1 Introduction

This paper has been inspired by growing interest in the dynamics in various settings, especially at low-Reynolds-number, of the combined effects of interfacial Marangoni stresses induced by thermocapillarity and those induced by evolving surfactant concentrations on an interface. The classic study of steady motion of a spherical droplet, without surfactants, in a linear temperature gradient is by Young et al. [1]. They showed analytically, in the limit of vanishing Reynolds, capillary and thermal Péclet numbers, how Marangoni stresses associated with thermocapillarity can work against the motion induced by a gravitational body force. They computed the strength of the temperature gradient needed to hold a bubble stationary. Homsy and Meiburg [2] used a two-dimensional setting to examine how insoluble surfactants affect the steady thermocapillary flow induced in a differentially heated slot, a problem previously investigated by Sen and Davis [3]. A key non-dimensional parameter introduced by Homsy and Meiburg [2] is a so-called elasticity number governing the relative sizes of the Marangoni stresses due to the two sources: surfactant concentration gradients and temperature gradients. They carry out a study of the interplay of this parameter with the surface Péclet number which quantifies the rate of surface diffusion of the insoluble surfactant. Other work on this problem was performed by Carpenter and Homsy [4] who used Wiener-Hopf techniques to study the formation of immobilized regions, or stagnant caps, that can arise when diffusive effects are small and Marangoni stresses associated with the presence of an immobilized surfactant are relatively weak compared to the thermocapillary stresses. Bickel [5] has built on the work of Carpenter and Homsy [4] with further studies of thermosolutal Marangoni effects due to a point source of heat in axisymmetric geometries involving both deep and shallow fluid regions. More recently, the coupling of temperature gradients and surfactant concentration gradients at a free surface has also been studied in the context of low-Reynolds-number locomotion with the moving entities on those surfaces given the designation Marangoni surfers [6]. Propulsion in viscous fluids is important across a wide range of areas, including targeted drug delivery, microsurgery, the study of biological microorganisms [7], and particle transport in microfluidic systems [8].

The present article is concerned with the unsteady dynamics of a two-dimensional bubble situated in a linear temperature gradient and whose surface is loaded with insoluble surfactant. The steady motion of a spherical droplet (where an inviscid bubble is included as a special case) in the same circumstances was studied both analytically and numerically by Kim and Subramanian [9] for the case of no surface diffusion, with diffusive effects considered later by the same authors in [10]. These studies also included the effect of a gravitational body force. The present study treats the closely analogous problem for a two-dimensional bubble but with no body forces and, in a departure from what was done by Kim and Subramanian [9, 10] for the spherical geometry, extends consideration to time-evolving flow so that spreading of an initial concentration of insoluble surfactant on the bubble boundary is captured.

In spite of the fundamental work on a two-dimensional differentially heated slot by Homsy and Meiburg [2] and Carpenter and Homsy [4], the thermosolutal Marangoni dynamics of a two-dimensional bubble appears not to have been tackled by previous authors, even in the steady case. This may be due to obstructions presented by the Stokes paradox [11]: it is not possible to solve the problem of a steadily translating bubble, droplet or particle subject to an external body force in two dimensions. However, while no solution for forced motion exists, there is no such obstruction to solving for force-free motion of a two-dimensional bubble induced simultaneously by thermocapillary and contamination-induced Marangoni effects. And, as will be shown here, this two-dimensional multiphysics problem happens to have a rich mathematical structure: the steady solutions at zero and infinite surface Péclet numbers can be found in closed analytical form, and the dynamical time-dependent problem can be reduced to a complex partial differential equation with a Burgers-type nonlinearity. Even more, it will be shown that this nonlinear partial differential equation can be linearized at any finite non-zero surface Péclet number, a surprising result of some theoretical importance given the nonlinear nature of the physical mechanisms. While it does not appear to be possible to find time-evolving analytical solutions to the resulting linear governing equation it can nevertheless be used as the basis for an efficient numerical method to calculate the dynamics of a bubble with some given initial surfactant concentration as it evolves to steady state. Such a method is described here.

The formulation generalizes, to the radial bubble geometry and to thermosolutal Marangoni dynamics, recent work by the author [12, 13] who has shown that, under the same modeling assumptions listed above, insoluble surfactant dynamics on the surface of a semi-infinite region of viscous fluid can be described by a complex Burgers equation at arbitrary surface Péclet number. Although the details are different, the main theoretical ramifications of that prior work carry over to the dynamics of a bubble experiencing Marangoni stresses with two physically distinct causes: thermocapillarity and surface contamination. A shared theoretical feature is the possibility to linearize the dynamical problem by a complex version of the Cole–Hopf transformation, a nonlinear change of variables used in compressible gas dynamics [12,13,14]. This paper therefore adds to mounting evidence that the theoretical study of Marangoni dynamics using complex partial differential equations of Burgers type is both valuable and fruitful. Crowdy et al. [15] have shown how the mathematical formulation in the half-plane fluid geometry considered by Crowdy [12, 13] can be extended to include a two-fluid scenario where the surfactant is soluble to the bulk. Other authors [5, 16] have recently built on the formulation set out in [12, 13] to reveal other new features of Marangoni spreading dynamics in the half-plane fluid geometry.

In a companion study [17] based on a formulation of the radial bubble geometry case akin to that given here, the author has examined viscous Marangoni migration of an inviscid bubble by surfactant spreading. In that problem, there is no additional thermocapillary stress caused by an imposed thermal gradient. As a consequence, the bubble motion is transient, and is arrested once the surfactant concentration has become uniform. Interest there is in quantifying the net displacement of the bubble after long times as a function of the initial surfactant distribution. When a temperature gradient is imposed, as in the present paper, the possibility now exists of steady migration at large times due to a balance between thermocapillary-induced and contamination-induced Marangoni stresses. As will be seen, in the absence of surface diffusion of surfactant such steady migration is not guaranteed but is contingent on the size of a non-dimensional parameter, herein called \(\nu \), measuring the relative sizes of these two surface stress contributions; it is the inverse of the elasticity number used by Homsy and Meiburg [2]. The presence of surface diffusion of surfactant is generally observed to promote net bubble migration at any value of \(\nu \) by limiting the potential of the surfactant to nullify thermocapillary effects. Of particular note is the fortuitous circumstance that many (but not all) of the important mathematical features that lead to the success of the analysis in [17] carry over when a background thermal gradient is introduced enabling key physical features of this multiphysics system to be exemplified quite explicitly.

The structure of this paper is as follows. After the coupled multiphysics problem is set out in Sect. 2 in a Cartesian (x, y) plane, the flow and temperature problems are reformulated in terms of three analytic functions of the complex variable \(z=x+\textrm{i} y\) in Sect. 3. It is then shown in Sect. 4 that the dynamical problem can be reduced to consideration of a single nonlinear complex partial differential equation (PDE) for a time-evolving function h(z, t) that is analytic and single-valued in the fluid region outside the bubble. This governing PDE has a nonlinearity characteristic of the Burgers equation. Steady solutions of this PDE are found explicitly for surface Péclet number \(Pe_s=0\) and \(Pe_s=\infty \) in Sects. 5 and 6, respectively. Some large \(Pe_s\) asymptotics are featured in Sect. 7. Section 8 then shows that a complex generalization of the classical Cole–Hopf transformation can be used to linearize the governing Burgers-type PDE derived in Sect. 4 and a numerical method based on the linearized PDE is formulated. Some illustrative calculations are given in Sect. 9 before the paper closes with a discussion of the main results in Sect. 10.

Fig. 1
figure 1

Dynamics of a two-dimensional inviscid bubble surrounded by viscous fluid in a linear temperature gradient \(T_\infty '\) given some initial concentration \(\Gamma _0(s)\) of insoluble surfactant. The challenge is to determine the unsteady bubble speed \(U_B(t)\) and the final steady migration speed of the bubble. There are Marangoni stresses due to both surfactant contamination and the presence of the temperature gradient

Full size image

2 Mathematical formulation of the multiphysics problem

A two-dimensional bubble of radius \(\mathcal{R}\) centered on the x axis in the (x, y) plane at time \(t=0\) is initially laden with insoluble surfactant having concentration \(\Gamma _0(s) = \Gamma (s, 0)\), where s denotes arclength around the bubble boundary, increasing in the clockwise direction. The bubble, which has constant pressure \(p_B\), sits in a uniform, time-independent, temperature gradient \(T_\infty '\) in the x-direction. Figure 1 shows a schematic. It will be supposed that the initial surfactant concentration is symmetric about the x axis passing through the center of the bubble; this stipulation is not necessary for the following analysis but it simplifies the presentation. The bubble is surrounded by fluid of viscosity \(\mu \) and its average absolute temperature is initially taken to be \(T_a\). The equation of state for the surface tension \(\sigma (s,t)\) on the bubble boundary has the form

$$\begin{aligned} \sigma = \sigma _c - \beta _\Gamma \Gamma - \beta _T T, \end{aligned}$$
(1)

where \(\sigma _c\) is the value of the clean-flow surface tension when \(\Gamma =0\) and \(T=0\); the quantity T in (1) is the departure of the temperature from the ambient value \(T_a\) and T is assumed to have zero average over the bubble surface. Under the assumption that \(T_\infty ' \mathcal{R} \ll T_a\) the parameter \(\beta _\Gamma \) is assumed to be well approximated by the constant value \(R T_a\) where R is the gas constant, The same approximation was made by Meiburg and Homsy [2]. With these assumptions, the equation of state (1) is linear in \(\Gamma \) and T. Since, subject to Marangoni stresses to be explained shortly, the bubble is expected to translate along the temperature gradient in the x direction with some speed \(U_B(t)\) the assumption that its average temperature equals \(T_a\) is only expected to be a good approximation for a limited window of time.

It is natural to move to a frame of reference cotravelling with the bubble at each instant with the a priori unknown speed \(U_B(t)\). The fluid velocity \(\textbf{u}=(u,v)\) outside the bubble in this frame satisfies the incompressible quasi-steady Stokes equations

$$\begin{aligned} - \nabla p + \mu \nabla ^2 \textbf{u} = 0, \qquad {\nabla }\cdot \textbf{u} = 0, \end{aligned}$$
(2)

where p(x, y, t) is the fluid pressure. One condition on the bubble boundary is

$$\begin{aligned} \textbf{u}\cdot \textbf{n} = 0, \end{aligned}$$
(3)

where n is the unit normal vector pointing into the viscous fluid from the bubble boundary. This states that the boundary must be a streamline in the cotravelling frame. The stress balance on the boundary of an inviscid bubble at constant pressure \(p_B\) is

$$\begin{aligned} - (p-p_B) n_\textrm{i} + 2 \mu \textrm{e}_\mathrm{{ij}} n_\textrm{j} = {\sigma \over \mathcal{R}} n_\textrm{i} - {\partial \sigma \over \partial s} t_\textrm{i}, \end{aligned}$$
(4)

where \(t_\textrm{i}\) and \(n_\textrm{i}\) denote components of the unit tangent t and normal n respectively and \(\textrm{e}_\mathrm{{ij}}\) is the usual fluid rate-of-strain tensor. The surface tension \(\sigma (s,t)\) is given by (1) and there are clearly two distinct physical sources of Marangoni stress, one due to surfactant contamination and another due to temperature variation over the bubble surface.

The surfactant concentration \(\Gamma (s,t)\) satisfies the surfactant evolution equation which, in a frame of reference comoving with the fixed-shape bubble, is [18]

$$\begin{aligned} {\partial \Gamma (s,t) \over \partial t} + {\partial (\Gamma (s,t) U(s,t)) \over \partial s} = {D_s} {\partial ^2 \Gamma (s,t) \over \partial s^2}, \end{aligned}$$
(5)

where U(s, t) is the surface slip in the tangential direction and \(D_s\) is the surface diffusion coefficient. On integrating (5) around the entire bubble boundary it follows that the average surfactant concentration over the interface is a constant of the motion, namely,

$$\begin{aligned} \langle \Gamma _0 \rangle \equiv {1 \over 2\pi \mathcal{R}} \int _{- \pi \mathcal{R}}^{\pi \mathcal{R}} \Gamma _0(s) \textrm{d} s = {1 \over 2\pi \mathcal{R}} \int _{- \pi \mathcal{R}}^{\pi \mathcal{R}} \Gamma (s,t) \textrm{d} s. \end{aligned}$$
(6)

As for the temperature, in the limit of zero thermal Péclet number the temperature field satisfies

$$\begin{aligned} \nabla ^2 T = 0, \end{aligned}$$
(7)

in the fluid region exterior to the bubble with

$$\begin{aligned} T \rightarrow T_\infty ' x, \qquad \textrm{as} \qquad |\textbf{x}| \rightarrow \infty . \end{aligned}$$
(8)

It is reasonable to assume that the inviscid, constant-pressure bubble is adiabatic, i.e.,

$$\begin{aligned} {\partial T \over \partial n} = 0, \qquad \textrm{on}~\quad |\textbf{x}|= \mathcal{R}. \end{aligned}$$
(9)

The condition on the fluid velocity in the far-field is that

$$\begin{aligned} \textbf{u} \rightarrow (-U_B(t),0), \qquad \textrm{as} \qquad |\textbf{x}| \rightarrow \infty , \end{aligned}$$
(10)

where the condition that the bubble must be free of net force will determine \(U_B(t)\). The assumed reflectional symmetry of the initial surfactant concentration about the real axis through the bubble center is expected to be dynamically preserved and ensures zero net torque on it.

3 Complex variable formulation

The analysis will be performed using a complex variable formulation of this multiphysics problem and considering analytic functions of the usual complex variable \(z=x+\textrm{i}y\).

In the cotravelling frame of reference the temperature field in this zero thermal Péclet number limit is time-invariant, and harmonic, so the following time-independent analytic function can be introduced:

$$\begin{aligned} q(z) = T+ \textrm{i} \chi , \end{aligned}$$
(11)

where \(\chi \) is the harmonic conjugate to T outside the bubble. By the Cauchy–Riemann equations it can be inferred without loss of generality from (9) that

$$\begin{aligned} \chi =0, \qquad \textrm{on} \qquad |z|=\mathcal{R}. \end{aligned}$$
(12)

It follows from (8) that \(q(z) \rightarrow T_\infty ' z\) as \(|z| \rightarrow \infty \). Standard arguments then lead to the explicit form

$$\begin{aligned} q(z) = T_\infty ' \left( z + {\mathcal{R}^2 \over z} \right) , \end{aligned}$$
(13)

where an additive constant has been set to zero to ensure the average of T over the bubble boundary is zero. Recall that T has been defined to be the departure of the temperature from the ambient value \(T_a\).

The quasi-steady Stokes flow generated by the Marangoni stresses can also be found using a complex variable formulation of two-dimensional Stokes flow of which an appendix of [19] sets out a derivation. On taking the curl of the Stokes equations (2) and introducing a streamfunction \(\psi \) associated with the incompressible two-dimensional flow

$$\begin{aligned} \textbf{u} = (u,v) = \left( {\partial \psi \over \partial y}, -{\partial \psi \over \partial x} \right) , \end{aligned}$$
(14)

it can be shown that \(\psi \) satisfies the biharmonic equation in the fluid region, namely,

$$\begin{aligned} \nabla ^4 \psi = 0, \qquad \nabla ^2 = {\partial ^2 \over \partial x^2} + {\partial ^2 \over \partial y^2}. \end{aligned}$$
(15)

This partial differential equation can be written as

$$\begin{aligned} {\partial ^4 \psi \over \partial z^2 \partial \overline{z}^2} = 0, \end{aligned}$$
(16)

which can be integrated [19] leading to a general representation of a real-valued biharmonic streamfunction given by

$$\begin{aligned} \psi = \psi (z,\overline{z},t) = \textrm{Im}[\overline{z}f(z,t) + g(z,t)], \end{aligned}$$
(17)

where \(\textrm{Im}[.]\) denotes the imaginary part of the complex quantity in square brackets and f(z, t) and g(z, t) are complex potentials, also known as Goursat functions. In general, all singularities of these analytic functions must be inside the bubble, or at infinity, and not in the viscous fluid. Moreover, a logarithmic singularity of f(z, t) inside the bubble can be identified with a Stokeslet singularity implying a non-zero net force on the bubble while a logarithmic singularity of g(z, t) inside the bubble can be identifed with a rotlet singularity implying a non-zero net torque on it. Since, in the present problem, the bubble is free of net force and torque both f(z, t) and g(z, t) will be analytic and single-valued in the viscous fluid region. This will be explained in more detail in what follows.

It can be shown [19] that p(x, y, t), the vorticity \(\omega = -\nabla ^2 \psi \), and the fluid rate-of-strain tensor \(\textrm{e}_\mathrm{{ij}}\) are related to f(z, t) and g(z, t) through the relations

$$\begin{aligned} \begin{aligned} 4 f'(z,t) = {p \over \mu } - \textrm{i} \omega ,&\qquad u - \textrm{i} v = - \overline{f(z,t)} + \overline{z} {f'(z,t)} + {g'(z,t)}, \\ \textrm{e}_{11} + \textrm{i} \textrm{e}_{12}&= z \overline{f''(z,t)} + \overline{g''(z,t)}, \end{aligned} \end{aligned}$$
(18)

where overbars denote complex conjugation and the prime notation denotes partial differentiation with respect to z. There is an additive degree of freedom in the choice of f(z, t) and \(g'(z,t)\) since if f(z, t) is changed to \( f(z,t) + c(t)\) and \(g'(z,t)\) is changed to \(g'(z,t) + \overline{c(t)}\), where c(t) is a complex function of time, then the complex velocity field \(u-\textrm{i}v\) given in (18) is unaltered. This degree of freedom will be eliminated shortly.

The complex form of the fluid stress at the boundary, or \(-p n_\textrm{i} + 2 \mu \textrm{e}_\mathrm{{ij}}n_\textrm{j}\), is [19]

$$\begin{aligned} - 2 \mu \textrm{i} {\partial H \over \partial s}, \qquad H(z,\overline{z},t) \equiv f(z,t) + z \overline{f'(z,t)} + \overline{g'(z,t)}. \end{aligned}$$
(19)

Since there is no net force on the bubble then integration of the fluid stress (19) with respect to arclength around its boundary must give zero. This shows that \(H(z,\overline{z},t)\) must be single-valued around the bubble. The single-valuedness of f(z, t) around the bubble then follows on noticing, from (18) and (19), that \(H(z,\overline{z},t) = 2f(z,t)+ (u+\textrm{i}v)\) and bearing in mind that \(u+\textrm{i} v\) is necessarily single-valued around the bubble.

On rearrangement the stress condition (4) on the bubble boundary can be written as

$$\begin{aligned} -p n_\textrm{i} + 2 \mu \textrm{e}_\mathrm{{ij}} n_\textrm{j} = {\sigma \over \mathcal{R}} n_\textrm{i} - {\partial \sigma \over \partial s} t_\textrm{i} - p_B n_\textrm{i}, \end{aligned}$$
(20)

and this will now be written in complex form. For this, note that the complex form of the unit tangent vector is

$$\begin{aligned} {\textrm{d}z \over \textrm{d}s} = - {\textrm{i} z \over \mathcal{R}}, \end{aligned}$$
(21)

and the complex form of the unit normal vector is

$$\begin{aligned} \textrm{i} {\textrm{d} z \over \textrm{d}s} = {z \over \mathcal{R}}. \end{aligned}$$
(22)

Consequently, (20) can be written as

$$\begin{aligned} \begin{aligned} - 2 \mu \textrm{i} {\partial H \over \partial s} = {\sigma \over \mathcal{R}} \textrm{i} {\textrm{d}z \over \textrm{d}s} - {\partial \sigma \over \partial s} {\textrm{d}z \over \textrm{d}s} - p_B \textrm{i} {\textrm{d} z \over \textrm{d}s}&= {\textrm{i} \sigma \over \mathcal{R}} {\textrm{d}z \over \textrm{d}s} + {\partial \sigma \over \partial s} {\textrm{i} z \over \mathcal{R}} - p_B \textrm{i} {\textrm{d} z \over \textrm{d}s} \\ &= {\textrm{i} \over \mathcal{R}} {\partial (\sigma z) \over \partial s} - p_B \textrm{i} {\textrm{d} z \over \textrm{d}s}, \end{aligned} \end{aligned}$$
(23)

where (19), (21) and (22) have been used. This can be integrated with respect to s:

$$\begin{aligned} - 2 \mu H(z, \overline{z},t) = {\sigma z \over \mathcal{R}} - p_B z, \end{aligned}$$
(24)

where an additive function of time has been set to zero without loss of generality by exploiting the aforementioned additive degree of freedom in the choice of f(z, t) and \(g'(z,t)\). On use of the equation of state (1), (24) implies that

$$\begin{aligned} {H(z, \overline{z},t) \over z} = {\beta _\Gamma \Gamma \over 2\mu \mathcal{R}} + +{\beta _T T \over 2\mu \mathcal{R}} + {1 \over 2\mu } \left( p_B - {\sigma _c \over \mathcal{R}} \right) . \end{aligned}$$
(25)

The next step is to introduce the decompositions

$$\begin{aligned} f(z,t) = {p_H \over 4 \mu } z + {\hat{f}}(z,t), \qquad g(z,t) = - {\mathcal{R}^2 {\hat{f}}(z,t) \over z}, \end{aligned}$$
(26)

where \(p_H\) is a real constant and \({\hat{f}}(z,t)\) is analytic and single-valued in the fluid region \(|z| > \mathcal{R}\) except for a simple pole at infinity, i.e.,

$$\begin{aligned} {\hat{f}}(z,t) \sim {{\hat{p}}(t) \over 4 \mu } z + U_B(t) + \mathcal{O}(1/z), \end{aligned}$$
(27)

as \(|z| \rightarrow \infty \) where \({\hat{p}}(t)\) is real-valued. The first term of f(z, t) in (26) encodes the hydrostatic pressure \(p_H\) of a clean bubble and the additional term \({\hat{f}}(z,t)\) will describe the quasi-steady Stokes flow generated by the Marangoni effects. The quantity \({\hat{p}}(t)\) represents the modification to the far-field fluid pressure due to the motion caused by the Marangoni effects and it is easy to check from (18) and (10) that the constant term in the far-field asymptotics (27) is the bubble speed \(U_B(t)\). The choice (26) satisfies the streamline condition (3) on the bubble surface for any choice of \({\hat{f}}(z,t)\). This is because, on \(|z| = \mathcal{R}\),

$$\begin{aligned} \psi (z, \overline{z}, t) = \textrm{Im}[\overline{z} f(z,t) + g(z,t)] = \textrm{Im} \left[ \overline{z} \left( {p_H \over 4 \mu } z + \hat{f}(z,t) \right) - {\mathcal{R}^2 {\hat{f}}(z,t) \over z} \right] = 0,\nonumber \\ \end{aligned}$$
(28)

where the fact that \(\overline{z}=\mathcal{R}^2/z\) on the bubble boundary has been used. The bubble boundary is therefore a streamline.

On substitution of the decompositions (26) into the definition of \(H(z, \overline{z}, t)\) given in (19) it follows that, on \(|z|= \mathcal{R}\),

$$\begin{aligned} H(z, \overline{z}, t)&= {p_H \over 4 \mu } z + {\hat{f}}(z,t) + {p_H \over 4 \mu } z + z \overline{{\hat{f}}'(z,t)} - \mathcal{R}^2 \overline{\left( {{\hat{f}}'(z,t) \over z} \right) } + \mathcal{R}^2 \overline{\left( {{\hat{f}}(z,t) \over z^2}\right) }\nonumber \\&= {p_H \over 2 \mu } z + {\hat{f}}(z,t) +\mathcal{R}^2 \overline{\left( {{\hat{f}}(z,t) \over z^2}\right) }, \end{aligned}$$
(29)

where the fact that \({z} = \mathcal{R}^2/\overline{z}\) on \(|z|= \mathcal{R}\) has been used. On division by z, this becomes

$$\begin{aligned} {H(z, \overline{z},t) \over z} = {p_H \over 2\mu } + 2 \textrm{Re} \left[ {{\hat{f}}(z,t) \over z} \right] = {p_H \over 2\mu } + \textrm{Re}[h(z,t)], \end{aligned}$$
(30)

where

$$\begin{aligned} h(z,t) \equiv {2 {\hat{f}}(z,t) \over z}. \end{aligned}$$
(31)

The function h(z, t) is analytic and single-valued in the fluid region, properties it inherits from \({\hat{f}}(z,t)\) in view of its definition (31) and the far-field asymptotics (27). Two expressions for H/z have now been derived in (25) and (30). Equating them leads to

$$\begin{aligned} {p_H \over 2\mu } + \textrm{Re}[h(z,t)] ={\beta _\Gamma \Gamma \over 2\mu \mathcal{R}} + +{\beta _T T \over 2\mu \mathcal{R}} + {1 \over 2\mu } \left( p_B - {\sigma _c \over \mathcal{R}} \right) , \end{aligned}$$
(32)

It turns out that the boundary slip velocity U(s, t) can also be conveniently expressed in terms of h(z, t). To see this, note that on \(|z|= \mathcal{R}\), where \(\overline{z} = \mathcal{R}^2/{z}\), it follows from (18) and (26) that

$$\begin{aligned} u - \textrm{i} v= & -\overline{{\hat{f}}(z,t)} + \overline{z} {\hat{f}}'(z,t) - \mathcal{R}^2 {{\hat{f}}'(z,t) \over z} + \mathcal{R}^2 {{\hat{f}}(z,t) \over z^2} \nonumber \\= & -\overline{{\hat{f}}(z,t)} + \mathcal{R}^2 {{\hat{f}}(z,t) \over z^2}. \end{aligned}$$
(33)

Using the fact that if \(a=a_x+\textrm{i} a_y\) and \(b=b_x+\textrm{i}b_y\) are the complex analogues of the vectors \(\textbf{a}=(a_x, a_y)\) and \(\textbf{b}=(b_x, b_y)\) then the complex analogue of the dot product \(\textbf{a}.\textbf{b}\) is \(\textrm{Re}[a \overline{b}]\) then the boundary slip velocity U(s, t) can be written, in complex form, as

$$\begin{aligned} \begin{aligned} U(s,t)&= \textbf{t}.\textbf{u} = \textrm{Re} \left[ - {\textrm{i} z \over \mathcal{R}} \left( -\overline{{\hat{f}}(z,t)} + \mathcal{R}^2 {{\hat{f}}(z,t) \over z^2} \right) \right] \\&= \textrm{Re} \left[ - {\textrm{i} \mathcal{R}} \left( {{\hat{f}}(z,t) \over z} - \overline{ \left( {{\hat{f}}(z,t) \over z} \right) } \right) \right] \\&= \mathcal{R} \textrm{Im} \left[ {{\hat{f}}(z,t) \over z} - \overline{ \left( {{\hat{f}}(z,t) \over z} \right) } \right] , \end{aligned} \end{aligned}$$
(34)

where (21) has been used. Therefore,

$$\begin{aligned} {U(s,t) \over \mathcal{R}} = 2 \textrm{Im} \left[ {{\hat{f}}(z,t) \over z} \right] = \textrm{Im}[h(z,t)]. \end{aligned}$$
(35)

4 A complex partial differential equation of Burgers type

A velocity scale associated with the presence of the surfactant is

$$\begin{aligned} U_0 = {\beta \langle \Gamma _0 \rangle \over 2 \mu }. \end{aligned}$$
(36)

It is natural to non-dimensionalize lengths using the bubble radius \(\mathcal{R}\), surfactant concentrations using \(\langle \Gamma _0 \rangle \), temperatures using \(T_\infty ' \mathcal{R}\), velocities induced by the presence of the surfactant using \(U_0\), and time using \(\mathcal{R}/U_0\). The hydrostatic pressure \(p_H\) and bubble pressure \(p_B\) are non-dimensionalized using \(\sigma _c/\mathcal{R}\). The bubble boundary is now \(|z|=1\) and it can be parametrized by \(z=\textrm{e}^{-\textrm{i}s}\) where s is the non-dimensional arclength.

With this non-dimensionalization the complex potential (13) becomes

$$\begin{aligned} q(z) = z + {1 \over z}. \end{aligned}$$
(37)

And noting that \({\hat{f}}(z,t)\) has the dimension of a velocity, the non-dimensional version of (35) is

$$\begin{aligned} {{\tilde{U}}(s,t)} = \textrm{Im}[{\tilde{h}}(z,t)], \end{aligned}$$
(38)

where a quantity decorated with a tilde denotes its non-dimensional counterpart. Equation (32) becomes

$$\begin{aligned} {\sigma _c \over 2 \mu \mathcal{R}} {\tilde{p}}_H + {U_0 \over \mathcal{R}} \textrm{Re}[{\tilde{h}}(z,t)] = {\sigma _c \over 2 \mu \mathcal{R}} (\tilde{p}_B-1) + {\beta \langle \Gamma _0 \rangle \over 2 \mu \mathcal{R}} {\tilde{\Gamma }} + {\beta _T T_\infty ' \mathcal{R} {\tilde{T}} \over 2\mu \mathcal{R}}, \end{aligned}$$
(39)

or

$$\begin{aligned} {\tilde{p}}_H + {2 \mu U_0 \over \sigma _c} \textrm{Re}[{\tilde{h}}(z,t)] = {\tilde{p}}_B -1 + {2 \mu U_0 \over \sigma _c} ({\tilde{\Gamma }} + \nu {\tilde{T}}), \end{aligned}$$
(40)

where (36) has been used and the non-dimensional parameter

$$\begin{aligned} \nu = {\beta _T T_\infty ' \mathcal{R} \over \beta _\Gamma \langle \Gamma _0 \rangle } \end{aligned}$$
(41)

introduced. This parameter \(\nu \), which measures the strength of the thermocapillary Marangoni stresses relative to those induced by the surfactant, is the inverse of the elasticity number used by Homsy and Meiburg [2]. The capillary number

$$\begin{aligned} \textrm{Ca}={2 \mu U_0 \over \sigma _c} \end{aligned}$$
(42)

can also now be introduced and taken to be small, i.e., \(\textrm{Ca} \ll 1\). In terms of it, (40) becomes

$$\begin{aligned} {\tilde{p}}_H + \textrm{Ca} ~\textrm{Re}[{\tilde{h}}(z,t)] = {\tilde{p}}_B -1 +\textrm{Ca}~ ({\tilde{\Gamma }} + \nu {\tilde{T}}). \end{aligned}$$
(43)

Hence, at leading order the Laplace–Young balance holds

$$\begin{aligned} {\tilde{p}}_B - {\tilde{p}}_H=1, \end{aligned}$$
(44)

and, at first order in Ca, it follows that

$$\begin{aligned} \textrm{Re}[{\tilde{h}}(z,t)] = {\tilde{\Gamma }} + \nu {\tilde{T}}. \end{aligned}$$
(45)

Henceforth, the tildes on any non-dimensionalized quantities will be dropped. Together, (38) and (45) imply that

$$\begin{aligned} h(z,t) = \Gamma + \nu T + \textrm{i} U. \qquad \textrm{on}~|z|=1. \end{aligned}$$
(46)

This is an important relation. A consequence of it is that the average of the real part of h(z, t) around the bubble boundary is unity since, by the choice of scaling, this is the value of the surface average of the \(\Gamma \) while the average of T around the boundary is assumed to be zero.

The next step is to rewrite the surfactant evolution equation (5) in terms of this complex variable formulation. The non-dimensional form of (5) is

$$\begin{aligned} {\partial \Gamma (s,t) \over \partial t} + {\partial (\Gamma (s,t) U(s,t)) \over \partial s} = {1 \over Pe_s} {\partial ^2 \Gamma (s,t) \over \partial s^2}, \end{aligned}$$
(47)

where the surface Péclet number \(Pe_s\) is defined by

$$\begin{aligned} Pe_s = {U_0 \mathcal{R} \over D_s}. \end{aligned}$$
(48)

It follows from (46) that, on the bubble boundary,

$$\begin{aligned} \Gamma = \textrm{Re}[h(z,t)] - \nu T = \textrm{Re}[ h(z,t) - \nu q(z)], \end{aligned}$$
(49)

where (11) and (12) have been used, and therefore that

$$\begin{aligned} {\partial \Gamma \over \partial t} = \textrm{Re} \left[ {\partial h(z,t) \over \partial t} \right] . \end{aligned}$$
(50)

This is the first term in (47). On the bubble boundary it is also true that

$$\begin{aligned} h^2(z,t) = (\Gamma +\nu T)^2 - U^2 + 2 \textrm{i} ( \Gamma + \nu T) U, \qquad \textrm{on}~|z|=1, \end{aligned}$$
(51)

where (46) has simply been squared. Consequently, on the boundary,

$$\begin{aligned} U ( \Gamma + \nu T) = \textrm{Re} \left[ - {\textrm{i} h^2(z,t) \over 2} \right] . \end{aligned}$$
(52)

But since, from (11) and (12), \(q(z)= T\) on \(|z|=1\) then

$$\begin{aligned} \nu U T = \textrm{Re} \left[ - {\textrm{i} \nu q(z) h(z,t)} \right] \end{aligned}$$
(53)

there. Hence, on combining (52) and (53),

$$\begin{aligned} \begin{aligned} \Gamma U&= \textrm{Re} \left[ - {\textrm{i} h^2(z,t) \over 2} \right] - \nu U T = \textrm{Re} \left[ - {\textrm{i} h^2(z,t) \over 2} + {\textrm{i} \nu q(z) h(z,t)} \right] . \end{aligned} \end{aligned}$$
(54)

On noting, from the chain rule, that since \(z=\textrm{e}^{-\textrm{i}s}\) then

$$\begin{aligned} {\partial \over \partial s} = - \textrm{i}z {\partial \over \partial z}, \end{aligned}$$
(55)

the second term in (47) is

$$\begin{aligned} \begin{aligned} {\partial (\Gamma U) \over \partial s}&= \textrm{Re} \biggl [ {\partial \over \partial s} \left( -{\textrm{i} h^2(z,t) \over 2} + {\textrm{i} \nu q(z) h(z,t)}\right) \biggr ] \\ &= \textrm{Re} \biggl [ - z h(z,t) {\partial \over \partial z} h(z,t) + z {\partial \over \partial z} (\nu q(z) h(z,t) ) \biggr ]. \end{aligned} \end{aligned}$$
(56)

Finally, on use of (49) and (55), the right hand side of (47) is \(1/Pe_s\) multiplied by

$$\begin{aligned} {\partial ^2 \Gamma \over \partial s^2} = \textrm{Re} \left[ {\partial ^2 \over \partial s^2} (h(z,t) - \nu q(z)) \right] = - \textrm{Re} \left[ z {\partial \over \partial z} \left( z {\partial (h(z,t) - \nu q(z)) \over \partial z} \right) \right] . \end{aligned}$$
(57)

On use of (37), the following equation is easily verified:

$$\begin{aligned} z {\partial \over \partial z} \left( \nu z {\partial q(z)) \over \partial z} \right) = \nu q(z). \end{aligned}$$
(58)

Putting all these observations together means that the surfactant evolution equation (47) can be written as

$$\begin{aligned} \begin{aligned} \textrm{Re} \biggl [ {\partial h(z,t) \over \partial t}&- z h(z,t) {\partial \over \partial z} h(z,t) + z {\partial \over \partial z} (\nu q(z) h(z,t) ) \\ &+{1 \over Pe_s} z {\partial \over \partial z} \left( z {\partial h(z,t) \over \partial z} \right) -{\nu \over Pe_s} q(z) \biggr ] = 0. \end{aligned} \end{aligned}$$
(59)

An important observation is that the function in square brackets in this equation is analytic in \(|z| > 1\) except possibly for singularities at infinity. At this point, the far-field boundary conditions must be taken into account. Since the flow in the far-field is expected to be a uniform flow with no vorticity it follows from (18) and (31) that, as \(|z| \rightarrow \infty \),

$$\begin{aligned} h(z,t) \sim 1 + {h_1(t) \over z} + {h_2(t) \over z^2} + \cdots . \end{aligned}$$
(60)

The first term ensures that the average of the real part of h(z, t) around the bubble boundary is unity, a requirement noted earlier. Note also, from (10) and (27) that the non-dimensional bubble speed is related to \(h_1(t)\) via

$$\begin{aligned} U_B(t) = {h_1(t) \over 2}. \end{aligned}$$
(61)

In view of (60) and (37) it follows that

$$\begin{aligned} h'(z,t) \sim - {h_1(t) \over z^2} - {2 h_2(t) \over z^3} + \cdots , \qquad q'(z) = 1- {1 \over z^2}, \end{aligned}$$
(62)

and the analytic function in square brackets in (59) can be seen to tend to \(\nu (1-1/Pe_s) z+ \textrm{o}(1)\) as \(|z| \rightarrow \infty \). Another function with this same far-field behavior, and also with vanishing real part on \(|z|=1\), is

$$\begin{aligned} \nu \left( 1-{1 \over Pe_s} \right) \left( z - {1 \over z} \right) . \end{aligned}$$
(63)

It follows, by analytic continuation off the curve \(|z|=1\), that the function in square brackets in (59) must equal this function so that

$$\begin{aligned} \begin{aligned} {\partial h(z,t) \over \partial t}&- z h(z,t) {\partial \over \partial z} h(z,t) + z {\partial \over \partial z} (\nu q(z) h(z,t) ) +{1 \over Pe_s} z {\partial \over \partial z} \left( z {\partial h(z,t) \over \partial z} \right) \\ &- {\nu \over Pe_s} q(z) = \nu \left( 1-{1 \over Pe_s} \right) \left( z - {1 \over z} \right) . \end{aligned} \end{aligned}$$
(64)

After some simplifications, this produces

$$\begin{aligned} \begin{aligned} {\partial h(z,t) \over \partial t}&- z h(z,t) {\partial \over \partial z} h(z,t) + z {\partial \over \partial z} (\nu q(z) h(z,t) ) +{1 \over Pe_s} z {\partial \over \partial z} \left( z {\partial h(z,t) \over \partial z} \right) \\ &\quad = \nu z q'(z) + {2 \nu \over Pe_s z}. \end{aligned} \end{aligned}$$
(65)

This complex nonlinear partial differential equation for h(z, t), which holds off the bubble boundary \(|z|=1\), governs the thermosolutal Marangoni dynamics of the bubble and its surface concentration of surfactant.

Equation (65) is a key result of this paper. It is the radial geometry analogue of a complex Burgers equation obtained for the half-plane geometry in [12, 13] where, in that case, the relevant lower-analytic function h(z, t) had a different functional connection to f(z, t). The reduction of the thermosolutal bubble problem to this complex partial differential equation of Burgers type (65) has significant theoretical ramifications to be explored in subsequent sections.

5 Steady solutions when \(Pe_s \rightarrow 0\)

Assuming a steady flow, as \(Pe_s \rightarrow 0\) equation (65) reduces to the linear ordinary differential equation

$$\begin{aligned} \begin{aligned} z {\textrm{d} \over \textrm{d} z} \left( z {\textrm{d} h(z) \over \textrm{d} z} \right) = {2 \nu \over z}. \end{aligned} \end{aligned}$$
(66)

This is readily integrated to give

$$\begin{aligned} h(z) = 1+{2 \nu \over z}, \end{aligned}$$
(67)

where a constant of integration has been chosen to ensure h(z) is single-valued in \(|z| > 1\). Hence, from (61), the non-dimensional speed of the bubble is \(U_B = \nu \) yielding the dimensional bubble speed

$$\begin{aligned} {\beta _\Gamma \langle \Gamma _0 \rangle \over 2\mu } \nu = {\beta _T T_\infty ' \mathcal {R} \over 2 \mu }, \end{aligned}$$
(68)

which is the pure thermocapillary-driven value. In this case, there is no additional Marangoni stress from the surfactant distribution which has become uniform at steady state in this diffusive limit.

6 Steady solutions when \(Pe_s \rightarrow \infty \)

For steady solutions in the opposite limit \(Pe_s \rightarrow \infty \) the governing equation (65) reduces to the nonlinear differential equation

$$\begin{aligned} \begin{aligned} - z h(z) {\textrm{d} \over \textrm{d} z} h(z)&+ z {\textrm{d} \over \textrm{d} z} (\nu q(z) h(z) ) = \nu z q'(z). \end{aligned} \end{aligned}$$
(69)

On dividing by z, this equation becomes

$$\begin{aligned} \begin{aligned} -{\textrm{d} \over \textrm{d} z} \left( {h(z)^2 \over 2} \right) +{\textrm{d} \over \textrm{d} z} (\nu q(z) h(z) ) = \nu q'(z), \end{aligned} \end{aligned}$$
(70)

which can be integrated with respect to z to produce

$$\begin{aligned} {h(z)^2 \over 2} - \nu q(z) h(z) = -\nu q(z) -{B \over 2}, \end{aligned}$$
(71)

where B is an integration constant. This rearranges to the quadratic equation

$$\begin{aligned} h^2 - 2 \nu q h + 2\nu q + B =0, \end{aligned}$$
(72)

which has solution

$$\begin{aligned} h = \nu q - \left( (\nu q)^2 - (B+2 \nu q) \right) ^{1/2}, \end{aligned}$$
(73)

where the minus sign associated with the square root has been selected to ensure that \(h \sim 1\) as \(z \rightarrow \infty \). This solution has square root branch points at values of z where

$$\begin{aligned} \nu ^2 q(z)^2- 2 \nu q(z) -B =0, \qquad \textrm{or} \qquad q(z) = {1 \pm \sqrt{1+B} \over \nu }. \end{aligned}$$
(74)

Each of these two equations can be reduced to a quadratic equation for the z-locations of any branch points. For generic values of B this can be expected to produce four solutions which will correspond physically to edges of immobilized zones, or stagnant caps. Yet only a single stagnant cap is expected in this problem corresponding to just two such branch points. For this, it is necessary to pick

$$\begin{aligned} 1 - \sqrt{1+B} =2\nu , \qquad \textrm{or} \qquad \sqrt{1+B} = 1 -2\nu , \end{aligned}$$
(75)

so that two of the four possible branch points merge at \(z=1\) producing a point of analyticity and leaving two other branch points at

$$\begin{aligned} z + {1 \over z} = \left( {1 + \sqrt{1+B} \over \nu } \right) = 2 \left( {1-\nu \over \nu } \right) . \end{aligned}$$
(76)

These correspond to the edges of a stagnant cap at \(z=\textrm{e}^{\pm \textrm{i} \phi _s}\) where \(\phi _s\) is a solution of

$$\begin{aligned} \cos \phi _s = {1 -\nu \over \nu }. \end{aligned}$$
(77)

This has real solutions for \(\phi _s\) when \(\nu \ge \nu _* = 1/2\). With the choice (75) the solution (73) is

$$\begin{aligned} \begin{aligned} h(z) = \nu q(z) - (\nu q(z)-1) \left( 1 - {(1- 2 \nu )^2\over (\nu q(z)-1)^2} \right) ^{1/2}, \end{aligned} \end{aligned}$$
(78)

where it has been written in the most convenient form for numerical evaluation. The speed of the bubble can be extracted from the far-field behavior:

$$\begin{aligned} \begin{aligned} h = \nu q(z) - (\nu q(z)-1) \left( 1 - {(1- 2 \nu )^2\over (\nu q(z)-1)^2} \right) ^{1/2} \sim 1+ {(1- 2\nu )^2\over 2 \nu } {1\over z} + \cdots , \end{aligned}\nonumber \\ \end{aligned}$$
(79)

as \(z \rightarrow \infty \). Hence, using (61), the non-dimensional speed of the bubble is

$$\begin{aligned} U_B = {(1- 2\nu )^2\over 4\nu }, \end{aligned}$$
(80)

yielding the dimensional bubble speed

$$\begin{aligned} {\beta _\Gamma \langle \Gamma _0 \rangle \over 2\mu } {(1- 2\nu )^2\over 4\nu ^2} \nu =\left( {1-2\nu \over 2\nu } \right) ^2 {\beta _T T_\infty ' \mathcal{R} \over 2 \mu }. \end{aligned}$$
(81)

There is no bubble migration when \(\nu = \nu _*\) and, from (77), this corresponds to \(\phi _s=0\) and the bubble being fully covered in surfactant so that its surface is fully immobilized. This situation is where the surfactant spreads in such a way as to cancel out the thermally induced Marangoni stress everywhere on the bubble. As \(\nu \rightarrow \infty \) the bubble speed (81) tends to the pure thermocapillary-driven value (68) which is to be expected since this limit of \(\nu \) signifies that the Marangoni stresses associated with thermal variations dominate over those due to surfactant concentration variations. This analysis suggests that the solution (78) is only valid for \(\nu \ge \nu _*\) and that for \(\nu < \nu _*\) the long-time steady solution of (73) is trivial, i.e.

$$\begin{aligned} h(z) = 1, \qquad B = -1, \end{aligned}$$
(82)

a suggestion borne out by the time-dependent simulations to follow. For \(\nu \ge \nu _*\) the mobile region is the sector \(\phi \in [-\phi _s, \phi _s]\); the stagnant cap is the rest of the surface. This mobile boundary portion gets small as \(\nu \rightarrow \nu _*^+\). As \(\nu \) increases above \(\nu _*\) more of the surface becomes mobile and, as \(\nu \rightarrow \infty \), \(\phi _s \rightarrow \pi ^-\). The stagnant cap vanishes in this limit, the thermocapillary Marangoni stress dominates, and the bubble speed (81) increases monotonically to the value (68). Figure 2 shows the normalized surfactant concentration \(\Gamma (s)/\Gamma (\pm \pi )\) as a function of s for \(\nu =2,1, 0.6, 0.51\) and \(\nu = \nu _* =0.5\) as calculated from (78). This graph should be compared with Figure 3 of Kim and Subramanian [9] where the same inflection point behavior is observed as the critical value \( \nu _*\) is approached.

Two final remarks are in order on this \(Pe_s \rightarrow \infty \) limit.

First, it is notable that the complex analysis approach adopted here leads naturally to a closed form expression (78) for the steady solution as \(Pe_s \rightarrow \infty \) without the need to solve any mixed boundary value problems that typically arise when tackling such problems using other methods [9]. Moreover, the expected square root branch points at the edges of the surfactant cap are explicitly manifest in the form of the solution (78).

Second, in previous studies [12, 13] using mathematical techniques akin to those used here but in the half-plane fluid geometry, the author has shown that the resulting unsteady governing nonlinear partial differential equations of Burgers type can profitably be analyzed using a complex version of the method of characteristics in the limit \(Pe_s \rightarrow \infty \); see also [15]. In principle, that approach can also be used to analyze (65) as \(Pe_s \rightarrow \infty \). While this is possible in principle, the details in this case turn out to be such that the approach offers no particular advantages over more direct numerical techniques to be explained in Sect. 9.

Fig. 2
figure 2

Normalized surfactant concentration \(\Gamma (s)/\Gamma (\pm \pi )\) as a function of s for \(\nu =2,1, 0.6, 0.51\) and \(\nu = \nu _* =0.5\) as calculated from the exact solution (78). For \(\nu > \nu _s\) the surfactant concentration is non-zero over the stagnant cap. The corresponding cap angles \(\phi _s\) follow from formula (77)

Full size image

7 Steady solutions for \(0< Pe_s < \infty \)

For general values of \(Pe_s\), i.e. for \(0< Pe_s < \infty \), the steady version of (65) is

$$\begin{aligned} \begin{aligned}&- z h(z) {\textrm{d} h(z) \over \textrm{d} z} + z {\textrm{d} \over \textrm{d} z} (\nu q(z) h(z) ) +{1 \over Pe_s} z {\textrm{d} \over \textrm{d} z} \left( z {\textrm{d} h(z) \over \textrm{d} z} \right) \\ &\quad = \nu z q'(z) + {2 \nu \over Pe_s z}. \end{aligned} \end{aligned}$$
(83)

On division by z this equation becomes integrable with respect to z to yield the first-order nonlinear differential equation

$$\begin{aligned} {1 \over Pe_s} z {\textrm{d}h(z) \over \textrm{d}z} = {h(z)^2 \over 2} - \nu q(z) h(z) + \nu q(z) - {2 \nu \over Pe_s} {1 \over z} + {B \over 2}, \end{aligned}$$
(84)

where B is an integration constant. This is a Riccati equation which can be linearized by an appropriate change of dependent variables. The resulting linear ordinary differential equation, which is of second order with non-constant coefficients, does not have any recognizable standard form for which analytical solutions are known. Moreover, this second-order equation can also be retrieved as the steady limit of a more general Cole–Hopf transformation of the full partial differential equation (65), to be discussed in the next section: it follows on setting the partial time derivative to zero in the forthcoming partial differential equation (101). In view of this, investigation of steady solutions for \(0< Pe_s < \infty \) will be carried out by solving the full time-dependent problem (101) and observing the long-time limits of the solutions as \(t \rightarrow \infty \). Such an approach will only capture stable steady solutions but those are the ones of principal physical interest in any case.

Before proceeding to the unsteady calculations, it is worth carrying out an analysis of the steady case for large \(Pe_s\) and for \(\nu < \nu _s\). Kim and Subramanian [10] performed a similar analysis for a spherical droplet. For \(Pe_s\) large but finite, perturbations to the \(Pe_s=\infty \) solution (82) can be supposed to have expansions of the form

$$\begin{aligned} h(z) = 1 + {h_1(z) \over Pe_s} + \cdots , \qquad B = -1 + {B_1 \over Pe_s} + \cdots , \end{aligned}$$
(85)

where \(h_1(z)\) and the constant \(B_1\) remain to be determined. On substitution of (85) into (84) it is found, at leading order in \(1/Pe_s\), that

$$\begin{aligned} h_1(z) - \nu q(z) h_1(z) - {2 \nu \over z} + {B_1 \over 2} =0, \end{aligned}$$
(86)

which has solution

$$\begin{aligned} h_1(z) = {1 \over 1-\nu q(z)} \left( {2 \nu \over z} - {B_1 \over 2} \right) . \end{aligned}$$
(87)

For \(\nu < \nu _s = 1/2\) the function \(1-\nu q(z)\) has a zero in the fluid region \(|z| > 1\) at

$$\begin{aligned} z = z_* = {1 + \sqrt{1-4 \nu ^2} \over 2\nu }, \end{aligned}$$
(88)

and this must be eliminated by a suitable choice of \(B_1\), namely, \(B_1 = {4 \nu / z_*}\), leading to

$$\begin{aligned} h_1(z) = {1 \over 1-\nu q(z)} \left( {2 \nu \over z} - {2 \nu \over z_*}\right) . \end{aligned}$$
(89)

The correction to the bubble speed is found by studying the large-z asymptotics of h(z), namely,

$$\begin{aligned} h(z) = 1 + {1 \over Pe_s(1-\nu q(z))} \left( {2 \nu \over z} - {2 \nu \over z_*}\right) + \cdots \sim 1 +\left( {2 \over Pe_s z_*}\right) {1 \over z} + \cdots , \nonumber \\ \end{aligned}$$
(90)

as \(|z| \rightarrow \infty \). Hence, from (61), the non-dimensional bubble speed is

$$\begin{aligned} U_B = {1 \over Pe_s z_*} ={2 \nu \over Pe_s (1+\sqrt{1-4\nu ^2})}. \end{aligned}$$
(91)

The dimensional bubble speed is therefore

$$\begin{aligned} {\beta _\Gamma \langle \Gamma _0 \rangle \over 2\mu } {2 \nu \over Pe_s (1+\sqrt{1-4\nu ^2})} ={1 \over Pe_s} \left( {2 \over 1+\sqrt{1-4\nu ^2}} \right) {\beta _T T_\infty ' \mathcal R \over 2 \mu }. \end{aligned}$$
(92)

This analysis shows that the presence of surface diffusion causes net bubble motion even for \(\nu < \nu _s\) when, in the absence of surface diffusion, the bubble would be stationary.

8 Unsteady Cole–Hopf type linearization at finite non-zero \(Pe_s\)

Remarkably, the full governing nonlinear partial differential equation (65) can be linearized by a change of both dependent and independent variables. The following transformation is a generalization of the classical Cole–Hopf transformation used to transform the (real) Burgers equation into the (real) heat equation [12, 14]. On introducing the decomposition

$$\begin{aligned} h(z,t) = 1 + {\hat{h}}(z,t), \end{aligned}$$
(93)

the governing equation (65) becomes

$$\begin{aligned} \begin{aligned}&{\partial {\hat{h}}(z,t) \over \partial t} - z {\partial \over \partial z} {\hat{h}}(z,t) - z {\hat{h}}(z,t) {\partial \over \partial z} {\hat{h}}(z,t) + z {\partial \over \partial z} (\nu q(z) ) + z {\partial \over \partial z} (\nu q(z) {\hat{h}}(z,t) ) \\&\quad +{1 \over Pe_s} z {\partial \over \partial z} \left( z {\partial {\hat{h}}(z,t) \over \partial z} \right) = \nu z q'(z) + {2 \nu \over Pe_s z}. \end{aligned} \end{aligned}$$
(94)

This simplifies to

$$\begin{aligned} \begin{aligned} {\partial {\hat{h}}(z,t) \over \partial t}&= z {\hat{h}}(z,t) {\partial \over \partial z} {\hat{h}}(z,t) - z {\partial \over \partial z} \left( (\nu q(z)-1) {\hat{h}}(z,t) \right) \\ &\quad -{1 \over Pe_s} z {\partial \over \partial z} \left( z {\partial {\hat{h}}(z,t) \over \partial z} \right) +{2 \nu \over Pe_s z}. \end{aligned} \end{aligned}$$
(95)

Now introduce the change of independent variable given by

$$\begin{aligned} \mathcal{Z} = \log z \qquad H(\mathcal{Z},t) \equiv {\hat{h}}(z,t), \qquad Q(\mathcal{Z}) \equiv q(z). \end{aligned}$$
(96)

Since h(z, t) must be single-valued as a function of z it is clear that admissible solutions for \(H(\mathcal{Z},t)\) must be \(2 \pi \textrm{i}\) periodic as functions of \(\mathcal{Z}\). Equation (95) becomes

$$\begin{aligned} {\partial H \over \partial t} = {\partial \over \partial \mathcal{Z}} \left( {H^2 \over 2} \right) - {\partial \over \partial \mathcal{Z}} \left( (\nu Q(\mathcal{Z}) -1 ) H \right) - {1 \over Pe_s} {\partial ^2 H \over \partial \mathcal{Z}^2} + {2 \nu \over Pe_s} \textrm{e}^{-\mathcal{Z}}. \end{aligned}$$
(97)

With a further change of dependent variable embodied in

$$\begin{aligned} H(\mathcal{Z},t) = -{2 \over Pe_s} {\partial \log \Phi (\mathcal{Z},t) \over \partial \mathcal{Z}}, \end{aligned}$$
(98)

equation (97) can be integrated with respect to \(\mathcal{Z}\) to yield

$$\begin{aligned} -{2 \over Pe_s} {1 \over \Phi } {\partial \Phi \over \partial t} = {H^2 \over 2} - (\nu Q -1 ) H - {1 \over Pe_s} {\partial H \over \partial \mathcal{Z}} - {2 \nu \over Pe_s} \textrm{e}^{-\mathcal{Z}}. \end{aligned}$$
(99)

An additive function of t has been set equal to zero without loss of generality because any non-zero choice just rescales \(\Phi (\mathcal{Z},t)\) by a function of t which does not affect the value of \(H(\mathcal{Z},t)\) given by the logarithmic \(\mathcal{Z}\) derivative (98). On substitution for \(H(\mathcal{Z},t)\) from (98), (99) becomes

$$\begin{aligned} \begin{aligned} -{2 \over Pe_s} {1 \over \Phi } {\partial \Phi \over \partial t}&= {2 \over Pe_s^2} \left( {\Phi ' \over \Phi } \right) ^2 - (\nu Q(\mathcal{Z}-1) \left( -{2 \over Pe_s} \right) {\Phi ' \over \Phi } \\ &\quad + {2 \over Pe_s^2} \left( {\Phi '' \over \Phi } - \left( {\Phi ' \over \Phi } \right) ^2 \right) - {2 \nu \over Pe_s} \textrm{e}^{-\mathcal{Z}}. \end{aligned} \end{aligned}$$
(100)

The nonlinear terms cancel, leaving the linear complex partial differential equation

$$\begin{aligned} {\partial \Phi \over \partial t} + (\nu Q(\mathcal{Z}) -1) {\partial \Phi \over \partial \mathcal{Z}} + {1 \over Pe_s} {\partial ^2 \Phi \over \partial \mathcal{Z}^2} - \nu \textrm{e}^{-\mathcal Z} \Phi = 0. \end{aligned}$$
(101)

By virtue of all these observations, this nonlinear dynamical problem for the thermosolutal Marangoni dynamics of a two-dimensional bubble can be linearized. The linear second-order ordinary differential equation discussed in Sect. 7 governing the steady state at finite, non-zero \(Pe_s\) follows on setting \({\partial \Phi / \partial t}=0\) in (101).

While it has not yet been possible to identify any analytical solutions to the linear partial differential equation (101), it serves as a convenient basis for a straightforward numerical scheme using a sum of separable solutions. Let

$$\begin{aligned} \Phi (Z,t) = \sum _{n \ge 0} A_n(t) \textrm{e}^{-n\mathcal{Z}} = A_0(t) + A_1(t) \textrm{e}^{-\mathcal{Z}} +A_2(t) \textrm{e}^{-2\mathcal{Z}} +A_3(t) \textrm{e}^{-3\mathcal{Z}} + \cdots ,\nonumber \\ \end{aligned}$$
(102)

which is a general expression for a time-evolving function that is \(2 \pi \textrm{i}\) periodic in \(\mathcal{Z}\) and is such that \(H(\mathcal{Z},t)\) as given in (98) decays as \(z \rightarrow \infty \). On substituting into (101) and equating coefficients, the following system of coupled linear ordinary differential equations is obtained:

$$\begin{aligned} \begin{aligned} \dot{A}_n - \nu (n+1)A_{n+1} -\nu n A_{n-1} + n \left( 1 + {n \over Pe_s} \right) A_n&= 0, \qquad n \ge 0, \end{aligned} \end{aligned}$$
(103)

where overdots are used to denote time derivatives. This system is to be solved for a given set of initial data \(\lbrace A_{n0} = A_n(0) | n \ge 0 \rbrace \). On truncating at some order N and letting

$$\begin{aligned} \textbf{x}(t) = (A_0(t), A_1(t), A_2(t), \ldots , A_N(t))^T, \end{aligned}$$
(104)

the set (103) can be written as a first-order matrix system

$$\begin{aligned} \dot{\textbf{x}} = \textbf{M} \textbf{x}, \end{aligned}$$
(105)

where \(\textbf{M}\) is the tridiagonal \((N+1)\)-by-\((N+1)\) matrix

$$\begin{aligned} \left( \begin{array}{cccccccc} 0 & \nu & 0 & 0 & . & . & \cdots & 0 \\ \nu & -1\big (1+{1 \over Pe_s}\big ) & 2 \nu & 0 & . & . & \cdots & 0 \\ 0 & 2 \nu & -2\big (1+{2 \over Pe_s}\big ) & 3 \nu & . & . & 0 & \cdots \\ . & . & . & . & . & . & . & . \\ . & . & . & . & . & ~~.~~ & (N-1)\nu & 0 \\ 0 & 0 & 0 & 0 & 0 & (N-1)\nu & ~~-(N-1)\big (1+{N-1 \over Pe_s}\big ) ~~& N \nu \\ 0 & 0 & 0 & 0& 0& 0 & ~~N \nu ~~& - N\big (1+{N \over Pe_s}\big ) \end{array} \right) . \end{aligned}$$
(106)

The system (105) is readily solved using standard methods for ordinary differential equations. Once the coefficients \(\lbrace A_n(t) | n \ge 0 \rbrace \) have been found it follows that

$$\begin{aligned} {\hat{h}}(z,t) = H(\mathcal{Z},t) = -{2 \over Pe_s} {\partial \log \Phi (\mathcal{Z},t) \over \partial \mathcal{Z}} ={2 \over Pe_s} { \left( {A_1 \over z} + {2 A_2 \over z^2} + {3 A_3 \over z^3} + \cdots \right) \over \left( A_0 + {A_1 \over z} + { A_2 \over z^2} + {A_3 \over z^3} + \cdots \right) }.\nonumber \\ \end{aligned}$$
(107)

Consequently, the required solution for h(z, t) is

$$\begin{aligned} h(z,t) = 1+ {2 \over Pe_s} { \left( {A_1(t) \over z} + {2 A_2(t) \over z^2} + {3 A_3(t) \over z^3} + \cdots \right) \over \left( A_0(t) + {A_1(t) \over z} + { A_2(t) \over z^2} + {A_3(t) \over z^3} + \cdots \right) }. \end{aligned}$$
(108)

The surfactant concentration and surface slip now follow easily from (46):

$$\begin{aligned} \Gamma (s, t) = \textrm{Re}[h(\textrm{e}^{-\textrm{i}s},t)] - 2\nu \cos s, \qquad U(s,t) = \textrm{Im}[h(\textrm{e}^{-\textrm{i}s},t)], \end{aligned}$$
(109)

and the streamfunction for the time-evolving flow in the bulk is given by

$$\begin{aligned} \psi (z,\overline{z},t) = {1 \over 2} \textrm{Im}[(z \overline{z}-1) h(z,t)]. \end{aligned}$$
(110)

9 Calculation of unsteady Marangoni dynamics at finite \(Pe_s\)

This section presents some numerical calculations made using the method just described. Suppose the initial condition is chosen to be

$$\begin{aligned} h(z,0) = 1+ {B \over z}, \end{aligned}$$
(111)

where B is a real constant then, on the bubble boundary where \(z=\textrm{e}^{- \textrm{i}s}\),

$$\begin{aligned} \textrm{Re}[h(\textrm{e}^{- \textrm{i}s},0)] = 1 + B \cos s = \Gamma (s,0) + \nu T(s,0)= \Gamma (s,0) + 2 \nu \cos s. \end{aligned}$$
(112)

This implies the initial surfactant concentration is

$$\begin{aligned} \Gamma (s,0) = 1 + (B- 2\nu ) \cos s. \end{aligned}$$
(113)

This is non-negative everywhere on the bubble boundary provided that \(B- 2\nu \le 1\). For initially uniform coverage of surfactant, the choice is

$$\begin{aligned} B = 2 \nu , \end{aligned}$$
(114)

and only this initial condition will be used in the calculations to follow since other choices are found to give qualitatively similar results. For the initial condition (111) it is necessary to pick

$$\begin{aligned} -{2 \over Pe_s} {\partial \log \Phi (\mathcal{Z},0) \over \partial \mathcal{Z}} = B \textrm{e}^{-\mathcal{Z}}, \qquad \textrm{or} \qquad \Phi (\mathcal{Z},0) = \textrm{e}^{Pe_s B \textrm{e}^{-\mathcal{Z}}/2}, \end{aligned}$$
(115)

from which the relevant initial data is read off as

$$\begin{aligned} A_{n0} = \left( {Pe_s B \over 2} \right) ^n {1 \over n!}, \qquad n \ge 0. \end{aligned}$$
(116)

The following calculations solve (105) with an implicit time-stepping scheme using the initial data (116).

Fig. 3
figure 3

Steady bubble speed \(U_B\) non-dimensionalized with respect to the pure thermocapillary speed (68) for \(\nu = 0.4, 0.75, 1, 2\) and 5. As \(Pe_s \rightarrow 0\) the curves tend to unity as thermocapillary effects dominate since any surfactant concentration gradients have been diffusively smoothed out. As \(Pe_s\) gets larger, the curves with \(\nu > \nu _s = 1/2\) tend to the value \(((1-2\nu )/2\nu )^2\) predicted by the steady \(Pe_s \rightarrow \infty \) result in (81); the \(\nu =0.4 < \nu _s\) curve tends to zero. The curves are found by solving the unsteady linear equations (105) with initial data (116) and finding the bubble speed as \(t \rightarrow \infty \)

Full size image

Figure 3 shows the bubble speed \(U_B\) non-dimensionalized with respect to the pure thermocapillary speed (68) for \(\nu = 0.4, 0.75, 1, 2\) and 5. The values reported in this graph are the bubble speeds after sufficiently long simulation times that it was judged that steady state has been reached; typically the calculations were performed up to \(t=10\). As \(Pe_s \rightarrow 0\) all curves tend to the pure thermocapillary speed (68) as predicted by the analysis in Sect. 5. As \(Pe_s\) increases, the curves for \(\nu = 0.75, 1, 2\) and 5 tend to the value \(((1-2\nu )/2\nu )^2\) as predicted by (81) from the analysis in Sect. 6 with asymptotes shown as horizontal dashed lines; the curve for \(\nu =0.4 < \nu _*\) tends to zero. The graphs in Fig. 3 not only provide validation of the analysis in the two limits \(Pe_s \rightarrow 0, \infty \) but also confirm the viability of using the linearized partial differential equation (101) to formulate the numerical method based on the linear system (105).

An interesting feature of Fig. 3 is that enhanced surface diffusion, or smaller \(Pe_s\), increases the migration velocity for a given \(\nu > \nu _s\). This is because surface diffusion smooths out the retarding Marangoni stresses due to the contamination-induced near-stagnant cap allowing the thermocapillary stress to more effectively provide locomotion.

Figure 4 shows graphs of \(\Gamma (s,10)\) and U(s, 10), by which time it was judged that steady state equilibrium had been reached, for fixed \(\nu =1\) and for gradually increasing values \(Pe_s = 1,2, 15\) and 100. The effect of surface diffusion weakens as \(Pe_s\) increases. These graphs corroborate that the locations of the edges of stagnant caps are tending, in the limit \(Pe_s \rightarrow \infty \), to the values predicted by (77) and indicated by red crosses; by \(Pe_s=100\), the edges of the immobilized portion of the bubble boundary are close to these crosses providing a consistency check on the analysis of the steady problem and the formulation of the unsteady dynamics.

Figure 5 shows superposed graphs of \(\Gamma (s,0)\) and U(s, 0) and of \(\Gamma (s,10)\) and U(s, 10) for \(\nu = 0.4 < \nu _s\) and \(Pe_s =50\). Owing to the fact that \(\nu < \nu _s\) there would be no net bubble migration after long times in the absence of surface diffusion, or \(Pe_s =\infty \), according to the analysis of Sect. 6. However the graph of U(s, 10) indicates small but non-zero surface slip that leads to the bubble migrating at a speed found to be well approximated by the perturbative formula (91) derived in Sect. 7. It is clear that the presence of surface diffusion reduces the efficacy of the surfactant to produce gradients that exactly nullify the Marangoni stresses induced by thermocapillarity leading to net bubble motion.

Fig. 4
figure 4

Graphs of \(\Gamma (s,10)\) and U(s, 10) for \(\nu =1\) and \(Pe_s=1, 2, 15, 100\); the evolution is close to the equilibrium by \(t=10\). The initial condition is given by (111) and (114). Red crosses show the locations of the edges of stagnant caps as predicted by (77) in the limit \(Pe_s \rightarrow \infty \). The curves for \(Pe_s=1, 2, 15\) are plotted by solving the unsteady linear equations (105) with initial data (116); the \(Pe_s=100\) curves make use of the numerical method based on the Laurent series (117)

Full size image
Fig. 5
figure 5

Graphs of \(\Gamma (s,10)\) and U(s, 10) for \(\nu =0.4 < \nu _s =1/2\) and \(Pe_s=50\); the evolution is close to the equilibrium by \(t=10\). The initial condition is one of uniform coverage given by (111) and (114). The numerical method based on the Laurent series (117) is used. Although \(\nu < \nu _s\) when, for \(Pe_s = \infty \), no bubble migration is expected, U(s, 10) is not exactly zero leading to slow bubble migration found to be well approximated by formula (91)

Full size image

The numerical method based on the linear system of ordinary differential equations (105) is used to compute the graphs in Fig. 4 for \(Pe_s=1, 2\) and 15. It performs well up to \(Pe_s=15\), by which time Fig. 3 shows that the bubble speeds are already close to their predicted values for \(Pe_s=\infty \) given in (81). However, this method fails for larger values of \(Pe_s\). Numerical difficulties might be anticipated from an examination of the initial coefficients given in (116): as \(Pe_s\) increases these initial coefficients become large with the coefficient \(A_{n0}\) of greatest magnitude corresponding to larger values of n suggesting that an increasing number of modes need to be retained as \(Pe_s\) increases. It is known in the context of the real-valued Burgers equation that numerical methods based on the classical Cole–Hopf transformation are susceptible to drawbacks at low values of the viscosity and attempts have been made to remedy these (see, for example, [20]). This limitation notwithstanding, the numerical method based on solving the linear system (105) using implicit time-stepping methods is found to be fast and efficient for \(0 < Pe_s \le 15\).

It is beyond the scope of this paper to carry out a numerical analysis of such methods. However, the results of the calculations made using (105) were compared against those obtained using an alternative numerical method based on the original nonlinear partial differential equation of Burgers type given in (65). In this second method, the function h(z, t) is represented as a Laurent series convergent in \(|z| \ge 1\) truncated at N terms, i.e.,

$$\begin{aligned} h(z,t) = 1 + \sum _{n \ge 1}^N {h_n(t) \over z^n}, \end{aligned}$$
(117)

with initial data

$$\begin{aligned} h_1(0) = B = 2 \nu , \qquad h_n(0) = 0, ~~ n \ge 2. \end{aligned}$$
(118)

In the calculations to follow N is chosen sufficiently large that, on increasing it further, negligible change in the computed results is observed. The governing equation (65) can be written as

$$\begin{aligned} \begin{aligned} {\partial h(z,t) \over \partial t}&= z h(z,t) {\partial \over \partial z} h(z,t) - z {\partial \over \partial z} (\nu q(z) h(z,t) ) \\ &\quad -{1 \over Pe_s} z {\partial \over \partial z} \left( z {\partial h(z,t) \over \partial z} \right) + \nu z q'(z) + {2 \nu \over Pe_s z}. \end{aligned} \end{aligned}$$
(119)

At any time t, given the current values of the coefficients in (117), the right hand side of (119), which contains only spatial derivatives, is evaluated using spectral differentiation and its Laurent expansion coefficients (valid for \(|z| \ge 1\)) computed using fast Fourier transforms. Then, by comparing Laurent coefficients with the left hand side of (119), namely,

$$\begin{aligned} {\partial h(z,t) \over \partial t} = \sum _{n \ge 1}^N {\dot{h}_n(t) \over z^n}, \end{aligned}$$
(120)

the time derivatives \(\lbrace \dot{h}_n(t) | n \ge 1 \rbrace \) of the coefficients \(\lbrace h_n(t) | n \ge 1 \rbrace \) can be extracted. It is then straightforward to integrate the resulting system of first-order nonlinear ordinary differential equations using standard numerical methods. This method is found to work well for surface Péclet numbers as large as \(Pe_s=10^3\), provided the truncation parameter N increases appropriately with increasing \(Pe_s\) to maintain accuracy. This method was used to calculate the \(Pe_s=100\) graphs in Fig. 4 and the \(Pe_s=50\) graph in Fig. 5. It faithfully reproduces the results of the method based on the linear system (105) in the range \(0 \le Pe_s \le 15\) providing a consistency check on both calculations.

Convergence of the Laurent series (117) is expected to deterioriate near \(|z|=1\) for very large \(Pe_s\) where the square root branch points evident from the analytical formula (65) for the steady solution must be approaching the boundary from inside the bubble. Improved numerical schemes that use this knowledge to promote improved convergence using singularity subtraction can be envisaged, but are left for a future study.

10 Discussion

This paper has unveiled a rich mathematical structure in a problem of thermosolutal viscous Marangoni dynamics of an inviscid, constant-pressure two-dimensional bubble. For steady motion, all the main features pointed out by Kim and Subramanian [9, 10] for a spherical droplet have been observed, including the formation of stagnant caps when thermocapillary stresses dominate over those due to surface contamination. However, this study has gone beyond what was done in [9, 10] and also resolved the unsteady evolution to the steady-state equilibria. Not only does the two-dimensional model allow the essential multiphysics system to be exhibited in a less cumbersome fashion than for a three-dimensional bubble (where even solving the steady problem at infinite surface Péclet number requires dual series methods based on Gegenbauer polynomials [9, 10]) but the approach allows the steady solutions at zero and infinite surface Péclet numbers to be found in closed analytical form, and without the need to solve any mixed boundary value problems. Moreover, the dynamical time-dependent problem can be reduced to a complex partial differential equation with a Burgers-type nonlinearity, an equation linearizable by the generalized Cole–Hopf transformation presented herein. A numerical method for the dynamics based on this linearized system has been formulated. It is not clear that any such linearization will be possible for unsteady surfactant evolution on a spherical droplet or bubble.

Beyond these important new mathematical features, Fig. 3 depicts the main physical lessons of this work: that the presence of surfactants serves to reduce the steady bubble migration from its pure thermocapillary-induced value (68). Moreover the explicit formula (81) for the bubble speed at \(Pe_s=\infty \) gives a lower bound on this reduced speed as a function of the parameter \(\nu \) that measures the relative strengths of Marangoni stresses due to thermocapillarity and surfactant contamination. Formula (81), which may provide a useful lower bound in practice, is valid provided \(\nu > \nu _s =1/2\). For \(\nu \le \nu _s\) the surfactant completely arrests any steady bubble motion at infinite surface Péclet number. For \(\nu > \nu _s\), as surface diffusion of surfactant increases, a steady bubble travels more quickly than the lower bound (81) because diffusion mollifies the surfactant gradients that act against the thermocapillary stresses. For \(\nu < \nu _s\) surface diffusion of surfactant means that a bubble that is stationary without surface diffusion begins to translate as the surfactant gradients become less effective at totally canceling the thermocapillary stresses; the explicit formula (92) provides good estimates of this migration speed, as a function of \(\nu \), for large but finite \(Pe_s\).

All these physical results are qualitatively shared by the spherical droplet studied by Kim and Subramanian [9, 10] but they are arguably showcased in a more amenable fashion for the two-dimensional case which enjoys the mathematical advantages set out here. It should be mentioned that similar speed reduction due to surfactant contamination of bubbles rising under body forces, rather than thermocapillary stresses, has been well studied in the case of a three-dimensional bubble [21, 22]. The prospect of any analogue of the present study for forced motion of a two-dimensional bubble is, unfortunately, disallowed by the Stokes paradox [11].

While the treatment has dealt with a nonlinear problem, the formulation has assumed an equation of state (1) that is linearized in the surfactant concentration \(\Gamma \). Another reasonable assumption in the equation of state is that \(\beta _\Gamma \), strictly speaking a function of temperature, is approximately constant. Additional effects can be expected for a nonlinear equation of state. Mayer and Crowdy [23] have carried out a combined theoretical and numerical study of the surfactant dynamics, assuming a nonlinear Langmuir equation of state and also in the presence of an ambient flow, on the menisci of a superhydrophobic surface. Near-stagnant caps, or immobilized regions, are found to form in that situation too, with a delicate interplay between the surface Péclet number, Marangoni number and the surfactant load determining their size. A complex variable formulation plays a key role in that study too [23].

Various generalizations of this work are possible. Recent work in the half-plane geometry [15] has shown how the formulation in terms of partial differential equations of Burgers type [12, 13] is extendible to a two-viscous-fluid scenario: a similar extension is possible here which would allow this study of inviscid bubbles to be extended to viscous droplets as considered by Kim and Subramanian [9, 10] in the spherical case. Reaction kinetics and the effects of surfactant solubility to the bulk have also been similarly modeled within this framework in other situations [13, 15] and those can be added in this radial bubble geometry setting too.

Finally, whether any analogous results in three dimensions are possible in light of these mathematical developments in two dimensions is an interesting open question.