface, the perturbations are
minimal. Besides, the one-dimensional geometry is easily imposed and the
concentration profile can be prescribed through the light intensity. We thus
hope that the exact solutions found in this work can be put to experimental
test, so as to provide, for the one-dimensional case at least, a complete
picture for the viscous Marangoni spreading of an insoluble surfactant.

##  Appendix A Steady release of surfactant

Figure 6:  Steady state around a source of surfactant. (a) Without
evaporation. The punctual source at the origin is surrounded by two punctual
sinks located at $x=l=\pm 1$. The dashed line is the local flux of surfactant
$j(x)=u(x)f(x)$. (b) With evaporation. The source at the origin is of Cauchy
type, with $a=1$ and $\alpha=1$. The dashed lines show the approximations at
small and large $x$.

We seek a steady state in the plane flow geometry when the surfactant is
continuously released at the surface with a distributed source profile $q(x)$.
Introducing the complex flux $\mathcal{Q}=\breve{q}-iq$ and neglecting
diffusion, the equation to solve is now

$\displaystyle\Psi\partial_{x}\Psi=-\alpha\Psi+\mathcal{Q}(x).$ (54)

As before, the source profile more susceptible to analytical treatment is a
Cauchy profile and we set $\mathcal{Q}=1/(x+ia)$.

##### No evaporation.

For a single Cauchy source, the solution to Eq. (54) diverges for
$x\rightarrow\infty$, suggesting that there is no steady state. Let us assume
then that some local surfactant sink exists, where surfactant molecules
disappear. This is physically realizable if a photoswitchable surfactant can
be instantaneously disabled with a certain light. We can thus consider a
source surrounded by two sinks at position $\pm l$, all of Cauchy type. The
resulting steady state is described by

$\displaystyle\Psi(x)$
$\displaystyle=-i\sqrt{\operatorname{Log}(l^{2}+(a-ix)^{2})-2\operatorname{Log}(a-ix)}.$
(55)

In the limit of a punctual source $a\rightarrow 0$, one gets
$\Psi(x)=-i\sqrt{\operatorname{Log}(1-l^{2}/x^{2})}$, where $x>0$ is assumed
for convenience. The concentration and velocity fields can be written
explicitly by using the relation
$\operatorname{Log}\mathrm{z}=\ln|\mathrm{z}|+i\operatorname{Arg}\mathrm{z}$
for a complex number $\mathrm{z}$.

The resulting steady state is illustrated in Fig. 6a. First, one can check
that the flux of surfactant is constant from the source to the sink. Second,
the concentration vanishes for $|x|>l$ whereas the velocity does not. This
illustrates the non-local relation between concentration and velocity. Third,
one can note the diverging slope of concentration and velocity profiles in the
vicinity of sources and sinks. Finally, the concentration is clearly distinct
from the linear profile expected with pure diffusion. The one-dimensional
steady source with sinks is another situations where Marangoni spreading is
not diffusive-like.

##### With evaporation.

Evaporation acts as a distributed sink and presumably ensures the existence of
a steady state. Even with the simplest case of a Cauchy source, the
integration of Eq. (54) leads to an equation on $\Psi$ that is untractable. We
can however look for an expansion of $\Psi$ at small and large $x$. For small
$x$, assuming $-i\Psi(x)=c_{0}+c_{1}ix+c_{2}x^{2}$ with all coefficients real
leads to an approximation that depends only on $c_{0}$. One finds
$c_{1}=1/ac_{0}-\alpha$ and $c_{2}=(\alpha
ac_{0}+c_{0}^{2}-1)/(2a^{2}c_{0}^{3})$. The coefficient $c_{0}$ is solution of
a transcendantal equation that can only be solved numerically. In the vicinity
of the source, the concentration profile is thus parabolic and the velocity
profile is linear. For large $x$, assuming a power series
$\Psi(x)=\sum_{n=1}^{\infty}c_{n}x^{-n}$ leads at lowest order to
$\Psi(x)=1/\alpha x-ia/\alpha x^{2}+...$ Far from the source, the velocity and
concentration fields decay as $x^{-1}$ and $x^{-2}$ respectively. Figure 6(b)
shows a solution of Eq. (54) obtained numerically, together with the
approximations above. The latter are quite satisfactory, except for position
around unity. Note finally that, consistent with the absence of steady
solution when $\alpha=0$, the limit of vanishing evaporation can not be taken.

##  Appendix B Riesz transform of a radial function

Obtaining the Riesz transform is not straightforward, even with the assumption
of radial symmetry. Accordingly, we give two expressions that we found useful
in this purpose. From its definition in Fourier space of Eq. (42), the Riesz
transform may be recast as

$\displaystyle\check{f}(r)=-\partial_{r}\left[\mathcal{F}^{-1}_{q\rightarrow
r}\left(\frac{1}{q}\mathcal{F}_{r\rightarrow q}(f(r))\right)\right],$ (56)

where $\mathcal{F}$ indicates a Fourier transform for a purely radial two-
dimensional function:

$\displaystyle\mathcal{F}_{r\rightarrow q}f$ $\displaystyle\equiv
2\pi\int_{0}^{\infty}f(r)J_{0}(qr)\,r\,\mathrm{d}r,$ (57a)
$\displaystyle\mathcal{F}^{-1}_{q\rightarrow r}f$
$\displaystyle\equiv\frac{1}{2\pi}\int_{0}^{\infty}f(q)J_{0}(qr)\,q\,\mathrm{d}q,$
(57b)

with $J_{m}$ the Bessel function of the first kind of order $m$. An equivalent
expression can be given in terms of Hankel transform 111111The expression
holds provided all involved integrals are convergent.

$\displaystyle\check{f}(r)=H^{-1}_{1,q\rightarrow r}\left[H_{0,r\rightarrow
q}f(r)\right],$ (58)

where $H_{\nu}$ is the Hankel transform of order $\nu$:

$\displaystyle H_{\nu,r\rightarrow q}f$
$\displaystyle\equiv\int_{0}^{\infty}f(r)J_{\nu}(qr)\,r\,\mathrm{d}r,$ (59a)
$\displaystyle H_{\nu,q\rightarrow r}^{-1}f$
$\displaystyle\equiv\int_{0}^{\infty}f(q)J_{\nu}(qr)\,q\,\mathrm{d}q.$ (59b)

As a side remark, we note that the Riesz transform may be interpreted by
analogy with electrostatic or gravitational field. If $f(r)$ is the radial
density of charge on an infinitely thin disk, $\check{f}(r)$ is the radial
component of the electric field in the plane of the disk. Let us then assume
that identically charged particles are restricted to move in a plane and in a
quiescent medium so that they have a fixed mobility, i.e. a linear
relationship between their velocity and the force acting on them. Then, if
non-electrostatic interaction can be neglected, the evolution of the charge
density is governed by Eq. (44) and the process is analogous to Marangoni
spreading.

##  Appendix C Long formulas

We give here some formulas that were too long to be included in the main body
of the text in Sec. III.2.

##### The Cauchy hole:

$\displaystyle 2t\Psi(x,t)=$ $\displaystyle
x+ia-i\bar{c}t-i\sqrt{a^{2}-a(4\mathcal{A}t-2\bar{c}t+2ix)+(\bar{c}t-ix)^{2}},$
(60a) $\displaystyle f(x,t)=$
$\displaystyle\frac{1}{2t}\left(\Upsilon^{+}-a+\bar{c}t\right),$
$\displaystyle\Upsilon^{\pm}$
$\displaystyle\equiv\sqrt{\sqrt{\xi^{2}+(2(a+\bar{c}t)x)^{2}}\pm\xi}/\sqrt{2},$
(60b) $\displaystyle u(x,t)=$
$\displaystyle\frac{1}{2t}\left(x-\Upsilon^{-}\right),$ $\displaystyle\xi$
$\displaystyle\equiv a^{2}+2a(-2\mathcal{A}+\bar{c})t+\bar{c}^{2}t^{2}-x^{2}.$
(60c)

##### The circular hole:

$\displaystyle\Psi(x,t)=$
$\displaystyle\frac{\mathcal{A}\left(\sqrt{-a^{2}+2a\mathcal{A}t-(\bar{c}t-ix)^{2}}+i\bar{c}t-x\right)-ia\bar{c}}{a-2\mathcal{A}t},$
(61a) $\displaystyle f(x,t)=$
$\displaystyle\frac{\mathcal{A}}{2\mathcal{A}t-a}\left(\Upsilon^{+}-a\bar{c}/\mathcal{A}+\bar{c}t\right),$
$\displaystyle\Upsilon^{\pm}$
$\displaystyle\equiv\sqrt{\sqrt{\xi^{2}+(2\bar{c}tx)^{2}}\pm\xi}/\sqrt{2},$
(61b) $\displaystyle u(x,t)=$
$\displaystyle\frac{\mathcal{A}}{2\mathcal{A}t-a}\left(x-\Upsilon^{-}\right),$
$\displaystyle\xi$ $\displaystyle\equiv
a^{2}-2a\mathcal{A}t+\bar{c}^{2}t^{2}-x^{2}.$ (61c)

## References

  * (1) Matar, O. K., Craster, R. V., Dynamics of surfactant-assisted spreading. Soft Matter 5, 3801 (2009).
  * (2) De Gennes, P., Brochard-Wyart, F., Quéré, D., Capillarity and wetting phenomena: drops, bubbles, pearls, waves, (Springer Verlag 2004).
  * (3) Scriven, L. E., Sternling, C. V., The Marangoni effects. Nature 187, 186 (1960).
  * (4) Manikantan, H., Squires, T. M., Surfactant dynamics: Hidden variables controlling fluid flows. J. Fluid Mech. 892, 1 (2020).
  * (5) Bel Fdhila, R. B., Duineveld, P. C., The effect of surfactant on the rise of a spherical bubble at high Reynolds and Peclet numbers. Phys. Fluids 8, 310 (1996).
  * (6) Takagi, S., Matsumoto, Y., Surfactant effects on bubbly flows. Annu. Rev. Fluid Mech. 43, 615 (2011).
  * (7) Palaparthi, R., Papageorgiou, D. T., Maldarelli, C., Theory and experiments on the stagnant cap regime in the motion of spherical surfactant-laden bubbles. J. Fluid Mech. 559, 1 (2006).
  * (8) Crowdy, D. G., Exact solutions for the formation of stagnant caps of insoluble surfactant on a planar free surface. J. Eng. Math. 131 (2021).
  * (9) Peaudecerf, F. J., Landel, J. R., Goldstein, R. E., Luzzatto-Fegiz, P., Traces of surfactants can severely limit the drag reduction of superhydrophobic surfaces. Proc. Natl. Acad. Sci. USA 114, 7254 (2017).
  * (10) Landel, J. R., Peaudecerf, F. J., Coleto, F. T., Gibou, F., Goldstein, R. E., Fegiz, P. L., A theory for the slip and drag of superhydrophobic surfaces with surfactant. J. Fluid Mech. 883, A18 (2020).
  * (11) Baier, T., Hardt, S., Influence of insoluble surfactants on shear flow over a surface in Cassie state at large Péclet numbers. J. Fluid Mech. 907, A3 (2021).
  * (12) Bickel, T., Effect of surface-active contaminants on radial thermocapillary flows. Eur. Phys. J. E 42, 16 (2019).
  * (13) Bickel, T., Loudet, J. C., Koleski, G., Pouligny, B., Hydrodynamic response of a surfactant-laden interface to a radial flow. Phys. Rev. F 4, 124002 (2019).
  * (14) Venerus, D. C., Nieto Simavilla, D., Tears of wine: New insights on an old phenomenon. Sci. Rep. 5, 16162 (2015).
  * (15) Trinschek, S., John, K., Thiele, U., Modelling of surfactant-driven front instabilities in spreading bacterial colonies. Soft Matter 14, 4464 (2018).
  * (16) Stetten, A. Z., Iasella, S. V., Corcoran, T. E., Garoff, S., Przybycien, T. M., Tilton, R. D., Surfactant-induced Marangoni transport of lipids and therapeutics within the lung. Curr. Opin. Colloid Interface Sci. 36, 58 (2018).
  * (17) Botte, V., Mansutti, D., Numerical modelling of the Marangoni effects induced by plankton-generated surfactants. J. Marine Syst. 57, 55 (2005).
  * (18) Morciano, M., Fasano, M., Boriskina, S. V., Chiavazzo, E., Asinari, P., Solar passive distiller with high productivity and Marangoni effect-driven salt rejection. Energy Environ. Sci. 13, 3646 (2020).
  * (19) Quéré, D., Fluid coating on a fiber. Annu. Rev. Fluid Mech. 31, 347 (1999).
  * (20) Sauleda, M. L., Hsieh, T. L., Xu, W., Tilton, R. D., Garoff, S., Surfactant spreading on a deep subphase: Coupling of Marangoni flow and capillary waves. J. Colloid Interface Sci. 614, 511 (2022).
  * (21) Cantat, I., Cohen-Addad, S., Elias, F., Graner, F., Hohler, R., Pitois, O., Rouyer, F., Saint-James, A., Foams - Structure and Dynamics, (Oxford 2013).
  * (22) Breward, C. J. W., Howell, P. D., The drainage of a foam lamella. J. Fluid Mech. 458, 379 (2002).
  * (23) Fei, W., Gu, Y., Bishop, K. J., Active colloidal particles at fluid-fluid interfaces. Curr. Opin. Colloid Interface Sci. 32, 57 (2017).
  * (24) Grosjean, G., Hubert, M., Collard, Y., Pillitteri, S., Vandewalle, N., Surface swimmers, harnessing the interface to self-propel. Eur. Phys. J. E 41, 137 (2018).
  * (25) Nakata, S., Pimienta, V., Lagzi, I., Kitahata, H., Suematsu, N. J., editors, Self-organized motion: Physicochemical design based on nonlinear dynamics, (RSC 2018).
  * (26) Rednikov, A. Y., Ryazantsev, Y. S., Velarde, M. G., On the development of translational subcritical marangoni instability for a drop with uniform internal heat generation. J. Colloid Interface Sci. 164, 168 (1994).
  * (27) Boniface, D., Cottin-Bizonne, C., Kervil, R., Ybert, C., Detcheverry, F., Self-propulsion of symmetric chemically active particles: Point-source model and experiments on camphor disks. Phys. Rev. E 99, 062605 (2019).
  * (28) Ender, H., Froin, A. K., Rehage, H., Kierfeld, J., Surfactant-loaded capsules as Marangoni microswimmers at the air–water interface: Symmetry breaking and spontaneous propulsion by surfactant diffusion and advection. Eur. Phys. J. E 44, 21 (2021).
  * (29) Suematsu, N. J., Nakata, S., Evolution of Self-Propelled Objects: From the Viewpoint of Nonlinear Science. Chem. Eur. J. 24, 6308 (2018).
  * (30) Nakata, S., Nagayama, M., Kitahata, H., Suematsu, N. J., Hasegawa, T., Physicochemical design and analysis of self-propelled objects that are characteristically sensitive to environments. Phys. Chem. Chem. Phys. 17, 10326 (2015).
  * (31) Dietrich, K., Jaensson, N., Buttinoni, I., Volpe, G., Isa, L., Microscale Marangoni surfers. Phys. Rev. Lett. 125, 098001 (2020).
  * (32) Boniface, D., Cottin-Bizonne, C., Detcheverry, F., Ybert, C., Role of Marangoni forces in the velocity of symmetric interfacial swimmers. Phys. Rev. F 6, 104006 (2021).
  * (33) Suematsu, N. J., Sasaki, T., Nakata, S., Kitahata, H., Quantitative estimation of the parameters for self-motion driven by difference in surface tension. Langmuir 30, 8101 (2014).
  * (34) Kitahata, H., Yoshinaga, N., Effective diffusion coefficient including the Marangoni effect. J. Chem. Phys. 148, 134906 (2018).
  * (35) Bickel, T., Spreading dynamics of reactive surfactants driven by Marangoni convection. Soft Matter 15, 3644 (2019).
  * (36) Jensen, O. E., Grotberg, J. B., Insoluble surfactant spreading on a thin viscous film: Shock evolution and film rupture. J. Fluid Mech. 240, 259 (1992).
  * (37) Afsar-Siddiqui, A. B., Luckham, P. F., Matar, O. K., The spreading of surfactant solutions on thin liquid films. Adv. Colloid Interface Sci. 106, 183 (2003).
  * (38) Dussaud, A. D., Matar, O. K., Troian, S. M., Spreading characteristics of an insoluble surfactant film on a thin liquid layer: Comparison between theory and experiment. J. Fluid Mech. 544, 23 (2005).
  * (39) Craster, R. V., Matar, O. K., Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131 (2009).
  * (40) Swanson, E. R., Strickland, S. L., Shearer, M., Daniels, K. E., Surfactant spreading on a thin liquid film: reconciling models and experiments. J. Eng. Math. 94, 63 (2015).
  * (41) Jensen, O. E., The spreading of insoluble surfactant at the free surface of a deep fluid layer. J. Fluid Mech. 293, 349 (1995).
  * (42) Roché, M., Li, Z., Griffiths, I. M., Le Roux, S., Cantat, I., Saint-Jalmes, A., Stone, H. A., Marangoni flow of soluble amphiphiles. Phys. Rev. Lett. 112, 208302 (2014).
  * (43) Le Roux, S., Roché, M., Cantat, I., Saint-Jalmes, A., Soluble surfactant spreading: How the amphiphilicity sets the Marangoni hydrodynamics. Phys. Rev. E 93, 013107 (2016).
  * (44) Bandi, M., Akella, V., Singh, D., Singh, R., Mandre, S., Hydrodynamic signatures of stationary Marangoni-driven surfactant transport. Phys. Rev. Lett. 119, 264501 (2017).
  * (45) Mandre, S., Axisymmetric spreading of a surfactant driven by self-imposed Marangoni stress under simplified transport. J. Fluid Mech. 832, 777 (2017).
  * (46) Thess, A., Spirn, D., Jüttner, B., Viscous flow at infinite Marangoni number. Phys. Rev. Lett. 75, 4614 (1995).
  * (47) Crowdy, D. G., Viscous Marangoni flow driven by insoluble surfactant and the complex Burgers equation. SIAM J. Appl. Math. 81, 2526 (2021).
  * (48) Shmyrov, A., Mizev, A., Surface diffusion in gaseous monolayers of an insoluble surfactant. Langmuir 35, 14180 (2019).
  * (49) Thess, A., Spirn, D., Jüttner, B., A two-dimensional model for slow convection at infinite Marangoni number. J. Fluid Mech. 331, 283 (1997).
  * (50) Kralchevsky, P. A., Danov, K. D., Chemical physics of colloid systems and interfaces, in Birdi, K. S., editor, Handbook of surface and colloid chemistry, (CRC Press 2016).
  * (51) Pawar, Y., Stebe, K. J., Droplet deformation in an extensional flow: the role of surfactant physical chemistry. Phys. Fluids 8, 1738 (1996).
  * (52) Thess, A., Stokes flow at infinite Marangoni number: exact solutions for the spreading and collapse of a surfactant. Phys. Scr. T67, 96 (1996).
  * (53) Pismen, L. M., Interaction of reaction-diffusion fronts and Marangoni flow on the interface of a deep fluid. Phys. Rev. Lett. 78, 382 (1997).
  * (54) Constantin, P., Lax, P. D., Majda, A., A simple one-dimensional model for the three-dimensional vorticity equation. Commun. Pure Appl. Math. 38, 715 (1985).
  * (55) King, F. W., Hilbert transforms, (Cambridge 2009).
  * (56) Burgers, J. M., The nonlinear diffusion equation, (Springer, Dordrecht 1974).
  * (57) Bonkile, M. P., Awasthi, A., Lakshmi, C., Mukundan, V., Aswin, V. S., A systematic literature review of Burgers’ equation with recent advances. Pramana - J. Phys. 90, 69 (2018).
  * (58) Hopf, E., The partial differential equation $u_{t}+uu_{x}=\mu\,u_{xx}$. Commun. Pure Appl. Math. 3, 201 (1950).
  * (59) Whitham, G. B., Linear and Nonlinear Waves, (Wiley, New York 1974).
  * (60) Arfken, G., Weber, H., Mathematical methods for physicists, volume 3, (Academic press New York 2005).
  * (61) To separate the real and imaginary part, one can use the relation $\sqrt{2}\sqrt{x+iy}=\sqrt{\rho+x}+i\operatorname{sign}(y)\sqrt{\rho-x}$, where $x$ and $y$ are reals and $\rho=\sqrt{x^{2}+y^{2}}$.
  * (62) Biler, P., Imbert, C., Karch, G., Barenblatt profiles for a nonlocal porous medium equation. C. R. Acad. Sci. Paris, Ser. I 349, 641 (2011).
  * (63) Barenblatt, G., Scaling, self-similarity and intermediate asymptotics, (Cambridge Univ Press 1997).
  * (64) Pattle, R. E., Diffusion from an instantaneous point source with a concentration-dependent coefficient. Quartely J. Mech. Appl. Math. 12, 407 (1959).
  * (65) The derivative of a Cauchy pulse leads to a third-order equation that is also solvable.
  * (66) In the sense that the norm of the difference between the solution and the circular solution tends toward zero at large time, that is $||f(x,t)-f_{\mathrm{c}}(x,t)||_{p}\to 0$ for $t\to\infty$, with $||\cdot||_{p}$ a $p$-norm.
  * (67) Xu, Y., Takayama, N., Komasu, Y., Takahara, N., Kitahata, H., Iima, M., Nakata, S., Self-propelled camphor disk dependent on the depth of the sodium dodecyl sulfate aqueous phase. Colloids Surf. A 635, 128087 (2022).
  * (68) For the concentration to remain positive, one should have $0\leqslant\mathcal{A}\leqslant\mathaccentV{bar}016{c}$.
  * (69) Corless, R. M., Gonnet, G. H., Hare, D. E., Jeffrey, D. J., Knuth, D. E., On the Lambert W function. Adv. Comput. Math. 5, 329 (1996).
  * (70) A Hubbard-Stratonovitch transform is first introduced for the exponential term. After switching the integration order, the double integral can also be obtained.
  * (71) Gradshteyn, I., Ryzhik, I., Table of integrals, series, and products, (Academic Press, London 2007).
  * (72) $C_{1}=-(\gamma_{\mathrm{E}}+\psi^{(0)}(1/2))/2\pi$, with $\gamma_{\mathrm{E}}$ Euler’s constant and $\psi$ a polygamma function.
  * (73) Remember that we set $a\mathcal{A}=1/\pi$ and take the limit $a\to 0$.
  * (74) Arangalage, M., Li, X., Lequeux, F., Talini, L., Dual Marangoni effects and detection of traces of surfactants. Soft Matter 14, 3378 (2018).
  * (75) Benouaguef, I., Musunuri, N., Amah, E. C., Blackmore, D., Fischer, I. S., Singh, P., Solutocapillary Marangoni flow induced in a waterbody by a solute source. J. Fluid Mech. 922, A23 (2021).
  * (76) Equation (42) depends on the convention chosen for the definition of Fourier transform.
  * (77) Bratukhin, Y. K., Maurin, L. M., Thermocapillary convection in a fluid filling a half-space. J. Appl. Math. Mech. 31, 605 (1967).
  * (78) Shtern, V. N., Hussain, F., Azimuthal instability of divergent flows. J. Fluid Mech. 256, 535 (1993).
  * (79) Whether in a specific experiment the surfactant may actually be considered as insoluble depends on a number of factors, including the characteristic time for desorption from the interface, the time over which spreading is observed and the initial concentration of surfactant [44].
  * (80) $\chi=1$ and $2$ for Cauchy and circular cases respectively.
  * (81) Grotberg, J. B., Halpern, D., Jensen, O. E., Interaction of exogenous and endogenous surfactant: Spreading-rate effects. J. Appl. Physiol. 78, 750 (1995).
  * (82) Sauleda, M. L., Chu, H. C., Tilton, R. D., Garoff, S., Surfactant-driven Marangoni spreading in the presence of predeposited insoluble surfactant monolayers. Langmuir 37, 3309 (2021).
  * (83) Mcnair, R., Jensen, O. E., Landel, J. R., Surfactant spreading in a two-dimensional cavity and emergent contact-line singularities. J. Fluid Mech. 930, A15 (2022).
  * (84) Shardt, O., Masoud, H., Stone, H. A., Oscillatory Marangoni flows with inertia. J. Fluid Mech. 803, 94 (2016).
  * (85) Fallest, D. W., Lichtenberger, A. M., Fox, C. J., Daniels, K. E., Fluorescent visualization of a spreading surfactant. New J. Phys. 12, 073029 (2010).
  * (86) Usma, C. L., Mariot, S., Goldmann, C., Roch, M., Salonen, A., Tresset, G., Fluorescent Marangoni flows under quasi-steady conditions. Langmuir 38, 9129 (2022).
  * (87) Eastoe, J., Vesperinas, A., Self-assembly of light-sensitive surfactants. Soft Matter 1, 338 (2005).
  * (88) Liu, X., Abbott, N. L., Spatial and temporal control of surfactant systems. J. Colloid Interface Sci. 339, 1 (2009).
  * (89) Chevallier, E., Saint-Jalmes, A., Cantat, I., Lequeux, F., Monteux, C., Light induced flows opposing drainage in foams and thin-films using photosurfactants. Soft Matter 9, 7054 (2013).
  * (90) Kavokine, N., Anyfantakis, M., Morel, M., Rudiuk, S., Bickel, T., Baigl, D., Light-driven transport of a liquid marble with and against surface flows. Angew. Chem., Int. Ed. 55, 11183 (2016).
  * (91) The expression holds provided all involved integrals are convergent.