2.1 Formulation

The growth rate of an isolated spherical gas bubble of radius $R(t)$ suspended in a quiescent, infinite liquid environment subject to a time-varying liquid ambient pressure, $P_{\infty }(t)$ , is to be determined. The analysis shall be restricted to monocomponent gas bubbles. The problem has spherical symmetry and consequently $r$ may be taken as the radial distance from the bubble centre, while $t$ is the time variable.

The molar concentration field $C(r,t)$ of dissolved gas in the liquid is to be solved for from the advection–diffusion equation with spherical symmetry,

where the dot notation stands for $\text{d}/\text{d}t$ and $D_{m}$ is the coefficient of mass diffusion. Notice that the factor $u_{r}(r,t)={\dot{R}}R^{2}/r^{2}$ corresponds to the radial velocity field in the liquid that, by virtue of the continuity equation, is induced by the rate of change of the bubble radius. The concentration field is subject to boundary and initial conditions

where $R_{0}=R(0)$ is the initial bubble radius. The initial concentration of dissolved gas is assumed uniform throughout the liquid and equal to the concentration at the far field, $C_{\infty }$ . The concentration boundary condition at the bubble surface is given by Henry’s law,

where $k_{H}$ is Henry’s coefficient (molar based) and $P_{g}(t)$ is the total gas pressure inside the bubble. The saturation pressure may thus be defined as $P_{\mathit{sat}}=C_{\infty }/k_{H}$ . The bubble pressure is determined considering the liquid–gas surface tension ${\it\gamma}_{lg}$ , but otherwise neglecting the liquid vapour pressure together with damping and inertial effects. Naturally, we will not consider the case of an inertially pulsating bubble, driven by strong periodic acoustic forcing (Louisnard & Gomez Reference Louisnard and Gomez2003) or by resonance-triggering frequencies often employed in rectified diffusion. Indeed, in most diffusion-driven processes, excluding the aforementioned scenario, the contributions of the inertial and viscous terms in the Rayleigh–Plesset equation are negligible since bubble radius accelerations and velocities are relatively small. Payvar (Reference Payvar1987) reported that for bubbles of 0.1–1 mm in diameter, $P_{\infty }(t)$ still remains the dominant term even for fast liquid decompressions of up to 100 bar taking place in less than 1 s. The bubble pressure is then simply obtained from the Young–Laplace equation,

An isothermal liquid at temperature $T_{\infty }$ , well below the boiling point, is considered. The bubble volume and pressure are related through the equation of state for an ideal gas,

where $n(t)$ is the number of moles inside the bubble and $R_{u}$ denotes the universal gas constant. Finally, Fick’s first law sets the molar flow rate of gas across the bubble surface to be

The governing equations (2.1)–(2.6) are best treated in dimensionless form. Consequently, let us define the dimensionless radius, ambient pressure, concentration and interfacial concentration as follows:

Here, $R_{c}$ denotes some characteristic bubble radius. Pressure $P_{c}$ is a characteristic liquid pressure, usually taken as the initial ambient pressure $P_{\infty }(0)$ . Additionally, let us introduce the following dimensionless parameters, which remain constant throughout the process:

The parameter ${\it\Upsilon}$ refers to the level of saturation of gas in the liquid at the characteristic pressure $P_{c}$ . In fact, the (dimensionless) saturation pressure is simply $p_{\mathit{sat}}=P_{\mathit{sat}}/P_{c}={\it\Upsilon}$ . Saturation conditions at $P_{c}$ are described by ${\it\Upsilon}=1$ , while ${\it\Upsilon}<1$ and ${\it\Upsilon}>1$ imply undersaturation and supersaturation respectively. Similarly, note that the saturation level at $P_{\infty }$ is given by ${\it\Upsilon}/p$ . Moreover, ${\it\sigma}$ represents the characteristic ratio of the Laplace pressure and $P_{c}$ , while ${\it\Lambda}$ serves as the solubility parameter. The latter represents the ratio between the bubble’s volume and the volume of liquid needed to dissolve, under saturation conditions, the gas it contains. Henry’s law yields a new expression for $c_{s}$ , which, when written in our notation, reads

Finally, we shall introduce two different dimensionless time variables, ${\it\tau}$ and $\tilde{{\it\tau}}$ , in addition to the radial coordinate ${\it\xi}$ ,

The identity

directly relates the ‘physical’ dimensionless time ${\it\tau}$ and nonlinear time $\tilde{{\it\tau}}$ . In the upcoming § 2.2, it will be shown that the mathematical nature of the problem encourages us to dispose of the ‘physical’ time ${\it\tau}$ and work with the nonlinear time $\tilde{{\it\tau}}$ instead. We advance that this is done in order to analytically solve for the concentration gradient at the bubble boundary from the diffusion equation, all the while accounting for large variations in the bubble radius. Finally, coordinate ${\it\xi}\in [1,\infty )$ scales with the instantaneous bubble radius $R(t)$ . Hence, the moving bubble boundary is advantageously always mapped by ${\it\xi}=1$ . Coordinate ${\it\xi}$ is an alternative to the Lagrangian coordinate ${\it\eta}=(r^{3}-R^{3}(t))/3$ which also eliminates the moving boundary problem. The latter transformation is usually employed in the treatment of rectified diffusion (Eller & Flynn Reference Eller and Flynn1965; Fyrillas & Szeri Reference Fyrillas and Szeri1994). However, in the diffusion-dominant regime of interest, the resulting advection–diffusion equation in ${\it\eta}$ is much harder, if not impossible, to treat analytically compared to the analogous equation in ${\it\xi}$ .

2.2 The history term on the concentration gradient at the bubble surface

The advection–diffusion equation (2.1) in dimensionless form becomes

Note that the nonlinear time $\tilde{{\it\tau}}$ is the time variable of choice for reasons that will soon become apparent. The boundary and initial conditions (2.2) of the concentration field $c({\it\xi},\tilde{{\it\tau}})$ become

Only the dimensionless advection term in (2.12) is explicitly dependent on the bubble dynamics through the prefactor $(\text{d}a/\text{d}\tilde{{\it\tau}})/a$ . We may characterize this prefactor as a time-dependent Péclet number based on the velocity of the bubble boundary,

whose magnitude is essentially associated with the instantaneous ratio of advective to diffusive transport. To illustrate the role of the history effect on the diffusive growth or dissolution rates of bubbles, we will restrict the analysis to mass transfer processes where the advection term in (2.12) is small compared to the diffusion term. We anticipate that the history effect is present in every transient diffusion–advection problem in which boundary conditions change in time. However, should the advective mass transport of species be dominant, the advective nature of the flow would then, at least partially, obscure the contribution of the history effect on the bubble dynamics as well as of any other effect of a diffusive nature.

When can we neglect the advection term? To find out, we can perform an order of magnitude analysis on (2.12), taking $O(\partial c)={\rm\Delta}c$ , $O({\it\xi})=1$ and denoting $O(\partial {\it\xi})$ as $l$ , a characteristic thickness of the boundary layer, defined as $\partial c/\partial {\it\xi}|_{{\it\xi}=1}={\rm\Delta}c/l$ . The analysis reveals that the diffusion term magnitude is ${\rm\Delta}c/l^{2}$ , while the advection term has magnitude $O(\mathit{Pe}){\rm\Delta}c$ . It follows from the Epstein–Plesset solution that the penetration length of the mass boundary layer will never be much greater than the size of the bubble: $l\lesssim 1$ . Thus, the dimensionless advection term may be neglected provided $|\mathit{Pe}(\tilde{{\it\tau}})|\ll 1$ at all times.

The negligible advection assumption is usually valid for small supersaturation or undersaturation ratios, defined as $C_{\infty }/[k_{H}P_{\infty }(t)]-1\equiv {\it\Upsilon}/p-1$ . Larger ratios are allowed if the gas solubility is very poor ( ${\it\Lambda}\ll 1$ ). In fact, it will be shown in a companion paper that, in the case of a growing or dissolving bubble at pressure $P_{\infty }$ , the Péclet number is related to the Jakob number, defined in (1.2b ), through: $|\mathit{Pe}|\approx Ja_{m}^{2}={\it\Lambda}|{\it\Upsilon}/p-1|$ . Thus, a highly soluble gas such as $\text{CO}_{2}$ gas in water ( ${\it\Lambda}\approx 0.8$ at room temperature) will require that $|{\it\Upsilon}/p-1|\ll 1$ for advection to be negligible.

It must be emphasized that the advection term has two components. The component containing $1/{\it\xi}^{2}$ is the (dimensionless) physical advection term, associated with the radial component of the velocity field, $u_{r}(r,t)=R^{2}{\dot{R}}/r^{2}$ . The component containing $-{\it\xi}$ is the artificially induced advection that compensates for the scaling nature of ${\it\xi}$ with $R(t)$ . It naturally arises from the non-dimensionalization process. Neglecting both terms essentially means that we are now rigorously solving the diffusion equation with an ever-present numerical advection term (associated to the non-physical velocity field $u_{r}(r,t)={\dot{R}}r/R$ ) that accounts for the moving boundary (Peñas López, Parrales & Rodríguez-Rodríguez Reference Peñas López, Parrales and Rodríguez-Rodríguez2015). Nonetheless, the effect of the artificial advection on the interfacial concentration gradient will still be small for $|\mathit{Pe}(\tilde{{\it\tau}})|\ll 1$ .

The concentration field may then be sought by solving

which has no direct dependency on the bubble dynamics. The choice of $\tilde{{\it\tau}}$ over ${\it\tau}$ is justified in that $\tilde{{\it\tau}}$ allows us to go from (2.1) to (2.15) based on the single assumption that the advection term is small. Had we chosen to perform an equivalent non-dimensionalization with physical time ${\it\tau}$ (cf. later (5.6)), it is only possible to treat the arising $a({\it\tau})$ -dependent diffusive term employing an additional approximation, namely treating the radius $a$ as a constant. The essence of the so-called quasi-stationary approximation (Weinberg & Subramanian Reference Weinberg and Subramanian1980) behind the Epstein–Plesset equation resides in these two approximations, namely (i) dropping the advective transport term arising from the interface motion and (ii) treating the bubble radius as a constant in the concentration boundary condition at the interface. Approximation (ii), hereon referred to as the quasi-static radius approximation, is only suitable when considering small or slow changes in the radius size from the equilibrium or initial size. Otherwise, this approximation will inherently decrease the accuracy of the solution. It is then concluded that the purpose of $\tilde{{\it\tau}}$ is to avert making use of the quasi-static radius approximation, thereby extending the parameter range in which the theory is valid. Additionally, writing the advection–diffusion equation in terms of $\tilde{{\it\tau}}$ allows us to point out an interesting conclusion regarding the appropriateness of neglecting advection effects. Indeed, the advective term in (2.12) is exactly zero at the bubble interface, as $(1/{\it\xi}^{2}-{\it\xi})=0$ there. Thus, in a region close to the bubble surface, it is reasonable to expect the advective term to play a small role even for moderate values of the Péclet number.

The solution to (2.15) for the concentration gradient across the bubble interface (see appendix A) reads

The initial interfacial concentration, $c_{s0}$ , is given by

where $p_{0}=p(0)$ and $a_{0}=a(0)$ denote the initial pressure and radius respectively. The first term in the right-hand side of (2.16) corresponds to the steady-state solution determined by the instantaneous interfacial concentration, $c_{s}(\tilde{{\it\tau}})$ . The second term is the transient component exclusively associated with the initial conditions. The third and final term is the history or memory integral term. It depends on the time history of $c_{s}(\tilde{{\it\tau}})$ caused by any prior variations in the ambient pressure $p(\tilde{{\it\tau}})$ . It addresses the temporal delay required for the concentration boundary layer to become fully developed (physically due to the finite diffusivity $D_{m}$ ) as the boundary condition, in this case $c_{s}$ , changes with time. In other words, the current concentration profile and therefore the mass flux across the bubble are conditioned by the preceding time history of the boundary condition.

The history integral may be evaluated analytically for the cases when $c_{s}$ varies in sudden (step-like) jumps and when $c_{s}$ varies harmonically in time. These will be addressed later in § 4 and § 5 respectively.

3.1 Recovering the Epstein–Plesset equation from the EPH

We now show that the Epstein–Plesset equation may be recovered from the EPH equation. In order to do so we revert back to (3.2). Since the original Epstein–Plesset equation is formulated using the linear time ${\it\tau}=D_{m}t/R_{c}^{2}$ , it is convenient for both time derivatives in $\tilde{{\it\tau}}$ to be replaced by derivatives in ${\it\tau}$ instead. Recalling that $\text{d}{\it\tau}=a^{2}\,\text{d}\tilde{{\it\tau}}$ , we may recast (3.2) as

The analytical expression for the interfacial concentration gradient is still a function of $\tilde{{\it\tau}}$ as given by (2.16). The Epstein–Plesset equation assumes a constant pressure history, i.e. $p=p_{0}$ and $\text{d}p/\text{d}{\it\tau}=0$ . Equation (3.5) then reduces to

The interfacial concentration gradient deserves special treatment. The history integral term vanishes, consequence of the constant-pressure condition. Moreover, unlike the EPH equations (3.3) or (3.4), the Epstein–Plesset equation is based on the quasi-static radius approximation (Weinberg & Subramanian Reference Weinberg and Subramanian1980). The concentration gradient is calculated for a fixed bubble boundary of radius $a$ . The solution is then coupled with the equation for mass conservation (3.5), where $a$ is now time dependent. This has two implications: (i) the bubble radius $a$ is treated as constant when solving for the concentration gradient. Consequently, (2.11) simplifies to ${\it\tau}=a^{2}\tilde{{\it\tau}}$ . Additionally, (ii) the initial interfacial concentration calculated for a static $a$ is in fact ‘reused’ for all its values. This amounts to setting $a_{0}=a$ in (2.17), hence $c_{s0}=c_{s}=p_{0}+{\it\sigma}/a-{\it\Upsilon}$ . Under the quasi-static radius approximation, the concentration gradient in (2.16) then becomes

Inserting this expression into (3.6), one finally recovers the Epstein–Plesset equation in dimensionless form (Epstein & Plesset Reference Epstein and Plesset1950, equation (34)):

An equivalent equation in $\tilde{{\it\tau}}$ , which does not make use of the quasi-static radius approximation, may be shown to be

In the absence of surface tension, ${\it\sigma}=0$ , (3.9) becomes a separable differential equation in $a$ and $\tilde{{\it\tau}}$ . Its analytical solution, with $a(0)=a_{0}$ , is easily found to be

The exact analytical solution of $a({\it\tau})$ to (3.10) exists in parametric form, with $a(\tilde{{\it\tau}})$ and ${\it\tau}(\tilde{{\it\tau}})$ . The physical time ${\it\tau}$ is related to $\tilde{{\it\tau}}$ through

which can be derived by solving (2.11).

4.1 An example to illustrate the history effect

Let us consider the radius history of a bubble with negligible Laplace pressure ( ${\it\sigma}=0$ ) initially in equilibrium at the saturation pressure. Setting $P_{c}=P_{\infty }(0)$ and $R_{c}=R(0)$ renders $p_{0}=1$ , $a_{0}=1$ and ${\it\Upsilon}=1$ . The bubble is then exposed to $N=6$ consecutive jumps in pressure. We prescribe coefficients $\tilde{T}_{n}$ and ${\rm\Delta}p_{n}$ , tabulated in table 1. These are entered into (4.1), resulting in the pressure time history plotted in figure 2(b). An iterative procedure was employed to establish coefficients $\tilde{T}_{n}$ such that the two expansion (growth) stages ( $T_{1}^{+}<{\it\tau}<T_{2}^{-}$ and $T_{4}^{+}<{\it\tau}<T_{5}^{-}$ ) have the same duration in ${\it\tau}$ (see figure 2 d). Moreover, both expansion stages are identical in the sense that $a(T_{1})=a(T_{4})=a_{0}=1$ , ${\rm\Delta}p_{1}={\rm\Delta}p_{4}$ and $p_{1}=p_{4}$ . The analytical solution for $a(\tilde{{\it\tau}})$ provided by (B 3) and (B 4) is plotted in figure 2(b). The solution is more naturally interpreted when presented in ${\it\tau}$ (figure 2 c). This requires numerical integration of (2.11) once the analytical solution for $a(\tilde{{\it\tau}})$ has been obtained.

At ${\it\tau}=T_{1}^{-}$ , right before the first growth stage, the history integral is identically zero, i.e. there is no previous history. This is obviously not the case at ${\it\tau}=T_{4}^{-}$ . Consequently, the memory effect must be behind the differences in bubble growth observed between these two pressure-wise identical expansion periods (figure 2 a,c).

Figure 2.

Analytical solution for (a) the evolution of the dimensionless bubble radius, $a$ , corresponding to (b) multiple jumps in the dimensionless ambient pressure $p$ , both plotted against the dimensionless, nonlinear time $\tilde{{\it\tau}}$ . Initial conditions correspond to perfect saturation. Additionally, equivalent plots of (c) the dimensionless bubble radius and (d) dimensionless ambient pressure are plotted against the dimensionless linear time ${\it\tau}$ . For reference, the physical parameters employed are those considering a $\text{CO}_{2}$ bubble of size $R_{c}=225~{\rm\mu}\text{m}$ in water ( $k_{H}=3.40\times 10^{-4}~\text{mol}~\text{N}^{-1}~\text{m}^{-1}$ , $D_{m}=1.92\times 10^{-9}~\text{m}^{2}~\text{s}^{-1}$ ) under conditions $P_{c}=4.9$ bar and $T_{\infty }=293~\text{ K}$ . Surface tension is neglected. The resulting solubility and saturation parameters are ${\it\Lambda}=0.828$ and ${\it\Upsilon}=1$ .

To highlight this effect, figure 3(a) compares the bubble radius history during both growth stages. At equal times, measured from the beginning of each expansion period, the bubble always exhibits a larger radius in the second growth cycle. This is consequence of the larger, history-augmented growth rate (figure 3 b). The second growth rate is most enhanced at the beginning. It then asymptotically converges in time with the first growth rate, by then confirming the complete dissipation of the history effect.

The physical explanation of the history effect lies in the concentration profiles near the bubble a short time after each jump (figure 3 b). Both concentration profiles are bounded by an identical value of $c_{s}({\it\tau}^{\ast })<0$ , and the far-field concentration $c(\infty ,{\it\tau})=0$ . The first concentration profile corresponds to a uniform initial condition $c({\it\xi},T_{1}^{-})=0$ . On the other hand, the second concentration profile remembers that $c_{s}>0$ during the previous dissolution period at $T_{2}^{+}<{\it\tau}<T_{3}^{-}$ . Consequently, there exists a supersaturation region ( $c>0$ ) near the bubble, containing the mass of gas that was transferred to the liquid during said dissolution period. The concentration boundary layer takes time to fully develop in response to any change in the interfacial concentration. This effective adaptation time, owing to the finite diffusivity, results in this case in a momentarily steeper interfacial concentration gradient that enhances the growth rate substantially.

Figure 3.

The first (dashed line) and second (solid line) growth cycles of the bubble (see figure 2) are compared through (a) the dimensionless radius $a$ and (b) its rate of change, computed numerically out of the analytical $a({\it\tau})$ . These are plotted against the dimensionless linear time ${\it\tau}$ after each of the two negative pressure jumps ( ${\rm\Delta}p_{1}$ and ${\rm\Delta}p_{4}$ ) that lead to undersaturation. Inset: dimensionless concentration radial profiles evaluated at a short time ${\it\tau}^{\ast }$ immediately after each of the two jumps.

The effect described here is also present in the experimental data shown in figure 1. As a matter of fact, both the size and growth rate time histories are qualitatively identical to those shown in figure 2, predicted by the analytical solution. The quantitative analysis of these experiments will be carried out in a future companion paper, as it involves taking into account the presence of the plate and natural convection, among other effects.

Another manifestation of the history effect is observed for ${\it\tau}>T_{6}^{+}$ , after the expansion–compression cycles have ended (see figure 2). Even though the pressure returns to the saturation value, $c_{s}({\it\tau}\geqslant T_{6}^{+})=0$ , static equilibrium is not instantaneously achieved. As before, a supersaturation region near the bubble ( $c>0$ ) remains from the preceding dissolution stage. This amounts to a positive interfacial concentration gradient. The radius of the bubble grows from $a({\it\tau}=T_{6}^{+})\approx 0.9$ to $a({\it\tau}=30)\approx 1.1$ , certainly a non-negligible increment. This growth is entirely provided by the history effect. In other words, the interfacial concentration gradient in (2.16) is non-zero due to the sole contribution of the history integral term.

4.2 Comparison with numerical simulation

An important question that arises now is how important are history effects compared to others that have been neglected in our discussion thus far, such as advection or surface tension. To shed light on this question, we numerically solve the full mass transfer problem (2.12), (2.13) and (3.2) which takes these effects into account. Details of the numerical treatment of the problem may be found in § B.2.

Figure 4.

Evolution of (a) the dimensionless bubble radius $a$ in time ${\it\tau}$ according to the simulation scenarios (i)–(iv) and the analytical solution in (B 3) and (B 4). The bubble is initially in a 105 % supersaturated liquid. It is exposed to (b) a prescribed pressure time history consisting of two particular jumps. Inset: the dissolution dynamics for ${\it\tau}>5$ according to the Epstein–Plesset solution (3.8) is compared with solutions (iii) and (iv). For reference, the physical parameters employed are those considering a $\text{CO}_{2}$ bubble of size $R_{c}=175~{\rm\mu}\text{m}$ in water ( $k_{H}=3.40\times 10^{-4}~\text{mol}~\text{N}^{-1}~\text{ m}^{-1}$ , $D_{m}=1.92\times 10^{-9}~\text{m}^{2}~\text{s}^{-1}$ , ${\it\gamma}_{\mathit{lg}}=0.07~\text{N}~\text{m}^{-1}$ ) under conditions $P_{c}=2$ bar and $T_{\infty }=293~\text{K}$ . The resulting surface tension, solubility and saturation parameters are ${\it\sigma}=0.004$ , ${\it\Lambda}=0.828$ and ${\it\Upsilon}=1.05$ .

As an example, let us consider a bubble in an initially 105 % supersaturated liquid. The initial concentration field of dissolved gas is assumed uniform. Setting $P_{c}=P_{\infty }(0)$ and $R_{c}=R(0)$ entails $p_{0}=1$ , $a_{0}=1$ and ${\it\Upsilon}=1.05$ . The bubble is then exposed to the pressure time history shown in figure 4(b). There are three constant-pressure stages, the first corresponding to liquid supersaturation (expansion stage), the second to undersaturation (compression stage) and the third to perfect saturation ( $p_{\mathit{sat}}={\it\Upsilon}=1.05$ ). The set of physical conditions are specified in the caption of figure 4. Four different versions of the problem are numerically solved, namely

1. (i) neglecting the dimensionless advection term, i.e. setting $\mathit{Pe}=0$ in (B 13);

2. (ii) neglecting surface tension, i.e. setting ${\it\sigma}=0$ in (B 12) and (B 14);

3. (iii) neglecting both the advection term and surface tension (the same scenario as the one analytically solved, i.e. $\mathit{Pe}=0$ , ${\it\sigma}=0$ ) and

4. (iv) taking into account both the advection term and Laplace pressure.

These solutions are compared in figure 4(a), along with the analytical solution (for which $\mathit{Pe}={\it\sigma}=0$ ). History effects are critical in order to properly describe this particular dissolution (compression) stage ( $5<{\it\tau}<10$ ). While growth is well determined by existing solutions (Epstein & Plesset Reference Epstein and Plesset1950; Scriven Reference Scriven1959), the dissolution experienced by this bubble is greatly affected by the low-concentration boundary layer left by the preceding growth stage. It stands to reason that for the initial instants of dissolution, during which history effects are most important, the dissolution rate observed is much faster than that obtained under the assumption of a historyless, uniform concentration field at ${\it\tau}=5^{+}$ . This is corroborated in the inset plot of figure 4, where we plot the Epstein–Plesset solution (3.8), which assumes a uniform initial concentration field (with $p_{0}=1.1$ ). To enable proper comparison, the initial radius $a_{0}$ at ${\it\tau}=5^{+}$ for the Epstein–Plesset solution with and without surface tension has been fitted to that predicted by curve (iv) and (iii) respectively. Note that curve (iii) is identical to the analytical solution.

When the pressure drops to the saturation value at ${\it\tau}=10$ , the bubble continues to grow for some time. As discussed in § 4.1, it is consequence of the high concentration boundary layer of dissolved gas in the nearby liquid that was left by the preceding dissolution stage. Since the bubble is nearly close to the equilibrium in this stage, both advection and surface tension are small compared to history, at least for a few time units. At later times however, surface tension breaks the diffusive equilibrium, and ultimately drives the bubble towards its total dissolution. Nonetheless, this occurs at times much longer than those of the pressure cycle considered here.

Under the conditions investigated here, the effect of advection and surface tension on the bubble radius dynamics nearly cancel each other out during the expansion–compression cycle. Consequently, it is observed that the full solution (iv) (which contains both surface tension and advection) is in better agreement with the analytical solution (iii) (which contains neither) than with curves (i) or (iii) during this period. In fact, a quantitative explanation may be obtained from a simple analysis. For small values of $\mathit{Pe}$ and constant ambient pressure, the contribution of advection towards the growth or dissolution rate may be quantified using the asymptotic solution of Duda & Vrentas (Reference Duda and Vrentas1969, equation (43)). To do so, we subtract the Epstein–Plesset solution (3.8) with ${\it\sigma}=0$ from (43). For our particular set of initial conditions, $p_{0}=a_{0}=1$ , we obtain

Conversely, a simple estimation of the effect of the Laplace pressure on the rate of change of the bubble radius can be done by subtracting (3.8) with ${\it\sigma}=0$ from itself. Provided $p_{0}=a_{0}=1$ and ${\it\sigma}\ll 1$ , this results in

We may use these formulas to evaluate the characteristic percentage contribution to the growth rate of each effect with respect to that given by Epstein–Plesset solution (3.8) with ${\it\sigma}=0$ . Halfway through the initial growth stage ( ${\it\tau}=2.5$ , $a=1.18$ ) we determine that advection enhances growth by approximately 5 % whereas surface tension slows it down by $-7\,\%$ . Thus, this small net difference of 2 % (which actually decreases as the bubble grows) justifies the close agreement between curves (iii) and (iv).

5.1 The oscillatory problem

The full advection–diffusion problem is usually tackled by splitting it into a smooth problem and an oscillatory problem following the work of Fyrillas & Szeri (Reference Fyrillas and Szeri1994). The smooth problem (diffusion time scale ${\it\tau}\sim 1$ ) accounts for the steady part of the boundary condition and yields the net flux of dissolved gas across the oscillating bubble. The oscillatory problem (diffusion time scale ${\it\tau}\sim {\it\Omega}^{-1}$ ) takes into account the unsteady part of the boundary condition and describes the zero-average-mass exchange occurring over one bubble oscillation. This approach is only valid on the assumption that the time scales of the two processes are well separated: ${\it\Omega}\gg 1$ . Since we are also considering frequencies ${\it\Omega}\sim 1$ , splitting the problem is no longer possible. Moreover, the smooth solution, whether following the approach of Eller & Flynn (Reference Eller and Flynn1965) or Fyrillas & Szeri (Reference Fyrillas and Szeri1994), is constructed from the mass transport equation in Lagrangian spherical coordinates (as opposed to (5.6)) and requires special time averaging of the radius dynamics to properly capture the effect of advection. The interested reader is referred to Ilinskii et al. (Reference Ilinskii, Wilson and Hamilton2008) for a brief review on these two forms of the smooth solution.

Fortunately, the history effect is only directly relevant in the oscillatory problem associated with the unsteady boundary condition. In other words, we are only interested in the solution during an individual period of oscillation about $\bar{a}$ . Solving the governing mass transport equation directly in the form

is suitable to this end. Note that $\mathit{Pe}({\it\tau})=(\text{d}a/\text{d}{\it\tau})a\equiv R{\dot{R}}/D_{m}$ is our ${\dot{R}}$ -based Péclet number (as previously defined in § 2.2). It follows that under the small-amplitude oscillation restriction, the magnitude of the dimensionless advection term in (5.6) is always smaller than the unsteady and diffusive terms by a factor of $O({\it\varepsilon})$ , where ${\it\varepsilon}\ll 1$ (see appendix D). Note that this condition holds true regardless of the magnitude of $\mathit{Pe}$ (cf. $O(\mathit{Pe})\sim {\it\varepsilon}{\it\Omega}\bar{a}^{2}/3$ ), which may not necessarily be smaller than unity. It is therefore not unreasonable to neglect the advection term when dealing just with the oscillatory problem consisting of an individual cycle. Note, nonetheless, that the small contribution of the advection term on the interfacial gradient (negligible over one individual cycle) is still crucial to providing net growth over many cycles.

Moreover, assumption (ii) allows us to make the approximation $a({\it\tau})=\bar{a}({\it\tau})+O({\it\delta})\approx \bar{a}$ , which is constant over an oscillation period according to (iii). While this greatly simplifies the problem, it only induces very small errors of $O({\it\varepsilon})\ll 1$ .

Additionally, we will consider saturation conditions ${\it\Upsilon}=1$ . This is justified in that the bulk concentration of the liquid comes into play in the smooth solution only (Fyrillas & Szeri Reference Fyrillas and Szeri1994). Similarly, surface tension and time-averaged bubble interface motion are mostly relevant in the smooth solution.

Implementing these notions, the solution to (5.6) for the concentration profile is found to be (see appendix D for the complete derivation)

which renders the following concentration gradient at the wall:

The first term on the right-hand side of (5.8) refers to the quasi-steady mass flux imposed by the current value of $c_{s}({\it\tau})={\it\varepsilon}\sin {\it\Omega}{\it\tau}$ . The last term in the right-hand side constitutes the contribution from the history integral (derived in appendix D). In the scenario of small-amplitude oscillations considered here, the mass flux coming from the contribution of the history term is thus observed to be proportional to the pressure amplitude ${\it\varepsilon}$ . The history effect imposes a phase shift in the sinusoidal interfacial concentration gradient, $0<{\it\phi}_{\mathit{grad}}({\it\Omega})\leqslant {\rm\pi}/4$ , with respect to the phase of $-c_{s}({\it\Omega}{\it\tau})$ or the oscillatory component of $a_{p}({\it\tau})$ .

As ${\it\Omega}\rightarrow 0$ , the history integral term vanishes and ${\it\phi}_{\mathit{grad}}\rightarrow 0$ . The mass flux is in phase with $a_{p}$ (cf. (5.5)). Conversely, as ${\it\Omega}\rightarrow \infty$ , the instantaneous mass flux is entirely provided by the history integral. The maximum possible phase shift with respect to $a_{p}$ is attained: ${\it\phi}_{\mathit{grad}}={\rm\pi}/4$ . In fact, the frequency dependency of this phase shift is independent of ${\it\varepsilon}$ and may be shown to be

Solution (5.7) may be described as a damped transverse wave with wavenumber $k=\bar{a}\sqrt{{\it\Omega}/2}$ and phase velocity $\sqrt{2{\it\Omega}}/\bar{a}$ attenuating over a penetration depth $l\sim 1/(\bar{a}\sqrt{{\it\Omega}})$ . Figure 5 shows the close agreement of this (advectionless) analytical solution with the full numerical computation (taking into account advection) involving (5.6) and the mass conservation equation (3.5) with no surface tension for ${\it\Omega}=1$ and ${\it\Omega}=1000$ (with equilibrium radius $\bar{a}=1$ ). This verifies the negligible qualitative impact of the advection term in the oscillatory problem even for this moderate pressure amplitude of ${\it\varepsilon}=0.01$ . On another note, the boundary layer thickness $h$ may be best estimated as the wavelength, $h=2{\rm\pi}/k$ . Over a distance $h$ , (horizontal span in figure 5) the concentration amplitude reduces by a factor of $\text{e}^{2{\rm\pi}}\approx 540$ .

Figure 5.

Radial concentration profiles at different times over an oscillation period for (a) ${\it\Omega}=1$ (b) ${\it\Omega}=1000$ taking $\bar{a}=1$ and ${\it\varepsilon}=0.01$ . The radial coordinate has been normalized by the wavenumber and spans one wavelength. The circle represents the relative bubble size (drawn to scale with the horizontal axis) for reference.

For ${\it\Omega}=1000$ , figure 5(b) hints that the shift in the phase of the concentration gradient with respect to $-c_{s}$ is practically that of the high-frequency limit, ${\it\phi}_{\mathit{grad}}={\rm\pi}/4$ . The interfacial gradient is flat at ${\it\Omega}{\it\tau}=3{\rm\pi}/4,7{\rm\pi}/4$ as opposed to $c_{s}({\it\tau})=0$ taking place at ${\it\Omega}{\it\tau}={\rm\pi},2{\rm\pi}$ . For ${\it\Omega}=1$ , direct evaluation of (5.9) yields ${\it\phi}_{\mathit{grad}}={\rm\pi}/8$ (as later observed in figure 6 a).

Figure 6.

Numerically obtained oscillation waveforms for ${\it\Omega}=1$ , with $\bar{a}=1$ , ${\it\varepsilon}=0.01$ and ${\it\Lambda}=0.828$ . Phase shifts are annotated in (a) showing the normalized waveforms for strains $(a-\bar{a})/{\it\delta}$ , $(a_{\mathit{corr}}-\bar{a})/{\it\delta}_{\mathit{corr}}$ , $(a_{p}-\bar{a}_{p})/{\it\delta}_{p}$ and for the interfacial concentration gradient $(\partial c/\partial {\it\xi}|_{{\it\xi}=1})/{\it\delta}_{\mathit{grad}}$ . (b) Unscaled strain waveforms $a/\bar{a}-1$ , $a_{\mathit{corr}}/\bar{a}-1$ and $a_{p}/\bar{a}-1$ .

It stands to reason that solutions (5.7) and (5.8) evaluated in the limit ${\it\Omega}\gg 1$ converge to those one may obtain following the approach of Fyrillas & Szeri (Reference Fyrillas and Szeri1994) (inherently valid for ${\it\Omega}\gg 1$ ) subject to the quasi-static radius approximation, i.e. taking $a({\it\tau})=\bar{a}$ as constant. The complete derivation may be found in § D.1.

5.2 Strain amplitudes and phase

The frequency-dependent ${\it\phi}_{\mathit{grad}}$ induced by the history effect implies that pressure-induced contraction of the bubble radius ( $p>1$ , $c_{s}>0$ , $\text{d}a_{p}/\text{d}{\it\tau}<0$ ) is not in phase with dissolution (negative mass diffusion rate, $\partial c/\partial {\it\xi}(1,{\it\tau})>0$ ). Likewise, the pressure-induced expansion is not in phase with growth. We would like to conclude this work by determining the dependency on ${\it\Omega}$ of the overall strain amplitude and phase lag with respect to $a_{p}({\it\phi}<0)$ of the isothermally pulsating bubble.

Let us then define the dimensionless pressure-corrected radius, $a_{\mathit{corr}}({\it\tau})$ , in an analogous manner to $R_{\mathit{corr}}$ in (1.3) as follows:

The pressure-corrected radius is associated purely with the mass transfer across the interface. It has the useful property that its rate of change, $\text{d}a_{\mathit{corr}}/\text{d}{\it\tau}$ , always takes the same sign as the interfacial concentration gradient. Consequently, ${\it\phi}_{\mathit{corr}}={\it\phi}_{\mathit{grad}}-{\rm\pi}/2$ , as shown in appendix E.

The characteristic amplitude of oscillation for the interfacial concentration gradient may be estimated from (5.8),

Equation (5.10) may be rewritten in terms of the different oscillation amplitudes about the equilibrium radius $\bar{a}$ ,

Linearization provides ${\it\delta}={\it\delta}_{p}+{\it\delta}_{\mathit{corr}}+O({\it\delta}_{p}{\it\delta}_{\mathit{corr}})$ . Amplitudes ${\it\delta}_{p}$ and ${\it\delta}_{\mathit{corr}}$ may be reasonably approximated as the product of the characteristic time derivative and the characteristic time scale, $1/{\it\Omega}$ . We obtain

which is consistent with (5.5). Likewise, making use of (E 3),

Two limiting cases arise. For small ${\it\Omega}\ll 1$ , we see that ${\it\delta}\approx {\it\delta}_{\mathit{corr}}\sim {\it\varepsilon}{\it\Lambda}/\bar{a}{\it\Omega}$ . The bubble oscillation amplitudes are provided entirely by mass transfer across the bubble surface, the pressure-induced expansion and contraction amplitude is negligible in comparison. The behaviour of the different phase shifts as ${\it\Omega}$ approaches this limit is:

Taking ${\it\Omega}=1$ , $\bar{a}=1$ (figure 6), the dominant contribution in ${\it\delta}$ and hence ${\it\phi}$ still come from ${\it\delta}_{\mathit{corr}}$ and ${\it\phi}_{\mathit{corr}}$ . The relative contributions of ${\it\delta}_{\mathit{corr}}$ and ${\it\delta}_{p}$ may be estimated from ${\it\delta}_{\mathit{corr}}/{\it\delta}_{p}\sim 6{\it\Lambda}\approx 5$ .

For large ${\it\Omega}\gg 1$ , we see that ${\it\delta}\approx {\it\delta}_{p}\sim {\it\varepsilon}\bar{a}/3$ . There is negligible mass transfer during an individual oscillation due to the short oscillation period. The phase shifts behave according to:

Taking ${\it\Omega}=100$ , $\bar{a}=1$ (figure 7), $a_{p}$ now provides the main contribution since ${\it\delta}_{\mathit{corr}}/{\it\delta}_{p}\sim 3{\it\Lambda}/10\approx 1/4$ .

Figure 7.

Oscillation waveforms for ${\it\Omega}=100$ , with $\bar{a}=1$ , ${\it\varepsilon}=0.01$ (see caption of figure 6).

Ultimately, the difference in the phase lags ${\it\phi}$ and ${\it\phi}_{\mathit{corr}}$ partly induced by the history effect has a direct impact on the relative contributions of the area effect and the so-called shell effect (Eller & Flynn Reference Eller and Flynn1965), the two driving mechanisms behind bubble growth in rectified diffusion. The area effect refers to the increase of the bubble interfacial area during expansion and the subsequent decrease during contraction. The velocity field generated by the bubble oscillations also influences the gas transport through the advection term in the advection–diffusion equation. This influence is referred to as the shell effect. More specifically, we can regard the shell effect as the advection-induced squeezing of the concentration boundary layer when the bubble radius expands and similarly the stretching when the radius contracts. In turn, this effect yields either a steeper or shallower concentration gradient at the bubble surface compared to that of a motionless bubble of the same size. As a result, the area effect increases mass transfer with respect to the motionless equilibrium bubble when $a>\bar{a}$ . Similarly, the shell effect has an amplifying effect on the interfacial mass flux when $\text{d}a/\text{d}{\it\tau}>0$ .

Figure 8.

Strain waveforms for ${\it\Omega}=5000$ , with $\bar{a}=1$ , ${\it\varepsilon}=0.01$ and ${\it\Lambda}=0.828$ . (a) Normalized strain waveforms $(a-\bar{a})/{\it\delta}$ and $(a_{\mathit{corr}}-\bar{a})/{\it\delta}_{\mathit{corr}}$ . SE and AE refer to the ‘shell effect’ and ‘area effect’, which may increase ( $+$ ) or hinder ( $-$ ) mass transfer across the bubble, be it growth ( $\text{d}a_{\mathit{corr}}/\text{d}{\it\tau}>0$ , grey background) or dissolution. (b) Unscaled strain waveforms $a/\bar{a}-1$ and $a_{\mathit{corr}}/\bar{a}-1$ .

Let us consider the high-frequency limit where ${\it\phi}=0$ , ${\it\phi}_{\mathit{corr}}={\rm\pi}/4$ and $a\approx a_{p}$ . The area and shell effects act non-symmetrically in the bubble dissolution ( $\text{d}a_{\mathit{corr}}/\text{d}{\it\tau}<0$ ) and growth ( $\text{d}a_{\mathit{corr}}/\text{d}{\it\tau}>0$ ) periods, which are of equal duration in time. Figure 8 shows that the area and shell effect increase mass transfer during three-quarters of the growth period and one-quarter of the dissolution period. Consequently, these effects reduce mass transfer during one-quarter of the growth period and three-quarters of the dissolution period. As a result, bubble growth is always promoted.