2 Theoretical derivation

The structure of the transitional film profile solution is shown in figure 1(b). The front of the bubble (region I) moves at the new speed  $U_{2}$ . In the lubrication limit, a film of thickness $h_{2}^{\infty }$ is selected according to (1.1), and a uniform film of this thickness is formed. Since there is no horizontal pressure gradient along the uniform film region, there is no motion in the film relative to the tube wall. In the rear part of the bubble (region IV), the film still has its thickness $h_{1}^{\infty }$ corresponding to the initial speed  $U_{1}$ . The two film regions are separated by a step (region III), which remains stationary in the frame of the tube, as we will show below. Thus the length of region II is determined by how much the front of the bubble has advanced since the change in speed at $t=0$ , which is $L_{II}\approx U_{2}t$ . Finally, there is a thin film region in the back of the bubble that exhibits the characteristic oscillations found by Bretherton (Reference Bretherton1961). However, owing to the incompressibility, the rear of the bubble moves at the same speed as the front, producing a mismatch between the speed $U_{2}$ and the film thickness  $h_{1}^{\infty }$ . As a result, the shape of the film at the back differs from that found for the classical steady-state Bretherton problem.

2.1 Uniform film thickness at steady state

We begin by summarizing Bretherton’s original steady solution, which describes the steady states shown in figure 1(a) and forms the basis for our description during the transition. The bubble surface is separated from the inner tube wall by a uniform thin liquid film of thickness  $h^{\infty }$ . Since the film thickness is much smaller than  $R$ , we can describe the film motion in an $(x,y)$ coordinate system as shown in figure 1(a). In the limit of small capillary numbers, it follows that $|\text{d}h/\text{d}x|\ll 1$ (as we confirm below), and the dynamics of the film are described by the lubrication equation (Eggers & Fontelos Reference Eggers and Fontelos2015)

(2.1) $$\begin{eqnarray}h_{t}+\frac{\unicode[STIX]{x1D6FE}}{3\unicode[STIX]{x1D707}}(h^{3}h_{xxx})_{x}=0,\end{eqnarray}$$

which describes the viscous motion inside the film in response to the gradient of the capillary pressure $-\unicode[STIX]{x1D6FE}h_{xx}$ (here subscripts denote derivatives). Moving with the bubble at the steady-state velocity  $U$ , the solution is of the form $h(x,t)=h(x-Ut)$ . Combining with (2.1) and integrating once over  $x$ , we obtain

(2.2) $$\begin{eqnarray}h^{3}h_{xxx}+3Ca(h^{\infty }-h)=0.\end{eqnarray}$$

Rescaling according to $H(X)=h(x)/h^{\infty }$ and $X=x(3Ca)^{1/3}/h^{\infty }$ , one obtains the similarity equation (or Landau–Levich–Derjaguin–Bretherton (LLDB) equation)

(2.3) $$\begin{eqnarray}H^{3}H_{XXX}+1-H=0,\end{eqnarray}$$

which describes the transition region between the film and the caps at either end.

We begin with the front of the bubble. Starting from the film, which corresponds to the boundary condition $H(-\infty )=1$ , we integrate (2.3) towards $X\rightarrow \infty$ , where the solution matches to a spherical cap of radius  $R$ , set by the tube radius. Integrating numerically, one finds that $H_{XX}(\infty )=0.643$ . The corresponding curvature $h_{xx}=H_{XX}(\infty )(3Ca)^{-2/3}/h^{\infty }$ is thus equal to the curvature of a spherical cap, $1/R$ , which yields Bretherton’s result (1.1), with a typical slope scale $|\text{d}h/\text{d}x|\sim Ca^{1/3}\ll 1$ . The LLDB equation takes a single boundary condition $H(-\infty )=1$ (apart from translations), as a result of which a unit film thickness (1.1) is selected at the front of the bubble.

In contrast, we have to integrate in the opposite direction in the rear of the bubble: starting from the film at $X=\infty$ , where $H=1$ , we now integrate towards $X\rightarrow -\infty$ , the rear spherical cap of radius  $R$ . Here we have used the fact that the film thickness $h^{\infty }$ is constant all along the middle of the bubble. Combining with (1.1), the similarity solution at the back is now determined by two boundary conditions

(2.4a ) $$\begin{eqnarray}\displaystyle & H(\infty )=1, & \displaystyle\end{eqnarray}$$

(2.4b ) $$\begin{eqnarray}\displaystyle & H_{XX}(-\infty )=h^{\infty }(3Ca)^{-2/3}/R=0.643. & \displaystyle\end{eqnarray}$$

Analysis of (2.3) shows that it has a unique solution subject to (2.4). While the profile has a monotonic behaviour at the front, it develops characteristic oscillations at the back.

2.2 Film thickness profile at the transitional state

Now we generalize the steady-state theory in the previous section to the time-dependent motion of a bubble in transition between two steady states, by imposing an instantaneous velocity change from $U_{1}$ to  $U_{2}$ , as sketched in figure 1(a).

2.2.1 The uniform film regions (II and IV)

As shown in the steady problem (§ 2.1), a specific film thickness is selected near the front of the bubble based on (1.1). Thus, at the initial velocity  $U_{1}$ , the front of the bubble lays down a film of thickness $h_{1}^{\infty }=0.643(3Ca_{1})^{2/3}R$ , where $Ca_{1}$ is the capillary number based on  $U_{1}$ . After the bubble velocity is switched to  $U_{2}$ , this film thickness becomes $h_{2}^{\infty }=0.643(3Ca_{2})^{2/3}R$ . We will show in the following subsection that, while the bubble front translates at speed  $U_{2}$ , the step connecting the two uniform film regions is effectively frozen in the laboratory frame. Taking the time of the velocity change to be $t=0$ , the length of region II is $L_{II}\approx U_{2}t$ (see figure 1 b). In the reference frame of the bubble, the step is seen to move towards the rear. Eventually, when the step has moved past the entire bubble, a new steady state is established (figure 1 a). This transitional process takes a period of time approximately $L/U_{2}$ , where $L$ is the length of the bubble.

2.2.2 Dynamics of the step (III)

We aim to find an approximate description for the dynamics of the step, after it has been created by an instantaneous velocity change at $t=0$ . We would like to know whether (i) the location of step changes in time, and whether (ii) there will be significant change in shape over the course of an experiment.

Following similar analyses in Boatto, Kadanoff & Olla (Reference Boatto, Kadanoff and Olla1993), McGraw et al. (Reference McGraw, Salez, Bäumchen, Raphaël and Dalnoki-Veress2012), and Bäumchen et al. (Reference Bäumchen, Benzaquen, Salez, McGraw, Backholm, Fowler, Raphaël and Dalnoki-Veress2013), we try an ansatz in the laboratory frame, in which (2.1) holds:

(2.5a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\hat{H}}(s)=\frac{h(x,t)}{h_{2}^{\infty }}, & \displaystyle\end{eqnarray}$$

(2.5b ) $$\begin{eqnarray}\displaystyle & \displaystyle s=\frac{x+U_{w}t}{f(t)}, & \displaystyle\end{eqnarray}$$

where $U_{w}$ is the travelling wave speed of the step and $f(t)$ describes the relaxation of the step region towards a flat state. As a result, equation (2.1) becomes

(2.6) $$\begin{eqnarray}\frac{3\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D6FE}(h_{2}^{\infty })^{3}}(U_{w}-sf_{t})f^{3}{\hat{H}}_{s}+({\hat{H}}^{3}{\hat{H}}_{sss})_{s}=0.\end{eqnarray}$$

Hence, $U_{w}$ must vanish for (2.6) to be consistent, and so the step remains stationary in the frame of the tube. This result is also consistent with the physical intuition that the thin film of fluid tends to be frozen in the lab frame due to the no-slip and shear-free boundary conditions at the tube wall and the bubble surface, respectively. Thus, the step relaxation function $f(t)$ satisfies

(2.7) $$\begin{eqnarray}f^{3}f_{t}=\unicode[STIX]{x1D6FE}(h_{2}^{\infty })^{3}/(3\unicode[STIX]{x1D707}).\end{eqnarray}$$

This means the local film profile relaxes in time according to

(2.8) $$\begin{eqnarray}f(t)=\left(\frac{4\unicode[STIX]{x1D6FE}(h_{2}^{\infty })^{3}}{3\unicode[STIX]{x1D707}}t+w^{4}\right)^{1/4},\end{eqnarray}$$

where $w=f(0)^{1/4}$ represents the initial width of the step, in agreement with Bäumchen et al. (Reference Bäumchen, Benzaquen, Salez, McGraw, Backholm, Fowler, Raphaël and Dalnoki-Veress2013). In addition, we expect the initial step width $w=O(R)$ . We assume that the velocity jump is triggered at $t=0$ , with a transition time $\unicode[STIX]{x0394}t\rightarrow 0$ , and information propagates rapidly in the liquid. Therefore, the curvature of the front spherical cap can adjust to the pressure signal almost instantaneously, but bubble translation is negligible. Thus, we can relate the fluid volume in the step region to the fluid volume entering/exiting the film region due to the sudden curvature change in the front spherical cap:

(2.9) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x03C0}w(R-h_{1}^{\infty })^{2}-{\textstyle \frac{1}{3}}\unicode[STIX]{x03C0}w[(R-h_{1}^{\infty })^{2}+(R-h_{2}^{\infty })^{2}+(R-h_{1}^{\infty })(R-h_{2}^{\infty })]\nonumber\\ \displaystyle & & \displaystyle \quad \approx {\textstyle \frac{2}{3}}\unicode[STIX]{x03C0}[(R-h_{1}^{\infty })^{3}-(R-h_{2}^{\infty })^{3}],\end{eqnarray}$$

where we approximate the step region of the bubble as a cone frustum on the left-hand side, and the front spherical cap as a hemisphere on the right-hand side, respectively. Assuming that $h_{1}^{\infty }$ , $h_{2}^{\infty }\ll R$ , the volume balance (2.9) simplifies to $w/R\approx 3/4=O(1)$ . This result is further confirmed in the experiments and simulations below.

Figure 2.

Similarity solution of the step-like region III. Results are obtained by solving (2.10) numerically for different values of $r=h_{1}^{\infty }/h_{2}^{\infty }=[0.2,0.5,1,1.5,2]$ .

With $s$ defined by (2.5b ), the similarity equation becomes

(2.10) $$\begin{eqnarray}({\hat{H}}^{3}{\hat{H}}_{sss})_{s}-s{\hat{H}}_{s}=0,\end{eqnarray}$$

with the boundary conditions ${\hat{H}}(\infty )=1$ , ${\hat{H}}(-\infty )=h_{1}^{\infty }/h_{2}^{\infty }\equiv r$ and ${\hat{H}}_{s}(\pm \infty )=0$ . The results corresponding to various values of $r$ are displayed in figure 2. Similar similarity profiles in thin films have been observed in the literature (e.g. McGraw et al. Reference McGraw, Salez, Bäumchen, Raphaël and Dalnoki-Veress2012; Bäumchen et al. Reference Bäumchen, Benzaquen, Salez, McGraw, Backholm, Fowler, Raphaël and Dalnoki-Veress2013), where the solutions develop small overshoots owing to the fourth-order structure of the equation.

It should be noticed that, based on the results above, the step relaxation in the transition region III is extremely slow. While the characteristic time for the translational dynamics is $\unicode[STIX]{x1D70F}_{trans}\sim L/U_{2}$ , the step relaxation time $\unicode[STIX]{x1D70F}_{relax}$ can be estimated using (2.8), $\unicode[STIX]{x1D70F}_{relax}\sim (\unicode[STIX]{x1D707}R/\unicode[STIX]{x1D6FE})(R/h_{2}^{\infty })^{3}$ , where the initial step width $w$ is approximated by the tube radius  $R$ . We expect to observe relaxation of the step-like profile when $\unicode[STIX]{x1D70F}_{trans}\gg \unicode[STIX]{x1D70F}_{relax}$ , which in the lubrication limit requires extremely long bubbles, $L/R\gg Ca_{2}^{-1}$ . Relaxation of the step is thus difficult to observe during a physical experiment, and the time duration over which a bubble transits from one steady state to another is mainly governed by the translational time scale $\unicode[STIX]{x1D70F}_{trans}$ . As a result, once the step with initial width $w$ is formed, the film profile $h(x,t)$ in region III translates at velocity $-U_{2}$ relative to the bubble surface, similar to a propagating shock.

2.2.3 The front and rear similarity solutions (I and V)

Given the existence of a uniform film in region IV, we now describe how Bretherton’s theory is modified near the back of the bubble. On one hand, the entire bubble moves at the new speed  $U_{2}$ . On the other hand, as discussed before, the film thickness $h_{1}^{\infty }$ is still associated with the initial speed  $U_{1}$ . Thus, in the reference frame of the bubble, the thin film equation is

(2.11) $$\begin{eqnarray}h^{3}h_{xxx}+Ca_{2}(h_{1}^{\infty }-h)=0.\end{eqnarray}$$

Next, we rescale to the LLDB equation (2.3) with the transformation $\tilde{H}=h/h_{1}^{\infty }$ and $\tilde{X}=x(3Ca_{2})^{1/3}/h_{1}^{\infty }$ . As in the Bretherton problem, requiring the curvature of the film solution to match the curvature $1/R$ of the rear cap, we obtain the boundary conditions

(2.12a ) $$\begin{eqnarray}\displaystyle & \tilde{H}(\infty )=1, & \displaystyle\end{eqnarray}$$

(2.12b ) $$\begin{eqnarray}\displaystyle & \tilde{H}_{\tilde{X}\tilde{X}}(-\infty )=h_{1}^{\infty }(3Ca_{2})^{-2/3}/R. & \displaystyle\end{eqnarray}$$

Combining with the expression (1.1) for $h_{1}^{\infty }$ , the dimensionless curvature for the rear spherical cap (2.12b ) is simplified to

(2.13) $$\begin{eqnarray}\tilde{H}_{\tilde{X}\tilde{X}}(-\infty )=0.643(Ca_{1}/Ca_{2})^{2/3},\end{eqnarray}$$

which reduces to the standard boundary condition (2.4b ) when $Ca_{1}=Ca_{2}$ .

Our numerical results for the similarity solutions in the front and the rear of the bubble are summarized in figure 3. By construction, the rescaled curvature of the front solution always approaches the universal value of 0.643 (figure 3 a). As a result, the front similarity solution is the universal solution of Bretherton. By contrast, according to (2.13) the rescaled curvature at the rear depends on the ratio of capillary numbers, producing a family of different solutions (figure 3 a). With the rear spherical cap curvature varying with the ratio $Ca_{1}/Ca_{2}$ , a small alternation to the pressure drop across the bubble rear interface may occur during the transient process. Furthermore, different film oscillations also appear on the rear air–liquid interface with various $Ca_{1}/Ca_{2}$ ratios (figure 3 b).

Figure 3.

Front and rear similarity solutions. (a) The dimensionless curvatures of the similarity solution. Unlike region I, whose solution agrees with Bretherton (Reference Bretherton1961) with a dimensionless curvature $\tilde{H}_{\tilde{X}\tilde{X}}(\infty )=0.643$ , a family of solutions and dimensionless curvatures $\tilde{H}_{\tilde{X}\tilde{X}}(-\infty )$ are obtained at the rear cap during the transition, depending on the ratio $Ca_{1}/Ca_{2}$ . (b) Different film undulations $\tilde{H}(\tilde{X})$ at the rear cap can be obtained as a result of the family of solutions.

3 Experimental setup and numerical methods

Experiments are performed in a refractive index matching setup (Yu et al. Reference Yu, Khodaparast and Stone2017, Reference Yu, Khodaparast and Stone2018), where a $5\times$ objective (Mitutoyo) is used in the imaging apparatus. Pure glycerol is used as the continuous phase, with viscosity $\unicode[STIX]{x1D707}=1.00$  Pa s (Anton Paar, Physica MCG 301) and surface tension $\unicode[STIX]{x1D6FE}=65.4~\text{mN}~\text{m}^{-1}$ (pendant drop method), respectively. A flexible tube connects the inlet of the glass capillary ( $R=566~\unicode[STIX]{x03BC}\text{m}$ ) to a glycerol reservoir, whose pressure is adjusted and controlled by an Elveflow^® OB1 MK3 pressure and vacuum controller, with a settling time as small as 40 ms (Elveflow 2009). The pressure and flow rate are linearly correlated based on calibration. The flexible tube is partially unfilled before being inserted in the glass capillary. A single bubble is formed in the glass capillary from the air column when a positive pressure $p_{1}$ is turned on. The bubble then translates at velocity  $U_{1}$ . The pressure $p_{2}$ is switched on with a step signal when the entire bubble enters the region of interest, which triggers the transition to a new steady state at  $U_{2}$ . Meanwhile, the transitional dynamics of the bubble are recorded by the imaging apparatus at a film rate of 30 f.p.s.

Image processing is performed for each experimental image sequence, where a Matlab program is written to identify the boundaries of the bubble and the inner tube wall, with the error controlled within $\pm 1$  pixel, or $\pm 2.44~\unicode[STIX]{x03BC}\text{m}$ . The film profile $h(x,t)$ is obtained from the difference between the position of the bubble surface and the tube wall, and the experimental bubble volume is calculated from a volume integral based on the film profile.

Numerical simulations are carried out using a commercial finite-element solver (COMSOL Multiphysics). The three-dimensional axisymmetric Stokes equations are solved numerically, and the deforming interface of the bubble is represented using the arbitrary Lagrangian–Eulerian (ALE) technique, which facilitates an accurate and sharp capture of the interface evolution. We impose a Poiseuille flow and zero pressure boundary condition at the inlet and outlet, respectively, and apply the Young–Laplace law at the bubble interface. The liquid phase is discretized and solved, assuming a zero gas–liquid viscosity ratio. The pressure on the external side of the interface is imposed by incorporating the constraint of constant bubble volume. This framework was developed by Balestra, Zhu & Gallaire (Reference Balestra, Zhu and Gallaire2018) and has been well validated against the classical theory (Bretherton Reference Bretherton1961). For more details, refer to Balestra et al. (Reference Balestra, Zhu and Gallaire2018) and Hadikhani et al. (Reference Hadikhani, Hashemi, Balestra, Zhu, Modestino, Gallaire and Psaltis2018).

A slight bubble volume variation is observed in experiments when the pressure jump is applied. Therefore, two simulations using different volumes are performed: we use the volume of the initial state at $U_{1}$ to obtain a steady-state bubble profile at $Ca_{1}$ in the first run, and that of the transitional process in the second, where we trigger the transition to $Ca_{2}$ by a sudden change of the underlying flux.

4 Comparison of experiments, numerics, and analysis

The experimental and numerical results of a bubble undergoing the transitional dynamics are shown in figure 4. A typical experimental image series is shown in figure 4(a), with the zoomed-in film profiles shown in figure 4(b). The transitional motion is triggered by a pressure jump from $p_{1}=150$  mbar to $p_{2}=950$  mbar, corresponding to $Ca_{1}=1.96\times 10^{-3}$ and $Ca_{2}=1.65\times 10^{-2}$ , respectively. The capillary numbers are calculated from the bubble speeds, which are measured experimentally by tracking the locations of the bubble nose, determined by the intersection between the tube centreline and the front cap of the bubble surface. Time $t=0$ is taken as the frame right before the pressure jump is applied, and the dynamics are shown in the consecutive frames with $\unicode[STIX]{x0394}t=0.467$  s between each panel. The boundary detection results are also displayed, with the red and blue curves representing the detected bubble surface and the tube wall, respectively. As is shown in figure 4(c), the experimental (red) and numerical (blue) bubble film profiles $h(x,t)$ are plotted in the laboratory frame within a distance $100~\unicode[STIX]{x03BC}\text{m}$ from the tube wall. The direct comparison between experiments and simulations shows excellent agreement. The simulation results in the first run are plotted for $t=0$  s, and those in the second run are used for the corresponding later times. The interior of the profile has the expected structure of two uniform film regions of different thicknesses, connected by a step, which remains stationary in the laboratory frame. The film thicknesses at regions II and IV measured experimentally, $41.5~\unicode[STIX]{x03BC}\text{m}$ and $13.4~\unicode[STIX]{x03BC}\text{m}$ , agree well with the theoretical prediction of $h_{2}^{\infty }$ and $h_{1}^{\infty }$ within the experimental resolution of $2.44~\unicode[STIX]{x03BC}\text{m}$ . In particular, note the difference between the front and rear of the bubble: while at the front the profile increases monotonically from a constant thickness towards the front cap, the behaviour at the back displays oscillations, as seen in the similarity solutions (figure 3).

Figure 4.

Experimental and numerical bubble profiles. (a) Experimental images and edge detection of the bubble profile. A pressure jump is applied at $t=0$ , which triggers the bubble to transit between steady states from $Ca_{1}=1.96\times 10^{-3}$ to $Ca_{2}=1.65\times 10^{-2}$ . The detected bubble surface and the tube wall are shown as red and blue curves, respectively. (b) Zoomed-in edge detection results in the thin film region. (c) Comparison of results in the laboratory frame among experiments (red), numerical simulations (blue) and the scaled similarity solution (2.10) (green). (d) Numerical results of the rescaled initial step width $w/R$ at different capillary number ratios $Ca_{1}/Ca_{2}$ . The red and blue dot symbols correspond to the cases shown in figure 4(c) and figure 5(b), respectively.

The step-like region III is considered in both figure 4(c) and figure 4(d). A rapid curvature adjustment is observed prior to the step generation. In the laboratory frame (figure 4 c), the lower left corner of the step is pinned while the front of the bubble moves forward at speed  $U_{2}$ , laying down the film to form the shape of a step. The step is seen to be stationary and not to change shape over the time scale of the experiment, which is consistent with the estimates of § 2.2.2. Based on the film thickness ratio $r=h_{1}^{\infty }/h_{2}^{\infty }=0.32$ , we computed the similarity solution for the step relaxation, which describes the relaxation of the step at long times, much longer than the duration of our experiment. Using the initial width $w$ in (2.8) as an adjustable parameter, we observe that the shape of the long-time similarity solution (green) fortuitously agrees very well with the initial step (red and blue) created by the sudden change in bubble speed. We find the initial step width to be $w=c_{1}R$ , with a fitting parameter $c_{1}=2$ . Furthermore, we repeated the numerical simulation with $Ca_{2}$ ranging from $8.6\times 10^{-4}$ to $2.0\times 10^{-2}$ , and measured the step width $w$ as the length over which the film thickness deviates from both $h_{1}^{\infty }$ and $h_{2}^{\infty }$ by at least 3 %. As is shown in figure 4(d), the rescaled step width $w/R=O(1)$ over different capillary number ratios $Ca_{1}/Ca_{2}$ , which validates the order of magnitude estimate for the step width  $w$ .

Figure 5.

Direct comparison for the front and rear spherical caps between the numerical simulation results (blue), and theoretical results both from § 2.2 (green) and Bretherton (Reference Bretherton1961) (red). The bubble is undergoing transition from one speed to another, and the profile at simulation time $t=0.88$ is shown. (b) The results are rescaled by the tube diameter $2R$ and plotted from the tube wall to the tube centreline. (a) Zoomed-in view near the rear spherical cap, where the results from the simulation and time-dependent theory agree, but both deviate from the classic steady-state theory at  $Ca_{1}$ . (c) Enlarged region near the front spherical cap, where the simulation results agree with the classic steady-state theory at  $Ca_{2}$ .

To perform a quantitative test of the theory developed in § 2.2 about the front and rear caps I and V, we performed numerical simulations of a bubble at much smaller capillary numbers, for the transition from $Ca_{1}=2.95\times 10^{-4}$ to $Ca_{2}=8.57\times 10^{-4}$ , as shown in figure 5. The profile of the entire bubble, obtained from the numerical simulation, is plotted in figure 5(b). In order to compare with the theory, the zoomed-in views of the rear and front similarity regions are displayed in panels (a) and (c), respectively. Starting in the front (figure 5 c), excellent agreement is shown between the numerical simulation (blue) and the theoretical similarity solution at $Ca_{2}$ (red), rescaled according to § 2.1. As explained before, the bubble front profile is identical to that of Bretherton’s steady-state theory. The rear region is shown in figure 5(a), comparing the numerical simulation (blue) to the time-dependent theory (green) with $Ca_{1}/Ca_{2}=0.35$ , where the results agree so well as to be nearly indistinguishable. The time-dependent similarity solution is rescaled based on § 2.2.3, and is one of the solutions shown in figure 3. For comparison, as the red curve, we also show the shape one would have obtained from the steady theory. This demonstrates that the unsteadiness introduces a significant change in the shape of the rear film.

5 Concluding remarks

Theoretical predictions are given for the time-dependent film profile of a confined bubble transitioning between two steady states, triggered by an instantaneous velocity change. The theory is validated with experiments and numerical simulations. Several film regions can be categorized during the transition, as shown in figure 1(b). The film profile near the front cap (I) and the thickness of the front uniform film region (II) are identical to those at the second steady state, with the length of region II extending in time. The rear uniform film region (IV) is of constant thickness $h_{1}^{\infty }$ , identical to that at the initial steady state. The two uniform film regions II and IV are connected by a step-like film region (III), which is stationary in the laboratory frame. A similarity solution is obtained for the film profile at region III. However, for the range of capillary numbers studied, the step relaxation is too slow to be observed in the experiments and simulations. During the transition, the rear film solution (V) is modified according to the ratio $Ca_{1}/Ca_{2}$ . This modification not only provides a family of solutions near the rear spherical cap (figure 3), but it also modifies the undulations at the rear transitional region near the spherical cap (figure 5). It has not escaped our notice that the present unsteady theory can be generalized to continuously varying bubble speeds, as well as continuously varying tube cross-sections. A fuller account of these cases is currently under preparation.