Film evolution of a spherical soap bubble

Abstract

We present a theoretical and numerical study of the time evolution of the film of a soap bubble. Bubbles are assumed to remain approximately spherical with axisymmetric evolution. Inertia is neglected, and the surfactant is assumed insoluble. Applying lubrication theory, we simplify the equations governing the film thickness evolution, surfactant concentration, and tangential fluid velocity. Solving the simplified equations numerically, we examine the long-term behavior of the film and find that most features follow a power law of the form $\alpha T^\beta$ where T is a dimensionless time parameter. We also predict the location (angle from the top) at which the film is thinnest and thus most likely to initiate bursting and present a similarity solution that predicts the decay rate of the minimum film thickness near the pinch-off location.

Appendices

Appendix A: Soluble surfactant

The surfactant concentration in the film, c, (in mass per volume) satisfies an advection-diffusion equation, given, in dimensional form, as

$$\begin{aligned} \frac{\partial c}{\partial t}+{\bar{\nabla }}\cdot ({\mathbf {u}}c)=k_\mathrm{c}{\bar{\nabla }}^2c, \end{aligned}$$

(52)

where $k_\mathrm{c}$ is the bulk diffusivity and ${\bar{\nabla }}$ is a dimensional gradient. In the presence of sorption, Eq. (4) is supplemented with a bulk exchange term [37, 38],

$$\begin{aligned} \frac{\partial \gamma }{\partial t}+{\bar{\nabla }}_\mathrm{s}\cdot ({\mathbf {u}}_\mathrm{s}\gamma ) =k_\Gamma {\bar{\nabla }}_\mathrm{s}^2\gamma +j(\gamma ,c_\mathrm{s}), \end{aligned}$$

(53)

where $k_\Gamma$ is a diffusivity of surface surfactant. Combining Eq. (52) with the exchange term, we obtain a boundary condition on the bulk concentration,

$$\begin{aligned} j(\gamma ,c_\mathrm{s})=-k_\mathrm{c}({\mathbf {n}}\cdot {\bar{\nabla }})c_\mathrm{s}, \end{aligned}$$

(54)

where ${\mathbf {n}}$ is the outward unit normal at h. To describe the bulk exchange term, we can apply the constitutive relation [37, 38]

$$\begin{aligned} -j(\gamma ,c_\mathrm{s})=k_{\mathrm {ad}}c_\mathrm{s}\bigg (1-\frac{\gamma }{\gamma _\infty }\bigg )-k_{\mathrm {de}}\gamma , \end{aligned}$$

(55)

where $k_{\mathrm {ad}}$ and $k_{\mathrm {de}}$ are kinetic constants of adsorption and desorption, with units of $\mathrm {m}/\mathrm {s}$ and $\mathrm {s}^{-1}$, respectively.

To obtain dimensionless equations, we define $C=c/c_0$ where $c_0$ is the equilibrium bulk concentration. The exchange term in Eq. (55) rescales to $J(\Gamma ,C_\mathrm{s})=j(\gamma ,C_\mathrm{s})/(\gamma _\infty u_0/r_0)$. To describe it more precisely, we write the gradient of the bulk concentration in coordinates, $(\varphi ,R)$:

$$\begin{aligned}{\bar{\nabla }}c=\left( \frac{1}{r}\frac{\partial c}{\partial \varphi },\frac{\partial c}{\partial r}\right) =\frac{c_0}{r_0}\left( \frac{\partial C}{\partial \varphi },\frac{1}{\varepsilon }\frac{\partial C}{\partial R}\right) ,\end{aligned}$$

so that in rescaled coordinate form, Eq. (54) becomes, on the outer film,

$$\begin{aligned} -J(\Gamma ,C_\mathrm{s})=\frac{1}{L\mathrm {Pe}_\mathrm {c}}\left( \frac{\partial C}{\partial R}\bigg |_\mathrm{s} -\varepsilon ^2\frac{\partial H}{\partial \varphi }\frac{\partial C}{\partial \varphi }\bigg |_\mathrm{s}\right) \left( 1+\varepsilon ^2\Big (\frac{\partial H}{\partial \varphi }\Big )^2\right) ^{-1/2}=\frac{1}{L\mathrm {Pe}_\mathrm {c}}\frac{\partial C}{\partial R}\bigg |_\mathrm{s}+{\mathcal {O}}(\varepsilon ), \end{aligned}$$

(56)

where L is a dimensionless surfactant depletion length $L=\gamma _\infty /h_0c_0$. The expression on the inner film is the same but with the opposite sign of the R derivative. Rescaling, we obtain

$$\begin{aligned} J(\Gamma ,C_\mathrm{s})=-\mathrm {Bi}\left( \mathrm {Ad}C_\mathrm{s}(1-\Gamma )-\Gamma \right) , \end{aligned}$$

(57)

where we have introduced a Biot number and an adsorption number

$$\begin{aligned} \mathrm {Bi}=\frac{r_0k_{\mathrm {de}}}{u_0}\quad \text {and}\quad \mathrm {Ad}=\frac{k_{\mathrm {ad}}c_0}{k_{\mathrm {de}}\gamma _\infty }. \end{aligned}$$

Then, if we discard higher order terms, Eqs. (52) and (53) rescale to

$$\begin{aligned} \frac{\partial C}{\partial T}+\frac{\partial }{\partial R}\left( VC\right) +\frac{1}{\sin \varphi }\frac{\partial }{\partial \varphi }(UC\sin \varphi ) =\frac{1}{\mathrm {Pe}_\mathrm {c}}\frac{\partial ^2 C}{\partial R^2}, \end{aligned}$$

(58)

and

$$\begin{aligned} \frac{\partial \Gamma }{\partial T}+\dfrac{1}{\sin \varphi }\frac{\partial }{\partial \varphi }(U_\mathrm{s}\Gamma \sin \varphi ) =\frac{1}{\mathrm {Pe}_\Gamma \sin \varphi }\frac{\partial }{\partial \varphi }\left( \sin \varphi \frac{\partial \Gamma }{\partial \varphi }\right) +J(\Gamma ,C_\mathrm{s}), \end{aligned}$$

(59)

where

$$\begin{aligned} \mathrm {Pe}_\mathrm {c}=\frac{\varepsilon ^2r_0u_0}{k_\mathrm{c}}\quad \text {and}\quad \mathrm {Pe}_\Gamma =\frac{r_0u_0}{k_\Gamma }, \end{aligned}$$

(60)

are Péclet numbers associated to the bulk and to the surface, respectively. Note that the factor of $\varepsilon ^2$ in $\mathrm {Pe}_\mathrm {c}$ ensures that $\mathrm {Pe}_\mathrm {c}$ is likely much smaller than $\mathrm {Pe}_\Gamma$. In general, surface diffusion, i.e., $1/\mathrm {Pe}_\Gamma$, tends to be very small. The symmetry of the system results in the condition $\frac{\partial C}{\partial R}\big |_{R=0}=0$ at the center of the film.

Proceeding in a manner similar to Sect. 2.5, we integrate the bulk surfactant equation, Eq. (58), across the film, using Eq. (17) and the boundary conditions (21) and (56). We obtain

$$\begin{aligned} \frac{\partial }{\partial T}\int _0^HC\,\mathrm{d}R+\dfrac{1}{\sin \varphi }\frac{\partial }{\partial \varphi }\bigg (\sin \varphi \int _0^HUC\,\mathrm{d}R\bigg )=-LJ(\Gamma ,C_\mathrm{s}). \end{aligned}$$

(61)

Appendix B: The derivation of equations (19) and (20)

Here, we derive equations (19) and (20) from Eq. (16). In axisymmetric spherical coordinates, the components of the stress tensor are given by

$$\begin{aligned} \tau _{rr}&=-p+2\mu \frac{\partial v}{\partial r}, \end{aligned}$$

(62)

$$\begin{aligned} \tau _{\varphi \varphi }&=-p+2\mu \left( \frac{1}{r}\frac{\partial u}{\partial \varphi } +\frac{v}{r}\right) \quad \text {and}\quad \end{aligned}$$

(63)

$$\begin{aligned} \tau _{r\varphi }&=\mu \left( \frac{1}{r}\frac{\partial v}{\partial \varphi }+\frac{\partial u}{\partial r} -\frac{v}{r}\right) . \end{aligned}$$

(64)

In dimensionless form, the components of the stress tensor rescale as

$$\begin{aligned} T_{rr}&=-P+{\mathcal {O}}(\varepsilon ^2), \end{aligned}$$

(65)

$$\begin{aligned} T_{\varphi \varphi }&=-P+{\mathcal {O}}(\varepsilon ^2)\quad \text {and}\quad \end{aligned}$$

(66)

$$\begin{aligned} T_{r\varphi }&=\varepsilon \frac{\partial U}{\partial R}+{\mathcal {O}}(\varepsilon ^2). \end{aligned}$$

(67)

Now we write the outward unit normal, ${\mathbf {n}}$, and the unit tangent, ${\mathbf {t}}$, in terms of ${\mathbf {r}}=r(\sin \varphi \cos \theta ,\sin \varphi \sin \theta ,\cos \varphi )$,

$$\begin{aligned}&{\mathbf {t}}=\left\| \frac{\partial {\mathbf {r}}}{\partial \varphi }\right\| ^{-1}\frac{\partial {\mathbf {r}}}{\partial \varphi }=\dfrac{\frac{\partial r}{\partial \varphi }{\hat{{\mathbf {r}}}}+r\hat{\varvec{\varphi }}}{\sqrt{\big (\frac{\partial r}{\partial \varphi }\big )^2+r^2}} =\varepsilon \frac{\partial H}{\partial \varphi }{\hat{{\mathbf {r}}}}+\hat{\varvec{\varphi }}+{\mathcal {O}}(\varepsilon ^2), \end{aligned}$$

(68)

$$\begin{aligned}&\quad \text {and}\quad {\mathbf {n}}=\left\| \frac{\partial {\mathbf {r}}}{\partial \varphi }\times \frac{\partial {\mathbf {r}}}{\partial \theta }\right\| ^{-1}\frac{\partial {\mathbf {r}}}{\partial \varphi }\times \frac{\partial {\mathbf {r}}}{\partial \theta } =\dfrac{-\frac{\partial {\mathbf {r}}}{\partial \varphi }\hat{\varvec{\varphi }}+r{\hat{{\mathbf {r}}}}}{\sqrt{\big (\frac{\partial r}{\partial \varphi }\big )^2+r^2}} ={\hat{{\mathbf {r}}}}-\varepsilon \frac{\partial H}{\partial \varphi }\hat{\varvec{\varphi }}+{\mathcal {O}}(\varepsilon ^2), \end{aligned}$$

(69)

where ${\hat{{\mathbf {r}}}}$ is the unit vector in the radial direction and $\hat{\varvec{\varphi }}$ is the unit vector in the angular direction. Now, we calculate the total curvature (the sum of the two principal curvatures), following [57]

$$\begin{aligned} \kappa =\dfrac{\det \left( \frac{\partial ^2 {\mathbf {r}}}{\partial \varphi ^2}\frac{\partial {\mathbf {r}}}{\partial \varphi }\frac{\partial {\mathbf {r}}}{\partial \theta }\right) \left| \frac{\partial {\mathbf {r}}}{\partial \theta }\right| ^2-2\det (\frac{\partial ^2 {\mathbf {r}}}{\partial \varphi \partial \theta }\frac{\partial {\mathbf {r}}}{\partial \varphi }\frac{\partial {\mathbf {r}}}{\partial \theta })\left( \frac{\partial {\mathbf {r}}}{\partial \varphi }\cdot \frac{\partial {\mathbf {r}}}{\partial \theta }\right) +\det (\frac{\partial ^2 {\mathbf {r}}}{\partial \theta ^2}\frac{\partial {\mathbf {r}}}{\partial \varphi }\frac{\partial {\mathbf {r}}}{\partial \theta })\left| \frac{\partial {\mathbf {r}}}{\partial \varphi }\right| ^2}{\left( \left| \frac{\partial {\mathbf {r}}}{\partial \varphi }\right| ^2\left| \frac{\partial {\mathbf {r}}}{\partial \theta }\right| ^2-\left( \frac{\partial {\mathbf {r}}}{\partial \varphi }\cdot \frac{\partial {\mathbf {r}}}{\partial \theta }\right) ^2\right) ^{3/2}}\quad . \end{aligned}$$

(70)

We obtain

$$\begin{aligned} K=r_0\kappa =-2+\varepsilon \left( \frac{\partial ^2 H}{\partial \varphi ^2}+\frac{\partial H}{\partial \varphi }\cot \varphi +2H\right) +{\mathcal {O}}(\varepsilon ^2). \end{aligned}$$

(71)

Next, we write Eq. (16) explicitly in normal and tangential components, using Eqs. (68) and (69), and keeping in mind the form of the Cauchy stress tensor,

$$\begin{aligned}{\mathbf {T}}=T_{rr}{\hat{{\mathbf {r}}}}\otimes {\hat{{\mathbf {r}}}}+T_{r\varphi }{\hat{{\mathbf {r}}}}\otimes \hat{\varvec{\varphi }} +T_{\varphi r}\hat{\varvec{\varphi }}\otimes {\hat{{\mathbf {r}}}}+T_{\varphi \varphi }\hat{\varvec{\varphi }}\otimes \hat{\varvec{\varphi }}\end{aligned}$$

to obtain

$$\begin{aligned} {\mathbf {n}}\cdot {\mathbf {T}}\cdot {\mathbf {n}}&=\bigg (\varepsilon ^2\Big (\frac{\partial H}{\partial \varphi }\Big )^2T_{\varphi \varphi }-2\varepsilon \frac{\partial H}{\partial \varphi }T_{r\varphi } +T_{rr}\bigg )\bigg /\left( 1+\varepsilon ^2\Big (\frac{\partial H}{\partial \varphi }\Big )^2\right) =\Sigma \frac{\partial ^2 H}{\partial \varphi ^2}+{\mathcal {O}}(\varepsilon )\quad \text {and}\quad \\ {\mathbf {t}}\cdot {\mathbf {T}}\cdot {\mathbf {n}}&=\bigg (\varepsilon \frac{\partial H}{\partial \varphi }(T_{rr}-T_{\varphi \varphi }) +T_{r\varphi }\bigg (1-\varepsilon ^2\Big (\frac{\partial H}{\partial \varphi }\Big )^2\bigg )\bigg )\bigg /\left( 1+\varepsilon ^2\Big (\frac{\partial H}{\partial \varphi }\Big )^2\right) =\frac{1}{\varepsilon }\frac{\partial \Sigma }{\partial \varphi }+{\mathcal {O}}(\varepsilon ). \end{aligned}$$

Substituting in Eqs. (65) and (67) and taking leading order terms, we obtain Eqs. (19) and (20).

Appendix C: Analysis of the large Marangoni number limit

For $\mathrm {El}={\mathcal {O}}(1)$ and $\mathrm {Mg}={\mathcal {O}}(\varepsilon ^{-2})\gg 1$, we expand $\Gamma$, $U_\mathrm{s}$, and H in powers of $\mathrm {Mg}^{-1}$,

$$\begin{aligned} \begin{aligned}&\Gamma (T,\varphi ,\mathrm {Mg})=\Gamma _{(0)}(T,\varphi )+\mathrm {Mg}^{-1}\Gamma _{(1)}(T,\varphi )+{\mathcal {O}}(\mathrm {Mg}^{-2}),\\&U_\mathrm{s}(T,\varphi ,\mathrm {Mg})=U_{\mathrm{s}(0)}(T,\varphi )+\mathrm {Mg}^{-1}U_{\mathrm{s}(1)}(T,\varphi )+{\mathcal {O}}(\mathrm {Mg}^{-2}),\\&\quad \text {and}\quad H(T,\varphi ,\mathrm {Mg})=H_{(0)}(T,\varphi )+\mathrm {Mg}^{-1}H_{(1)}(T,\varphi )+{\mathcal {O}}(\mathrm {Mg}^{-2}). \end{aligned} \end{aligned}$$

(72)

Substituting this result into Eq. (28) and Taylor expanding the logarithm, we obtain

$$\begin{aligned} \mathrm {Mg}^{-1}\frac{\partial U}{\partial R}\bigg |_{R=H}=\frac{\partial }{\partial \varphi }\left( \log (1-\Gamma _{(0)})-\mathrm {Mg}^{-1}\dfrac{\Gamma _{(1)}}{1-\Gamma _{(0)}}\right) +{\mathcal {O}}(\mathrm {Mg}^{-2}). \end{aligned}$$

(73)

The leading order term in this equation implies that $\Gamma _{(0)}$ is independent of $\varphi$. In fact, surfactant conservation, combined with our assumption of insolubility, additionally implies that $\Gamma _{(0)}$ is constant in time, so $\Gamma _{(0)}={\bar{\Gamma }}$, where ${\bar{\Gamma }}$ is the average surfactant concentration on the film.

The ${\mathcal {O}}(\mathrm {Mg}^{-1})$ terms in Eq. (73) can be substituted into the radial derivative of Eq. (25), to obtain

$$\begin{aligned} \dfrac{1}{1-{\bar{\Gamma }}}\frac{\partial \Gamma _1}{\partial \varphi }=H_{(0)}\left( \frac{\partial P}{\partial \varphi }-\mathrm {Bo}\sin \varphi \right) . \end{aligned}$$

(74)

Substituting Eq. (72) into Eq. (14), we find that

$$\begin{aligned} \Sigma =1+\mathrm {El}\log (1-{\bar{\Gamma }})+\dfrac{\mathrm {El}\Gamma _1(T,\varphi )}{\mathrm {Mg}(1-{\bar{\Gamma }})}+{\mathcal {O}}(\mathrm {Mg}^{-2})=\Sigma _{\mathrm {eq}}+{\mathcal {O}}(\mathrm {Mg}^{-1}) \end{aligned}$$

where $\Sigma _{\mathrm {eq}}=1+\mathrm {El}\log (1-{\bar{\Gamma }})$ is the equilibrium surface tension of the film. Thus, we can substitute Eq. (19) into Eq. (74), to find

$$\begin{aligned} \frac{\partial \Gamma _{(1)}}{\partial \varphi }=-(1-{\bar{\Gamma }})H_{(0)}\left( \Sigma _{\mathrm {eq}}\frac{\partial }{\partial \varphi }\left( \frac{\partial ^2 H_{(0)}}{\partial \varphi ^2}+\frac{\partial H_{(0)}}{\partial \varphi }\cot \varphi +2H_{(0)}\right) +\mathrm {Bo}\sin \varphi \right) . \end{aligned}$$

(75)

Then, the ${\mathcal {O}}(\mathrm {Mg}^{-1})$ terms yield the equation

$$\begin{aligned} \frac{\partial \Gamma _{(1)}}{\partial T}+\dfrac{\Gamma _{(0)}}{\sin \varphi }\frac{\partial }{\partial \varphi }(U_{\mathrm{s}(1)}\sin \varphi )=0. \end{aligned}$$

(76)

By defining a new rescaled Bond number, $\mathrm {Bo}_{\mathrm {eq}}$ (32) in terms of the equilibrium surface tension $\Sigma _{\mathrm {eq}}$ rather than the surfactant-free surface tension, and a new timescale, ${\bar{T}}$, based on the equilibrium surface tension, we obtain an equation similar to Eq. (30). Substituting Eq. (72) into Eq. (13), and taking the leading order terms only, we obtain $U_{\mathrm{s}(0)}=0$. Dropping the subscript (0) in $H_{(0)}$, we have (33). Finally, although our analysis assumes $\mathrm {Mg}={\mathcal {O}}(\varepsilon ^{-2})$, our numerical results indicate that $U_\mathrm{s}$ is negligible even for moderate values of $\mathrm {Mg}$ (see Fig. 10).

Appendix D: Exact root of Eq. (47)

If we define $\xi =1+\cos \varphi _\mathrm{c}$, then Eq. (47) takes the form

$$\begin{aligned} \dfrac{24}{\mathrm {Bo}}=\dfrac{\xi ^3}{2-\xi }\quad \text {or}\quad \xi ^3+\dfrac{24}{\mathrm {Bo}}\xi -\dfrac{48}{\mathrm {Bo}}=0. \end{aligned}$$

This is a depressed cubic with coefficients $24/\mathrm {Bo}$ and $-48/\mathrm {Bo}$. The discriminant of the equation is then

$$\begin{aligned} \Delta =-4\left( \dfrac{24}{\mathrm {Bo}}\right) ^2\left( 27+\dfrac{96}{\mathrm {Bo}}\right) . \end{aligned}$$

For $\mathrm {Bo}>0$, this discriminant is always negative, guaranteeing a single real root. That root then yields

$$\begin{aligned} \cos \varphi _\mathrm{c}=\left( \dfrac{24}{\mathrm {Bo}}\right) ^{1/3}\left( \left( \sqrt{1+\dfrac{8}{9\mathrm {Bo}}}+1\right) ^{1/3}-\left( \sqrt{1+\dfrac{8}{9\mathrm {Bo}}}-1\right) ^{1/3}\right) -1 \end{aligned}$$

(77)

Note that this is confined to the interval $(-1,1)$, and is monotonic.