2.1.1. The static cap

In the outer region, the fluid around the cap of the bubble can be considered at rest (Bretherton Reference Bretherton1961; Lamstaes & Eggers Reference Lamstaes and Eggers2017) so that in the cylindrical reference frame co-rotating with the tube and translating with the bubble, the pressure $P$ in the surrounding fluid satisfies

(2.2)\begin{equation} \boldsymbol{0}={-}\boldsymbol{\nabla} P +\rho \omega^2 r\boldsymbol{e}_r-\rho g \boldsymbol{e}_z. \end{equation}

By integrating the radial component of (2.2), we obtain

(2.3)\begin{equation} \left.\begin{array}{c} \displaystyle P(r,z)=\dfrac{1}{2}\rho \omega^2(r^2-r_1(z)^2)+\gamma \kappa +P_{air},\\ \displaystyle \kappa={-}\dfrac{1}{r_1(z)(1+r'_1(z)^2)^{1/2}}+\dfrac{r''_1(z)}{(1+r'_1(z)^2)^{3/2}}, \end{array}\right\} \end{equation}

where $r_1(z)$ and $\kappa$ denote the location of the air–liquid interface measured from the central axis (oriented upwards) and its curvature, respectively.

From the axial component of the momentum conservation equation (2.2), it follows that

(2.4)\begin{equation} \gamma \kappa-\tfrac{1}{2} \rho \omega^2r_1(z)^2+\rho g z =\text{cst}. \end{equation}

By denoting as $s$ the arc-length of the interface profile measured from the tip of the bubble and $\theta$ its tangent angle with respect to the horizontal (see figure 2), the static interface profile is given by

(2.5)\begin{equation} -\gamma \left[\frac{{\rm d} \theta}{{\rm d}s} +\frac{\sin\theta}{r_1(s)}\right]-\frac{1}{2} \rho \omega^2r_1(s)^2+\rho g z(s)=\text{cst}, \end{equation}

where the coordinates $(r_1(s), z(s))$ locate the position of the gas–liquid interface at $s$ in the $(r, z)$ plane. Using that ${\textrm {d}r_1}/{\textrm {d}s} = \cos (\theta )$ and ${\textrm {d}z}/{\textrm {d}s} = -\sin (\theta )$, differentiating with respect to the curvilinear coordinate $s$ gives

(2.6)\begin{equation} \gamma \left[\frac{{\rm d}^2\theta}{{\rm d}s^2}+\frac{\cos\theta}{r_1(s)}\frac{{\rm d}\theta}{{\rm d}s}-\frac{\cos\theta\sin\theta}{r_1(s)^2}\right]={-}\rho \omega^2 r_1(s) \cos\theta-\rho g\sin\theta. \end{equation}

Figure 2. Sketch of a bubble in a vertical tube that rotates around its central axis with angular velocity $\omega$. Here $s$ is the arc-length of the interface measured from the tip of the bubble at $r=0$. In the static cap region, the air–liquid interface is located at a distance $r_1(s)$ from the central axis, and the angle its tangent makes with the horizontal axis is denoted as $\theta (s)$. In the inner region, where a two-dimensional Cartesian system $(x,y)$ is used, the interface is located instead by its distance from the solid wall $y_1(x)$.

Finally, with the dimensionless variables $\bar {r}_1=r_1/R$ and $\bar {s}=s/R$, (2.6) becomes

(2.7)\begin{equation} \frac{{\rm d}^2\theta}{{\rm d}\bar{s}^2}+\frac{\cos(\theta)}{\bar{r}_1}\frac{{\rm d}\theta}{{\rm d}\bar{s}}-\frac{\cos(\theta)\sin(\theta)}{\bar{r}_1^2}={-}Bo\sin(\theta)-\bar{r}_1\,Ce\cos(\theta), \end{equation}

where the Bond number $Bo={\rho g R^2}/{\gamma }=(R/\ell _c)^2$ is introduced as the square of the ratio between the tube radius and the capillary length. The centrifugal number $Ce={\rho \omega ^2 R^3}/{\gamma }$ can be seen as a rotational Bond number where the centrifugal acceleration $R\omega ^2$ plays the role of the gravitational acceleration.

For a given set of parameters $(Bo, Ce)$, two boundary conditions are required to integrate (2.7) from the position $(r_1(s=0)=0, z(s=0)=0)$. A first condition is provided by the symmetry of the problem, which imposes $\theta (0)=0$ at the top of the static cap. The second boundary condition will be determined upon matching of this static profile with that of the inner region.

2.1.2. The thin-film region

In the inner region where the bubble is surrounded by a thin lubricating film, the film's thickness is extremely small compared with the tube's radius. Following Bretherton (Reference Bretherton1961), we thus neglect the azimuthal curvature of the air–liquid interface and consider the thin-film region as planar instead of annular.

Under these assumptions, we introduce the two-dimensional, stationary, Cartesian coordinate system $(x,y)$, where $x=z-U_b t$ opposes gravity, and $y=R-r$ represents the distance to the solid wall. In the framework of the lubrication approximation, the viscous flow in the thin film is driven by a pressure gradient resulting from a combination of gravity, capillarity and centrifugal force. The axial velocity accordingly is written as (see Appendix A.1 for a detailed derivation)

(2.8)\begin{equation} u(x,y)= \frac{\gamma}{2 \mu} \left({-}y_1''' +\frac{\rho \omega^2 R}{\gamma} y_1' +\frac{\rho g}{\gamma}\right)(y^2-2y_1y)-U_b, \end{equation}

where $y_1(x)$ denotes the distance of the air–liquid interface to the solid wall of the tube and a prime denotes the derivative with respect to $x$. In (2.8), the first term of the right-hand side stems from surface tension effects, the second from the centrifugal force and the third from gravity. Upon integration within the thin film, the volume flux reads

(2.9)\begin{equation} Q={-}2{\rm \pi} RU_by_1 -2{\rm \pi} R \frac{\gamma}{3 \mu} \left({-}y_1''' +\frac{\rho \omega^2 R}{\gamma} y_1' +\frac{\rho g}{\gamma}\right)y_1^3. \end{equation}

This flux must equate the volume of fluid displaced per unit time by the top of the bubble, which is equal to ${\rm \pi} R^2 U_b$. Since $y_1/R \ll 1$, the $-2{\rm \pi} RU_by_1$ term in the expression of the flow rate is a negligible correction. Finally, by imposing flux continuity with the region far away from the tip, where the film thickness can be considered as uniform and equal to a constant $b$, we obtain the following thin-film equation:

(2.10)\begin{equation} y_1'''=\frac{\rho g}{\gamma}\left(1-\frac{b^3}{y_1^3}\right)+\frac{\rho \omega^2R}{\gamma}y_1'. \end{equation}

Since $b$ is the length scale governing the flow in the inner region, we non-dimensionalize, as in Bretherton (Reference Bretherton1961), with

(2.11a,b)\begin{equation} y_1=\eta b,\quad x=\zeta b(\rho g b^2/\gamma)^{{-}1/3}. \end{equation}

This leads to the following ordinary differential equation:

(2.12)\begin{equation} \eta'''=\frac{\eta^3-1}{\eta^3}+a\eta', \end{equation}

where $a=({Ce}/{Bo^{2/3}})({b}/{R})^{2/3}$. In (2.12), the left-hand side represents the surface tension term, while on the right-hand side, the first term accounts for gravity, the second for viscous dissipation and the third for the effect of centrifugation.

2.1.3. Matching

We aim at matching the inner solution, which is described by (2.12), with the static cap solution provided by (2.6). Given that $b\ll R$, this requires taking the limit $\eta \rightarrow \infty$ in (2.12) for the inner solution. In this limit, the equation behaves as

(2.13)\begin{equation} \eta'''=1+a\eta', \end{equation}

whose general solution reads

(2.14)\begin{equation} \eta = c_1 \,{\rm e}^{\sqrt{a}\zeta}+c_2\,{\rm e}^{-\sqrt{a}\zeta}+c_3-\frac{\zeta}{a}. \end{equation}

The values of $c_1$, $c_2$ and $c_3$ can be found through interpolation using the numerical solution of the complete equation (2.12), whose initial conditions are obtained from the uniform film solution. Indeed, when $\eta \rightarrow 1$, (2.12) becomes

(2.15)\begin{equation} \eta'''=3(\eta-1)+a\eta', \end{equation}

for which the only non-oscillating solution is $\eta _{0}=1 + \mathcal {C}\exp (\mathcal {F}\zeta )$, where $\mathcal {C}$ is an integration constant and

(2.16)\begin{equation} \mathcal{F}=\frac{\left(\dfrac{2}{3}\right)^{1/3}a}{(27+\sqrt{3}\sqrt{243-4a^3})^{1/3}} +\frac{(27+\sqrt{3}\sqrt{243-4a^3})^{1/3}}{2^{1/3}\,3^{2/3}}. \end{equation}

Since the value of $\mathcal {C}$ can be adjusted by shifting the origin of $\zeta$, we can set $\mathcal {C}=1$ and a large, negative initial value $\zeta _0$ to initialize the integration. This procedure yields initial conditions for the full nonlinear equation (2.12), i.e.

(2.17a–c)\begin{equation} \eta(\zeta_0)=1+\exp(\mathcal{F}\zeta_0), \quad \eta'(\zeta_0)=\mathcal{F}\exp(\mathcal{F}\zeta_0), \quad \eta''(\zeta_0)=\mathcal{F}^2\exp(\mathcal{F}\zeta_0). \end{equation}

We solve (2.12) (using the built-in Matlab ODE solver ode45) to obtain the inner solution $\eta$ for various values of $a$, and fit the outer profile $\eta = c_1 \,\textrm {e}^{\sqrt {a}\zeta }+c_2\,\textrm {e}^{-\sqrt {a}\zeta }+c_3-{\zeta }/{a}$ in the region where $\eta \gg 1$. This allows us to retrieve the coefficients $c_1(a)$, $c_2(a)$ and $c_3(a)$. From figure 3(a), it is evident that the outer profile $\eta$ of the inner solution exhibits an inflection point. We thus translate the origin to the position where $\eta '' =0$, located at the coordinate

(2.18)\begin{equation} \zeta^*=\frac{1}{2\sqrt{a}}\log\left(-\frac{c_2}{c_1}\right). \end{equation}

Using the shifted variable $\chi =\zeta -\zeta ^*$, we can now define $\eta (\chi )=c_1^*\,\textrm {e}^{\sqrt {a}\chi }+c_2^*\,\textrm {e}^{-\sqrt {a}\chi }+c_3^*-{\chi }/{a}$, where the new coefficients $c_1^*$, $c_2^*$ and $c_3^*$ are expressed as

(2.19a)$$\begin{gather} c_1^*=c_1\,{\rm e}^{\sqrt{a}\zeta^*}=c_1\sqrt{-\frac{c_2}{c_1}}=\text{sgn}(c_1)\sqrt{-c_1c_2}, \end{gather}$$

(2.19b)$$\begin{gather}c_2^*=c_2\,{\rm e}^{-\sqrt{a}\zeta^*}=c_2\sqrt{-\frac{c_1}{c_2}}=\text{sgn}(c_2)\sqrt{-c_1c_2}, \end{gather}$$

(2.19c)$$\begin{gather}c_3^*=c_3-\frac{\zeta^*}{a}=c_3-\frac{1}{2a^{3/2}}\log\left(-\frac{c_2}{c_1}\right)=\eta(\chi=0). \end{gather}$$

Figure 3. (a) The inner region profile $\eta$ as a function of the dimensionless height $\zeta$. The circles represent the solution $\eta$ of the full equation (2.12) for various values of $a$, while the black solid lines represent the outer profiles of the inner region $\eta = c_1 \,\textrm {e}^{\sqrt {a}\zeta }+c_2\,\textrm {e}^{-\sqrt {a}\zeta }+c_3-{\zeta }/{a}$, where $c_1(a)$, $c_2(a)$ and $c_3(a)$ are obtained by fitting with the full inner solution, in the $\eta \gg 1$ region. The outer profiles clearly exhibit an inflection point, at a distance referred to as $\zeta ^*(a)$. For each value of $a$, the origin is then shifted so that $\eta ''(0)=0$. (b) Shifted coefficient $c_1^*=-c_2^*$ as a function of $a$ (red circles). The black solid line corresponds to $c_1^*=0.500 a^{-3/2}+0.286 a^{-1/2}$. Inset: shifted coefficient $c_3^*=\eta (0)\equiv \eta ^*$ as a function of $a$. This coefficient does not vary significantly with $a$.

The new coefficients are well fitted (see figure 3b) by

(2.20a)$$\begin{gather} c_1^*={-}c_2^* \approx 0.500 a^{{-}3/2}+0.286 a^{{-}1/2}, \end{gather}$$

(2.20b)$$\begin{gather}c_3^*\approx 1.10, \text{ independently of the value of } a. \end{gather}$$

Thus, the distance from the wall at which the outer profile exhibits an inflection point can be evaluated as

(2.21)\begin{equation} y_1(0)=\eta(0)b=(c_1^*+c_2^*+c_3^*)b \approx 1.10b \ll R. \end{equation}

Furthermore, the slope of the profile at the inflection point is given by

(2.22)\begin{align} y_1'(0)&=\eta'(0)(\rho g b^2/\gamma)^{1/3}=\left(c_1^* \sqrt{a}-c_2^*\sqrt{a}-\frac{1}{a} \right)(\rho g b^2/\gamma)^{1/3}\nonumber\\ &=0.572(\rho g b^2/\gamma)^{1/3}>0. \end{align}

Remarkably, the conditions on film thickness (2.21) and slope (2.22) at the inflection point are the same as described in Bretherton (Reference Bretherton1961). Therefore, the centrifugal force alters the inner region solution and the static cap profile, but the matching conditions (and thus the boundary conditions for the static cap solution) remain surprisingly unchanged from those of Bretherton (Reference Bretherton1961).

The above analysis provides the missing information required to solve the static cap profile described by (2.7). Specifically, the static profile for a given set of parameters $(Bo, Ce)$ is obtained through a shooting method, searching for the first derivative $\dot \theta _0$ at the tip of the cap, which is such that the integration of (2.7) from initial conditions $(\theta _0=0, \dot \theta _0)$ results in a profile where the inflection point $\ddot r_1(z)=0$ is reached for $r_1(z)=R$. The numerical integration of (2.7) is performed using the Matlab built-in ODE solver ode23t, with a spacing along the curvilinear coordinate of $0.01R$, while the shooting method is implemented by means of the nonlinear Matlab system solver fsolve. The resulting slope $r'_1(z)\vert _{r=R}$ at the inflection point is then computed from the generated profile.

For fixed $Bo< Bo_{c,0}=(R_c/\ell _c)^2$ and varying $Ce$, it appears that some profiles are unphysical: below a critical value $Ce_c$ that depends on the Bond number $Bo$, the static cap shape exhibits a positive slope at the inflection point at the solid wall, causing the upper profile to extend beyond the fluid domain $r< R$, as illustrated in figure 4, which is not compatible with the matching condition $0< y_1'(0)=-r'_1(z)\vert _{r=R}$. By progressively increasing the centrifugal number $Ce$, the slope at the inflection point at the wall decreases, leading to a reduction of the bulge outside the fluid domain, as shown in figure 4(c). Ultimately, for $Ce>Ce_c$, the slope becomes negative, causing the entire static cap to reside within the fluid domain; see figure 4(b,c).

Figure 4. Static cap profile for a Bond number $Bo=0.55$ and a centrifugal number $Ce$ that is (a) below and (b) above the threshold $Ce_c(Bo)$. Below the threshold, the slope $r'_1(z)\vert _{r=R}$ is positive at the inflection point, causing the upper profile to escape the fluid domain $r< R$. Angle $\phi$ is the angle between the tangent to the static cap profile at the wall and the vertical axis: $\phi =\tan ^{-1}(-r'_1(z)\vert _{r=R})$ and is thus negative in (a) and positive in (b). (c) Evolution of the static cap profile at fixed $Bo=0.55$, when increasing $Ce$ from a value below the threshold $Ce_c\approx 1$ to a value slightly above threshold.

In the following, we denote as $\phi$ the resulting angle between the liquid–air interface and the vertical axis at the inflection point, i.e. $\phi =\tan ^{-1}(-r'_1(z)\vert _{r=R})=\tan ^{-1}(y_1'(0))$. A closer inspection reveals that within a small range around $Ce_c$, i.e. for $\vert Ce-Ce_c(Bo)\vert <0.2$, $\phi$ varies linearly with $Ce$, as shown in figure 5(a). Within this range, the angle $\phi$ (in radians) is well approximated by

(2.23)\begin{equation} \phi(Ce,Bo)\approx 0.144(Ce-Ce_c(Bo)), \end{equation}

where the factor 0.144 is independent of $Bo$ (up to variations less than 0.001 radians). Interestingly, this prediction is consistent with the occlusion criterion derived by Manning et al. (Reference Manning, Collicott and Finn2011) under weightlessness (i.e. $Bo=0$), reported in the introduction of this section (equation (2.1)). Indeed, $\phi$ represents the slope of the gas–liquid interface at the inflection point, located at a distance $1.10b$ from the solid wall of the tube (equation (2.21)). In this sense, it can be seen as the contact angle of a meniscus occluding a virtual channel of radius $(R-1.10b)$, rotating at angular velocity $\omega$. Thus, in the absence of gravity, and in the limit of small contact angle $\phi$, the occlusion criterion (2.1) becomes

(2.24)\begin{equation} Ce(\phi, Bo=0)=\frac{\rho\omega^2 (R-1.10b)^3}{\gamma}=32\sin^3\left(\frac{{\rm \pi} + 2\phi}{6}\right) \approx 4(1+\sqrt{3} \phi), \end{equation}

which can be recast as $\phi (Ce, Bo=0)\approx 0.144 (Ce-Ce_{c,0})$, where $Ce_{c,0} \equiv Ce_c(Bo=0)=4$.

Figure 5. (a) Angle $\phi$ of the static cap profile between the vertical axis and the tangent to the static cap profile (obtained by integrating (2.7) while requiring that the interface reaches the solid wall with an inflection point) as a function of the centrifugal number $Ce$, for various Bond numbers $Bo$. The circles are the values computed from the shape of the interface, while the solid lines are the best linear fit. For each $Bo$, the critical centrifugal number $Ce_c$ is defined as the value of $Ce$ for which the best linear fit cancels out. (b) The critical centrifugal number $Ce_c$ as a function of the Bond number $Bo$. At a given value of $Bo$, the matching with the inner region is only possible if $Ce > Ce_c(Bo)$. The circles are the values of $Ce$ that cancel the linear approximation of $\phi (Ce, Bo)$ for each Bond number, while the back line represents the approximation (2.25). The red circle with coordinates $(Bo_{c,0}=0.842, Ce_c=0)$ locates the threshold in the absence of centrifugation.

Thus, at fixed $Bo$, the critical centrifugal number $Ce_c(Bo)$ for vanishing angle $\phi$ (or equivalently for vanishing slope) is retrieved as the value of $Ce$ at which the best linear fit of $\phi (Ce, Bo)$ cancels out. Its dependency on the Bond number is depicted in figure 5(b). Note that we performed a convergence analysis and observed no further variations of $Ce_c$, within a tolerance of 0.07 %, when increasing 10 times the resolution on the spacing along the curvilinear coordinate used to integrate the static cap profiles. For $Ce< Ce_c$, the geometrical constraint $\phi = \tan ^{-1}(y_1(0)) \approx y_1'(0) >0$ cannot be satisfied, so that this value corresponds to the threshold for the onset of motion.

A second-order polynomial approximation of $(Ce_c(Bo), Bo)$ is $Bo \approx 0.842-0.295 \, Ce_c(Bo) + 0.020\, Ce_c^2(Bo)$, whose solution, shown in figure 5(b), reads

(2.25)\begin{equation} Ce_c=\frac{0.295-\sqrt{0.295^2-0.080(Bo_{c,0}-Bo)}}{0.040}, \end{equation}

where $Bo_{c,0}=0.842$ is the critical Bond number in the absence of centrifugation ($Ce=0$). Note that in the limit $Bo \rightarrow 0$, the approximation (2.25) yields $Ce_c(Bo=0)\approx 3.9$, which is close to the threshold $Ce_{c,0}=4$ computed by Manning et al. (Reference Manning, Collicott and Finn2011) under weightlessness. Conversely, in the limit $(Ce_c \rightarrow 0$, $Bo\rightarrow Bo_{c,0})$, (2.25) simplifies into

(2.26)\begin{equation} Ce_c\approx \frac{1}{0.295}(Bo_{c,0}-Bo). \end{equation}

By injecting (2.26) into (2.23), we obtain, in the limit $Ce \rightarrow 0$:

(2.27)\begin{equation} \phi \approx 0.49(Bo-(Bo_{c,0}-0.295 Ce)), \end{equation}

reminiscent of the expression derived by Bretherton for a non-rotating capillary tube ($\phi =0.49(Bo-Bo_{c,0})$). The similarity of these expressions highlights the role of the centrifugation as a downward shift in the critical Bond number for the onset of motion.

In addition, this analysis provides a prediction for the rising velocity of the bubble for $Ce > Ce_c(Bo)$. Indeed, for $Ce$ close enough to the threshold $Ce_c(Bo)$, the slope of the inner solution at the inflection point should verify, using (2.21), (2.22) and (2.23):

(2.28)\begin{align} y_1'(0)&=0.572(\rho g b^2/\gamma)^{1/3}\nonumber\\ &=\tan(\phi)\approx \phi \approx 0.144\left(\frac{\rho \omega^2 (R-1.10.b)^3}{\gamma}-Ce_c(Bo)\right). \end{align}

Since the volume of fluid displaced per unit time by the tip of the bubble ${\rm \pi} R^2 U_b$ should be equal to to the volume flux in the uniform film region, we can relate the thickness $b$ to the inner radius $R$ and the velocity $U_b$ through $\rho g b^3/3\mu U_b=R/2$. Expanding (2.28) up to first order in $(b/R)$ yields the following expression of $Ca$ as an implicit function of $Ce$ and $Bo$:

(2.29)\begin{equation} Ce-Ce_{c}=3.78 Ce\left(\frac{Ca}{Bo}\right)^{1/3}+4.35 Bo^{1/3}\left(\frac{Ca}{Bo}\right)^{2/9}, \end{equation}

where $Ce_c$ is the function of $Bo$ described above. The ratio $Ca/Bo$ compares the bubble velocity $U_b$ with the settling velocity $U^*=\rho g R^2/\mu$. Note that (2.29) admits an analytical solution: by setting the unknown $x=(Ca/Bo)^{1/9}$, (2.29) can indeed be recast as an equation of the type $x^3 + a_1 x^2 + a_2 x + a_3 =0$.

2.2.1. Experimental set-up and procedure

Cylindrical borosilicate capillary tubes (Hilgenberg GmbH) are partially filled with silicone oil (Sigma Aldrich, $\rho =964$ kg m$^{-3}$, $\mu =9.64\times 10^{-2}$ Pa s, $\gamma =2.09\times 10^{-2}\ {\rm N}\ {\rm m}^{-1}$), leaving an air bubble with a length $L$ greater than 10 times the radius $R$ of the tube. Both ends of the tubes are sealed with epoxy resin. The inner radii of the tubes used in our experimental campaign vary between 0.8 and 1.3 mm, corresponding to Bond numbers $Bo\equiv \rho g R^2/\gamma$ in the range $[ 0.29, 0.76]$. Note that the uncertainty in the inner diameters of the tubes is of 0.05 mm.

The tube attachment system, presented in figure 6(a), consists of two circular mounts made of PETG, rigidly connected together via two vertical steel rods, and linked by bearings to a fixed aluminium frame (not represented in the figure). On each mount, a central, threaded circular mouthpiece accommodates a hollow cylinder whose inner radius matches the outer radius of the capillary tube. The extremities of the tube are then inserted into these cylinders, and securely clamped to the mounts using a clamping chunk. By this means, the tube can be easily replaced by a capillary of a different size with minimal adjustments of the set-up. The lower mount is also connected to the shaft of a DC motor that imposes the rotation of the tube attachment system. While the tube, the mounts and the rods rotate collectively, the verticality and stability of the whole set-up are ensured by the fixed aluminium frame.

Figure 6. Experimental set-up and post-processing for rotating bubbles. (a) Tube attachment system. The capillary tube is clamped on both extremities to mounts connected together by two metal rods. The bottom mount is linked to the shaft of a DC voltage-controlled motor that imposes the rotation of the system around its central, vertical axis. (b) Photographs of a long bubble inside a tube filled with silicone oil at different and equally spaced time steps within the transient regime. In the red frame, the motor has been switched on and the upper cap starts rising while the bottom cap remains still. Along with the resulting bubble elongation, the surrounding liquid film gets progressively thicker from the top to the bottom part of the bubble. The dotted line roughly locates the position of the propagation front. Once the front has reached the lower cap, it starts rising. (c) Intensity profile as a function of time along the tube axis. To produce this image, a column of pixels aligned with the central axis of the tube is extracted from each frame of the movie. The columns are then juxtaposed to each other. The locations of the upper and lower cap as a function of time are easily identified as the two roughly parallel black curves limiting a darker domain that corresponds to the position of the bubble itself. At time $t_1$, the motor is switched on. At $t_2$, the bubble dynamics reaches a stationary state: the upper and lower caps rise at same constant velocity, as highlighted by the parallel blue solid lines that are superimposed on the position of the caps as a function of time. For (b,c), $R=1.2$ mm and $Ce=1.97$. The transient duration is approximately equal to $t_2-t_1\approx 130$ s and the capillary number computed from the steady state is $Ca \approx 3.03 \times 10^{-4}$.

The motor is voltage-controlled to achieve the desired rotation speed, measured with less than 1 % error using a tachometer. To enhance visualization, a light-emitting-diode panel is positioned behind the tube attachment system.

A Basler camera records the evolution of the bubble in the tube, and the velocities of the upper and lower caps of the bubble are obtained through image post-processing performed via a custom Matlab script. Specifically, a column of pixels aligned with the tube that crosses the upper and bottom profiles of the bubble is extracted from each frame. These slices are then juxtaposed to each other into an image where the horizontal axis represents time. The displacement of the bubble extremities with time are clearly visible on the resulting image, as shown in figure 6(c).

Once the motor is switched on, a transient regime occurs where the upper cap of the bubble rises while the bottom cap remains immobile, resulting in bubble elongation, reminiscent of spinning bubble experiments (Vonnegut Reference Vonnegut1942). This is accompanied by the progressive thickening of the surrounding film that propagates from the top to the lower cap of the bubble, as seen in figure 6(b). Once the propagation front reaches the bottom extremity, the lower cap starts its ascent at the same (constant) velocity as the upper cap; see figure 6(c). The rising regime is assumed to be stationary if the difference between the caps’ velocities is less than 5 %. The bubble velocity is computed as the mean velocity between the upper and the bottom cap velocities.

For a given inner radius $R$ and a fixed rotational speed $\omega$, both $Bo$ and $Ce$ are fixed. The capillary number $Ca=\mu U_b/\gamma$ is then derived from the measurement of the bubble velocity at steady state $U_b$. For the experiments reported in this section, we specify that the Bond number remains smaller than the threshold $Bo_c=0.842$. Thus, the bubble does not move at all if the rotational speed is zero. To avoid excessively long working time for the motor, we did not operate it more than 8 hours consecutively. Considering that with our experimental set-up we cannot precisely detect a motion smaller than 1 mm between the start and the end of an experiment, the smallest capillary number that is experimentally measurable is $Ca_{min}=1.6 \times 10^{-7}$. A value inferior to this limit will be accordingly set equal to zero. The maximal rotational speed achieved by the DC motor is $\omega _{max}=400$ rad s$^{-1}$. For a given tube inner radius, this sets a limit on the maximal centrifugal number $Ce_{max}$ that is experimentally reachable.

2.2.2. Experimental results and comparison with the theoretical threshold

For comparison with the theoretical threshold for the onset of motion, we present our experimental findings in the $(Bo, Ce)$ diagram featured in figure 7(a). Overall, the theoretical prediction is quantitatively consistent with the experimental results, which reveal a rapid decay in the bubble velocity as the centrifugal number $Ce$ approaches the theoretical threshold $Ce_c(Bo)$. As it was challenging to precisely determine the experimental threshold, we endeavoured to establish a narrow range by identifying the highest $Ce\equiv Ce_{c,exp}$ for which the bubble displacement fell below our detection limit. This lower bound is denoted by red crosses in figure 7(a), and closely aligns with the theoretical threshold. However, the prediction is less precise for the smallest values of $Bo$: the experimental threshold is downward-shifted with respect to the theoretical prediction.

Figure 7. (a) Diagram $(Bo, Ce)$, where each circle corresponds to a measurement of $Ca>0$ for a given set of parameters $(Bo, Ce)$. The red crosses indicate the couples $(Bo, Ce=Ce_{c,exp}(Bo))$ for which the bubble displacement fell below our detection limit. The black circles indicate the theoretical threshold for the onset of motion and the black solid line represents the approximation (2.25) of $Ce_c(Bo)$. Below this line, the grey area indicates the region of parameters where the steady rising of a bubble is not possible according to our theoretical analysis. Finally, the shaded area corresponds to the region of parameters that is not accessible with our set-up, due to the constraints on the maximal angular velocity provided by the motor. (b) Capillary number ${Ca}$ as a function of the centrifugal number $Ce$ measured for various Bond numbers. The circles are the experimental points, and for each Bond number $Bo$, the solid line is the theoretical prediction (2.29), computed using the corresponding experimental value of $Bo$ indicated in the legend. The dotted lines also represent the prediction (2.29), but for $Bo \pm \Delta Bo$, where $\Delta Bo$ accounts for the $\pm$0.05 mm uncertainty on the tube inner diameters. The errorbars represent the measurement uncertainty on the bubble velocity.

Figure 7(b) reports measurements of the bubble velocity as a function of the rotational speed. The trend is satisfyingly captured by (2.29). We note, however, that close to the threshold, the measured velocities are in general higher than predicted, consistent with the downward shift of the experimental threshold mentioned above. We believe that this can be ascribed to horizontal vibrations of the tube attachment system observed while operating the motor. As observed by Kubie (Reference Kubie2000), the rising velocity of a Taylor bubble within a vertical tube is indeed larger when the tube is oscillated in the horizontal plane, and increases with the oscillation acceleration. This hypothesis is backed up with the photographs of rotating bubbles along their ascent, that show some asymmetry of the bubble profile with respect to the tube axis, as can be seen for instance in figure 6(b). This is compatible with the observations of Kubie (Reference Kubie2000) under horizontal oscillations: the relative position of the bubble moves periodically from one side of the tube to the other, which thickens the lubricating film on one or the other side of the bubble, resulting in a more efficient drainage and thus in faster bubble ascent.

Despite these discrepancies, our theoretical analysis seems to provide a good estimation of the threshold for the onset of motion and a satisfying prediction for the general trend of the rising velocity as a function of the rotational speed.

In summary, rotation reduces the critical tube radius for the onset of motion and facilitates bubble ascent. From a theoretical point of view, the most appreciable effect of centrifugation is the modification of the static cap profile, while geometrical constraints stemming from the matching with the thin-film region appear remarkably unchanged from the classical case without rotation. At the same time, experiments demonstrate the thickening of the thin film surrounding the elongated part of the bubble, for increasing rotational speed. This thickening is caused by the centrifugal acceleration which induces a radial, ‘gyrostatic’ pressure gradient that pushes liquid towards the solid wall. We can thus interpret centrifugation as a means to tune the thickness, and thus the flow rate within the gap between the tube wall and the bubble, resulting in the lowering of the critical Bond number for the onset of motion.

An alternative and simple strategy to modify the hydrostatic pressure gradient is to tilt the tube with respect to gravity, whose effect is investigated in the next section.

3.1.1. The three-dimensional static cap

We first introduce the equilibrium equation for the static three-dimensional shape of the upper cap of the bubble. We define a Cartesian coordinate system $(x,y,z)$, where $z$ is the direction aligned with the central axis of the tube. The gravity vector reads $\boldsymbol {g} = (g\cos (\alpha ), 0, -g\sin (\alpha ))$, where $\alpha$ is the tilt angle of the tube ($\alpha =90^\circ$ corresponds to a vertical tube); see figure 8(a).

Figure 8. (a) Sketch of the static cap in a tube tilted with angle $\alpha$ with respect to the horizontal plane. A Cartesian coordinate system $(x,y,z)$ is used, where $z$ is the direction aligned with the central axis of the tube. The height of the liquid–air interface is denoted as $h(x,y)$ and the gas–liquid interface meets the wall with an angle $\phi$, in all radial directions. (b) Sketch of the thin-film region, assumed to be axisymmetric. A two-dimensional Cartesian coordinate system $(\tilde {x},\tilde {y})$ is used, where $\tilde {x}$ is the direction aligned with the central axis of the tube. The distance of the liquid–air interface from the solid wall is denoted by $y_1(\tilde {x})$. (c) Comparison between the solution of (3.1) (red solid line) and the axisymmetric solution of (2.6) with no rotation ($\omega =0$) (pink dotted line), for tilt angle $\alpha =90^\circ$ (vertical tube), $\phi =0.50^\circ$ and Bond number $Bo=0.86$.

The evaluation of the static interface profiles of the upper cap is based on the two-dimensional Young–Laplace equation, where length scales are non-dimensionalized with the tube radius (Manning et al. Reference Manning, Collicott and Finn2011; Rascón, Parry & Aarts Reference Rascón, Parry and Aarts2017):

(3.1)\begin{equation} \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\frac{\boldsymbol{\nabla} \bar{h}}{\sqrt{1+(\boldsymbol{\nabla} \bar{h})^2}}\right)=Bo(\cos(\alpha)\bar{x}-\bar{h}\sin(\alpha)), \end{equation}

where $\bar {h}(\bar {x},\bar {y})$ denotes the height of the static cap; the quantity on the left-hand side is the curvature $\kappa$ of the liquid–gas interface while the term on the right-hand side corresponds to the hydrostatic contribution. Equation (3.1) is complemented with the boundary condition at the solid wall: $({\boldsymbol {\nabla } h}/{\sqrt {1+(\boldsymbol {\nabla } h)^2}})\boldsymbol {\cdot}\boldsymbol {n}=-\cos (\phi )$, where $\boldsymbol {n}$ is the outwards-oriented vector normal to the tube, and the angle $\phi$ is defined similarly to that in the first part of this study, as the angle between $\boldsymbol {e}_z$ and the tangent at the wall to the intersection of the liquid–gas interface with the plane $(\boldsymbol {n}, \boldsymbol {e}_z)$. We thus require the interface to reach the solid wall with a specified slope, which is assumed to be the same for all directions $\boldsymbol {n}$, for consistency with the matching with the thin-film profile, assumed in the following to be radially symmetric. Note that this differs from the first part of this study, where we imposed a vanishing radial curvature at the wall and computed the angle $\phi$ a posteriori. Here, the the slope at the wall is specified as a boundary condition, with no requirement on the curvature. We acknowledge that requiring the gas–liquid interface to reach the wall for all directions $\boldsymbol {n}$ is somehow counterintuitive, given that for small tilt angles $\alpha$ (i.e. for a strongly inclined tube with respect to gravity), we expect the film surrounding the bubble to be thicker in the direction $x>0$, and the liquid–gas interface to be relatively far from the solid wall in this region. However, we focus here on the vicinity of the threshold for the onset of motion, where surface tension is dominant and causes the bubble to expand in the entire fluid domain $\sqrt {x^2+y^2}=r< R$ in all directions, as experimentally observed (see for instance figure 11d). The derivation of (3.1) can be found in Appendix B.

3.1.2. The thin-film region and matching

Here, we opt for a simplified description of the inner region, which is assumed to be radially symmetric. Neglecting the azimuthal curvature allows us to describe the bottom thin film with a two-dimensional, stationary Cartesian reference frame $(\tilde {x}=z-U_bt,\tilde {y})$, where $\boldsymbol {e}_{\tilde {x}}$ is aligned with the tube central axis and points upwards (such that $\boldsymbol {e}_{\tilde {x}}\boldsymbol {\cdot}\boldsymbol {g}=-g\sin (\alpha )$), and $\boldsymbol {e}_{\tilde {y}}$ is the inward vector normal to the inner solid wall, such that $\boldsymbol {e}_{\tilde {y}}\boldsymbol {\cdot}\boldsymbol {g}=-g\cos (\alpha )$; see figure 8(b).

Within the lubrication framework, the viscous flow in the thin film is driven by Laplace and hydrostatic pressure gradients. The axial velocity in the bottom thin film is accordingly written as (see Appendix A.2 for a detailed derivation)

(3.2)\begin{equation} u(\tilde{x},\tilde{y})=\frac{\gamma}{2\mu}\left[{-}y_1'''+\frac{\rho g \cos(\alpha)}{\gamma}y_1'+\frac{\rho g \sin(\alpha)}{\gamma}\right](\tilde{y}^2-2y_1\tilde{y})-U_b, \end{equation}

where $y_1$ denotes the distance of the air–liquid interface from the solid wall of the tube. Upon integration within the thin film and neglecting azimuthal variations of curvature and film thickness, the volume flux reads

(3.3)\begin{equation} Q \approx{-}2{\rm \pi} RU_b y_1-2{\rm \pi} R\frac{\gamma}{3\mu}\left({-}y_1'''+\frac{\rho g\cos(\alpha)}{\gamma}y_1'+\frac{\rho g\sin(\alpha)}{\gamma}\right)y_1^3. \end{equation}

Owing to mass conservation, this flux must be equal to the volume of fluid displaced per unit time by the top of the bubble, which is ${\rm \pi} R^2 U_b$. Given that $y_1/R \ll 1$, the term $-2{\rm \pi} RU_by_1$ in the flow rate expression is a negligible correction. Finally, by enforcing flux continuity with the region far from the tip, where the film thickness is uniformly constant and equal to $b$, we derive the following thin-film equation:

(3.4)\begin{equation} y_1'''=\frac{\rho g \sin(\alpha)}{\gamma}\left(1-\frac{b^3}{y_1^3}\right)+\frac{\rho g \cos(\alpha)}{\gamma}y_1'. \end{equation}

We non-dimensionalize with

(3.5a,b)\begin{equation} y_1=\eta b, \quad \tilde{x}=\zeta b(\rho g b^2\sin(\alpha)/\gamma)^{{-}1/3}, \end{equation}

which leads to the following ordinary differential equation:

(3.6)\begin{equation} \eta'''=\frac{\eta^3-1}{\eta^3}+a\eta', \end{equation}

where $a=\cos (\alpha )\sin (\alpha )^{-2/3}Bo^{1/3}({b}/{R})^{2/3}$.

Upon introduction of the parameter $a$, this equation is exactly the same as (2.12) describing the inner region in a centrifugated tube, which has been solved in § 2.1. Thus, shifting the origin to the position where $\eta ''=0$, the distance from the wall at which the inner solution exhibits an inflection point is

(3.7)\begin{equation} y_1(0)=\eta(0)b=1.10 b \ll R, \end{equation}

and the slope of the inner region profile at the inflection point is given by

(3.8)\begin{equation} y_1'(0)=\eta(0)(\rho g b^2\sin(\alpha)/\gamma)^{1/3}=0.572 \, Bo^{1/3}\sin(\alpha)^{1/3}\left(\frac{b}{R}\right)^{2/3} >0. \end{equation}

For the matching of the thin-film region with a two-dimensional cap, we would need to determine the value of (positive) angle $\phi$ which leads to zero curvature at the wall. For the matching with the previously introduced three-dimensional shape of the static cap, we extend this analysis by searching for the angle $\phi$ that gives rise to zero radial curvature in at least one point of the matching boundary. Note that although we imposed as a boundary condition a constant angle $\phi$ at the interface when reaching the wall, the height of the interface at the wall, and so the curvature, varies along the azimuthal direction. Since the angle $\phi$ at the point of vanishing curvature should be positive according to the matching condition (3.8), we identify the critical Bond number as the $Bo$ value for which $\phi =0$, and the radial curvature at the wall vanishes in at least one point. For smaller Bond numbers, the geometrical constraint on the slope cannot be satisfied in at least one point of the domain.

Equation (3.1), together with its boundary conditions, is implemented in the finite-element solver Comsol Multiphysics. We exploit fourth-order Lagrangian shape functions, solving for the height $h$ and the mean curvature $\kappa$ in a grid composed of quadrangular elements, with 10 boundary layers of 1.3 stretching factor to properly capture the curvature at the boundary. For each tilt angle $\alpha$, solutions are obtained for different values of the angle $\phi$ and Bond number $Bo$ using the built-in Newton algorithm, initialized with the zero solution. We then perform a continuation study by gradually decreasing the angle $\phi$ from 90$^\circ$. Note that the boundary condition $\phi =0^\circ$ cannot be imposed in this framework, as it implies infinite directional derivatives for the thickness. We thus study solutions in the close vicinity of $\phi =0^\circ$ and extrapolate the retrieved behaviour for $\phi \rightarrow 0$; however, this limitation will not significantly affect the evaluation of the threshold for the bubble rise. For fixed Bond number, a convergence analysis from a characteristic size of 0.05$R$ to 0.01$R$ (i.e. from 3200 to 33 012 elements) showed variations of ${\sim }10^{-4}$ rad in the value of $\phi$ resulting in a zero radial curvature. The numerical code for $\alpha =90^\circ$ (i.e. for a vertical tube), $\phi =0.5^\circ$ and $Bo=0.86$ is compared against the axisymmetric solution of (2.6) with no rotation ($\omega =0$). The result of the comparison is reported in figure 8(c) and a good agreement is observed.

Figure 9 shows the static cap of the bubble for different inclination angles and Bond numbers, for same angle $\phi =0.5^\circ$. For $\alpha <90^\circ$, the static cap is not axisymmetric: as the tilt angle increases, the apex of the cap moves towards negative $x$. Conversely, an elongated region (tongue) develops in the vicinity of the $x$ axis, in the direction $x>0$ (i.e. in the direction of positive gravitational acceleration) and becomes longer as the tilt angle increases. The elongated region presents abnormal values of mean curvature with respect to the rest of the cap. The highest (negative) curvature is observed to be localized at the tip point of this tongue.

Figure 9. (a) Static cap profiles computed as solutions of (3.1) for various tilt angles $\alpha$, with $\phi =0.5^\circ$ and $Bo\gtrapprox Bo_c(\alpha )$. The colourbar represents the radial curvature $\kappa _r$ computed along the height profiles in the plane $y=0$. The heights of the profiles for various $\alpha$ have been translated for visualization purposes. (b) Three-dimensional static cap shape close to critical conditions $\phi =0.5^\circ$, for various tilt angles. The colourbar represents the profile height $\bar {h}-\bar {h}_{max}$.

To obtain the critical conditions, we fix the tilt angle and the Bond number and progressively decrease the angle $\phi$. A preliminary analysis showed that the highest radial curvature is obtained at the tip point of the tongue (of coordinates $(\bar {x}=1,\bar {y}=0)$), in agreement with the above observations, and increases with decreasing $\phi$. For each angle $\phi$, we thus compute the radial curvature $\kappa _r=({\partial ^2 \bar {h}}/{\partial \bar {x}^2})/{(1+({\partial \bar {h}}/{\partial \bar {x}})^2)^{3/2}}$ at the extremity of the tongue. Note that $({\partial \bar {h}}/{\partial \bar {y}}) (\bar {x},\bar {y}=0)=0$ because of symmetry. The angle $\phi$ is then decreased until $\kappa _r$ vanishes. This limit value (which is obtained through linear interpolation, when a change of sign is detected, of the values of curvature between two successive values of $\phi$, with a step of $9 \times 10^{-5}$ rad) is denoted $\phi _{lim}(\alpha, Bo)$. For the set of parameters $(Bo, \alpha, \phi =\phi _{lim}(\alpha, Bo))$, the liquid–air interface exhibits then an inflection point at the wall, and its tangent plane makes an angle $\phi _{lim}(\alpha, Bo)$ with the $z$ direction.

Repeating the same procedure varying the Bond number while fixing the tilt angle $\alpha$, we can retrieve $\phi _{lim}$ as a function of $Bo$. In the range $\phi _{lim} \in [0.5^\circ, 2 ^\circ ]$, $\phi _{lim}(\alpha, Bo)$ varies linearly with the Bond number $Bo$, as shown in figure 10(a). For each tilt angle $\alpha$ we interpret the Bond number value at which $\phi _{lim}(\alpha, Bo_c)=0$ as the threshold $Bo_c(\alpha )$ for the onset of motion. We retrieve this value by performing for each tilt angle $\alpha$ a linear fit of $\phi _{lim}(\alpha, Bo)$ for $\phi _{lim}$ varying between 0.5$^\circ$ and 2$^\circ$, and by extrapolating the value $Bo_c(\alpha )$ that corresponds to $\phi _{lim}=0^\circ$. The fit is performed by considering at least eight points within the declared range. We verified that the threshold and slope do not vary appreciably by decreasing the number of points while keeping a constant distance between the remaining points. The slope of the fit is also a function of $\alpha$, so that overall, $\phi _{lim}(\alpha,Bo)$ is approximated by

(3.9)\begin{equation} \phi_{lim}(\alpha, Bo)=\beta(\alpha)(Bo-Bo_c(\alpha)). \end{equation}

Figure 10. (a) Angle $\phi _{lim}$ for which the radial curvature of the liquid–air interface vanishes at one point at the wall as a function of $Bo$ for various values of $\alpha$ (coloured circles). For each panel, the black solid line is the linear fit $\phi _{lim}(\alpha, Bo)=\beta (\alpha )[Bo-Bo_c(\alpha )]$ performed in order to retrieve the threshold for the onset of motion $Bo_c(\alpha )$, that corresponds to $\phi _{lim}=0$. (b) Threshold $Bo_c$ as a function of the tilt angle $\alpha$ (filled circles). The maximum extrapolation error is of the order of 0.001 and is smaller than the marker size. The black solid line represents the polynomial approximation (3.10). The open circles represent instead the threshold $Bo^{2D}_c(\alpha )$ retrieved from the matching of the thin film with a two-dimensional static cap profile. The black dotted line is a guide for the eyes. Inset: coefficient $\beta$ as a function of the tilt angle $\alpha$. The errorbars represent the 95 % confidence interval. The black solid line represents the polynomial approximation (3.11).

The result of this procedure is displayed in figure 10. Our study clearly indicates that the threshold for the onset of motion is lowered by tilting the tube, with a minimum that is reached for a tilt angle of $45^\circ <\alpha _{opt}<50^\circ$. Overall, the critical Bond number and the slope $\beta$ as a function of the tilt angle are well approximated by

(3.10)$$\begin{gather} Bo_{c}(\alpha)\approx 0.54\left[1 + 0.5\left(\frac{\rm \pi}{180}\right)^2(\alpha -48^\circ)^2 + \left(\frac{\rm \pi}{180}\right)^4(\alpha-48^\circ)^4\right], \end{gather}$$

(3.11)$$\begin{gather}\beta(\alpha) \approx{-}\frac{2}{3}\left(\frac{\rm \pi}{180}\right)^2(\alpha-51^\circ )^2+0.85 \quad [\text{rad}], \end{gather}$$

as shown in figure 10(b). Note that in the vertical case, we retrieved values for the critical Bond number $Bo_{c}(\alpha =90^\circ )$ and for the slope $\beta (\alpha =90^\circ )$ that match Bretherton's values ($Bo_c=0.842$, $\beta =0.49$ rad) within 0.03 % and 2 % of relative error, respectively, thus validating further the procedure.

We now aim at providing a prediction for the ascent velocity. The matching of the two-dimensional thin-film region with the static cap shape at the inflection point amounts to enforcing

(3.12)\begin{align} &0.572 \, Bo^{1/3}\sin(\alpha)^{1/3}\left(\frac{b}{R}\right)^{2/3}\nonumber\\ &\quad =\phi_{lim}(\alpha, Bo)=\beta(\alpha)\left[\frac{\rho g}{\gamma}(R-1.10b)^2-Bo_c(\alpha)\right]. \end{align}

The volume of fluid displaced per unit time by the tip of the bubble ${\rm \pi} R^2U_b$ being equal to the volume flux in the uniform film region, $(b/R)=({3Ca}/{2Bo\sin (\alpha )})^{1/3}$. Expanding (3.12) up to first order in $(b/R)$ finally yields the following implicit function for the bubble velocity:

(3.13)\begin{align} (Bo-Bo_c(\alpha))\sin(\alpha)&=2.52 (Bo\sin(\alpha))^{2/3}\,Ca^{1/3}\nonumber\\ &\quad +\frac{0.63 \sin(\alpha)}{\beta(\alpha)}(Bo\,\sin(\alpha))^{1/9}\,Ca^{2/9}. \end{align}

As for the centrifugated case, (3.13) can be recast as an equation of the type $x^3 + a_1 x^2 + a_2 x + a_3 =0$, by setting $x=Ca^{1/9}$.

3.2.1. Experimental set-up and procedure

The same silicone oil used in § 2.2 is employed to partially fill capillary tubes, that are then sealed on both ends, trapping a long air bubble inside. The inner radii of the tubes vary between 1.08 and 1.96 mm, corresponding to Bond numbers in the range $Bo \in [0.53, 1.73]$. As in the previous section, the uncertainty in the tube inner diameters is 0.05 mm.

The experimental set-up is depicted in figure 11(a). The tube is attached on an aluminium arm that can be tilted by an angle $\alpha \in [0^\circ, 180^\circ ]$ with respect to the horizontal plane. A light-emitting-diode panel is positioned behind the set-up for visualization purposes. Once the tilt angle is fixed, a camera records the rising motion of the bubble along the central axis of the tube. From the recorded footage, we can then retrieve the bubble velocity as previously described in § 2.2, and as illustrated in figure 11(b,c). For the narrowest tubes where the bubble velocity is the smallest (if not zero), we use time lapses instead of movies.

Figure 11. (a) Sketch of the experimental set-up. (b) Photographs of a long bubble inside a tube filled with silicone oil, at different and equally spaced time steps. Here, $Bo=1$ and the tube is tilted by $\alpha =35^\circ$ with respect to the horizontal axis. (c) Intensity profile as a function of time along the tube axis. To produce this image, a column of pixels aligned with the central axis of the tube is extracted from each frame of the movie. The columns are then juxtaposed to each other. The locations of the upper and lower cap as a function of time are easily identified as the two roughly parallel black curves limiting a slightly darker domain that corresponds to the position of the bubble itself. The rising velocity is given by the slope of these black lines. As in (b), $Bo=1$ and $\alpha =35^\circ$. (d) Photograph of a bubble in a tube tilted by $\alpha =50^\circ$, with $Bo=0.7$. For these parameters, the system is close to critical conditions for the onset of motion.

Unlike the case of motor-driven rotating tubes, there is in principle no limitation on the observation time, allowing the detection of much slower bubble displacements. In practice, we consider the bubble velocity to be zero if the displacement of the bubble over a week is smaller than our resolution limit of 1 mm. This implies that the smallest experimentally measurable capillary number is $Ca_{min}=7.6 \times 10^{-9}$.

3.2.2. Experimental results and comparison with the theoretical prediction

Our experimental findings are summarized and compared with our theoretical predictions in figure 12. Firstly, we observe that the bubble velocity strongly depends on the tilt angle $\alpha$ and reaches its maximum at $\alpha \approx 50^\circ$, a value independent of $Bo$ within the range of Bond numbers investigated here, as shown in figure 12(b). Furthermore, for $Bo_c(\alpha =50^\circ )=0.5401< Bo<0.842=Bo_c(\alpha =90^\circ )$, tilting the tube by the appropriate angle actually enables the motion of a bubble that would otherwise be stuck in a vertical configuration, as illustrated for instance by the cases $Bo=0.71$ and $Bo=0.65$ reported in figure 12(b). No motion at all is observed below the threshold $Bo_c(\alpha \approx 50^\circ )$. Those observations align well with our theoretical analysis.

Figure 12. (a) Diagram $(Bo, \alpha )$ where each circle corresponds to a measurement of $Ca>0$ for a given set of parameters $(Bo, \alpha )$. The red crosses indicate the couples $(Bo, \alpha )$ for which the bubble displacement fell below our detection limit. The black circles indicate the theoretical threshold for the onset of motion and the black solid line represents the polynomial approximation (3.10). Below this line, the grey area corresponds to the region of parameters where the steady rising of a bubble is not possible according to our theoretical analysis. (b) Velocity of the bubble as a function of the tilt angle $\alpha$ for various Bond numbers. The circles are the experimental points while for each Bond number, the solid line is described by (3.13), using the corresponding experimental value of $Bo$ reported above each panel. No fit parameter is used here: the values of $\beta (\alpha )$ and $Bo_c(\alpha )$ in (3.13) are those displayed in figure 10. The dotted lines also represent the prediction (3.13), but for $Bo \pm \Delta Bo$, where $\Delta Bo$ accounts for the $\pm$0.05 mm uncertainty on the tube inner diameters. Here, the marker size represents the maximal measurement uncertainty on the bubble velocity.

Finally, the bubble velocity as a function of the tilt angle $\alpha$ at low Bond numbers seems to be well described by (3.13), without any fitting parameter; see figure 12(b). We note that the agreement with the theoretical prediction appears to slightly deteriorate at larger Bond numbers. Indeed, several assumptions made in the theoretical analysis, reasonable in the vicinity of the threshold, are expected to fail in the large-$Bo$ regime. Notably, Atasi et al. (Reference Atasi, Khodaparast, Scheid and Stone2017) measured the top and bottom film thickness around long bubbles in horizontal tubes and observed that the film asymmetry grows with the Bond number. Thus, in large capillaries, the thin-film thickness cannot be considered as uniform along the azimuthal direction: the lubricating film is indeed much thicker in the direction $x>0$ (Zukoski Reference Zukoski1966). Similarly, requiring the static cap profile to expand in the entire fluid domain $r< R$ is likely to become inadequate as $Bo$ increases. All together, however, the comparison tends to validate the relevance of a two-dimensional analysis to describe the thin lubricating film surrounding the bubble, even in a tilted configuration, in the low-$Bo$ regime.

It is worth mentioning that as a first attempt to describe the phenomenon, we opted for a fully two-dimensional description of the air–liquid interface. By matching the two-dimensional static cap with the thin-film profile, we obtained the threshold $Bo_c^{2D}(\alpha )$ reported in figure 10(b). This threshold exhibits the same non-monotonic trend as a function of the tilt angle, with a minimum reached for $\alpha _{opt} \lessapprox 50^\circ$. However, it is downward-shifted with respect to the critical Bond number relying on a three-dimensional description of the static cap, which provides a much better agreement with experimental measurements; see figure 12(a). From this comparison, we conclude that while a simplified, two-dimensional description of the thin-film region is acceptable, a proper characterization of the phenomenon requires accounting for the three-dimensional shape of the static cap.

We can now rationalize the theoretical and experimental results: the increase in transversal acceleration due to the tilt angle tends to increase the film thickness at the tongue of the static cap, enabling higher velocities within the tube for the same axial gravity. However, tilting the tube decreases the driving buoyancy force, which in turn reduces the bubble velocity. The interplay between these two effects leads to the observed non-monotonic dependency of the rising velocity on the tilt angle. In the limit case $\alpha \rightarrow 0^\circ$ (horizontal tube), there is no motion within the tube since the driving force disappears.