3.1. Phenomenology

To describe how spinning twisted ribbons are formed, we begin by considering a curved liquid sheet that has developed two holes, as illustrated in figure 2(a). It is well known that the holes expand at a constant speed, known as the Taylor–Culick velocity, given by $u_c=\sqrt{2\gamma / (\rho \delta) }$ , where $\gamma$ is the surface tension, $\rho$ is the density of the liquid, and $\delta$ is the thickness of the sheet. Because the radius of curvature of the crown film, $R_f$ , is much larger than the travelled distance of the rim, $d$ , it has negligible impact on the measured Taylor–Culick velocity $u_c$ . Specifically, in our experiments, $d/R_f\lt 0.1$ , so that the projection factor $\cos \theta _p \approx 1-((d/R_f)^2/2) \approx 1$ , where $\theta _p$ is the angle between the image plane and the tangential plane at the hole. As a hole expands, the displaced fluid accumulates at the circular rim of the liquid sheet, while the thickness of the rest of the sheet remains constant (Savva & Bush Reference Savva and Bush2009).

As the two holes expand, their edges will eventually ‘meet’, while the subsequent development depends on whether the liquid sheet is planar or curved. On a planar liquid sheet, it has been shown that the rims will collide head-on, producing lamella and ejecta when the collision Weber number is sufficiently high (Néel et al. Reference Néel, Lhuissier and Villermaux2020; Tang et al. Reference Tang, Adcock and Mostert2024). On a curved liquid sheet, in contrast, the rims cross each other laterally, as demonstrated in supplementary movie 2 and figure 2(a,b). This occurs because the trajectories of the rims deviate from the initial curved surface: since surface tension acts tangentially, initially, there is no centripetal force to maintain the rims on a curved path, as illustrated in figure 2(a) and supplementary movie 5. Note that at later times, as the rims deviate from the initial surface, the liquid film bends outwards, and surface tension provides centripetal acceleration for curvilinear motions. In the case of a single-hole rupture on a bubble, it has been observed that the centripetal acceleration destabilises the rim via Rayleigh–Taylor instability (Lhuissier & Villermaux Reference Lhuissier and Villermaux2012; Jiang et al. Reference Jiang, Rotily, Villermaux and Wang2022). However, in the current case of multiple-hole ruptures, the rims meet one another ( $\sim$ 0.3 ms) before the instability becomes prominent ( $\sim$ 50 ms) (figure 19 of Lhuissier & Villermaux Reference Lhuissier and Villermaux2012).

The spinning twisted ribbon is formed after the rims cross laterally. Due to the attractive surface tension of the connecting liquid sheet, the rims rotate around the axis where they cross laterally, indicated by the dashed red line in figure 2(b,c), forming the spinning twisted ribbon, as shown in supplementary movie 2 and figure 2(c). We define $t = 0$ as the moment that the rims cross laterally for the first time.

Note that not every pair of neighbouring holes produces a twisted ribbon. We find that the average number of ribbons per hole is $0.8 \pm 0.1$ , based on a total count of 198 holes across 21 drop-impact experiments. Qualitatively, even on a curved film, head-on collisions between rims can occur in the following scenarios, preventing the ribbon formation. First, when the two holes are very close to each other, the effect of curvature is insignificant. Second, the two holes rupture simultaneously, resulting in a mirror symmetric system. This symmetry leads to an angled head-on collision.

We identify several key features of the twisted ribbon from the video images. The ribbon looks similar to a helicoid, but strictly speaking it is not. The ribbon exhibits mirror symmetry with respect to the central point. There are several points of conjunction (red arrows in figure 2 c), where the two rims align along the same line of sight, visually overlapping but not physically touching. The number of conjunction points increases over time, indicating more twisting of the sheet, with the points moving outwards along the axis.

Figure 3. Schematic drawings for the derivation of (3.2).

3.2. Kinematic model

To explain the key features of the spinning twisted ribbon, we propose a kinematic model with several idealised assumptions. The predicted structures from the model are shown in figure 2(d) and supplementary movie 6, and compared with the data in figure 2(c). The parameters involved are defined in figure 2(a,b) and elaborated by the schematic drawings in figure 3.

In this kinematic model, we consider two horizontal circular holes of equal size. Let the origin of our coordinate system be the midpoint between the rupture points (centres) of the two holes. The rims of the holes expand horizontally at speed $u_{c}$ from their rupture points along two planes parallel to the $yz$ -plane and separated by a distance $2R$ . The distance between the $xz$ -plane and rupture points is $d$ . At time $t=0$ , the rims reach the $xz$ -plane and conjunct (relative to an observer at $x\rightarrow \infty )$ for the first time. Thus we call the $xz$ -plane the conjunction plane.

Next, we consider an arbitrary segment of the rim, represented by an azimuthal angle $\beta$ , as shown in figure 3. It reaches the conjunction plane at time $\tau = d/(u_c \cos \beta )- d/u_c$ and at $z$ -position $z_0=d \tan \beta =(u_c\tau +d)\sin \beta$ . After reaching the conjunction plane, it rotates around a rotation axis ( $z$ -axis) with rotation radius $R$ , angular speed $\omega = u_c \cos \beta /R$ and axial speed $u_z= u_c \sin \beta$ . For simplicity, here we take the rotation radius $R$ as a constant, implying a circular closed orbit. The actual non-circular open orbit is presented later with the central force model. Therefore, the motion of a segment of the rim is described by

(3.1) \begin{align} x &= R \cos \left (\omega \left ( t-\tau \right )\right ),\nonumber\\ y &= R \sin \left (\omega \left ( t-\tau \right )\right ),\nonumber\\ z &= u_z(t-\tau ) +z_0. \end{align}

Recall that $\omega ,\tau ,u_z,z_0$ can be expressed in terms of the variable $\beta$ and constants $u_c,R,d$ .

Finally, we assume that the ribbon surface is a ruled surface formed by connecting two rims by introducing a parameter $\alpha$ , and each rim is a collection of rim segments described by (3.1) with parameter $\beta$ and constants $u_c,R,d$ . The surface of the twisted ribbon is described by the parametric equations

(3.2) \begin{align} x(\alpha ,\beta )&=\alpha R \cos \left ( \frac {u_c t+d}{R}\cos \beta - \frac {d}{R} \right ),\nonumber\\ y(\alpha , \beta )&=\alpha R \sin \left ( \frac {u_c t+d}{R}\cos \beta - \frac {d}{R} \right ),\nonumber\\ z(\beta )&=(u_c t+d)\sin \beta, \end{align}

where $-1 \leqslant \alpha \leqslant 1$ , $-\beta _0 \leqslant \beta \leqslant \beta _0$ . Here the range of $\beta$ is bounded by $\beta _0$ that satisfy $t\geqslant \tau$ in (3.1), given by $\cos \beta _0= d/(u_c t+d)$ .

This surface is plotted in figure 2(d) using measured parameters $u_c$ = 2.69 m s⁻¹, $d$ = 1.08 mm and $R=82.8$ $\unicode{x03BC}$ m, obtained from the video frames in figure 2(c). The velocity $u_c$ and distance $d$ are measured with correction for parallax error, as described in Appendix B. The radius $R$ is obtained by measuring the central angular frequency $\omega _0\equiv u_c/R=32\,500$ s $^{-1}$ or 5170 Hz. The calculated surfaces resemble the ribbons observed in the experiment, reproducing the key features discussed earlier. No fitting parameters are involved. Note that the rims’ surfaces are not included in the equations, and only added in figure 2(d,e) for the sake of clarity. The model only considers two holes, while in the experiments more than two holes frequently occur, affecting the shape of the twisted ribbon. For example, see the third rim on the right-hand side in figure 2(c).

Furthermore, to verify the kinematic model, we compare the positions of the measured and calculated conjunction points at different times in figure 4. By (3.1) and (3.2), the $z$ -position of conjunction points are given by $z_n=(u_c t+d)\sin \beta _n$ , where the angles $\beta _n$ have to satisfy the condition $\omega (t-\tau )=n\pi$ with $n=0,1,2,\dots$ . Simplifying, the $z$ -positions of conjunction points are

(3.3) \begin{equation} \frac {z_n}{d}=\sqrt {\left (\frac {u_c t}{d}+1\right )^2-\left (\frac {n \pi R}{d}+1\right )^2}, \end{equation}

for $t \geqslant n\pi R/u_c$ and $n=0,1,2,\dots$ . They are plotted in figure 4 using the same values of $d$ and $R$ employed in plotting figure 2(d). The measured data and calculated results show reasonably good agreement.

New conjunction points emerge at regular time intervals of $\pi R/u_c$ . By (3.3), the speeds of conjunction points are

(3.4) \begin{equation} \frac {1}{u_c}\frac {\mathrm{d}z_n}{\mathrm{d}t} \approx 1+\frac {1}{2}\left (\frac {n\pi R/d+1}{u_c t/d+1}\right )^2+\cdots . \end{equation}

At large time, their speed approaches a constant value $u_c$ , which is the Taylor–Culick speed in our case.

Figure 4. Positions of conjunction points at different times. The measured data (dots) agree well with the calculation results (curved lines) predicted by the kinematic model and (3.3). Some data points overlap due to the mirror symmetry with respect to the $xy$ -plane.

Next, we calculate the mean curvature of the ribbon by

(3.5) \begin{equation} H=\frac {eG-2fF+gE}{2(EG-F^{2})} , \end{equation}

where $H$ is the mean curvature, $\{E, F, G\}$ and $\{e, f, g\}$ are the coefficients of the first and second fundamental form of the surface described by (3.2). We get

(3.6) \begin{equation} H(\alpha ,\beta )=\left (\frac {1}{u_{c}t+d}\right )\left ( \frac {\alpha }{2\left (\cos ^{2}\beta + \alpha ^2\sin ^{2}\beta \right )^{3/2}}\right ) . \end{equation}

Numerically, in the range of interest $|\beta |\lt \pi /4$ and $|\alpha |\leqslant 1$ ,

(3.7) \begin{equation} H\lt \frac {0.55}{u_c t+d}\lt \frac {1}{d} . \end{equation}

This means the upper bound is $1/d$ . Typically, the hole–axis distance $d$ ( $\sim$ 1 mm) is much larger than the rotation radius $R$ ( $\sim$ 0.1 mm), which represents the size of the ribbon. Therefore, the calculated mean curvature of the ribbon is small.

The small mean curvature obtained agrees with our expectations. First, physically, it suggests a pressure equilibrium across the film, which is known to be established rapidly after film rupture (Bird et al. Reference Bird, De Ruiter, Courbin and Stone2010). Second, geometrically, it aligns with the visual similarity between the twisted ribbon and the helicoid, which is a minimal surface.

3.3. Asymmetric kinematic model

We extend the kinematic model to account for asymmetric cases where the sizes of the two holes are different. This scenario happens when a larger hole, which is formed earlier, interacts with a smaller hole. When the holes are the same size, it is obvious that their rims meet along a straight line at the midpoint. When the hole sizes are different, their rims meet along a hyperbola, as illustrated in figure 5. This is because the difference in radius between the two expanding holes is a constant. This is analogous to the interference of two circular waves (Pain Reference Pain2005, p. 356). The hyperbola is described by the equation in polar coordinate as

(3.8) \begin{equation} r_{h} (\beta ) = \frac {d(1 - e^{-2})}{e^{-1} + \cos \beta } , \end{equation}

where $e= ({d_1+d_2})/({d_2-d_1})$ is the eccentricity, $2d=d_1+d_2$ is the distance between the centres of the two holes, $d_{1,2}$ are the shortest distances from the centres of the holes to the ribbon and $d_1\lt d_2$ , as defined in figure 5(a). If the system is symmetric, $e^{-1}=0$ and thus $r_h=d/\cos \beta$ becomes a straight line as expected.

Figure 5. The asymmetric model. (a) Schematic diagram consisting of two holes of different sizes (dotted circles), with their rims meeting along a hyperbola (red) and forming a ribbon. (b) Snapshot of a small hole (left) interacting with a large hole (right). (c,d) Magnified video frames and the corresponding predicted surfaces of the spinning twisted ribbon formed by a small hole (top) and a large hole (bottom). The surfaces are plotted by (3.14), using the measured parameters $u_c$ = 2.17 m s⁻¹, $d_1$ = 1.19 mm, $d_2$ = 1.92 mm, $\omega _0=19\,500$ s $^{-1}$ or 3100 Hz.

In the asymmetric case, the momenta of the two rims are also different. The rim of the larger hole is thicker due to the larger hole size and volume conservation, as the rim collects the film liquid during its motion. Its normal component of the velocity is also larger because $\beta ' \lt \beta$ , as shown in figure 5(a). Therefore, the rim of the larger hole should have a larger momentum than that of the smaller hole, driving the ribbons to drift in one direction. Nevertheless, in our experiments, the liquid sheet itself is also moving, making it difficult to measure this momentum mismatch directly.

We now express the ribbon surface for the asymmetric case in a form similar to the symmetric case given in (3.2). For simplicity, we assume that their rim thicknesses are the same, so that

(3.9) \begin{align} \omega &= \frac {u_c}{R}\frac {\cos \beta + \cos \beta '}{2} =f_c(\beta ) \omega _s,\nonumber\\ u_z &= u_c\frac {\sin \beta + \sin \beta '}{2} =f_s(\beta ) u_c \sin \beta , \end{align}

where $\omega _s= u_c \cos \beta /R$ is the angular speed for the symmetric case, $f_c(\beta )$ and $f_s(\beta )$ are geometric factors that account for the asymmetry, given by

(3.10) \begin{align} f_c(\beta ) &= \frac {e^{-1}\cos \beta +1}{2e^{-1}\cos \beta +r_h(\beta )\cos \beta /d},\nonumber\\ f_s(\beta ) &= \frac {e^{-1}+r_h(\beta )/d}{2e^{-1}+r_h(\beta )/d}. \end{align}

If the configuration is symmetric ( $e^{-1}=0$ ), $f_c(\beta )=f_s(\beta )=1$ as expected. The phase lag $\tau$ for the asymmetric case is

(3.11) \begin{align} \tau &= \frac {r_h (\beta )}{u_c}-\frac {r_h (0)}{u_c} =g(\beta ) \tau _s , \end{align}

where $\tau _s= d/(u_c \cos \beta )- d/u_c$ is the phase lag of the symmetric case, the geometric factor $g(\beta )$ is given by

(3.12) \begin{align} g(\beta ) &= \frac {(1-e^{-1})\cos \beta }{e^{-1}+\cos \beta }. \end{align}

Therefore, similar to (3.2), the parametric equations are

(3.13) \begin{align} x(\alpha ,\beta )&=\alpha R \cos \left (f_c \omega _s \left ( t-g\tau _s \right ) \right ),\nonumber\\ y(\alpha , \beta )&=\alpha R \sin \left (f_c \omega _s \left ( t-g\tau _s \right ) \right )-y_0(\beta ),\nonumber\\ z(\beta )&= f_s u_c \sin \beta (t-g\tau _s))+z_0(\beta ), \end{align}

or, equivalently,

(3.14) \begin{align} x(\alpha ,\beta )&=\alpha R \cos \left ( \left ( \frac {u_c t+d g(\beta )}{R}\cos \beta - \frac {d g(\beta )}{R} \right ) f_c(\beta ) \right ),\nonumber\\ y(\alpha , \beta )&=\alpha R \sin \left ( \left ( \frac {u_c t+d g(\beta )}{R}\cos \beta - \frac {d g(\beta )}{R} \right ) f_c(\beta ) \right )-y_0(\beta ),\nonumber\\ z(\beta )&= f_s(\beta ) \sin \beta \left (u_c t - g(\beta ) \left (\frac {d}{\cos \beta }-d\right )\right )+z_0(\beta ), \end{align}

where $y_0(\beta )=r_{h}(\beta ) \cos \beta -r_{h}(0)$ and $z_0(\beta )=r_{h} (\beta ) \sin \beta$ . An example involving two holes of different sizes is shown in figure 5(b,c). The predicted ribbon surface is plotted in figure 5(d) using measured parameters $u_c$ = 2.17 m s⁻¹, $d_1$ = 1.19 mm, $d_2$ = 1.92 mm and $R=111$ $\unicode{x03BC}$ m. From $d_1$ and $d_2$ , we get $d$ = 1.55 mm and eccentricity $e$ = 4.24. The calculated surfaces resemble the ribbons observed in the experiment.

In this case, $d_2/d_1=1.62$ . For comparison, the earlier case presented in figure 2 has $d_2/d_1=1.07$ , and its slight asymmetry can therefore be neglected.

3.4. Dynamics: two-body central force model

We elucidate the ‘orbit’ of the rims of the spinning ribbon by using the classical two-body central force model and direct measurements (Goldstein, Poole & Safko Reference Goldstein, Poole and Safko2001). We assumed a circular orbit in the last section, but in fact it is open and non-circular. Consider a thin strip of the ribbon with a dumbbell-shaped cross-section, as shown in figure 6(a), where two circular rims are connected by a thin liquid string. The surface tension on the liquid string exerts an attractive central force between the two circular rims. For simplicity, we consider a thin strip at the centre plane ( $z=0$ ).

Figure 6. Explanation of the ribbon’s orbit by the two-body central force model. (a) Sketch of a thin strip of the ribbon with a dumbbell-shaped cross-section. Two circular rims are connected by a thin liquid string, which exerts an attractive central force. (b–d) Measured orbits, showing the data (dots) and interpolation (line). The colours indicate time progression, ranging from red ( $t=0$ ) to purple ( $t=550$ , 650 and 390 $\unicode{x03BC}$ s). (e–g) The corresponding orbits calculated by solving (3.15) with liquid properties $\gamma = 20.8$ mN m⁻¹, $\rho = 960$ kg m⁻ $^3$ , and initial conditions approximated from measurements: (e) $r(0)= 841$ $\unicode{x03BC}$ m, $v_{\theta }(0)=4.45$ m s⁻¹, $v_{r}(0)= 0$ , $m_{\mu }= 586$ $\unicode{x03BC}$ g m⁻¹; (f) $r(0)= 1071$ $\unicode{x03BC}$ m, $v_{\theta }(0)= 4.01$ m s⁻¹, $v_{r}(0)=$ $-0.40$ m s⁻¹, $m_{\mu } = 449$ $\unicode{x03BC}$ g m⁻¹; (g) $r(0) = 1003$ $\unicode{x03BC}$ m, $v_{\theta }(0) = 4.94$ m s⁻¹, $v_{r}(0)=$ $-0.33$ m s⁻¹, $m_{\mu } = 397$ $\unicode{x03BC}$ g m⁻¹. The data agree with the model semi-quantitatively.

Following the classical treatment, the two-body system is reduced to a one-body system with a relative position $\textbf {r}(t)$ and a reduced mass per length $m_{\mu }=\rho \pi r_{1}^2 r_{2}^2/(r_{1}^2+r_{2}^2)$ , where $\rho$ is the density of the liquid sheet, $r_{1,2}$ are the radii of the rims. The magnitude of the central force per length is twice the surface tension $\gamma$ , accounting for both the upper and lower surfaces of the liquid string. The orbit is obtained by solving the following differential equations numerically by Mathematica (Taborek Reference Taborek2010),

(3.15) \begin{align} r''(t)-r(t)\theta '(t)^2+\frac {2 \gamma }{m_{\mu} }=0, \nonumber\\ r(t)\theta ''(t)+2 r'(t)\theta '(t)=0, \end{align}

where $r(t)$ and $\theta (t)$ are the relative distance and angle at time $t$ in the polar coordinate system. The initial conditions, $v_r(t)\equiv r'(t)$ and $v_\theta (t)\equiv r(t)\theta '(t)$ , are measured from experiments. The value of $v_\theta (0)$ is the measured Taylor–Culick speed, i.e. the constant film retraction speed before spinning. Open non-circular orbits are obtained as expected, according to Bertrand’s theorem.

We experimentally measure the orbits using two high-speed cameras positioned at orthogonal angles, and then compare them with the theoretical orbits. The measured orbits are shown in figure 6(b–d). The dots with error bars represent measured data, and the line shows an interpolated orbit. The colours indicate time progression, ranging from red ( $t=0$ ) to purple ( $t=550$ , 650 and 390 $\unicode{x03BC}$ s). We calculate the orbits, as shown in figure 6(e–g), by solving (3.15) numerically with real liquid properties and measured initial conditions (see figure captions). The experiments agree with numerical calculations semiquantitatively, revealing similar non-circular open orbits.

We compare the measured and predicted orbits by their periods, as shown in figure 7(a). Although bounded open orbits cannot be characterised by rotation period, they can instead be characterised by the angular period $T_\theta$ and the radial period $T_r$ . We calculate the angular period by $T_\theta =\Delta t_\theta 2\pi / \Delta \theta$ , where $\Delta \theta$ is the angular distance travelled during a time interval $\Delta t_\theta$ . The radial period is calculated by $T_r=\Delta t_r/N_{cc}$ , where $N_{cc}$ is the number of crest-to-crest cycles observed in the plot of $r(t)$ over a time interval $\Delta t_r$ (Arya Reference Arya1997, p. 258). The measured and predicted periods of three different experiments are shown in figure 7(a), which shows reasonably good agreement.

Figure 7. Comparison of the periods and radii between the central force model and the experimental results. (a) The angular period (red) and radial period (blue) from three different experiments are shown. The measured periods agree with the predicted periods. (b) The plots of $r(t)$ of the orbits in figures 6(b) and 6(e) are shown. The measured orbit is shrinking over time.

The main discrepancy is that the size of the measured orbit is smaller and shrinks over time, as highlighted by the downward trend of $r(t)$ in figure 7(b), which is inconsistent with the model. We believe the shrinking of the orbit is due to the air resistance and the variation in mass of the rims over time, which in turn results from the instabilities and the axial flow that will be discussed in the next section. Further studies on the internal fluid flow could improve the model’s accuracy.

Although the analysis considers only the centre plane ( $z=0$ , or equivalently, $\beta =0$ ), the central force model should remain applicable to off-centre planes by recognising that the corresponding initial velocity is $v_\theta (0)=u_c \cos \beta$ . Recall that the kinematic model in (3.2) and figure 2 has already assumed the ribbon is a ruled surface by connecting two rims, with a satisfactory result. However, it is challenging to compare with experimental results because the orbits in off-centre planes drift outward along the axial direction, which will be discussed in the next section.

This system presents an unusual scenario that the central force is distance-independent (potential is linear), in contrast to the typical celestial orbits (inverse-square law) and harmonic oscillators (Hooke’s law). Other examples of systems with linear potential include the triangular quantum well in high-electron-mobility transistors (Harrison & Valavanis Reference Harrison and Valavanis2016, p. 116) and the quark confinement in mesons (Griffiths Reference Griffiths2020, p. 173).

4.1. Phenomenology

We observe that the rims become unstable during spinning at later times, forming corrugations along their lengths that grow into ligaments, which pinch off and eject secondary droplets, as shown in supplementary movie 7 and figure 8(a). The corrugations are marked by red dots. The ligaments are marked by cyan arrows. The instability is not uniquely associated with spinning, as similar corrugations are also observed without spinning. However, in the latter case, they do not grow into ligaments, as shown in figure 8(b).

We can distinguish this phenomenon from the previously discovered rim-splashing phenomenon by inspecting supplementary movie 7 and figure 8(a), even though they appear similar in still images (Néel et al. Reference Néel, Lhuissier and Villermaux2020; Tang et al. Reference Tang, Adcock and Mostert2024). First, unlike in rim splashing, the two rims of the ribbon have not coalesced when the ligaments develop. Second, in rim splashing, the ligaments are straight and perpendicular to the film. For the spinning ribbon, in contrast, the ligaments are rotating with the ribbon. For example, the centre ligament in supplementary movie 7 has been rotated by $\sim$ 180 $^\circ$ . Third, under spinning conditions, ligaments that break up into droplets are observed at a Weber number lower than that in rim splashing on a planar surface. For comparison, under the current spinning conditions, the local Weber number is $We_{loc}=\rho (2 u_c)^2 (2 R_{rim})/\gamma \sim 58$ , where $R_{rim}$ is the rim radius. In rim splashing, ligaments that break up into droplets are observed at $We_{loc}\gt 120$ (Néel et al. Reference Néel, Lhuissier and Villermaux2020; Tang et al. Reference Tang, Adcock and Mostert2024). Although we have not yet explored the threshold Weber number, the data suggest that ligaments and droplets can be produced at a lower Weber number in this rim-spinning case compared with the rim-splashing case.

Figure 8. Corrugations and ligaments. (a) The corrugations on the rims are marked by the red dots. The ligaments are marked by the cyan arrows. Corrugations grow into ligaments, pinch off and then eject secondary droplets due to the spinning. (b) For the non-spinning case, the corrugations do not grow into ligaments. (c) The measured average wavelength $\lambda$ agrees with the Plateau–Rayleigh instability (solid line) for both spinning (solid dots) and non-spinning (open dots) conditions at different rim radius $R_{rim}$ and surface tensions $\gamma$ . (d) The measured average wavelength $\lambda$ agrees with the Rayleigh–Taylor instability (solid line) at different capillary length $l_c$ and surface tensions $\gamma$ . The number of segments used in the averaging is denoted by $N$ . Because $l_c$ is calculated based on the centre rotational speed $\omega _0$ , the average wavelength based on fewer segments is more reliable.

4.2. Plateau–Rayleigh and Rayleigh–Taylor instabilities

We plot the wavelength of the instability, $\lambda$ , against the rim radius, $R_{rim}$ , as shown in figure 8(c). The plot includes data for both spinning and non-spinning conditions for two different surface tensions. The average wavelength $\lambda$ is calculated by dividing the length of a segment by the number of corrugations $N$ it contains. The rim radii are measured either just before spinning begins or, for the non-spinning case, just after the two rims coalesce. For all conditions studied here, the data agree with the Plateau–Rayleigh instability that $\lambda_{\textit{PR}} =9.0 R_{rim}$ (Rayleigh Reference Rayleigh1878; Wang et al. Reference Wang, Dandekar, Bustos, Poulain and Bourouiba2018). Therefore, it is very likely that the Plateau–Rayleigh instability leads to the corrugations, while the spinning motion is also essential for the formation of ligaments that pinch off into droplets.

On the other hand, the corrugation may also be induced by Rayleigh–Taylor instability under the centripetal acceleration caused by spinning (Eisenklam Reference Eisenklam1964). The wavelength of Rayleigh–Taylor instability is $\lambda _{RT} = 2\sqrt {3}\pi l_{c}$ , where $l_{c}=\sqrt {{\gamma }/(\rho a)}$ is the capillary length, $\gamma$ is the surface tension, $\rho$ is the liquid density, $a=R \omega ^2$ is the centripetal acceleration, $R$ is the rotation radius, $\omega$ is the rotational speed.

However, the rotational speed, and thus the centripetal acceleration, is not a constant but position dependent, along $z$ . It is highest at the centre plane ( $z=0$ ) and decreases towards the two ends. Focusing on the region near the centre plane, we take $\omega \rightarrow \omega _0$ as the angular speed at $z=0$ measured before the corrugations are observed. Correspondingly, the measured wavelengths are calculated using one or two segments nearest to the centre. The plot of the measured wavelengths versus the capillary length $l_{c}=\sqrt {{\gamma }/(\rho R \omega _0^2)}$ is shown in figure 8(d) and is compared with the theoretical expression (solid line). The experimental data is close to the expected values for the Rayleigh–Taylor instability.

In addition, we calculate another set of average wavelengths that includes more segments that are farther from the centre plane, as shown in figure 8(d) (open symbols). The average number of segments used is $\langle N \rangle =4.0$ . Nearly all of the obtained wavelengths increase. This agrees with the model that the rotational speed of the ribbon is highest at the centre plane and decreases towards the two ends, given that $\lambda _{RT}\sim 1/\omega$ .

We have shown that the measured wavelengths agree with both the Plateau–Rayleigh and Rayleigh–Taylor instabilities. Comparing the wavelengths of Plateau–Rayleigh and Rayleigh–Taylor instabilities, one can show that $\lambda _{RT} / \lambda _{PR}=1.2/\sqrt {\textit {Bo}}$ where $\textit {Bo}={\rho a R^2_{rim}}/{\gamma }$ is the local rim Bond number, in which the rim radius $R_{rim}$ is taken as the characteristic length, and $a$ is the acceleration. It is commonly found that $\textit {Bo}\sim O(1)$ in different breakup scenarios (Wang et al. Reference Wang, Dandekar, Bustos, Poulain and Bourouiba2018), so that $\lambda _{RT} \sim \lambda _{PR}$ . In our experiments, calculated from the rotational speed at the centre plane, $Bo$ ranges from 1.5 to 4.6 so that the ratio $\lambda _{RT} / \lambda _{PR}$ ranges from 0.6 to 1.0. Therefore, the wavelengths of Plateau–Rayleigh and Rayleigh–Taylor instabilities are similar in magnitude under current experimental conditions.

Figure 9. Outward axial flow. (a) Snapshots of the twisted ribbon at different times. The corrugations and ligaments (red dots) are moving outward with speed $u_z$ relative to the centre ( $z=0$ ) of the twisted ribbon, confirming the existence of the outward axial flow. The blue arrows highlight the newly emerged corrugations at later times. (b) The speed of the outward axial flow $u_z$ measured at different conditions as shown in the legend. See also supplementary movie 7.

4.3. Axial flow

The corrugations and ligaments are moving outward with speed $u_z$ relative to the centre ( $z=0$ ) of the twisted ribbon, as indicated in figure 9(a), confirming the existence of the outward axial flow. It originates from the rims’ non-zero velocity component parallel to the rotation axis, as the rims move radially from the points of rupture before reaching the rotation axis. By (3.2), the axial flow speed $u_z$ is given by

(4.1) \begin{equation} \frac {u_z (z, t)}{u_c} = \sin \beta = \frac {|z|}{u_c t+d} . \end{equation}

This relation agrees with experimental data as shown in figure 9(b). Experimentally, $u_z$ is measured by tracking the positions of corrugations over time, and $d$ is measured as the distance between the ribbon and the smaller hole without considering the asymmetry.

It is interesting that, by volume conservation, the outward flow would reduce the thickness of the sheet and the rim, adding complexity to the understanding of the rims’ orbit. Unfortunately, the resolution in our experiments is insufficient to quantify this change in thickness directly.

The axial flow also causes the emergence of new corrugations over time, as shown in the last row of figure 9(a) and supplementary movie 7. Due to the axial flow, the spacing between neighbouring corrugations or ligaments increases over time. When the spacing becomes sufficiently large relative to the wavelength of the instability, new corrugations begin to emerge. Such self-sustained population of corrugations has been studied previously in the rim of an expanding liquid sheet (Gordillo, Lhuissier & Villermaux Reference Gordillo, Lhuissier and Villermaux2014).