Dispersion of a passive scalar around a Taylor bubble

Abstract

The motion of Taylor bubbles in capillaries is typical of many engineering and biological systems, ranging from subsurface flows to small-scale reactors. Although the hydrodynamics of elongated bubbles has been the object of several studies, the case where a solute is transported in the surrounding liquid and surface mass-transfer mechanisms act on the solid wall or the bubble–fluid interface is much less understood. To fill this gap, we investigate the transport problem around a confined Taylor bubble to access the competition between advection, diffusion and surface mass transfer in the different regions of the bubble. With this aim, we derive a one-dimensional advection–diffusion–mass-transfer equation where the transport mechanisms are described through an effective velocity, an effective diffusion coefficient and an effective Sherwood number. Our model generalises the Aris–Taylor dispersion to the case of a Taylor bubble and clarifies the impact of surface mass transfer in the advection and diffusion dominated regimes for both the front and rear menisci. The model recovers the typical Pèclet square relationship of the effective diffusion coefficient, which also depends on the film thickness. Also, when the Pèclet number balances with the Sherwood number, there exist conditions that lead to the formation of hot spots of concentration. We show that the typical shape oscillations of the bubble rear locally enhance superficial mass transfer. Finally, we study the transport problem in the uniform film region where the concentration field can be found analytically.

1. Introduction

The study of mass transfer in confined geometries is extremely important in many engineering and biological systems. In the context of geological ${\rm CO}_{2}$ sequestration, carbon dioxide is injected into subsurface reservoirs, leading to the formation of elongated bubbles that can either be trapped, move or interact with the solid matrix (e.g. Rathnaweera, Ranjith & Perera 2016; Soulaine et al. 2018; Li, Garing & Benson 2020). The presence of ${\rm CO_2}$ has the effect of increasing the acidity of the in situ brine, boosting a series of chain reactions that enhance rock dissolution (Steefel, Molins & Trebotich 2013). This may threaten the long-term integrity of the storage process due to the formation of leakage pathways for carbon dioxide.

In medicine, a good understanding of the coupling between the hydrodynamics of confined bubbles and mass transfer is fundamental for treating air embolism (e.g. Grotberg 1994; Eckmann & Lomivorotov 2003; Suzuki & Eckman 2003; Barak & Yeshayahu 2005; Li et al. 2021), microcirculation and oxygen transport in blood vessels (e.g. Berg et al. 2020; Vadapalli, Goldman & Popel 2002) and targeted microbubbles for drug delivery (e.g. Bull 2005). In the context of microfluidics, small-scale reactors, cell cultures and cooling devices rely on an efficient control of transport processes in microchannels, (e.g. Ajaev & Homsy 2006; Lynn 2016; Khodaparast et al. 2017). All the aforementioned applications involve the motion of elongated bubbles (also known as Taylor bubbles) through capillaries where surface mass-transfer mechanisms take place between the surrounding liquid, the solid and the bubble interface. Exchanges of mass could be driven either by chemical reactions or surface phenomena, such as evaporation or dissolution. Thus, studying how mass transfer is enhanced or retarded due to the presence of a confined bubble is highly important for a correct interpretation of practical problems.

So far, many researchers have focused primarily on understanding the hydrodynamics of Taylor bubbles in capillary tubes in regimes where viscous forces and surface tension dominate over buoyancy and inertia. The seminal work of Bretherton (1961) revealed that bubble characteristics (i.e. the film thickness, the bubble speed) can be expressed as a function of the capillary number and that the film profile can be described with a similarity solution typical of Landau–Levich–Derjaguin–Bretherton problems (see de Gennes, Brochard-Wyart & Quéré 2003; Stone 2010). The theory is supported by several experimental measurements (Fairbrother & Stubbs 1935; Taylor 1961; Schwartz, Princen & Kiss 1986; Aussillous & Quéré 2000) and, in the last decades, it has been extended to include the effect of viscous forces and weak inertia (Cox 1962; Reinelt & Saffman 1985; Aussillous & Quéré 2000; Heil 2001; de Ryck 2002; Khodaparast et al. 2015; Magnini et al. 2017), unsteady flow, (Yu et al. 2018), buoyancy, (Leung et al. 2012; Atasi et al. 2017; Lamstaes & Eggers 2017), bubble viscosity and non-Newtonian effects (Chen 1986; Hodges, Jensen & Rallinson 2004; Balestra, Zhu & Gallaire 2018; Shukla et al. 2019; Picchi et al. 2021).

However, the case where a passive scalar (i.e. a solute) is transported in the liquid surrounding the Taylor bubble is still an object of research. Motivated by the discrepancies between experiments and Bretherton's theory, Schwartz et al. (1986), Hirasaki & Lawson (1985) and Ginley & Radke (1988) accounted for the effect of variation in the surface tension due to the presence of surface-active contaminants (i.e. the Marangoni effect) in the uniform film region. Ratulowski & Chang (1990) have shown that the Marangoni effect can explain the increased film thickness of the experiments; more recent studies (Park 1992; Stebe & Barthés-Biesel 1995; Olgac & Muradoglu 2013; Yu, Khodaparast & Stone 2017) confirm the importance of accessing the transport problem in the surroundings of the bubble to properly describe solute driven mechanisms.

Unfortunately, the problem of solute transport by advection and diffusion in confined geometries is a long standing issue. The seminal works of Taylor (1953) and Aris (1956) showed that the flow of a passive scalar in a circular pipe enhances axial diffusion. Although the Aris–Taylor dispersion theory was later extended to cases where the solute can be also absorbed from the solid walls (e.g. Gupta, Gupta & Taylor 1972; Sankarasubramanian, Gill & Benjamin 1973; Ng 2006; Mikelić, Devigne & van Duijn 2006), to the best of our knowledge, a comprehensive approach to generalise the transport problem in the presence of moving Taylor bubbles is still missing in the literature.

Existing studies on the topic are primarily numerical and experimental. Shim et al. (2014), Michelin, Guérin & Lauga (2018) and Rivero-Rodriguez & Scheid (2019) studied mass transfer around a series of spherical bubbles. Other works focus on quantifying mass-transfer fluxes around gaseous Taylor bubbles in horizontal (e.g. Yue et al. 2009; Shao, Gavriilidis & Angeli 2010; Sobieszuk et al. 2011; Cubaud, Sauzade & Sun 2012; Ganapathy, Al-Hajri & Ohadi 2013; Ganapathy et al. 2014; Jia & Zhang 2016; Svetlov & Abiev 2016; Zhu et al. 2017; Silva, Campos & Araújo 2019; Zhou et al. 2020) and vertical (e.g. Hayashi et al. 2014; Kastens et al. 2015; Hori et al. 2020) microchannels. Despite the amount of data available, in most of the cases experiments are analysed only in terms of empirical correlations. More sophisticated theories for modelling transport processes in complex geometries based on averaging and homogenisation techniques are available in the field of flow in porous materials (e.g. Brenner & Stewartson 1980; Rubinstein & Mauri 1986; Mauri 1991; Battiato & Tartakovsky 2011; Parmigiani et al. 2011; Picchi & Battiato 2018), but an attempt to adapt those theories to the case of a Taylor bubble has not been proposed yet.

To fill this gap, the goal of this paper is to study the transport problem around a moving Taylor bubble in the presence of surface mass transfer and to clarify the competition between diffusion, advection and superficial mass transfer in the different regions of the bubble. We account for the transport of a solute in the bulk of the fluid surrounding the Taylor bubble and surface mass-transfer effects at the wall and at the bubble–fluid interface. Our goal is to derive a theoretical framework to describe in a rigorous way the transport problem, including transient effects. The model generalises the Aris–Taylor dispersion theory to the case of a Taylor bubble and allows for the identification of the dominant transport regime in the front and rear menisci and in the uniform film region (i.e. the bubble centre).

With this aim, starting from the transport equations (§ 2.2), we derive an upscaled model for the average concentration by means of two-scale asymptotic expansions (§ 2.3). Specifically, we derive analytical expressions for the effective velocity, effective diffusion and effective mass-transfer coefficient in an advection–diffusion–mass-transfer equation that describes the evolution of the averaged concentration as a function of space and time (§ 2.4). Our approach (the full derivation is presented in Appendix B) allows us to determine the theoretical bounds of validity of the model, expressed in terms of the governing dimensionless numbers. The analysis is complemented by a classification of the dominant transport regimes depending on the magnitude of the Pèclet and Sherwood numbers of the problem (§ 2.5). Then, we solve numerically the advection–diffusion–mass-transfer equation coupled with the bubble profile (Bretherton 1961) to study the transport problem in the bubble front and the bubble rear (§§ 3.1 and 3.2). In a separate section (§ 3.3), we present the analytical solution of the transport problem in the uniform film region. The results shed light on the mechanisms that control the transport of a passive scalar around a Taylor bubble in the presence of superficial mass transfer.

2. Theoretical derivation

2.1. Viscous flow in the film

We consider a Taylor bubble confined in an horizontal planar channel that advances through an incompressible Newtonian fluid at a steady velocity $U$, as sketched in figure 1. The motion is driven by a Poiseuille flow with a constant average velocity far ahead of the bubble, see figure 1. The bubble is sufficiently long so that a region with uniform film thickness $h_\infty$ exists and $h_\infty /R\ll 1$. In the limit of small capillary number, the slope of the interface in the film is small ${\rm d}\hat {h}/{\rm d}\hat {x}\ll 1$ and the dynamics of $\hat {h}(\hat {x},\hat {t})$ is described by the lubrication equation (Eggers & Fontelos 2015)

(2.1)\begin{equation} \frac{\partial \hat{h}}{\partial \hat{t}}+ \frac{\sigma}{3\mu}\frac{\partial}{\partial \hat{x}}\left( \hat{h}^{3} \frac{\partial^{3}\hat{h}}{\partial \hat{x}^{3}}\right)=0, \end{equation}

where $\mu$ and $\sigma$ are the liquid viscosity and surface tension, respectively. Equation (2.1) describes the motion in the film region in response to the gradient of the capillary pressure $\hat {p}=-\sigma \,{\rm d}^{2} \hat {h}/{\rm d}\hat {x}^{2}$. In a moving reference frame attached to the bubble $\hat {h}(\hat {x},\hat {t})=\hat {h}(\hat {x}-U\hat {t})$, the integration of (2.1) yields to the Landau–Levich–Derjaguin–Bretherton similarity equation (see de Gennes et al. 2003; Stone 2010)

(2.2)\begin{equation} \frac{{\rm d}^{3}\eta}{{{\rm d} x}^{3}} = \frac{\eta - 1}{\eta^{3}}, \end{equation}

where

(2.3a–c)\begin{equation} x = \frac{\hat{x}}{h_\infty(3Ca)^{{-}1/3}}, \quad \eta=\frac{\hat{h}}{h_\infty}, \quad Ca=\frac{\mu U}{\sigma}. \end{equation}

This is the film equation derived by Bretherton (1961) that describes the transition region between the film and the caps at either end. At the bubble front, (2.2) can be integrated numerically starting from the uniform film, which corresponds to the boundary condition $\eta (-\infty )=1$, towards $x\rightarrow \infty$ where the solution matches to a spherical cap of radius $R$. In the vicinity of the uniform film, the solution has an exponential behaviour, while, for $\eta \gg 1$ a parabolic region with constant dimensionless curvature ${\rm d}^{2} {\eta }/{\rm d}x^{2}$ is established. To get the profile at the bubble rear, (2.2) is integrated in the opposite direction towards $x\rightarrow -\infty$ starting from the uniform film region. Imposing that the rear curvature far from the uniform has the same curvature of the front, the typical oscillating profile at the back is obtained. All the details for the computation of the bubble profile are described in Bretherton (1961).

Figure 1. Sketch of the confined bubble that moves at speed $U$ within a channel of half-width $R$. The film regions of length $L$ of the front and rear menisci are depicted. The Taylor bubble is sufficiently long so that a region of uniform film thickness $h_\infty$ exists.

2.2. Governing equations for transport

We consider the transport of a scalar of concentration $c(\hat {x},\hat {y},\hat {t})$ within the film region of a confined Taylor bubble. The scalar is advected by the velocity field and diffuses with a constant diffusion coefficient $D$. In a reference frame attached to the bubble, the governing equation for the concentration reads

(2.4)\begin{equation} \frac{\partial \hat{c}}{\partial \hat{t}} + \left[ \hat{u} -U \right] \frac{\partial \hat{c}}{\partial \hat{x}} +{\hat{v}\frac{\partial \hat{c}}{\partial \hat{y}} } = D\frac{\partial^{2} \hat{c}}{\partial \hat{x}^{2}}+D\frac{\partial^{2} \hat{c}}{\partial \hat{y}^{2}}, \quad \hat{y}\in[0, \hat{h}], \end{equation}

where $\hat {u}$ and $\hat {v}$ are the local velocity in the $x$ and $y$ directions, respectively. According to Bretherton (1961), the flow in the film region is expressed using a lubrication approximation with $\hat {u}(\hat {x},\hat {y})$ and $\hat {v}(\hat {x},\hat {y})$ given in (A1) and (A3), respectively. We assume that at the bottom wall, $\hat {y}=0$, and the bubble–fluid interface, $\hat {y}=\hat {h}$, the scalar is transferred with first-order mass-transfer coefficients $k_w$ and $k_i$, respectively, as described by the following boundary conditions:

(2.5)$$\begin{gather} \left. -D \boldsymbol{\nabla} \hat{c}\right|_0 \boldsymbol{\cdot} \boldsymbol{n} = k_w \hat{c} \quad {\rm at} \ \hat{y}=0, \end{gather}$$

(2.6)$$\begin{gather}\left. -D \boldsymbol{\nabla} \hat{c}\right|_{\hat h} \boldsymbol{\cdot} \boldsymbol{n} = k_i \hat{c} \quad {\rm at} \ \hat{y}=\hat{h}, \end{gather}$$

where $\boldsymbol {n}$ is the unit vector perpendicular to the boundary pointing out of the fluid domain, see figure 1. At the wall, the normal unit vector is $\boldsymbol {n}=(0,-1)$ while at the bubble–fluid interface the unit vector depends on the meniscus profile as

(2.7)\begin{equation} \boldsymbol{n}=\left( -\dfrac{\partial \hat{h}/\partial \hat{x}}{\sqrt{1+(\partial \hat{h}/\partial \hat{x})^{2}}}, \dfrac{1}{\sqrt{1+(\partial \hat{h}/\partial \hat{x})^{2}}} \right)^{\rm T}. \end{equation}

Before recasting the transport equation in terms of dimensionless variables, it is worth identifying the characteristic scales of the problem. The two relevant length scales in the $\hat {x}$ and $\hat {y}$ directions are the characteristic length of the film region, $L=h_\infty (3Ca)^{-1/3}$ from Bretherton (1961), and the uniform film thickness $h_\infty$. Based on these, we define the following scale parameter:

(2.8)\begin{equation} \varepsilon=\frac{h_\infty}{{L}}=(3Ca)^{{1}/{3}}\ll 1. \end{equation}

The determination of the relevant time scale is not unique (e.g. Mauri 1991; Auriault & Adler 1995; Mikelić et al. 2006; Battiato & Tartakovsky 2011; Bourbatache, Millet & Moyne 2020). Assuming that the characteristic scale for the velocity is the bubble speed $U$, we can define the axial advective time $\tau _a=L/U$. Molecular diffusion introduces two additional time scales, the axial, $\tau _L=L^{2}/D$, and the transverse, $\tau _h=h_\infty ^{2}/D$, diffusion times. Since we are primarily interested in transport dynamics in a time frame much larger than the transverse diffusion time and close to the advection time, we choose $\tau _a$ as the reference scale for the time variable.

The presence of superficial mass transfer introduces additional time scales in the axial, $\tau _{Lw}={L}/{k_w}$ and $\tau _{Li}={L}/{k_i}$, and the transverse, $\tau _{hw}={h_\infty }/{k_w}$ and $\tau _{hi}={h_\infty }/{k_i}$, directions. As discussed later, we focus on the most general case when the transversal mass transfer balances with axial advection.

We make (2.4) dimensionless with

(2.9a–f)\begin{equation} x=\frac{\hat{x}}{{L}}, \quad {y=\frac{\hat{y}}{{h_\infty}}}, \quad t=\frac{\hat{t}}{\tau_a},\quad u=\frac{\hat{u}}{U}, \quad {v=\frac{\hat{v}}{\varepsilon U}}, \quad c=\frac{\hat{c}}{c_{ref}}, \end{equation}

where $c_{ref}$ is a reference value for the concentration. This procedure yields

(2.10)\begin{equation} \frac{\partial c}{\partial t} + [u-1]\frac{\partial c}{\partial x} + {{v}\frac{\partial {c}}{\partial {y}} } = \frac{1}{Pe}\frac{\partial^{2} c}{\partial x^{2}} + \frac{1}{Pe \varepsilon^{2} }\frac{\partial^{2} c}{\partial y^{2}}, \quad {{y}\in[0, \eta]}. \end{equation}

where $u(x,y)$ and $v(x,y)$ are given in (A7) and (A8), respectively. The Pèclet number $Pe$ is defined as

(2.11)\begin{equation} Pe = \frac{{L} U}{D}=\frac{\tau_{L}}{\tau_a}, \end{equation}

and can be interpreted as the ratio between advection and diffusion in the axial direction or, alternatively, between the diffusion and the advection time scales. When the Pèclet number is small, $Pe\ll 1$, the problem is diffusion dominated; when $Pe={O}(1)$ diffusion and advection play similar roles; for $Pe\gg 1$ the problem is advection dominated. The dimensionless boundary conditions at the wall and at the bubble–fluid interface read

(2.12)$$\begin{gather} \frac{\partial c}{\partial y} = \varepsilon^{2}\, Sh_w\, c, \quad {\rm at} \ y=0, \end{gather}$$

(2.13)$$\begin{gather}\dfrac{\varepsilon^{2}\partial {\eta}/\partial {x}}{\sqrt{1+\varepsilon^{2}(\partial {\eta}/\partial {x})^{2}}}\frac{\partial c}{\partial x}- \dfrac{1}{\sqrt{1+\varepsilon^{2}(\partial {\eta}/\partial {x})^{2}}}\frac{\partial c}{\partial y}=\varepsilon^{2}\,Sh_i\, c, \quad {\rm at} \ y=\eta , \end{gather}$$

where

(2.14a,b)\begin{equation} Sh_w=\frac{{L}^{2}k_w}{h_\infty D}=\frac{\tau_{{L}}}{\tau_{hw}} \quad Sh_i=\frac{{L}^{2}k_i}{h_\infty D}=\frac{\tau_{{L}}}{\tau_{hi}}, \end{equation}

are the Sherwood numbers for the wall and interfacial mass-transfer mechanisms. Both Sherwood numbers are the ratio between transversal mass transfer and axial diffusion: for finite $Sh$, superficial mass transfer competes with advection and diffusion; for $Sh\ll 1$ superficial mass transfer becomes negligible. Under the assumption that the lubrication approximation for the film, equation (2.2) holds, i.e. $\varepsilon \ll 1$, (2.13) simplifies to

(2.15)\begin{equation} -\frac{\partial c}{\partial y}= \varepsilon^{2}\,Sh_i c, \quad {\rm at} \ y=\eta . \end{equation}

In other words, the slope of the bubble–fluid interface is sufficiently small, ${\rm d}\hat {h}/{\rm d}\hat {x}\ll 1$, that the unit vector (2.7) almost aligns with the $y$ axis in figure 1, i.e. $\boldsymbol {n}=(0,1)$ at the bubble–fluid interface.

2.3. Two-scale asymptotic expansion of the transport equation

The goal of this work is to obtain a one-dimensional approximation of the transport equation in the film region. We seek to account for advection, diffusion and superficial mass-transfer mechanisms through an effective velocity, effective diffusion and effective mass-transfer coefficient in an advection–diffusion–mass-transfer equation. To do so, we proceed using a two-scale asymptotic expansion (Hornung 1997; Bensoussan, Lions & Papanicolaou 2011; Boutin, Auriault & Geindreau 2010):

1. (i) The starting point is the local description of the transport problem in dimensionless form, see (2.10), (2.12) and (2.15), where we define the scale parameter $\varepsilon =(3Ca)^{{1}/{3}}\ll 1$, see (2.8). Specifically, the proposed asymptotic expansion holds in the limit of small capillary numbers, $Ca\ll 1$, as constrained by Bretherton's theory.

2. (ii) The governing dimensionless numbers are evaluated with respect to the scale parameter – or the capillary number – as

   (2.16a–c)\begin{equation} Pe = \varepsilon^{-\alpha},\quad Sh_w = \varepsilon^{\beta}, \quad Sh_i = \varepsilon^{\gamma}, \end{equation}
   where the exponents $\alpha$, $\beta$ and $\gamma$ characterise the system behaviour.

3. (iii) The concentration field $c(x,y,t)$ is expanded in (2.10), (2.12) and (2.15) as a perturbation series in the scale parameter as follows:

   (2.17)\begin{equation} c = c_0(x,y,t) + \varepsilon c_1(x,y,t) + \varepsilon^{2} c_2(x,y,t) + \ldots . \end{equation}

4. (iv) Then, after collecting the terms with the same order, the successive boundary-value problems are solved to obtain the effective description of the transport problem in terms of the film-averaged concentration

   (2.18)\begin{equation} \langle c \rangle = \frac{1}{\eta}\int_0^{\eta} c(x,y,t)\, {{\rm d} y} , \end{equation}
   plus higher-order corrections. Specifically, the zeroth-order term in (2.17) is independent on $y$ and, therefore, $c_0(x,t)=\langle c \rangle (x,t)$.

5. (v) The last step is checking if the aforementioned procedure yields the desired result (the existence of an effective description in terms of the average concentration) or conditions where the coupling between scales prevents the existence of an equivalent effective description. The parameter space where this result holds is rigorously determined and the applicability conditions of the model can be mapped in terms of $Pe$, $Sh_w$ and $Sh_i$.

The full derivation is provided in Appendix B, distinguishing between three main transport regimes: advection dominated, competition between advection and diffusion and diffusion dominated. The final result of this procedure is given in the next section.

2.4. The one-dimensional advection–diffusion–mass-transfer equation

The advection–diffusion–mass-transfer equation describing the transport of the averaged concentration in the film region of a Taylor bubble is given by

(2.19)\begin{equation} \frac{\partial \langle c \rangle }{\partial t }+ (u^{{\star}}-1)\, \frac{\partial \langle c \rangle}{\partial x } + \frac{Sh^{{\star}}}{Pe}\langle c \rangle= \frac{ D^{{\star}}}{Pe} \frac{\partial^{2} \langle c \rangle}{\partial x^{2}}, \end{equation}

where $u^{\star }$, $Sh^{\star }$ and $D^{\star }$ are the effective velocity, the effective Sherwood number and the effective diffusion coefficient, respectively, defined as

(2.20)\begin{align} u^{{\star}}&=\frac{\eta-1}{3\eta}-\varepsilon^{2}Pe\,\frac{(73\eta-8)(\eta-1)}{3780\eta}\dfrac{{\rm d}\eta}{{\rm d} x}- \varepsilon^{2}Sh_i\dfrac{14\eta(\eta-1)+(73\eta+47)\,{\rm d}\eta/{{\rm d} x}}{360\eta}\nonumber\\ &\quad + \varepsilon^{2}Sh_w\dfrac{16\eta(\eta-1)+(47\eta+13)\,{\rm d}\eta/{{\rm d} x}}{360\eta}, \end{align}

(2.21)\begin{align} Sh^{{\star}}&=\frac{Sh_w+Sh_i}{\eta} + \dfrac{\varepsilon^{2} }{3}\left(Sh_w\, Sh_i-Sh_w^{2}-Sh_i^{2} \right) \nonumber\\ &\quad-\varepsilon^{2}Pe\dfrac{Sh_i(7\eta-18)+Sh_w(7-3\eta)}{120 \eta} \dfrac{{\rm d}\eta}{{\rm d} x} , \end{align}

(2.22)\begin{align} D^{{\star}}&= 1+\varepsilon^{2}Pe^{2} \frac{2{(\eta-1)^{2}}}{945}. \end{align}

The effective coefficients of (2.19) incorporate the impact of shear flow in the film and the changes in the bubble shape. Specifically, the shear flow spreads the concentration distribution in the axial direction, enhancing the axial diffusion coefficient $D^{\star }$ at sufficiently large Péclet numbers. At the same time, changes in the curvature of the bubble along the axial direction affect the pressure gradient (that drives the flow), and both $u^{\star }$ and $Sh^{\star }$.

The effective advection accounts for the contribution of three mechanisms: the velocity in the axial direction (the first term in (2.20)), the coupled effect of bubble shape and the velocity field in both the axial and transverse directions (the second term in (2.20)), superficial mass-transfer mechanisms at the boundaries (the last two terms in (2.20)).

The presence of surface mass transfer translates to a source term in (2.19) whose intensity is controlled by the effective Sherwood number. The source term acts to diminish the averaged concentration ($Sh^{\star }$ is always positive) as a consequence of how the boundary conditions (2.12) and (2.15) are formulated, i.e. assuming outflow fluxes at the boundaries. By inspection of (2.21), we can see that the effective mass transfer accounts for the contribution of the bubble shape (the first term in (2.21)) and the coupled effect of local advection and changes in the bubble shape (the last term in (2.21)).

The evolution of the effective coefficients in different regions of the bubble and their physical origins will be discussed § 3. In Appendix C, we will show that only the leading term in (2.20) and (2.21) contributes to the overall $u^{\star }$ and $Sh^{\star }$ in both the front and the rear menisci.

Concerning the applicability of the effective equation, (2.19) holds only if the following conditions are met:

1. (i) $\varepsilon \ll 1$;

2. (ii) $Pe\ll {O}(\varepsilon ^{-2})$;

3. (iii) $Sh_w\ll {O}(\varepsilon ^{-2})$ and $Sh_i\ll {O}(\varepsilon ^{-2})$;

4. (iv) $Sh_i/Pe\ll {O}(\varepsilon ^{-1})$ and $Sh_w/Pe\ll {O}(\varepsilon ^{-1})$.

Condition (i) ensures the $x-y$ scale separation holds, i.e. the film region is much longer compared with thickness of the uniform film. Conditions (ii) and (iii) provide the upper bounds to the Pèclet and the Sherwood numbers, identifying the parameter space for which it is possible to derive an effective transport equation for the averaged concentration. Condition (iv) ensures that mass-transfer mechanisms do not prevail over advection.

2.5. Identification of transport regimes

In this section we clarify the competition between advection, diffusion and mass transfer in the film surrounding a Taylor bubble, defining different transport regimes for (2.19). To do so, we summarise the regimes in a phase diagram based on the magnitude of the Pèclet and Sherwood numbers, see figure 2. The coloured regions of figure 2 refer to physical conditions where the transport equation for the film-averaged concentration can be formally written by means of two-scale expansions. Outside the coloured regions the scales are coupled. In the phase diagram, the roles of the Sherwood number at the wall, $Sh_w$, and the interface, $Sh_i$, are equivalent since they appear with the same functional dependence in (2.20) and (2.21).

Figure 2. Phase diagrams of transport regimes in the $Pe$–$Sh_w$ or $Pe$–$Sh_i$ planes. Coloured areas correspond to regimes where the effective transport equation for the film-averaged concentration can be formally written by means of two-scale expansions. We identify the following transport regimes: I, effective advection, effective diffusion and effective mass transfer; II, effective advection and effective diffusion; III, effective advection, molecular diffusion and effective mass transfer; IV, effective advection and molecular diffusion; V, molecular diffusion and effective mass transfer; VI, molecular diffusion only.

When the Pèclet number is large ($Pe={O}(\varepsilon ^{-1}$)) and at least one of the two Sherwood numbers is of the order of ${O}(\varepsilon ^{-1})$, the transport is driven by the competition between effective advection, effective diffusion and effective mass transfer as

(2.23)$$\begin{gather} \text{Regime I:}\quad \dfrac{\partial \langle c \rangle }{\partial t }+ (u^{{\star}}-1)\, \frac{\partial \langle c \rangle}{\partial x } + \varepsilon\, {Sh^{{\star}}_I}\langle c \rangle= \varepsilon\, {D^{{\star}}_I} \frac{\partial^{2} \langle c \rangle}{\partial x^{2}}, \end{gather}$$

(2.24)$$\begin{gather}\text{with}\quad {D^{{\star}}_I} \sim 1+ \frac{2{(\eta-1)^{2}}}{945},\quad Sh^{{\star}}_I\sim \frac{1}{\varepsilon\eta}, \quad u^{{\star}}\sim\frac{\eta-1}{3\eta}, \end{gather}$$

where we have used the approximations given in Appendix C. Near the uniform film where $\eta ={O}(1)$, shear and shape induced diffusion is negligible since the fluid is nearly at rest. Therefore, the effective diffusion coefficient reduces to molecular diffusion, $D^{\star }_I \approx 1$. Far from the uniform film, instead, shear-flow and shape induced diffusion plays a role augmenting $D^{\star }_I$. The effective mass-transfer coefficient $Sh^{\star }_I$ plays a role only in the proximity of the uniform film where the surface available to mass transfer is high compared with the amount of fluid. If $\eta \gg 1$, $Sh^{\star }_I$ becomes negligible. For smaller Sherwood numbers, $Sh_i \ll {O}(\varepsilon ^{-1})$ and $Sh_w\ll {O}(\varepsilon ^{-1})$, the source term in (2.23) can be dropped; we refer to this case as regime II.

When the Pèclet number is finite ($Pe={O}(1$)) and at least one of the Sherwood numbers is of the order of ${O}(1)$, effective advection competes with effective mass transfer and molecular diffusion leading to

(2.25)$$\begin{gather} \text{Regime III:} \quad \dfrac{\partial \langle c \rangle }{\partial t }+ (u^{{\star}}-1)\, \frac{\partial \langle c \rangle}{\partial x } + {Sh^{{\star}}_{III}}\langle c \rangle= { D^{{\star}}_{III}} \frac{\partial^{2} \langle c \rangle}{\partial x^{2}}, \end{gather}$$

(2.26)$$\begin{gather}{\rm with} \quad {D^{{\star}}_{III}} \approx 1,\quad \quad Sh^{{\star}}_{III}\sim \frac{1}{ \eta}. \end{gather}$$

In this regime, shear and shape induced diffusion becomes negligible. In case the Sherwood numbers are smaller, the mass-transfer term in (2.25) can be dropped (regime IV).

When the Pèclet number is small ($Pe={O}(\varepsilon$)) and at least one of the Sherwood numbers is of the order of ${O}(\varepsilon )$, transport is purely driven by molecular diffusion and advection plays a negligible role. Specifically,

(2.27)\begin{equation} \text{Regime V:} \quad Pe\dfrac{\partial \langle c \rangle }{\partial t }+ {Sh^{{\star}}_{V}}\langle c \rangle= { D^{{\star}}_{V}} \frac{\partial^{2} \langle c \rangle}{\partial x^{2}}, \quad {\rm with} \ {D^{{\star}}_{V}} = 1, \end{equation}

where the mass-transfer term can be dropped when $Sh_w$ and $Sh_i$ assumes smaller values (regime VI), see figure 2.

2.6. Numerical solution

The general solution of (2.19) can be found numerically for the front and rear menisci if the film profile and the boundary and initial conditions for the averaged concentration are provided.

The film profile, $\eta (x)$, is obtained solving (2.2) for the rear and the front menisci separately, as discussed in § 2.1, using the differential equation solver ode45 of Matlab. The code used in this work for this purpose has been developed in a previous publication (Picchi et al. 2021), where all the details concerning the boundary conditions are discussed.

The concentration field $\langle c \rangle (x,t)$ is obtained solving (2.19) using the solver pdepe of Matlab. A different set of initial and boundary conditions is considered and will be discussed in the following sections. The front and the rear menisci are treated separately and, since the effective coefficients depend on the local film thickness $\eta (x)$, we developed a Matlab function that provides the film thickness at a specific location to evaluate the coefficients of (2.19).

3. Results and discussion

3.1. The concentration field in the front meniscus

In this section, we investigate how the solute is transported within the film region in the bubble front. Figure 3(a) shows the typical profile of the front meniscus (black line): close to the uniform film region the interface grows slowly (exponential region with $\eta \sim \exp (x)$) while, for $\eta \gg 1$, the profile follows a parabolic trend ($\eta \sim 0.321 x^{2}$ as in Bretherton 1961). The following analysis assumes that the scale parameter is sufficiently small so that the concentration field can be expanded as in (2.17), i.e. $\varepsilon =0.01$.

Figure 3. (a) Contours of the magnitude of the velocity, $\sqrt {u(x,y)^{2}+v(x,y)^{2}}$ and flow streamlines with respect to a fixed reference frame in the bubble front; (b) effect of Pèclet number on the effective diffusion coefficient, $D^{\star }$, in the front meniscus; right, profile of the front meniscus; $\varepsilon =0.01$.

Diffusion is controlled by the effective diffusion coefficient $D^{\star }$ defined in (2.22). In the uniform film region, $\eta =1$, the fluid is at rest and only molecular diffusion plays a role, $D^{\star }=1$. In the front meniscus, instead, the shear flow tends to spread the scalar along the axial direction (see the streamlines in figure 3a) enhancing the effective diffusion at sufficiently large Péclet numbers. This effect is the combination of the local axial velocity field and changes in bubble shape, as shown in figure 3(b). For the sake of physical interpretation, we can recast (2.22) in terms of the Pèclet number of the film, $Pe_f$,

(3.1)\begin{equation} D^{{\star}}= 1+Pe^{2}_f \frac{2}{945}\quad {\rm where} \ {Pe_f=\frac{U(\hat{h}-h_\infty)}{D},} \end{equation}

recovering the same scaling law of the Aris–Taylor dispersion coefficient. In other words, the effective diffusion coefficient follows a Pèclet square behaviour.

The effective velocity given in (2.20) is negligible only in the uniform film region. As the magnitude of the velocity becomes relevant, see figure 3(a), the averaged concentration experiences additional advection, as plotted in figure 4. The major contribution comes from the axial velocity which follows a $(\eta -1)/3\eta$ behaviour and reaches the limiting value of $u^{\star } \rightarrow 1/3$. The effective velocity slightly diminishes as an effect of Pe and $Sh_i$ when $\eta$ is high; the evolution of $u^\star$ with respect to $Sh_w$ is not shown for the sake of brevity. Interestingly, the contribution of the wall and interfacial Sherwood numbers, $Sh_w$ and $Sh_i$, appear with opposite sign in (2.20). This can be explained with differences in the local velocity between the channel wall (where the velocity is zero due to the no-slip condition) and the bubble–fluid interface (where the velocity is non-zero according to the free-surface boundary condition).

Figure 4. Effective velocity in the front meniscus. (a) Left, effect of $Pe$; right, profile of the front meniscus. (b) Left, effect of mass transfer at the bubble–fluid interface; right, profile of the front meniscus. (c) Left, effect of the Sherwood number at the wall; right, profile of the front meniscus. The evolution of $Sh_i$ is not shown since there is no discernible difference from $Sh^{\star }$. In all cases $\varepsilon =0.01$.

The effective Sherwood number, (2.21), is maximum in the uniform film region. There, since the fluid is at rest, the solute cannot escape the film and, therefore, it is rapidly consumed from the boundaries. Outside of the uniform film region, $Sh^{\star }$ decays as $\eta ^{-1}$ since the non-zero velocity causes the scalar to escape the surface, see figure 4(c). Note that the second and the third terms in (2.21) make only small contributions to the overall $Sh^{\star }$, meaning that the effective mass transfer is mostly driven by the proximity to the boundaries (i.e. the bubble shape) via the variable $\eta$.

In order to identify which mechanism dominates, we construct the following representative test case. We solve (2.19) imposing a concentration front $\langle c \rangle (x_0,t)=1$ at $x\geqslant x_0$ and initialising $\langle c \rangle (x,0) =0$ in the entire domain. Since the location of the concentration front is purely arbitrary, we chose to place it at $x_0=20$. This choice ensures that the concentration front is sufficiently far from the transition between the exponential region (where $\eta \approx 1$) and the parabolic region. In this way, we inject the solute into a region where the dimensionless curvature is constant. This set-up mimics the case where the bubble advances through a ‘dirty’ (i.e. sharp discontinuity in the concentration field) channel. Specifically, we show the transient dynamics of the most general case, regime I defined in § 2.5, and the transitions to regimes II, III and IV.

Figure 5 shows the time evolution of the average of the concentration along the front meniscus for a regime where advection dominates over diffusion and superficial mass transfer balances with advection. Specifically, we refer to a case where $Pe=100$, $Sh_w=100$ and $Sh_i=0$ (regime I). At $t=1$, see figure 5(a), the concentration front is sharp, while at later times, the front gets more and more diffuse. Interestingly, as the front gets closer to the uniform film region, the concentration rapidly decreases since the effective Sherwood number $Sh^{\star }$ in (2.19) becomes dominant, see figure 4(c); the movie of this case is available as supplementary Movie 1 available at https://doi.org/10.1017/jfm.2022.829. To get a more quantitative picture, we present the breakthrough curves in figure 6(e): the solute is entirely consumed even before reaching the uniform region. In this regime, even though the bubble advances through a ‘dirty’ channel, we expect the uniform film region to be ‘clean’ from the solute at later times.

Figure 5. Time evolution of the average of the concentration, $\langle c \rangle (x,t)$, in the bubble front for $Pe=100$, $Sh_w=100$, $Sh_i=0$ and $\varepsilon =0.01$. Note that the quantity plotted is a one-dimensional approximation of a two-dimensional field. See online supplementary Movie 1 for a simulation of this case.

Figure 6. Breakthrough curves for the averaged concentration in the front meniscus as a function of $Pe$ and $Sh_w$; in all the cases $\varepsilon =0.01$. On the left of each panel the averaged concentration at different times is plotted, on the right the front meniscus. Cases (a,c,e) are dominated by advection; in (b,d, f) diffusion compete with advection.

When the surface mass transfer is reduced to $Sh_w/Pe=0.1$, the decay of the passive scalar in the film region is less intense and, at later times, the front propagates inside the uniform film region, see figure 6(c). In case $Sh_w=0$ (regime II), the front propagates almost rigidly except for the effect of weak diffusion, see figure 6(a).

When diffusion and advection compete for $Pe=1$ (regimes III–IV), the breakthrough curves assume a different shape, see figure 6(b,d, f). At early times, the front is rapidly diffused independently of the magnitude of the Sherwood number and an increase of $Sh_w$ results in a more intense decay in the film region. Also for this regime, when $Pe\sim Sh_w$ the solute is consumed even before reaching the uniform film. Note that we only show the impact of the wall mass-transfer mechanism, but analogous results are obtained for mass transfer at the bubble–fluid interface imposing $Sh_i \neq 0$.

The effect of the Pèclet and Sherwood numbers is summarised in figure 7, where we show the breakthrough curves at the steady state, $\langle c \rangle (x,t\rightarrow \infty )$. Each steady state curve represents the envelope of the transient behaviour described in figure 6. As expected, when $Sh_w=0$, the mass-transfer mechanism does not play any role and, therefore, the steady state is the trivial solution $\langle c \rangle =1$. As $Sh_w$ increases, instead, the solute is quickly consumed: the location where $\langle c \rangle \approx 0$ shifts closer to the concentration source, see figure 7(a). The Pèclet number affects the steady state behaviour by shifting the breakthrough curve in the direction of the uniform film (i.e. enhancing the effect of advection), see figure 7(b).

Figure 7. Breakthrough curves for the averaged concentration at the steady state, $\langle c \rangle (x,t\rightarrow \infty )$, in the front meniscus. In all the cases $\varepsilon =0.01$ and $Sh_i=0$. (a) Effect of the Sherwood number; (b) effect of the Pèclet number.

Another aspect of interest for applications is the formation of hot spots in the concentration field, which may determine localised mass-transfer fluxes. To see this, we initialise the simulations imposing a concentration field $\langle c \rangle (x,0)=1$ in the entire fluid domain and $\langle c \rangle (x_0,t)=0$ at the right boundary of the domain.

Figure 8(a) shows a significant case where the formation of a hot spot can be observed in the advection dominated regime ($Pe=Sh_w=100$). At early times, $t=0.2$, the concentration starts decaying in the fluid domain due to surface mass transfer according to the magnitude of $Sh^{\star }$, see figure 4(c). The solute is quickly consumed in the uniform film region (where $Sh^{\star }$ is maximum) while it is mainly advected into the parabolic region of the front meniscus (where $u^{\star }$ is maximum and $Sh^{\star } \approx 0$). Since the effective velocity vanishes in the uniform film, an accumulation of solute forms. Finally, at later times, the hot spot is consumed by the surface mass transfer.

Figure 8. Breakthrough curves during the formation of the hot spot of the average of the concentration, $\langle c \rangle (x,t)$, for (a) regime I and (b) regime III. See online supplementary Movies 2 and 3 for simulations of these cases.

The hot spot is the result of the transition between the parabolic region, where the effective velocity dominates, and the uniform film, where the effective mass transfer dominates over the advection coefficient. The formation of the hot spot is observed only when the Pèclet number balances with the Sherwood number, i.e. $Pe\sim Sh_w$, otherwise one of the two mechanisms prevails. A similar behaviour occurs for $Pe=Sh_w=1$, as shown in figure 8(b) and in supplementary Movies 2 and 3.

3.2. The concentration field in the rear meniscus

In this section, we investigate how the solute is transported in the rear of the bubble. Differently from the front, the meniscus shows the typical oscillations and, therefore, the effective coefficients in (2.19) are non-monotonic with respect to $x$.

The effective diffusion coefficient $D^{\star }$, is marginally influenced by the oscillations and the enhancement due to shear flow is significant only far from the uniform film, see figure 9(b). Specifically, the shape oscillations induce changes in the sign of the driving force (when $\eta <1$ then ${\rm d}p/{{\rm d} x}>0$ while $\eta >1$ gives ${\rm d}p/{{\rm d} x}<0$) and regions of backflow appear. This leads to the formation of recirculating vortices in the bubble rear (see the streamlines in figure 9a) and the effective velocity changes sign in correspondence of such oscillations. The evolution of $u^{\star }$ is presented in figure 10 where we can observe that the effect of $Pe$ and $Sh_i$ on the effective velocity is negligible.

Figure 9. (a) Contours of the magnitude of the velocity, $\sqrt {u(x,y)^{2}+v(x,y)^{2}}$ and flow streamlines with respect to a fixed reference frame in the bubble rear. (b) Effective diffusion coefficient as function of $Pe$ in the rear meniscus; the scale parameter is set to $\varepsilon =0.01$.

Figure 10. Effective velocity in the rear meniscus as a function of $Pe$, (a) and $Sh_i$, (b). (c) Effective mass-transport coefficient in the rear meniscus as a function of $Sh_w$; the evolution of $Sh_i$ is not shown since there is no discernible difference in plotting $Sh^{\star }$. In all cases the scale parameter is set to $\varepsilon =0.01$.

The presence of the recirculating vortices drives the evolution of the effective Sherwood number. Differently from the bubble front, where $Sh^{\star }$ is maximal in the uniform film region, in the rear, mass transfer is locally enhanced in proximity to the minimum of the bubble profile, see figure 10(c). There, a recirculating vortex forms and tends to squeeze out the liquid from the narrow film (see the streamlines in figure 9a). The scalar is concentrated in a narrow gap in proximity to the boundaries, enhancing the mass-transfer mechanism. This effect is more pronounced as $Sh_w$ (or $Sh_i$) increases.

The fact that shape oscillations enhance mass transfer at the bubble–fluid interface is a known phenomenon for the case of unconfined bubbles (Beek & Kramers 1962; Angelo, Lightfoot & Howard 1966) and this seems to be confirmed for the case of Taylor bubbles. Hayashi et al. (2014) demonstrated that fluctuations of the bubble surface area at the bubble rear account for the main contribution to the oscillation of the effective Sherwood number. The authors arrived at this conclusion after an analysis of the velocity and the concentration fields around a Taylor bubble rising in a microchannel. Our theoretical results support this evidence.

Figure 11 shows the time evolution of the averaged concentration in the bubble rear for the advection dominated regime with $Pe=100$, $Sh_w=10$ and $Sh_i=0$. The case has been constructed solving (2.19) with initial concentration $\langle c \rangle (x,0)=0$ in the entire domain and $\langle c \rangle (0,t)=1$ at the right boundary of the domain. The concentration front is sharp at $t=1$, see figure 11(a), and it gets more and more diffuse at later times, see figure 11(d). As the front moves away from the uniform film region, the concentration progressively decays as an effect of the surface mass-transfer mechanism; the breakthrough curves for this case are presented in figure 12(c). The movie of this case is available as supplementary Movie 5.

Figure 11. Time evolution of the average of the concentration, $\langle c \rangle (x,t)$, for $Pe=100$, $Sh_w=10$, $Sh_i=0$ and $\varepsilon =0.01$. Note that the quantity plotted is a one-dimensional approximation of a two-dimensional field. See online supplementary Movie 5 for a simulation of this case.

Figure 12. Breakthrough curves for the averaged concentration in the rear meniscus as a function of $Pe$ and $Sh_w$; in all the cases $\varepsilon =0.01$. On the left of each panel the averaged concentration at different times is plotted, on the right the rear meniscus. Cases (a,c,e) are dominated by advection; in (b,d, f) advection competes with diffusion.

When the surface mass-transfer mechanism is augmented to $Sh_w=Pe=100$, the decay of the solute is more intense and the concentration front cannot escape the film region, figure 12(e). On the other hand, if mass transfer is turned off, $Sh_w=0$, the front advances towards the bubble rear almost undisturbed, figure 12(a).

When diffusion and advection compete for $Pe=1$, the front is rapidly diffused and, as $Sh_w$ increases, the concentration decay in the film region becomes more intense. The effect of the Sherwood number is presented in figure 12(b,d, f). In this regime, when $Pe\sim Sh_w$, the solute is consumed before exiting the uniform film region, figure 12( f).

3.3. The concentration field in the uniform film

In this section, we study the mechanisms that govern the transport problem in the uniform film region, i.e. where $\eta =1$ and ${\rm d}\eta /{{\rm d} x}=0$ in figure 1. In this region, the fluid is at rest since the curvature is infinite and consequently ${\rm d}p/{{\rm d} x}=0$, see (A6). The advection–diffusion–mass-transfer equation (2.19) reduces to

(3.2)\begin{equation} \frac{\partial \langle c \rangle }{\partial t } - \frac{\partial \langle c \rangle}{\partial x } + \frac{Sh_f^{{\star}}}{Pe} \langle c \rangle= \frac{1}{Pe} \frac{\partial^{2} \langle c \rangle}{\partial x^{2}}, \end{equation}

where

(3.3)\begin{equation} Sh^{{\star}}_f={Sh_w+Sh_i} + \dfrac{\varepsilon^{2} }{3}(Sh_w\, Sh_i-Sh_w^{2}-Sh_i^{2}). \end{equation}

Differently from the general case, the effective velocity is identically zero, $u^{\star }_f= 0$, and the effective diffusion coefficient is equal to unity, $D^{\star }_f= 1$. The concentration distribution is controlled by the competition between diffusion and superficial mass transfer, while the advection term in (3.2) is there only because we consider a reference frame attached to the bubble. In the film, the effective Sherwood number $Sh_f^{\star }$ does not depend on the bubble geometry and, therefore, it is representative of the total mass transfer.

The main advantage in studying the transport problem in the uniform film region is that (3.2) admits analytical solutions. In fact, the desired solution can be constructed from the following Green function (i.e. the solution to a point-source initial condition centred at $x=0$):

(3.4)\begin{equation} \mathcal{G}(x,t)=\dfrac{\exp\left(-\dfrac{Sh^{{\star}}_f}{Pe}t\right)}{\sqrt{4 {\rm \pi}t /Pe}}\exp \left( -\dfrac{(x-t)^{2}}{4t/Pe}\right). \end{equation}

Since the Green function is general and it is applicable to different sets of initial and boundary conditions, we focus on the study of a simplified case. We solve (3.2) initialising the concentration film with a pulse of height equal to unity, centred at $x=0$, in a regime where advection dominates over diffusion, $Pe=100$, and competes with superficial mass transfer, $Sh_f^{\star }=100$. The solution is a Gaussian function that diffuses and decays in time while being advected along $x$, as shown in figure 13(a). Interestingly, the total mass, i.e.

(3.5)\begin{equation} \int_{-\infty}^{\infty }\mathcal{G}(x,t) \,{{\rm d} x} =\exp\left(-\dfrac{Sh_f^{{\star}}}{Pe}t\right), \end{equation}

exponentially decays, controlled by the competition between $Sh_f^{\star }$ and $Pe$. Specifically, when $Sh_f^{\star } /Pe\rightarrow 0$, axial diffusion has an effect only on the concentration maximum, but the total mass remains constant. When $Sh_f^{\star } /Pe$ is finite, instead, the total mass diminishes, as shown in figure 13(b) and the residence time of the solute in the uniform film gets progressively shorter (it scales as $Pe/Sh_f$).

Figure 13. (a) Evolution of the averaged concentration in response to a pulse of height equal to unity, centred at $x=0$. (b) Decay of the total mass as a function of time and the ratio $Sh_f^{\star }/Pe$.

The analytical solution of the transport problem in the uniform film can be employed in the construction of mechanistic or more sophisticated models where knowing the spatial and time evolution of the concentration field is of interest.

4. Conclusions

In this paper, we studied the transport of a passive scalar around a confined Taylor bubble in the presence of surface mass transfer. We derived a one-dimensional advection–diffusion–mass-transfer equation that applies to the front and rear menisci of the bubble. Specifically, advection, diffusion and superficial mass-transfer mechanisms are described through an effective velocity, effective diffusion and effective mass-transfer coefficients, expressed analytically as a function of the Pèclet and Sherwood numbers, and the film thickness. Our analysis helped in quantifying the contributions that shear flow in the film, changes in the meniscus shape and the presence of mass transfer at the domain boundaries make to the effective coefficients.

Concerning the bubble front, shear flow in the film acts to smear out the concentration distribution in the axial direction, enhancing the effective diffusion coefficient at sufficiently large Pèclet number. Interestingly, the diffusion coefficient scales with the square of the film Pèclet number, recovering the same scaling law typical of the of Aris–Taylor dispersion. Superficial mass transfer is maximum in the uniform film region, where the fluid is at rest and the solute is rapidly consumed. Also, there exist regimes where the solute is entirely consumed even before reaching the uniform film region and, when the Pèclet number balances with the Sherwood number, there exists the condition that leads to the formation of hot spots of concentration around the nose of the bubble.

The analysis of the transport problem in the bubble rear reveals that the mass transfer is locally enhanced in correspondence with the typical meniscus oscillations. There, the presence of recirculating vortices enhances the solute availability for superficial mass transfer. The analysis of the uniform film region is facilitated since the analytical solution can be obtained starting from Green functions of the presented advection–diffusion–mass-transfer equation.

Despite the fact that the motivation of our work is oriented to microfluidics applications that involve solute transport and mass transfer, its ramifications are relevant also in scenarios where the presence of a solute affects the surface tension (i.e. Marangoni effect) or even drives the flow (i.e. diffusioosmosis). In fact, cases where the gradient in the concentration of colloidal particles spontaneously drives the flow are typical of emerging fields, such as micro-filtration and bio-medical engineering: an understanding of those coupling mechanisms in the surroundings of elongated bubbles could be of help for the design of such applications. Thanks to the similar mathematical structure, the results obtained may also offer a new interpretation of heat transfer in the proximity of the Taylor bubble. More generally, several practical problems that involve thin films share a lot of similarities from a fluid mechanics perspectives (Stone 2010), and, therefore, the proposed model may serve as a starting step to generalise the well-known Landau–Levich–Derjaguin–Bretherton problem in the presence of an active scalar.