2.1 Hypotheses

It is assumed that (i) the bubble size in a dissolving train of bubbles decreases much more slowly than the mass transfer time scale. It allows us to freeze the bubble diameter, and consequently also the flow field, to characterise the mass transfer. The hydrodynamics and the geometry are quasi-steady, and quasi-periodic as a consequence of the quasi-steadiness. In addition, we assume that (ii) the difference between the transferred mass of two consecutive bubbles, which is due to the difference in time they have been transferring mass, is small compared to the mass to be dissolved. This allows us to consider that the concentration of gas in the liquid around the bubble is also quasi-periodic. These hypotheses are referred hereafter as (i) quasi-steadiness and (ii) quasi-periodicity. They allow us to decouple the mass transfer (i.e. the mass transfer coefficient) and the dissolution process (i.e. the change of bubble size), as already done in work by Mikaelian et al. (Reference Mikaelian, Haut and Scheid2015a ,Reference Mikaelian, Haut and Scheid b ). In the present work, we focus on the mass transfer.

Additionally, since the mass transfer is decoupled from the dissolution of the bubble, different physical phenomena can be studied in addition to the bubble dissolution, such as the polymerisation of monodisperse beads by solvent diffusion from droplets (Hung et al. Reference Hung, Teh, Jester and Lee2010), nutrient intake by cells (Michelin & Lauga Reference Michelin and Lauga2011) independently of its metabolism, cell breath, drug delivery, bioreactions, etc.

2.2 Scaling

In this work, quantities with tildes are dimensional and quantities without tildes are their dimensionless counterparts considering the characteristic length, velocity and pressure as the hydraulic diameter of the channel $d_{h}$ , the mean flow velocity $\tilde{J}$ and the viscous stress $\tilde{\unicode[STIX]{x1D707}}\tilde{J}/\tilde{d}_{h}$ , respectively. Note that $\tilde{J}(\unicode[STIX]{x03C0}/4)\tilde{d}_{h}^{2}=\tilde{Q}_{L}+\tilde{Q}_{G}$ , where $\tilde{Q}_{L}$ and $\tilde{Q}_{G}$ are the liquid and gas flow rates, respectively. We then have the dimensionless numbers

(2.1a-c ) $$\begin{eqnarray}Pe=\frac{\tilde{J}\tilde{d}_{h}}{\tilde{D}},\quad d=\frac{\tilde{d}}{\tilde{d}_{h}},\quad L=\frac{\tilde{L}}{\tilde{d}_{h}},\end{eqnarray}$$

where $Pe$ is the Péclet number. The choice of the convective velocity as the characteristic one does not allow the numerical limit of $Pe\rightarrow 0$ . However, the results are independent of the $Pe$ numbers for sufficiently small $Pe$ which then represents the limit $Pe\rightarrow 0$ .

It is worth noting that $t$ is the time made dimensionless with the convective time scale whereas $tPe^{-1}$ is the time made dimensionless with the diffusive time scale. The dimensional mass transfer coefficient can be written as $\tilde{k}_{\ell }=Sh\,\tilde{D}/\tilde{d}$ , where $Sh$ is the Sherwood number, defined later by (2.12) in § 2.4.

2.3 Modelling of the hydrodynamics

According to the previous description, the liquid flow around the bubble in a channel segment of length $L$ , $L$ being the distance between two consecutive bubbles from centre to centre, can be analysed by solving the continuity and the steady Stokes equations,

(2.2a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}=0\quad \text{at }{\mathcal{V}}, & \displaystyle\end{eqnarray}$$

(2.2b ) $$\begin{eqnarray}\displaystyle & \displaystyle \mathbf{0}=-\unicode[STIX]{x1D735}p+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D749}\quad \text{at }{\mathcal{V}}, & \displaystyle\end{eqnarray}$$

$p$

$\boldsymbol{v}$

$\unicode[STIX]{x1D749}=\unicode[STIX]{x1D735}\boldsymbol{v}+(\unicode[STIX]{x1D735}\boldsymbol{v})^{\text{T}}$

In the reference frame attached to the bubble moving at the velocity $V$ in the axial direction, the velocity of the liquid at the wall is written as

(2.3) $$\begin{eqnarray}\boldsymbol{v}(\boldsymbol{x})=-V\boldsymbol{e}_{x}\quad \text{at }\unicode[STIX]{x1D6F4}_{W}.\end{eqnarray}$$

The velocity $V$ is the one leading to the hydrodynamic equilibrium in the axial direction for the bubble,

(2.4) $$\begin{eqnarray}\int _{\unicode[STIX]{x1D6F4}_{B}}(-p\boldsymbol{n}+\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D749})\boldsymbol{\cdot }\boldsymbol{e}_{x}\,\text{d}\unicode[STIX]{x1D6F4}=0,\end{eqnarray}$$

where $\boldsymbol{n}$ and $\text{d}\unicode[STIX]{x1D6F4}$ are the outer normal and the differential surface, respectively.

The velocity profile and pressure gradient are periodic in the axial direction with period $L$ . Consequently, the pressure field between two different cross-sections differs by a constant pressure, and the boundary conditions are

(2.5a ) $$\begin{eqnarray}\displaystyle & \displaystyle p(\boldsymbol{x}+L\boldsymbol{e}_{x})=p(\boldsymbol{x})-\unicode[STIX]{x0394}p+L\unicode[STIX]{x2202}_{x}p_{P}\quad \text{at }\unicode[STIX]{x1D6F4}_{cross}, & \displaystyle\end{eqnarray}$$

(2.5b ) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}(\boldsymbol{x}+L\boldsymbol{e}_{x})=\boldsymbol{v}(\boldsymbol{x})\quad \text{at }\unicode[STIX]{x1D6F4}_{cross}, & \displaystyle\end{eqnarray}$$

(2.5c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{x}\boldsymbol{v}(\boldsymbol{x}+L\boldsymbol{e}_{x})=\unicode[STIX]{x2202}_{x}\boldsymbol{v}(\boldsymbol{x})\quad \text{at }\unicode[STIX]{x1D6F4}_{cross}, & \displaystyle\end{eqnarray}$$

$\unicode[STIX]{x1D6F4}_{cross}$

$L$

$\unicode[STIX]{x2202}_{x}p_{P}=-32$

$\unicode[STIX]{x0394}p$

(2.6) $$\begin{eqnarray}\int _{\unicode[STIX]{x1D6F4}_{cross}}(\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{e}_{x}+V-1)\,\text{d}\unicode[STIX]{x1D6F4}=0,\end{eqnarray}$$

where it is worth noting that since the mean flow velocity has been chosen as the characteristic velocity, it corresponds to the unity term. Note that $\unicode[STIX]{x0394}p$ is unknown and that (2.5)–(2.6) must be imposed only at one arbitrary cross-section because of the periodicity. The solution is independent of the convenient choice of the position of $\unicode[STIX]{x1D6F4}_{cross}\equiv \unicode[STIX]{x1D6F4}_{OUT}$ and the centre of the bubble at $x=0$ .

The stress-free boundary conditions involve no stresses in the tangential direction and the impermeability condition in the normal direction, which are written as

(2.7a ) $$\begin{eqnarray}\displaystyle & \displaystyle -p\boldsymbol{n}+\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D749}=-p_{S}\boldsymbol{n}\quad \text{at }\unicode[STIX]{x1D6F4}_{B}, & \displaystyle\end{eqnarray}$$

(2.7b ) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{v}=0\quad \text{at }\unicode[STIX]{x1D6F4}_{B}, & \displaystyle\end{eqnarray}$$

$p_{S}$

$\boldsymbol{n}$

$p_{S}$

In the system (2.2)–(2.7), $p$ and $\boldsymbol{v}$ are domain-dependent variables, $p_{S}$ is a boundary-dependent variable and $V$ and $\unicode[STIX]{x0394}p$ are global variables, whereas the rest are parameters, such as $d$ and $L$ .

2.4 Modelling of the mass transfer

The concentration of the gas dissolved in the liquid, $C(\boldsymbol{x},t)=[\tilde{C}_{sat}-\tilde{C}(\boldsymbol{x},t)]/[\tilde{C}_{sat}-\tilde{C}_{0}]$ , is governed by the unsteady advection–diffusion equation

(2.8) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}C+\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}C=Pe^{-1}\unicode[STIX]{x1D6FB}^{2}C\quad \text{at }{\mathcal{V}},\end{eqnarray}$$

together with the impermeability of the gas through the wall,

(2.9) $$\begin{eqnarray}-\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735}C=0\quad \text{at }\unicode[STIX]{x1D6F4}_{W},\end{eqnarray}$$

the saturation concentration at the bubble surface,

(2.10) $$\begin{eqnarray}C=0\quad \text{at }\unicode[STIX]{x1D6F4}_{B},\end{eqnarray}$$

where we assume that $\unicode[STIX]{x1D6F4}_{B}$ is steady, and the periodicity conditions

(2.11a,b ) $$\begin{eqnarray}C(\boldsymbol{x},t)=C(\boldsymbol{x}+L\boldsymbol{e}_{x},t)\quad \text{and}\quad \unicode[STIX]{x2202}_{x}C(\boldsymbol{x},t)=\unicode[STIX]{x2202}_{x}C(\boldsymbol{x}+L\boldsymbol{e}_{x},t)\quad \text{at }\unicode[STIX]{x1D6F4}_{cross}.\end{eqnarray}$$

The Sherwood number depends on time $Sh(t)$ and can be defined from the mass transfer rate $\dot{{\mathcal{M}}}$ ,

(2.12) $$\begin{eqnarray}\dot{{\mathcal{M}}}Pe\equiv \unicode[STIX]{x03C0}d_{\unicode[STIX]{x1D6F4}}Sh(t)\overline{C}(t)=\int _{\unicode[STIX]{x1D6F4}_{B}}-\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735}C(\boldsymbol{x},t)\,\text{d}\unicode[STIX]{x1D6F4},\end{eqnarray}$$

where $\dot{{\mathcal{M}}}Pe$ is the mass transfer rate based on the diffusive time scale and the overline represents mean value obtained by the volume average $\overline{(\cdot )}=(1/{\mathcal{V}})\int _{{\mathcal{V}}}(\cdot )\,\text{d}{\mathcal{V}}$ , hence $\overline{C}(t)$ is the mean concentration.

In the system (2.8)–(2.11), $C$ is a domain-dependent variable and $Sh$ is obtained from its definition (2.12).

4.1 Small $Pe$

In this subsection we study both small $Pe$ number regimes, PD and CD. For this purpose, we observe that if the distance between bubbles is much larger than the channel diameter, $L\gg 1$ , the diffusion is mainly in the axial direction in the far field, i.e. sufficiently far from the bubble, and two-dimensional in the region close to the bubble, i.e. near field. Thus, the concentration is uniform within any cross-section located at $x$ except in the vicinity of the bubbles, as quantified later. The mass transfer modes are therefore governed by the one-dimensional counterpart of (3.3) in the axial direction and together with the boundary condition at the bubble surface (3.4a ) and periodicity (3.4c ). Note that (3.4b ) is automatically fulfilled since the mass transfer mode is considered as uniform in the cross-section. In addition, it is worth noting that the convection is imposed by the difference between the mean flow velocity and the velocity of the bubble such that the relevant Péclet number becomes $Pe(V-1)$ , whose sign is chosen to be positive. Consequently, if the bubble travels at the mean flow velocity, the mass transfer is purely diffusive. Then the one-dimensional model can be written as

(4.1a,b ) $$\begin{eqnarray}-\breve{\unicode[STIX]{x1D70E}}_{n}Pe\,\breve{\unicode[STIX]{x1D703}}_{n}-Pe(V-1)\unicode[STIX]{x2202}_{x}\breve{\unicode[STIX]{x1D703}}_{n}=\unicode[STIX]{x2202}_{xx}\breve{\unicode[STIX]{x1D703}}_{n}\quad \text{and}\quad \breve{\unicode[STIX]{x1D703}}_{n}(0)=\breve{\unicode[STIX]{x1D703}}_{n}(L)=0,\end{eqnarray}$$

where $\breve{\unicode[STIX]{x1D703}}_{n}(x)$ is the axial mass transfer mode at the cross-section located at $x$ and the velocity of the bubble, $V$ , is obtained from the hydrodynamics. The domain in this model is the one between two bubbles, which are then located at $x=0$ and $x=L$ . Although the periodicity (3.4c ) is not imposed, it is automatically fulfilled because of the continuity of the solution. However, in model (4.1) there is a discontinuity of fluxes which is regularised by the near field which is not modelled by (4.1). Model (4.1) is referred to hereafter as the ‘axial model’, as opposed to the ‘full model’ corresponding to (3.3)–(3.4).

The analytical solution of the eigenvalue problem (4.1) is written as

(4.2a,b ) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D70E}}_{n}Pe\frac{L^{2}}{\unicode[STIX]{x03C0}^{2}}=n^{2}+\left[Pe(V-1)\frac{L}{2\unicode[STIX]{x03C0}}\right]^{2},\quad \breve{\unicode[STIX]{x1D703}}_{n}=\text{e}^{-Pe(V-1)(x/2)}\sin \left(\frac{n\unicode[STIX]{x03C0}x}{L}\right).\end{eqnarray}$$

Since the long-time mass transfer is governed by the first mode (3.8), it is depicted for $n=1$ in figure 7. The eigenvalue is depicted in figure 7(a) where the cross-over between the PD and CD regimes is marked with 

$Pe(V-1)=2\unicode[STIX]{x03C0}/L$

$\breve{\unicode[STIX]{x1D70E}}_{1}Pe(L^{2}/\unicode[STIX]{x03C0}^{2})=2$

$L$

${\mathcal{V}}\approx \unicode[STIX]{x03C0}L/4$

$Sh_{\infty }$

$\unicode[STIX]{x1D70E}_{1}$

$(4/\unicode[STIX]{x03C0}^{2})Sh_{\infty }Ld\approx \unicode[STIX]{x1D70E}_{1}Pe(L^{2}/\unicode[STIX]{x03C0}^{2})$

$\breve{\unicode[STIX]{x1D703}}_{1}$

$x$

$L$

$\ell _{CD}$

$Pe(V-1)\ll 2\unicode[STIX]{x03C0}/L$

$Sh_{\infty }$

$Sh_{\infty }\sim L^{-1}$

$Pe(V-1)\gg 2\unicode[STIX]{x03C0}/L$

$x\sim \ell _{CD}\sim Pe^{-1}$

Figure 7.

Transition between the PD and CD regimes of the ‘axial model’ (4.1). (a) Eigenvalues and (b,c) eigenmodes represented with two different scalings suitable for PD and CD, respectively.

In figure 8 we compare the $Sh_{\infty }$ obtained from the ‘axial model’ and the ‘full model’. For this purpose, in figure 8(a) we conveniently rescale the data of figure 6, i.e.  $Pe$ and $Sh_{\infty }$ with $(V-1)L/2\unicode[STIX]{x03C0}$ and $4Ld/\unicode[STIX]{x03C0}^{2}$ , respectively. In figure 8(a), we can observe that the transition in the small $Pe$ regime is governed by the far field, as in the Stokes drag problem, provided $L/d$ is sufficiently large, and then is recovered by the ‘axial model’. In figure 8(b) we can observe that, again, the ‘axial model’ is recovered by varying the bubble diameter, hence the bubble velocity. Indeed, the effect of the bubble diameter is considered through the bubble velocity as determined by solving the hydrodynamics, which is reported in our previous work (Rivero-Rodriguez & Scheid Reference Rivero-Rodriguez and Scheid2018a ). In figure 8(c) we observe that the minimum value of $L$ to reproduce the ‘axial model’ depends on $Pe$ . We observe that the near-field effect is more important for small values of $L$ and $d$ , whose influence can be either to increase or decrease the mass transfer.

Figure 8.

Transition between the PD and CD regimes of the ‘full model’ and —— ‘axial model’ (4.2) for (a) bubbles of diameter $d=0.45$ and several distance between bubbles or (b) for distances between bubbles $L/d=1.2\times 10^{2.5}$ and several bubble diameters. (c) Influence of $d$ and $L$ in the PD regime, $Pe\rightarrow 0$ .

4.2 Large $Pe$

In this subsection we study the two large $Pe$ number regimes, BL and PC. In figure 9 we can observe the influence of $Pe$ , $d$ and $L$ on $Sh_{\infty }$ and the transition between both regimes. In figure 9(a) we depict the effect of the $Pe$ number and bubble size for a sufficiently large distance between bubbles, $L=1.2\times 10^{2.5}$ , to have the transition of the PC regime at a larger $Pe$ such that $Pe=10^{6}$ still lies in the BL regime. For this value of $L$ , the BL regime is exhibited for values of $Pe$ over a few decades. We can observe a $1/2$ slope for the scaling of $Sh_{\infty }$ as a function of $Pe$ similarly to the case of an infinite medium (Haas, Schmidt-Traub & Brauer Reference Haas, Schmidt-Traub and Brauer1972), which differs from the work by Mikaelian et al. (Reference Mikaelian, Haut and Scheid2015a ) who reported a slope of $2/5$ . However, in this model we can observe that the effect of the confinement through the dependency on $d$ of $Sh_{\infty }$ is by the group $Pe\,d^{3}$ which emerges from the balance of convection and diffusion in the BL region. Actually, in shear flows where the velocity is of order $\tilde{J}$ and changes within distances of order $\tilde{d}$ , such a balance is written as

(4.3) $$\begin{eqnarray}\frac{\tilde{J}}{\tilde{d}_{h}}\frac{\tilde{d}}{\tilde{d}_{h}}\sim \frac{\tilde{D}}{\tilde{\unicode[STIX]{x1D6FF}}^{2}},\end{eqnarray}$$

where $\tilde{J}/\tilde{d}_{h}$ represents the far-field velocity gradient and $\tilde{d}/\tilde{d}_{h}$ represents the change of the velocity in the boundary layer due to the shear that occurs within the direction tangential to the bubble of size $\tilde{d}$ , since in the normal direction there is no change of velocity due to the stress-free boundary conditions (2.7), and diffusion occurs within the boundary layer thickness $\tilde{\unicode[STIX]{x1D6FF}}$ . Given that $\tilde{k}_{\ell }\sim \tilde{D}/\tilde{\unicode[STIX]{x1D6FF}}$ , or equivalently $Sh_{\infty }\sim (\tilde{\unicode[STIX]{x1D6FF}}/\tilde{d})^{-1}$ , (4.3) can be rewritten as $Sh_{\infty }\sim (Pe\,d^{3})^{1/2}$ .

In figure 9(b) we conveniently rescale the data of figure 6, taking into consideration the $1/2$ slope in the BL regime obtained from (4.3), i.e.  $Pe$ and $Sh_{\infty }$ rescaled with $L^{-2}$ and $L^{-1}$ , respectively. This results in a collapse of the curves for sufficiently large $L$ , where we can observe that the transition between the BL and PC regimes occurs for $Pe\,L^{-2}\sim 10^{3}$ . For larger values of $Pe\,L^{-2}$ , the mass transfer becomes independent of the $Pe$ number and depends only on $L$ as $Sh_{\infty }\sim L$ . This can be explained by the fact that the bubble mainly dissolves radially through the wake whose length is $L$ in the PC regime. The small discrepancy might be due to near-field effects close to the bubble, i.e. thicker wakes in this region.

Figure 9.

Influence of the $Pe$ number on the $Sh_{\infty }$ number in regimes with large $Pe$ number. (a) Scaling for the BL regime for a large distance between bubbles $L/d=1.2\times 10^{2.5}$ and several bubble diameters and (b) transition to the PC regime for a bubble of diameter $d=0.45$ , and different lengths $L/d$ .

In effect, the concentration pattern in the PC regime is mainly uniform in the axial direction, except in the vicinity of the bubble as shown in figure 5(e). In figure 10(a) we depict the concentration profile of the slowest mode $\unicode[STIX]{x1D703}_{1}$ in an arbitrary cross-section far from the bubble where the wake is uniform, $r$ being the cylindrical radial coordinate. We can observe that the core of the wake is saturated, i.e.  $\unicode[STIX]{x1D703}_{1}=0$ , and diffusion is maximal at the contour of the wake of diameter $d_{w}$ , decaying as it gets closer to the wall where it vanishes, i.e.  $-\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}_{1}=0$ . The radial counterpart of (3.3)–(3.4), corresponding to uniform velocity and concentration in the axial direction, can be written as

(4.4a,b ) $$\begin{eqnarray}-\hat{\unicode[STIX]{x1D70E}}_{n}Pe\,\hat{\unicode[STIX]{x1D703}}_{n}=\frac{1}{r}\unicode[STIX]{x2202}_{r}(r\unicode[STIX]{x2202}_{r}\hat{\unicode[STIX]{x1D703}}_{n})\quad \text{and}\quad \hat{\unicode[STIX]{x1D703}}_{n}\left(\frac{d_{w}}{2}\right)=\unicode[STIX]{x2202}_{r}\hat{\unicode[STIX]{x1D703}}_{n}\left(\frac{1}{2}\right)=0,\end{eqnarray}$$

where $\hat{\unicode[STIX]{x1D703}}_{n}(r)$ is the radial mass transfer mode at $r$ . Model (4.4) is referred to hereafter as the ‘radial model’. The solution of this equation is written as

(4.5) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D703}}_{n}(r)=\text{Y}_{0}\left(\sqrt{\hat{\unicode[STIX]{x1D70E}}_{n}Pe}\,\frac{d_{w}}{2}\right)\text{J}_{0}\left(\sqrt{\hat{\unicode[STIX]{x1D70E}}_{n}Pe}\,r\right)-\text{J}_{0}\left(\sqrt{\hat{\unicode[STIX]{x1D70E}}_{n}Pe}\,\frac{d_{w}}{2}\right)\text{Y}_{0}\left(\sqrt{\hat{\unicode[STIX]{x1D70E}}_{n}Pe}\,r\right),\end{eqnarray}$$

where $\text{J}_{n}$ and $\text{Y}_{n}$ are the Bessel function of the first and second kind and $\hat{\unicode[STIX]{x1D70E}}_{n}$ are obtained from the zeros of the expression

(4.6) $$\begin{eqnarray}\text{J}_{0}\left(\sqrt{\hat{\unicode[STIX]{x1D70E}}_{n}Pe}\,\frac{d_{w}}{2}\right)\text{Y}_{1}\left(\sqrt{\hat{\unicode[STIX]{x1D70E}}_{n}Pe}\frac{1}{2}\right)-\text{Y}_{0}\left(\sqrt{\hat{\unicode[STIX]{x1D70E}}_{n}Pe}\frac{d_{w}}{2}\right)\text{J}_{1}\left(\sqrt{\hat{\unicode[STIX]{x1D70E}}_{n}Pe}\frac{1}{2}\right)=0.\end{eqnarray}$$

As shown in figure 3(b)(iv), the diameter of the wake $d_{w}$ coincides with the diameter of the recirculation region schematised in figure 2. Then the ‘radial model’, $\hat{\unicode[STIX]{x1D703}}_{1}$ , reproduces the ‘full model’, $\unicode[STIX]{x1D703}_{1}$ , as we show by dashed lines in figure 10(a). The value of $d_{w}$ depends on $d$ as shown in figure 10(b) where we can see that it is always smaller than $d$ . It also corresponds to the size of the wake defined by the width of the recirculating regions shown by the streamlines in figure 5(PC). In figure 10(c) we depict the relation between $\hat{\unicode[STIX]{x1D70E}}_{1}Pe$ and $d_{w}$ . In effect, the decay rate increases for larger bubbles as $d_{w}$ increases with $d$ . Since $\unicode[STIX]{x1D70E}_{1}Pe$ does not depend on $L$ , the $Sh_{\infty }$ number, related to $\unicode[STIX]{x1D70E}_{1}Pe$ by (3.8), is thus proportional to $L$ as chosen in the scaling of the PC regime shown in figure 9(b), i.e.

(4.7) $$\begin{eqnarray}Sh_{\infty }(d,L)=\frac{L}{4d}\hat{\unicode[STIX]{x1D70E}}_{1}[d_{w}(d)]Pe.\end{eqnarray}$$

It can be also observed that since the recirculation zone is saturated in the slowest mode, the mass transfer rate at the bubble surface (2.12) is the same as the mass transfer rate at the surface of the recirculation zone. For the latter, the concentration gradient is independent of $L$ whereas the surface is proportional to $L$ , which explains that $Sh_{\infty }\sim L$ in the PC regime as shown in (4.7).

It is worth noting the different dependency of $Sh_{\infty }$ on $L$ in the different ‘interacting’ regimes, PD and PC, because of the different mechanisms. In the PD regime, the concentration gradient along the axial distance is modified while the bubble surface is kept constant, and thus $Sh_{\infty }$ is proportional to $L^{-1}$ , whereas in the PC regime, the surface of the recirculation zone is modified while the radial gradient is not affected, and hence $Sh_{\infty }$ is proportional to $L$ . In both cases, the total transferred mass is independent of $Sh_{\infty }$ and proportional to ${\mathcal{V}}$ , which increases as $L$ increases.

Figure 10.

Radial diffusion in the PC regime for $L=3$ . (a) Concentration in a cross-section far from the bubble for different bubble diameters. (b) Relation between the bubble diameter and the width of the wake $d_{w}$ . (c) Influence of the diameter $d_{w}$ on the decay rate $\hat{\unicode[STIX]{x1D70E}}_{1}$ .

The BL regime was previously studied by Mikaelian et al. (Reference Mikaelian, Haut and Scheid2015a ). Despite the fact that in the model by Mikaelian the mass is steadily advected behind the bubble and in our model the concentration field is periodic and the transferred mass is stored at the liquid in a transient way, there is good agreement between both models in the BL regime, as shown in figure 11. In this figure, we compare our solution in the BL regime with the results of Mikaelian et al. (Reference Mikaelian, Haut and Scheid2015a ). We observe good agreement for large $Pe\,d^{3}$ where the bubble is in the BL regime, whereas the power law slightly overestimates the results of Mikaelian et al. (Reference Mikaelian, Haut and Scheid2015a ) for smaller $Pe\,d^{3}$ in the T regime, especially for stress-free boundary condition as expected outside the BL regime (see figure 9 a). It also explains the $2/5$ power found by Mikaelian et al. (Reference Mikaelian, Haut and Scheid2015a ) by fitting their data in a wider range than the BL regime for which the $1/2$ power should rigorously apply in the case of bubbles with stress-free interface. The cross-over between the T regime and the BL regime depends on $d$ , and we have identified with a double-arrowed line the region of $Pe\,d^{3}$ where the BL regime occurs independently of the value of $d$ within the considered range $0.15<d<0.75$ .

Figure 11.

Comparison with Mikaelian et al. (Reference Mikaelian, Haut and Scheid2015a ) in the BL regime for (a) stress-free and (b) rigid bubble surfaces (see the model extension in § 6.1). Solid lines —— correspond to the best fit in the BL regime of (a) figure 9(a) and (b) figure 15(b).

4.3 Dominating near-field effects: small $L$

In this subsection we quantify the near-field effects which dominate for small distance between bubbles, i.e.  $L\sim d$ , already shown in figure 8(c) for the PD regime. We observed in figure 6 that the range of $Pe$ for ‘non-interacting’ regimes vanishes for sufficiently small $L$ , showing almost no influence of the $Pe$ number. Consequently, we can restrict the analysis to the PD regime without loss of generality. In figure 12 we depict the slowest eigenvalue of the full model (3.3)–(3.4) in the PD regime for several diameters and small values of $L$ . In figure 12(a) we observe that for sufficiently separated bubbles the ‘axial model’ (4.2) is recovered (dashed line). However, when the size of the gap between bubbles $L-d$ is smaller than the bubble diameter, there is no more influence of the distance between bubbles. This is because the diffusion takes place mainly in the radial direction, as happens in the mass transfer of the wake in the PC regime, which leads to very similar $Sh_{\infty }$ in both regimes. The effect of the bubble diameter is depicted in figure 12(b), where we observe that $\unicode[STIX]{x1D70E}_{1}$ exponentially increases with $d$ as $\unicode[STIX]{x1D70E}_{1}Pe=\exp (1.1966+3.8318d)$ . In this situation, diffusion occurs at a distance that decreases as the bubble diameter increases, thus $\unicode[STIX]{x1D70E}_{1}$ increases. The solution of the ‘radial model’ (4.4), with boundary conditions $\hat{\unicode[STIX]{x1D703}}_{n}(d/2)=\unicode[STIX]{x2202}_{r}\hat{\unicode[STIX]{x1D703}}_{n}(1/2)=0$ , is depicted by a dashed line. Examples of the mass transfer modes for small and large bubbles are given in figure 12(b). We observe that for small bubbles the diffusion is mainly radial (cylindrically) since near-field effects are smaller than the distance between the bubble and the wall, thus showing a good agreement with the radial model. However, for large bubbles near-field effects induce axial concentration gradients, increasing the diffusion length and decreasing more significantly the $Sh_{\infty }$ number with respect to the ‘radial model’ (4.4).

Figure 12.

Near-field effect in the PD regime. Influence on the decay rate of (a) the distance between bubbles for several diameters $d$ , and (b) $d$ in the limit of small distances $L-d\ll 1$ . Two examples of the mass transfer modes are also provided.

6.1 Influence of a rigid interface

Tailoring the surface rheology is known to modify the mass transfer, either because of mechanical modification of the surface by adding surfactants (Atasi et al. Reference Atasi, Haut, Pedrono, Scheid and Legendre2018) or solid particles (Beltramo et al. Reference Beltramo, Gupta, Alicke, Liascukiene, Gunes, Baroud and Vermant2017) or because of chemical modification of the mass transfer through the interface. In this subsection we consider the effect of the former in the limit of spherical bubble with rigid interface, which is exhibited in the presence of sufficiently large Marangoni stresses (Atasi et al. Reference Atasi, Haut, Pedrono, Scheid and Legendre2018). For this purpose, the hydrodynamics is governed by (2.2) together with (2.3)–(2.6) and substituting (2.7) by the non-slip condition at the bubble surface,

(6.1) $$\begin{eqnarray}\boldsymbol{v}(\boldsymbol{x})=\mathbf{0}\quad \text{at }\unicode[STIX]{x1D6F4}_{B}.\end{eqnarray}$$

It is worth noting that (2.7b ), which is used in the derivation of (3.8), is fulfilled, thus (3.8) also holds in this case.

In figure 15 we depict the influence of $Pe$ on $Sh_{\infty }$ . First, we compare it with the stress-free case (dashed lines) in figure 15(a). We observe that the small $Pe$ regime is not affected by the boundary conditions of the bubble surface, and thus it is also in agreement with the ‘axial model’, which would also describe the PD and CD regimes for other boundary conditions such as those corresponding to viscous droplets or finite Marangoni stresses. Nevertheless, for large $Pe$ the mass transfer decreases with respect to the stress-free case due to smaller velocities close to the bubble surface as a consequence of the no-slip condition. In particular, the slope is $1/3$ in the BL regime for large $L$ as shown in figure 15(b), as already reported by Mikaelian et al. (Reference Mikaelian, Haut and Scheid2015a ) and in general agreement with mass transfer in shear flow (Clift Reference Clift1978). It is worth noting that the interaction of bubbles reduces the slope in the BL regime and makes the PC regime disappear at least in the range of $Pe$ considered. In figure 15(b) it can also be observed that in the BL regime, $Sh_{\infty }$ depends on $d$ proportionally, $Sh_{\infty }\sim Pe^{1/3}\,d$ . In effect, in shear flows where the velocity is of order $\tilde{J}$ and changes within distances of order $\tilde{\unicode[STIX]{x1D6FF}}$ , such a balance is written as

(6.2) $$\begin{eqnarray}\frac{\tilde{J}}{\tilde{d}_{h}}\frac{\tilde{\unicode[STIX]{x1D6FF}}}{\tilde{d}_{h}}\sim \frac{\tilde{D}}{\tilde{\unicode[STIX]{x1D6FF}}^{2}},\end{eqnarray}$$

where $\tilde{J}/\tilde{d}_{h}$ represents the far-field velocity gradient and $\tilde{\unicode[STIX]{x1D6FF}}/\tilde{d}_{h}$ represents the change of the velocity in the boundary layer due to the shear that occurs in the normal direction among the boundary layer thickness $\tilde{\unicode[STIX]{x1D6FF}}$ due to the rigid boundary condition (6.1), and diffusion occurs within the boundary layer thickness $\tilde{\unicode[STIX]{x1D6FF}}$ . Given that $\tilde{k}_{\ell }\sim \tilde{D}/\tilde{\unicode[STIX]{x1D6FF}}$ , or equivalently $Sh_{\infty }\sim (\tilde{\unicode[STIX]{x1D6FF}}/\tilde{d})^{-1}$ , (6.2) can be rewritten as $Sh_{\infty }\sim (Pe\,d^{3})^{1/3}$ .

In order to understand the interaction between bubbles it is convenient to depict the concentration patterns in the large $Pe$ regime. In figure 16, we depict the concentration pattern of bubbles with $d=0.5$ , and distance between bubbles $L=2$ for different $Pe$ numbers. We can observe the strong influence of the flow pattern on the concentration pattern. A mass transfer boundary layer appears with very narrow concentration wakes around the streamlines originating from stagnation points and rings. When wakes extend up to the following bubble, the boundary layers are fed with mass coming from the previous bubble, but the recirculation region is not as saturated as in the case of bubbles with stress-free interface as shown in figure 16(b–e). Consequently, the influence of interaction between bubbles weakens the mass transfer, decreasing $Sh_{\infty }$ . It is worth noting that in the case of bubbles with stress-free interfaces, the wakes are of a similar size to the bubble, hence having a stronger impact on the boundary layer. Additionally, the interaction between bubbles does not impose a concentration pattern independently of the $Pe$ number as it occurs in the PC regime, at least for the $Pe$ numbers considered in the present work.

Figure 15.

Influence of the $Pe$ number on the $Sh_{\infty }$ number for bubbles with rigid interface. (a) Comparison with respect to stress-free bubbles (dashed lines) for diameter $d=0.45$ and several $L$ . (b) Influence of the diameter for $L=379.43d$ .

Figure 16.

Streamlines and mass transfer mode $\unicode[STIX]{x1D703}_{1}$ (dark red corresponds to saturation $\unicode[STIX]{x1D703}_{1}=0$ ) in the case of bubbles with rigid interface, diameter $d=0.45$ and distance between bubbles $L=2$ .

6.2 Influence of a deformable interface

In this subsection we consider the effect of deformable bubbles governed by surface tension. For this purpose, we consider the Young–Laplace contribution for the stress balance in the normal direction. Thus, the hydrodynamics is governed by (2.2) together with (2.3), (2.5), (2.6) and substituting (2.4) and (2.7a ) by the equations for the bubble size and position, i.e.

(6.3a ) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{{\mathcal{V}}_{B}}\text{d}{\mathcal{V}}={\mathcal{V}}_{B}, & \displaystyle\end{eqnarray}$$

(6.3b ) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{{\mathcal{V}}_{B}}\boldsymbol{x}\,\text{d}{\mathcal{V}}=0, & \displaystyle\end{eqnarray}$$

(6.4) $$\begin{eqnarray}-p\boldsymbol{n}+\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D749}+p_{G}\boldsymbol{n}=-\frac{1}{Ca}\boldsymbol{n}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\quad \text{at }\unicode[STIX]{x1D6F4}_{B},\end{eqnarray}$$

where $p_{G}$ is the pressure of the gas. In (6.4), $Ca=\tilde{\unicode[STIX]{x1D707}}\tilde{J}/\tilde{\unicode[STIX]{x1D6FE}}$ is the capillary number. This system of equations is solved using the boundary arbitrary Lagrangian–Eulerian method developed by Rivero-Rodriguez, Perez-Saborid & Scheid (Reference Rivero-Rodriguez, Perez-Saborid and Scheid2018) for the solution of partial differential equations on deformable domains, and outlined in appendix A.

Figure 17.

Influence of the deformability of bubbles, as gauged by $Ca$ for bubbles with diameter $d=0.45$ and distance between bubbles $L=12d$ , on (a) the $Sh_{\infty }$ number and (b) the bubble surface.

In figure 17 we plot the influence of the deformability of the bubble gauged by the $Ca$ number. In figure 17(a) the influence of $Ca$ number for the different regimes is depicted, conveniently scaling the $Pe$ number by the relative flow velocity $V-1$ . We observe that the $Sh_{\infty }$ number does not depend on $Ca$ except in the PC regime. It is worth noting that the surface of the bubble $\unicode[STIX]{x1D6F4}_{B}$ and its equivalent diameter $d_{\unicode[STIX]{x1D6F4}}$ , on which the mass transfer coefficient and $Sh_{\infty }$ numbers are based, as well as the velocity of the bubble significantly depend on the $Ca$ number and volume of the bubble, ${\mathcal{V}}_{B}=\unicode[STIX]{x03C0}d^{3}/6$ , as depicted in figure 17(b), respectively. The bubble surface increases (see figure 17 b) as well as the velocity of the bubble which can even increase above the maximum velocity of the Poiseuille flow for sufficiently large $Ca$ as already shown in our previous work (Rivero-Rodriguez & Scheid Reference Rivero-Rodriguez and Scheid2018a ), i.e.  $V>2$ . It is worth noting that the deformable case reduces, for $Ca\rightarrow 0$ , to the bubble with stress-free spherical interface.

Figure 18.

Streamlines and mass transfer mode $\unicode[STIX]{x1D703}_{1}$ (dark red corresponds to saturation $\unicode[STIX]{x1D703}_{1}=0$ ) for deformable bubbles with $d=0.5$ , $L=1.5$ and in the PC regime, with $Pe=10^{5}$ .

To understand the influence of $Ca$ in the PC regime, we depict the concentration pattern in this regime in figure 18. We can observe how bubbles and wakes stretch as $Ca$ increases. Since we have seen that $Sh_{\infty }$ diminishes in the PC regime when wakes stretch, $Sh_{\infty }$ thus diminishes as $Ca$ increases.

6.3 Influence of inertia

In this subsection we study the effect of inertia in the mass transfer around the bubble. To consider inertia, the hydrodynamics is governed not by the Stokes equations (2.2), but by the Navier–Stokes equations

(6.5a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}=0\quad \text{at }{\mathcal{V}}, & \displaystyle\end{eqnarray}$$

(6.5b ) $$\begin{eqnarray}\displaystyle & \displaystyle Re\,\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{v}=-\unicode[STIX]{x1D735}p+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D749}\quad \text{at }{\mathcal{V}}, & \displaystyle\end{eqnarray}$$

$Re=\tilde{\unicode[STIX]{x1D70C}}\tilde{J}\tilde{d}_{h}/\tilde{\unicode[STIX]{x1D707}}$

After simulations for $Re$ up to $100$ , we observed that changes in the hydrodynamics have no influence on $Sh_{\infty }$ within a $1\,\%$ error. This is due to the fact that for small $Pe$ the system behaviour is well described by the ‘axial model’ which only considers the effect of the hydrodynamics through the velocity of the bubble. In regimes with large $Pe$ , the Schmidt number $Sc=\unicode[STIX]{x1D707}/(D\unicode[STIX]{x1D70C})$ is much larger than unity, $Sc\gg 1$ . Thus, the viscous boundary layer is much thicker than the diffusive one, and hence the details of the viscous boundary layers tuned by the $Re$ do not affect the diffusive one.

Thus, the influence of inertia is through the inertial migration force, responsible for the off-centre position of the bubbles. This effect was first reported by Segré & Silberberg (Reference Segré and Silberberg1962). In the next subsection we consider the effect of off-centred positions of the bubble on the mass transfer.

6.4 Effect of off-centred positions

In our recent work (Rivero-Rodriguez & Scheid Reference Rivero-Rodriguez and Scheid2018a ), the equilibrium positions of deformable bubbles and rigid particles in inertial flows in the presence of an outer uniform force as a function of the diameter and the $Re$ and the $Ca$ numbers are reported. In this subsection we consider the effect on mass transfer of off-centred positions of the bubble, $\unicode[STIX]{x1D700}$ , measured from the centre of the microchannel. In the case of stress-free bubbles the governing equations remain the same. However, in the case of rigid spheres in an off-centred position, the velocity at the bubble interface (6.1) is modified as

(6.6a ) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}(\boldsymbol{x})=\unicode[STIX]{x1D734}\times (\boldsymbol{x}-\unicode[STIX]{x1D73A})\quad \text{at }\unicode[STIX]{x1D6F4}_{B}, & \displaystyle\end{eqnarray}$$

(6.6b ) $$\begin{eqnarray}\displaystyle & \displaystyle \mathbf{0}=\int _{\unicode[STIX]{x1D6F4}_{B}}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D749}\times (\boldsymbol{x}-\unicode[STIX]{x1D73A})\,\text{d}\unicode[STIX]{x1D6F4}, & \displaystyle\end{eqnarray}$$

$\unicode[STIX]{x1D734}$

$\unicode[STIX]{x1D73A}=\unicode[STIX]{x1D700}\boldsymbol{e}_{y}$

In figure 19 we plot the influence of off-centred positions for bubbles with both stress-free and rigid interfaces. On the one hand, we can observe that in regimes with small $Pe$ the $Sh_{\infty }$ is affected by the eccentricity through the velocity of the bubble. On the other hand, the velocity field slightly affects the $Sh_{\infty }$ in the regime with large $Pe$ . In particular, the effect on bubbles with stress-free interfaces is non-monotonic. For bubbles with rigid interface the effect is monotonic, tending to decrease the slope and even revealing the PC regime for large $Pe$ in the range of $Pe$ considered.

Figure 19.

Effect on $Sh_{\infty }$ of the off-centre position for bubbles with diameter $d=0.45$ and distance between bubbles $L=20d$ . (a) Stress-free and (b) rigid interfaces.

In figure 20, we can observe the influence of the eccentricity $\unicode[STIX]{x1D700}$ on the flow structure and hence on the mass transfer modes, which we depicted in the region with large $Pe$ number, $Pe=10^{4}$ . We can observe how the stagnation regions, either points or lines, detach from the bubble surface. Observe that the divergent stagnation points/lines extract concentration from the boundary layer at the bubble surface. This intricate effect on the hydrodynamics might explain the non-monotonicity of the mass transfer, which makes it difficult to quantify by analytical means.

Figure 20.

Streamlines and mass transfer mode $\unicode[STIX]{x1D703}_{1}$ (dark red corresponds to saturation $\unicode[STIX]{x1D703}_{1}=0$ ) in the case of bubbles with stress-free interface, diameter $d=0.45$ , distance between bubbles $L=20d$ and $Pe=10^{4}$ .

7.1 Steady state

Different scenarios arise depending of the comparison between the amount of gas that the liquid can potentially absorb, $\tilde{{\mathcal{V}}}(\tilde{C}_{sat}-\tilde{C}_{0})$ , which is written as ${\mathcal{V}}$ in dimensionless form, and the initial amount of gas in the bubble, $\unicode[STIX]{x1D70C}_{B}(\unicode[STIX]{x03C0}/6)d_{0}^{3}$ , available for dissolution, $\unicode[STIX]{x1D70C}_{B}=\tilde{\unicode[STIX]{x1D70C}}_{B}/(\tilde{C}_{sat}-\tilde{C}_{0})$ being the dimensionless counterpart of the density of the gas inside the bubble, $\tilde{\unicode[STIX]{x1D70C}}_{B}$ , and $d_{0}$ its dimensionless initial diameter. On the one hand, if the liquid cannot absorb all the gas, $\unicode[STIX]{x1D70C}_{B}(\unicode[STIX]{x03C0}/6)d_{0}^{3}>{\mathcal{V}}$ , there is partial dissolution and the amount $\unicode[STIX]{x1D70C}_{B}(\unicode[STIX]{x03C0}/6)d_{0}^{3}-{\mathcal{V}}$ will remain undissolved in a bubble of size $d(t\rightarrow \infty )/d_{0}=\sqrt[3]{1-{\mathcal{V}}/(\unicode[STIX]{x1D70C}_{B}(\unicode[STIX]{x03C0}/6)d_{0}^{3})}$ . The amount ${\mathcal{V}}$ dissolves first according to the short-time behaviour by an amount $(1-\unicode[STIX]{x1D6F9}_{1}){\mathcal{V}}$ and then according to the long-time behaviour by an amount $\unicode[STIX]{x1D6F9}_{1}{\mathcal{V}}$ . During the short-time behaviour, a small portion of $\unicode[STIX]{x1D6F9}_{1}{\mathcal{V}}$ is also dissolved. On the other hand, if the liquid can absorb all the gas, the dissolution is complete and bubbles vanish. In figure 21 we schematise this process. The symbol ● represents the initial conditions, and the vertical line from 0 to $\unicode[STIX]{x1D70C}_{B}(\unicode[STIX]{x03C0}/6)d_{0}^{3}$ represents the initial amount of gas in the bubble. This amount is the total amount of undissolved and dissolved gas. The latter can be divided in short- and long-time behaviour. If the operating conditions are below the complete–incomplete dissolution line $\cdot$  –  $\cdot$  –, the bubble completely dissolves. If the operating conditions are within the short-time region, only short-time dissolution matters. However, this is only found for gas with high solubility, $\unicode[STIX]{x1D70C}_{B}(\unicode[STIX]{x03C0}/6)d_{0}^{3}<(1-\unicode[STIX]{x1D6F9}_{1}){\mathcal{V}}$ . The threshold for partial or complete dissolution is $\unicode[STIX]{x1D70C}_{B}(\unicode[STIX]{x03C0}/6)d_{0}^{3}={\mathcal{V}}$ , whereas the threshold for short- and long-time dissolution is $\unicode[STIX]{x1D70C}_{B}(\unicode[STIX]{x03C0}/6)d_{0}^{3}=(1-\unicode[STIX]{x1D6F9}_{1}){\mathcal{V}}$ .

Figure 21.

Map on the dissolution process. After the dissolution process, the amount $\unicode[STIX]{x1D70C}_{B}\frac{\unicode[STIX]{x1D70B}}{6}d_{0}^{3}-{\mathcal{V}}$ remains undissolved, the amount $\unicode[STIX]{x1D6F9}_{1}{\mathcal{V}}$ dissolves in the long-time behaviour and the amount $(1-\unicode[STIX]{x1D6F9}_{1}){\mathcal{V}}$ dissolves in the short-time behaviour.

It is worth noting that for $\unicode[STIX]{x1D70C}_{B}(\unicode[STIX]{x03C0}/6)d_{0}^{3}>{\mathcal{V}}$ , bubbles do not completely dissolve through the mechanism studied in this work, especially for a finite train length for which there is a collective dissolution behaviour (Michelin, Guérin & Lauga Reference Michelin, Guérin and Lauga2018) such as the gas diffuses in the liquid out of the train of bubbles. This collective mode is much slower than that presented in this work since the distance that the gas needs to be transported becomes much longer and of the order of the bubble train. An increase in the distance between bubbles can lead to a decrease or increase in the $Sh$ number in the PD and PC regimes, respectively, and has no influence on the other regimes. In addition, it is associated with an increase in ${\mathcal{V}}$ and hence a decrease in $\unicode[STIX]{x1D70C}_{B}(\unicode[STIX]{x03C0}/6)d_{0}^{3}-{\mathcal{V}}$ , and it might lead to an increment of the undissolved amount of gas which remains to be dissolved in the collective mode. For instance, in the PD regime, it initially leads to a faster saturation of the liquid inside the train, decreasing the dissolved amount and an increment of the amount to be dissolved in the collective mode. It thus yields a large increment of the total dissolution time.

Figure 22.

Influence of bubble size changes in the (a,b) PD regime for $Pe\rightarrow 0$ and (c,d) PC regime, $Pe=10^{4}$ , for $d_{0}=0.6$ and $L_{0}=3$ . $(\unicode[STIX]{x03C0}/6)d^{3}/{\mathcal{V}}=0.050\,42$ .

7.2 Transient dissolution

Although in many situations the change in size is very small as compared to its initial value as ${\mathcal{V}}\ll \unicode[STIX]{x1D70C}_{B}(\unicode[STIX]{x03C0}/6)d_{0}^{3}$ , changes in bubble size can be important. Nonetheless, its effect might be limited, as shown in what follows. To study its influence on the dissolution process, changes in bubble size must be taken into account. For this purpose, the mass balance of gas is applied:

(7.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}_{B}\frac{4}{3}\unicode[STIX]{x03C0}d^{3})}{\unicode[STIX]{x2202}t}=-\dot{{\mathcal{M}}},\quad d(0)=d_{0},\end{eqnarray}$$

Since $(1/4)\unicode[STIX]{x03C0}L(t)={\mathcal{V}}+(4/3)\unicode[STIX]{x03C0}d^{3}(t)$ and the volume ${\mathcal{V}}$ does not change in time, the distance between bubbles can be obtained from

(7.2) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\left(\unicode[STIX]{x03C0}L/4\right)}{\unicode[STIX]{x2202}t}=-\frac{\dot{{\mathcal{M}}}}{\unicode[STIX]{x1D70C}_{B}},\quad L(0)=L_{0},\end{eqnarray}$$

where $L_{0}$ is the initial distance between bubbles. In addition, $C_{sat}$ depends on the Henry’s constant, $H$ , and the pressure of the bubble, $p_{B}$ , through Henry’s law, $C_{sat}=Hp_{B}$ . In this work, we consider it as constant as well as $\unicode[STIX]{x1D70C}_{B}$ . However, it might change because of change of Laplace pressure, as shown in Mikaelian et al. (Reference Mikaelian, Haut and Scheid2015b ).

In figure 22 we show the influence of the change of bubble size on the mass transfer. We depict the solution of the full transient problem considering bubble size changes, for values of $\unicode[STIX]{x1D70C}_{B}$ in the partial dissolution region of figure 21, and in the PD and PC regimes. In figure 22(a) we depict $Sh(t)$ and $Sh_{\infty }$ as depending on time because $d$ and $L$ also do. We can observe in the PD regimes that for short times the $Sh$ decreases towards the $Sh_{\infty }$ value in the case where the bubble does not change size, $1/\unicode[STIX]{x1D70C}_{B}=0$ , as shown in figure 22(b). As $1/\unicode[STIX]{x1D70C}_{B}$ increases, changes in bubble size becomes relevant and the $Sh_{\infty }$ changes, whereas $Sh(t)$ tries to follow such changes with a small discrepancy, around $10\,\%$ in the worst case. This behaviour reveals that the change in bubble size triggers or excites faster modes but its effect is limited and smaller than the influence of other geometrical quantities. In the PC regime, we can observe that for short times the mass transfer through the boundary layer is not affected, as compared to $Sh_{BL}$ , whereas the discrepancy with $Sh_{\infty }$ is very small, as shown in figure 22(c), even for large changes in the bubble size, shown in figure 22(d).

The governing equations for the full transient problem are constituted by the system of (2.2)–(3.1) together with (7.1)–(7.2). Note that the impermeability condition has not been changed. It has no effect in the PD regimes, since the convection is negligible. In the PC regime it should not affect the boundary layer much since it is very thin, and then the normal gradients of normal velocity are very small and the boundary layer is simply advected with the bubble surface and not squeezed.