1. Introduction
When a heavier fluid is accelerated by a lighter one, the interface becomes susceptible to Rayleigh–Taylor instability (RTI) (Rayleigh Reference Rayleigh1883; Taylor Reference Taylor1950). This instability, driven by density mismatch and effective acceleration, is of fundamental importance in inertial confinement fusion (ICF) (Lindl et al. Reference Lindl, Landen, Edwards, Moses and NIC Team2014; Zhou et al. Reference Zhou, Sadler and Hurricane2025), astrophysical phenomena such as supernovae (Müller Reference Müller2020), and various industrial processes in cylindrical and spherical geometries. In the present study, we consider a divergent cylindrical configuration, i.e. the interfaces move radially outwards, in contrast to convergent (imploding) geometries often studied in ICF. More specifically, radially expanding cylindrical liquid layers are encountered in inkjet printing (where annular jets undergo break-up), industrial spray atomisation, liquid curtains for dust suppression, and pressure-driven flows in liquid-lined cylindrical vessels. However, in many realistic situations, material layers have finite thickness, and two interfaces exist in close proximity. The perturbation on one interface can influence the evolution of the other through the pressure and velocity fields in a finite-thickness layer – a coupling mechanism known as feedthrough (Mikaelian Reference Mikaelian2025). Understanding feedthrough in such divergent liquid–air interfaces can guide the control of interfacial instabilities in these applications.
Taylor (Reference Taylor1950) first studied the linear development of RTI for a single fluid layer accelerated in vacuum. Goncharov et al. (Reference Goncharov, McKenty, Skupsky, Betti, McCrory and Cherfils-Clérouin2000) considered compressibility and analysed RTI growth in finite-thickness fluid layers across different geometries. Ryutov & Dorf (Reference Ryutov and Dorf2014) investigated RTI growth in a cylindrical shell geometry, including the effect of a magnetic field. Weis et al. (Reference Weis, Zhang, Lau, Schmit, Peterson, Hess and Gilgenbach2015) also studied RTI development at a shell interface in a cylindrical geometry in the presence of a magnetic field. Mikaelian (Reference Mikaelian1982a
,
Reference Mikaelianb
) examined the linear RTI growth rates at
$N-1$
interfaces in a system of
$N$
fluids, and argued that any arbitrary density transition layer can be approximated by a large number of finite-thickness fluid layers. However, as the number of fluid interfaces increases, obtaining an analytical expression for the perturbation growth rate becomes difficult. Wang et al. (Reference Wang, Guo, Wu, Ye, Liu, Zhang and He2014) developed a weakly nonlinear model, and showed that the layer thickness dominates the perturbation evolution when the normalised thickness is
$kd\lt 1$
, where
$k$
and
$d$
are the wavenumber and layer thickness, respectively. More recently, Mikaelian (Reference Mikaelian2025) defined feedthrough efficiency factors for RTI and Richtmyer–Meshkov instability (RMI), demonstrating that the feedthrough is stronger in RMI than in RTI, and further exploring how it can be exploited to suppress instabilities. Recently, Yuan et al. (Reference Yuan, Zhao, Liu, Wang, Liu and Lu2023, Reference Yuan, Zhao, Liu, Wang, Liu and Lu2024) extended linear stability analyses for cylindrical fluid layers in vacuum to arbitrary Atwood numbers, considering thin-shell corrections and interface coupling effects in shock-accelerated configurations. They found that the Atwood number and the initial phase difference between the two interfaces of a thin shell have a non-negligible influence on interfacial instabilities. These studies collectively show that finite-thickness effects not only modify the growth rate of a single interface, but more importantly, allow perturbations on one interface to feed through the layer to the other.
The geometry of the interface plays a crucial role on the instability development. The ICF capsules are spherical, supernova ejecta are shell-like, and many experimental set-ups adopt cylindrical configurations to combine convergence effects with optical accessibility. Bell (Reference Bell1951) first analysed Taylor instability on cylinders and spheres in the small-amplitude regime, and Plesset (Reference Plesset1954) systematically studied interface stability in converging geometries, laying the foundation of the Bell–Plesset effect. This effect indicates that even in the absence of classical RTI acceleration, radial convergence or divergence of an interface amplifies (or suppresses) initial perturbations. Epstein (Reference Epstein2004) noted that ‘undriven growth’ more accurately describes convergence and compression effects in the absence of buoyancy-driven RTI. Recent high-energy-density experiments have provided valuable insights into cylindrical RTI and feedthrough. Sauppe et al. (Reference Sauppe2020) demonstrated scale-invariant RTI growth in laser-driven convergent cylindrical implosions, showing that the instability evolves self-similarly for targets scaled by a factor of three, highlighting the importance of cylindrical convergence effects on RTI. In the context of feedthrough, Merritt et al. (Reference Merritt, Doss, Levesque, Rasmus, Desjardins, Di Stefano, Flippo and Schmidt2024) developed a quantitative radiography technique to measure density fluctuations and mixing parameters in a thin-layer RMI experiment under planar shock–reshock conditions. Together, these studies underscore the broad relevance of interface coupling and cylindrical geometry to ICF, and they complement our investigation by covering different acceleration regimes and geometries. Most existing studies have focused on the RTI and RMI of gaseous interfaces (Weir et al. Reference Weir, Chandler and Goodwin1998; Ding et al. Reference Ding, Si, Yang, Lu, Zhai and Luo2017), while studies on the RTI of multiphase interfaces remain scarce. The presence of a multiphase interface, such as a liquid–gas interface, introduces surface tension and viscosity effects (Liang & Luo Reference Liang and Luo2025, Reference Liang and Luo2026). Furthermore, a natural and important question arises: how does feedthrough behave when a finite-thickness layer possesses two curved liquid–air interfaces in a cylindrical configuration?
(a) Experimental set-up schematic. (b) Initial interface configuration.
This study fills this gap through combined experimental and theoretical investigation of a cylindrically divergent water layer. We utilise hydrophobic-coated rings to create a thin water layer with controllable initial conditions. We derive coupled evolution equations for the two interface perturbations, analyse how cylindrical divergence modulates feedthrough efficiency, and quantify its dependence on physical parameters.
2. Experimental method
The experiment was conducted in a device comprising two transparent acrylic sheets (spacing
$h=3.0$
mm, radius
$R_L=200.0$
mm), as shown in figure 1(a). A 4.0 mm diameter hole was drilled at the centre of the top sheet. Two pairs of precisely machined micro-channels (width 1.0 mm, depth 2.0 mm) with single-mode patterns were engraved on opposing surfaces (marked in red and blue), into which hydrophobic-coated acrylic rings (width 0.9 mm, height 2.0 mm) were embedded to constrain the inner and outer water–air interfaces. The single-mode micro-channels are positioned at inner radius
$R_i^0=25.0$
mm and outer radius
$R_o^0=33.0$
mm. The hydrophobic-coated rings increase the water–air interface contact angle (typically
$\gt 90^{\circ }$
), enabling precise control of the interface geometry. The hydrophobic rings serve not only to pin the contact lines, but also to prescribe the initial conditions on the two interfaces. Changing the ring geometry (e.g. width or shape) would alter the achievable initial radii and wavenumbers, which in turn significantly affect the instability evolution (Liang & Luo Reference Liang and Luo2025, Reference Liang and Luo2026). Contact line pinning inevitably introduces a slight three-dimensionality, including a thicker interface contour and a weak Poiseuille-type flow in the direction perpendicular to the acrylic plates. However, the width of the interface contour caused by pinning is much smaller than the expanding radius of the interface. Hence the two-dimensional governing equations remain a good approximation for the dominant instability dynamics. Before experiments, the space between the inner and outer single-mode rings was filled with pure water (dyed in cyan). At the start of the experiments, high-pressure air was suddenly injected through the top hole at 25 l min−1 regulated by a flowmeter via a cable.
The initial single-mode perturbations on the inner interface (
$\eta _i$
) and outer interface (
$\eta _o$
) in polar coordinates are given by
\begin{equation} \eta _i^0=R_i^0+a_i^0\cos (n\theta ),\quad\eta _o^0=R_o+a_o^0\cos (n\theta ), \end{equation}
where
$a_i^0$
and
$a_o^0$
are the initial amplitudes of the inner and outer interfaces, respectively,
$n$
is the azimuthal wavenumber, and
$\theta$
is the angle. Table 1 lists the initial parameters for all cases, where the initial conditions are interfaces initially in-phase (IP) and anti-phase (AP), and an initially unperturbed outer interface (UP). The initial amplitude-to-wavelength ratios are all within 0.1, ensuring that the evolution remains in the linear regime initially. A high-speed camera (FASTCAM MINI WX; 1000 fps, 0.25 ms shutter) captured the flow at resolution 0.067 mm pixel
$^{-1}$
, with backscatter illumination. The ambient pressure and temperature are 101.3 kPa and
$293.5\pm 0.5$
K, respectively. The surface tension (
$\sigma$
) of the water–air interfaces is 67.5 mN m
$^{-1}$
(measured with a Kruss K20 tensiometer). The densities of air (
$\rho _l$
) and water (
$\rho _h$
) are 1.2 kg m
$^{-3}$
and 997.0 kg m
$^{-3}$
, respectively. The viscosities of air (
$\mu _l$
) and water (
$\mu _h$
) are
$1.8\times 10^{-2}$
mPa s and 1.0 mPa s, respectively. The Atwood numbers of the inner interface
$A_i = (\rho _h-\rho _l)/(\rho _h+\rho _l)$
and outer interface
$A_o = (\rho _l-\rho _h)/(\rho _h+\rho _l)$
are 0.9976 and
$-0.9976$
, respectively. The weighted viscosity coefficient
$\nu = (\mu _h+\mu _l)/(\rho _h+\rho _l)$
is
$1.02\times 10^{-6}$
N s m
$^{-2}$
.
Initial interface parameters, where
$n$
denotes the azimuthal wavenumber,
$R_i^0$
and
$R_o^0$
denote the initial radii of the inner and outer interfaces, respectively, and
$a_i^0$
and
$a_o^0$
denote the initial amplitudes of the inner and outer interfaces, respectively.
Table 1. Long description
A table comparing initial interface parameters for different cases. The table has four rows and eight columns. The columns are labeled Case, n, R_i^0 (mm), R_o^0 (mm), a_i^0 (mm), a_o^0 (mm), |n a_i^0 / 2 pi R_i^0|, and |n a_o^0 / 2 pi R_o^0|. The rows are labeled IP, UP, and AP. Row 1: IP, 12, 25.0, 33.0, 1.0, 1.0, 0.076, 0.058. Row 2: UP, 12, 25.0, 33.0, 1.0, 0, 0.076, 0. Row 3: AP, 12, 25.0, 33.0, 1.0, -1.0, 0.076, 0.058.
Figure 2(a) shows the time-dependent radii of the inner interface
$R_i$
and outer interface
$R_o$
measured from experiments. The uncertainty in radius measurements is approximately
$\pm 1$
pixel (0.067 mm) due to the edge detection algorithm. The inner and outer radii of the water layer in the three cases exhibit good repeatability and display a linear growth trend over the experimental duration (20 ms). Dimensionless analysis uses time-dependent Reynolds numbers (
${\textit{Re}}_{i/o}$
) and Weber numbers (
${\textit{We}}_{i/o}$
) for the inner/outer interface:
\begin{equation} {\textit{Re}}_{i/o}=\frac {\dot {R}_{i/o}R_{i/o}}{n\nu },\quad {\textit{We}}_{i/o}=\frac {(\rho _h+\rho _l)R_{i/o}}{\sigma n}A_{i/o}^2\dot {R}_{i/o}^2. \end{equation}
The measured time-dependent Reynolds and Weber numbers are presented in figure 2(b). The Reynolds numbers range from 500 to 800, and the Weber numbers range from 2 to 5. Although
${\textit{Re}} \gg 1$
generally indicates negligible viscous effects, our measured Reynolds numbers are well above the critical threshold 256, below which viscosity becomes important for interfacial instability, confirming that the system operates in the inviscid regime (Walchli & Thornber Reference Walchli and Thornber2017). Conversely, while
${\textit{We}} \gg 1$
would typically suggest negligible surface tension effects, our observed Weber numbers 2–5 fall within the transitional regime, where surface tension significantly influences interface dynamics (Hsiang & Faeth Reference Hsiang and Faeth1992). To further assess the influence of boundary layers developing on the acrylic plates, we estimate the laminar displacement thickness
$\delta ^*=1.72\sqrt {\nu x/U}$
, where
$x=0.0043$
m is the radial displacement of the outer interface at 20 ms in the IP case,
$U=\dot {R}\approx 0.28$
m s−1, and
$\nu =\mu _h/\rho _h=1.0\times 10^{-6}$
m
$^2$
s−1. This gives
$\delta ^*\approx 0.21$
mm, which is only
$7\,\%$
of the gap height (3.0 mm). Therefore, the boundary layers are negligible, and the flow can be treated as inviscid for the purpose of interfacial instability dynamics. In contrast, surface tension plays a significant role in the development of the cylindrical water layer.
(a) Time-dependent radii of the inner interface
$R_i$
and outer interface
$R_o$
measured from experiments. (b) Time-dependent Reynolds numbers (
${\textit{Re}}_{i/o}$
) and Weber numbers (
${\textit{We}}_{i/o}$
) for the inner/outer interfaces.
3. Results and discussion
Figure 3 illustrates the evolution of successive cylindrically divergent water–air interfaces under various initial conditions. The inner water–air interface is marked in red, and the outer one in blue. During expansion, the perturbation on the inner interface grows because the instability occurs when the lighter fluid (air) pushes against the heavier fluid (water). In contrast, the perturbation on the outer interface is suppressed because the configuration is stable when the heavier fluid (water) pushes against the lighter fluid (air). In the IP and AP cases, the perturbation on the outer interface almost disappears by approximately 12 ms, similar to the UP case where the outer interface is initially unperturbed. Subsequently, the perturbation on the outer interface reverses and becomes prominent at later times, as seen at 20 ms in the IP case. Notably, the bubble–spike symmetry on the inner interface remains largely preserved, whereas the outer interface becomes polygonal, with spikes becoming narrow and sharp, and bubbles becoming wide and flat. This well-defined interface morphology enables precise amplitude measurements via contour analysis.
Evolution of a cylindrically divergent water layer under various initial conditions, where the outer edges of the inner and outer interfaces are marked with red and blue dashed lines, respectively.
Time-varying dimensionless amplitudes of water–air interfaces in various cases. Symbols represent the experimental results, while solid and dashed lines represent the linear model predictions for the inner and outer interfaces, respectively.
Owing to the large difference in refractive index between water and air, the interface contour is clearly observable, and the outer edges of the bubble and spike of the outer and inner interfaces can be easily distinguished, as indicated by the blue and red dashed lines in figure 3, respectively. The perturbation amplitude is defined as half the difference in radius between bubbles and spikes. Figure 4 shows the time-varying dimensionless amplitudes of the inner interface (
$\alpha _i$
) and outer interface (
$\alpha _o$
) for the three experimental cases (see symbols). Time is scaled as
$\tau = t\dot {R}_i / R_i^0$
, and the amplitude is scaled as
$\alpha _{i/o} = n (a_{i/o} - a_{i/o}^0) / R_{i/o}^0$
. The uncertainty in the perturbation amplitude is
$\pm 0.067$
mm, and the corresponding dimensionless amplitude has uncertainty approximately
$\pm 0.03$
. These uncertainties are smaller than the symbol sizes in figure 4.
For the inner interface in the IP case,
$\alpha _i$
grows continuously, peaks at
$\tau \approx 0.1$
, then decreases due to surface tension suppression (Liang & Luo Reference Liang and Luo2025). In contrast, after
$\tau \approx 0.1$
,
$\alpha _i$
remains nearly constant in the UP case, and increases slightly in the AP case. For the outer interface in the UP case,
$\alpha _o$
grows slightly due to the weak feedthrough between the two interfaces. In the IP and AP cases,
$\alpha _o$
increases continuously, with the IP case showing a slightly higher
$\alpha _o$
after
$\tau \approx 0.1$
. It is noted that the definition of the dimensionless amplitude causes
$\alpha _o$
to grow from zero, making the phase reversal not directly evident, and leading to an apparent monotonic increase. Overall, the initial phase difference between the two interfaces has a non-negligible influence on the instability evolution, as explained in figure 5.
Sketch of the development of inner and outer interfaces before and after the reversal of the outer interface in cases (a) IP and (b) AP. Straight arrows illustrate the growth direction of the interface perturbation, and arcs with arrows illustrate the vorticity deposition.
In the IP case, where the two interfaces are initially in-phase, before the perturbation on the outer interface reverses, the bubble/spike of the inner interface grows opposite to the spike/bubble of the outer interface. Because the water layer is incompressible, the outer interface suppresses perturbation growth on the inner interface. After the reversal of the outer interface, the bubble/spike of the inner interface grows in the same direction as that of the outer interface, so the inner interface promotes the growth of the outer interface. In contrast, in the AP case, where the two interfaces are initially anti-phase, before the reversal of the outer interface, the bubble/spike of the inner interface grows in the same direction as that of the outer interface; thus the outer interface promotes growth on the inner interface. After the reversal of the outer interface, the inner interface grows opposite to the outer interface, so the inner interface suppresses growth on the outer interface.
We consider a system of three incompressible, inviscid and initially irrotational fluids separated by two interfaces with small perturbations
$\eta _i$
and
$\eta _o$
(see (2.1)) in cylindrical geometry (Guo et al. Reference Guo, Wang, Ye, Wu and Zhang2017). The densities of the inner, middle and outer fluids are, respectively,
$\rho _1$
(
$0 \lt R \lt R_i$
),
$\rho _2$
(
$R_i \lt R \lt R_o$
) and
$\rho _3$
(
$R_o \lt R \lt \infty$
). A small single-mode perturbation on the inner interface (
$\eta _i$
) and outer interface (
$\eta _o$
) is written as
\begin{equation} \eta _i(\theta ,t)=R_i+a_i\cos (n\theta ),\quad \eta _o(\theta ,t)=R_o+a_o\cos (n\theta ). \end{equation}
The velocity potentials for the inner fluid (
$\phi _1$
), middle fluid (
$\phi _2$
) and outer fluid (
$\phi _3$
) satisfy the kinematic continuity condition for normal velocity components across the inner interface at
$R=R_i$
,
\begin{equation} \frac {\partial \eta _i}{\partial t}+\frac {1}{R^2}\frac {\partial \eta _i}{\partial \theta }\frac {\partial \phi _1}{\partial \theta }-\frac {\partial \phi _1}{\partial R}= \frac {\partial \eta _i}{\partial t}+\frac {1}{R^2}\frac {\partial \eta _i}{\partial \theta }\frac {\partial \phi _2}{\partial \theta }-\frac {\partial \phi _2}{\partial R}=0, \end{equation}
and across the outer interface at
$R=R_o$
,
\begin{equation} \frac {\partial \eta _o}{\partial t}+\frac {1}{R^2}\frac {\partial \eta _o}{\partial \theta }\frac {\partial \phi _2}{\partial \theta }-\frac {\partial \phi _2}{\partial R}= \frac {\partial \eta _o}{\partial t}+\frac {1}{R^2}\frac {\partial \eta _o}{\partial \theta }\frac {\partial \phi _3}{\partial \theta }-\frac {\partial \phi _3}{\partial R}=0. \end{equation}
Then we can obtain the perturbed velocity potentials as
\begin{equation} \phi _1=d_1(t)\,R^n\cos (n\theta ), \end{equation}
\begin{equation} \phi _2=[d_2(t)\,R^n+d_3(t)\,R^{-n}]\cos (n\theta ), \end{equation}
\begin{equation} \phi _3=d_4(t)\,R^{-n}\cos (n\theta ). \end{equation}
At the inner interface, normal stress continuity including surface tension (
$\sigma _i$
) yields
\begin{equation} \begin{aligned} &-\rho _1\left [\frac {\partial \phi _1}{\partial t}+\frac {1}{2}\left (\frac {\partial \phi _1}{\partial R}\right )^2+\frac {1}{2R^2}\left (\frac {\partial \phi _1}{\partial \theta }\right )^2+gR_i\right ]+\sigma _i\kappa _i\\ &\quad +\rho _2\left [\frac {\partial \phi _2}{\partial t}+\frac {1}{2}\left (\frac {\partial \phi _2}{\partial R}\right )^2+\frac {1}{2R^2}\left (\frac {\partial \phi _2}{\partial \theta }\right )^2+gR_i\right ]=0. \end{aligned} \end{equation}
At the outer interface, normal stress continuity including surface tension (
$\sigma _o$
) yields:
\begin{equation} \begin{aligned} &-\rho _2\left [\frac {\partial \phi _2}{\partial t}+\frac {1}{2}\left (\frac {\partial \phi _2}{\partial R}\right )^2+\frac {1}{2R^2}\left (\frac {\partial \phi _2}{\partial \theta }\right )^2+gR_o\right ]+\sigma _o\kappa _o\\ &\quad {}+\rho _3\left [\frac {\partial \phi _3}{\partial t}+\frac {1}{2}\left (\frac {\partial \phi _3}{\partial R}\right )^2+\frac {1}{2R^2}\left (\frac {\partial \phi _3}{\partial \theta }\right )^2+gR_o\right ]=0. \end{aligned} \end{equation}
The terms
$ ({\partial \phi }/{\partial R} )^2$
are of second order and are therefore neglected. The curvatures of the inner interface (
$\kappa _i$
) and outer interface (
$\kappa _o$
) can be approximated as
\begin{equation} \kappa _i\approx \frac {1}{R_i}-\frac {a_i(n^2-1)\cos (n\theta )}{R_i^2},\quad \kappa _o\approx \frac {1}{R_o}-\frac {a_o(n^2-1)\cos (n\theta )}{R_o^2}. \end{equation}
Collecting terms proportional to
$\cos (n\theta )$
in (3.7) and (3.8) gives the governing equations for cylindrical RTI of successive interfaces considering surface tension:
\begin{equation} \ddot {a}_i-\frac {(1+A_i)\zeta ^{-n+1}}{1+A_i\zeta ^{-2n}}\ddot {a}_o+2\frac {\dot {R}_i}{R_i}\dot {a}_i-\frac {1-\zeta ^{-2n}}{1+A_i\zeta ^{-2n}}\left [\frac {nA_ig}{R_i}-\frac {\sigma _i(n^2-1)n}{(\rho _1+\rho _2)R_i^3}\right ]a_i=0, \end{equation}
\begin{equation} \!\! \ddot {a}_o-\frac {(1-A_o)\zeta ^{-n-1}}{1-A_o\zeta ^{-2n}}\ddot {a}_i+2\frac {\dot {R}_o}{R_o}\dot {a}_o-\frac {1-\zeta ^{-2n}}{1-A_o\zeta ^{-2n}}\left [\frac {nA_og}{R_o}-\frac {\sigma _o(n^2-1)n}{(\rho _2+\rho _3)R_o^3}\right ]a_o=0, \end{equation}
with
$\zeta =R_o/R_i$
. The present model is consistent, in the incompressible, zero-acceleration, zero-surface-tension limit, with the linear model for thin-shell cylindrical RTI proposed by Yuan et al. (Reference Yuan, Zhao, Liu, Wang, Liu and Lu2024).
Figure 4 shows the linear theory predictions for the inner interface (solid lines) and outer interface (dashed lines). The initial amplitude growth rates of the inner and outer interfaces,
$\dot {a}_i^0 = nA_i a_i^+ \dot {R}_i/R_i^0$
and
$\dot {a}_o^0 = nA_o a_o^+ \dot {R}_o/R_o^0$
, follow cylindrical RMI theory (Mikaelian Reference Mikaelian2005), where
$a_i^+$
and
$a_o^+$
are the suddenly-injected-flow compressed amplitudes of the inner and outer interfaces measured from experiments. The theoretical predictions agree well with experiments in all near-unity Atwood number cases, confirming that under
$|A_i|=|A_o|=1.0$
conditions: (i) when the two interfaces are initially in-phase, the feedthrough suppresses the instability on the inner interface and promotes it on the outer interface; (ii) when initially anti-phase, the feedthrough promotes the instability on the inner interface and suppresses it on the outer interface; and (iii) the inner interface instability in the case where the outer interface is initially unperturbed lies between the in-phase and anti-phase cases. A small but noticeable discrepancy is observed for the inner interface in the IP case at
$\tau \approx 0.17$
, where the experimental amplitude is slightly lower than the linear prediction. This deviation is likely caused by minor fluctuations in the injected air flow rate (approximately
$\pm 2\,\%$
around the nominal 25 l min−1).
For cylindrical RTI, assuming exponential form for the perturbation amplitudes, (3.10) and (3.11) give the growth factors of the inner interface (
$\gamma _i$
) and outer interface (
$\gamma _o$
) satisfying
\begin{equation} \gamma _i^2-\frac {a_o^0}{a_i^0}\frac {(1+A_i)\zeta ^{-n+1}}{1+A_i\zeta ^{-2n}}\gamma _o^2+2\frac {\dot {R}_i}{R_i}\gamma _i-\frac {1-\zeta ^{-2n}}{1+A_i\zeta ^{-2n}}\left [\frac {nA_ig}{R_i}-\frac {\sigma _i(n^2-1)n}{(\rho _1+\rho _2)R_i^3}\right ]=0, \end{equation}
\begin{align} \gamma _o^2-\frac {a_i^0}{a_o^0}\frac {(1-A_o)\zeta ^{-n-1}}{1-A_o\zeta ^{-2n}}\gamma _i^2+2\frac {\dot {R}_o}{R_o}\gamma _o-\frac {1-\zeta ^{-2n}}{1-A_o\zeta ^{-2n}}\left [\frac {nA_og}{R_o}-\frac {\sigma _o(n^2-1)n}{(\rho _2+\rho _3)R_o^3}\right ]=0. \end{align}
Explicit expressions for
$\gamma _i$
and
$\gamma _o$
can only be achieved under the conditions
$\dot {R}_i=\dot {R}_o=0$
. Physically, this condition corresponds to a quasi-static state where the mean radii of the interfaces do not change with time, i.e. the interfaces are stationary or the radial motion is momentarily zero. Under this condition, the expression for
$\gamma _i$
and
$\gamma _o$
can be derived as
\begin{equation} \gamma _i^2=\epsilon _i(\gamma _i^\infty )^2,\quad \gamma _o^2=\epsilon _o(\gamma _o^\infty )^2, \end{equation}
where
$\gamma _i^\infty$
and
$\gamma _o^\infty$
represent the growth factors of the independent inner and outer interfaces (
$\zeta \rightarrow \infty$
), respectively:
\begin{equation} (\gamma _i^\infty )^2=\frac {nA_ig}{R_i}-\frac {\sigma _i(n^2-1)n}{(\rho _1+\rho _2)R_i^3},\quad (\gamma _o^\infty )^2=\frac {nA_og}{R_o}-\frac {\sigma _o(n^2-1)n}{(\rho _2+\rho _3)R_o^3}. \end{equation}
Here,
$\epsilon _i$
and
$\epsilon _o$
are the feedthrough efficiency factors of the inner and outer interfaces, respectively:
\begin{equation} \epsilon _i=\frac {1-A_o\zeta ^{-2n}}{1+A_iA_o\zeta ^{-2n}}+\frac {a_o^0}{a_i^0}\frac {(1+A_i)\zeta ^{-n+1}}{1+A_iA_o\zeta ^{-2n}}\left (\frac {\gamma _o^\infty }{\gamma _i^\infty }\right )^2, \end{equation}
\begin{equation} \epsilon _o=\frac {1+A_i\zeta ^{-2n}}{1+A_iA_o\zeta ^{-2n}}+\frac {a_i^0}{a_o^0}\frac {(1-A_o)\zeta ^{-n-1}}{1+A_iA_o\zeta ^{-2n}}\left (\frac {\gamma _i^\infty }{\gamma _o^\infty }\right )^2. \end{equation}
The feedthrough efficiency factors
$\epsilon _i$
and
$\epsilon _o$
are defined as the ratios
$\gamma _i^2/(\gamma _i^\infty )^2$
and
$\gamma _o^2/(\gamma _o^\infty )^2$
. This definition is natural: efficiency 1 corresponds to no feedthrough (the growth rate equals that of an independent interface); values different from 1 indicate that coupling either enhances (
$\epsilon \gt 1$
) or suppresses (
$\epsilon \lt 1$
) the instability relative to the independent case. Since
$\epsilon _i\gt 0$
and
$\epsilon _o\gt 0$
, the perturbation growth trend of the two successive interfaces is solely determined by the instability growth of the independent inner and outer interfaces: if
$ [{nA_ig}/{R_i}-( {\sigma _i(n^2-1)n})/({(\rho _1+\rho _2)R_i^3}) ]\lt 0$
for the inner interface (or
$ [{nA_og}/{R_o}-( {\sigma _o(n^2-1)n})/({(\rho _1+\rho _2)R_o^3} )]\lt 0$
for the outer interface), then the perturbation growth presents an oscillatory solution; if it equals zero, then we have a freeze-out solution; if it is positive, then we have an unstable solution. Because the interface acceleration achievable in the laboratory is limited or even zero (as in the present study), the oscillatory solution is observed in our experiments (e.g. case IP in figure 4).
Now we discuss the influence of physical parameters such as the radius ratio of two interfaces, interface acceleration, perturbation wavenumber on the feedthrough between the two interfaces according to (3.16) and (3.17) under near-unity Atwood number (
$A_i=1.0$
,
$A_o=-1.0$
), same initial perturbation amplitude (
$a_i^0=a_o^0$
) and same surface tension coefficient (
$\sigma _i=\sigma _o$
) conditions. First, for the zero acceleration case
$g=0$
, the feedthrough efficiency factors reduce to
\begin{equation} \epsilon _i|_{g=0}=\frac {1+\zeta ^{-2n}+2\zeta ^{-n-2}}{1-\zeta ^{-2n}},\quad \epsilon _o|_{g=0}=\frac {1+\zeta ^{-2n}+2\zeta ^{-n+2}}{1-\zeta ^{-2n}}. \end{equation}
Second, for an extremely large acceleration case (e.g. ICF and supernovae) where
$g\rightarrow \infty$
, the feedthrough efficiency factors reduce to
\begin{equation} \epsilon _i|_{g\rightarrow \infty }=\epsilon _o|_{g\rightarrow \infty }=\frac {1+\zeta ^{-2n}-2\zeta ^{-n}}{1-\zeta ^{-2n}}. \end{equation}
Figure 6 shows the variation of
$\epsilon _i$
and
$\epsilon _o$
versus
$\zeta$
in the range 1.0–2.0, for
$g=0$
and
$g\rightarrow \infty$
. As
$\zeta \rightarrow 1$
, the feedthrough strength between the two interfaces increases, making the two interfaces more unstable under
$g=0$
(since
$\epsilon _i\gt 1$
and
$\epsilon _o\gt 1$
), but more stable under
$g\rightarrow \infty$
(since
$\epsilon _i\lt 1$
and
$\epsilon _o\lt 1$
). Moreover, as the azimuthal wavenumber
$n$
increases, the values of
$\epsilon _i$
and
$\epsilon _o$
approach 1.0, indicating weaker feedthrough. Furthermore, the feedthrough effect on the outer interface is stronger than that on the inner interface under
$g=0$
, and the difference diminishes as
$n$
increases. In contrast, under
$g\rightarrow \infty$
, the feedthrough effect is the same for both interfaces. It is encouraging to see that the feedthrough suppresses instabilities on both interfaces under
$g\rightarrow \infty$
conditions, especially at lower wavenumbers. This is crucial for ICF experiments, where suppressing hydrodynamic instability is key to minimising mixing between the hot fuel and the cold pusher, and improving nuclear ignition efficiency.
Feedthrough efficiency factors
$\epsilon _i$
and
$\epsilon _o$
versus radius ratio
$\zeta$
for various azimuthal wavenumber
$n$
under
$A_i=1.0$
,
$A_o=-1.0$
,
$a_i=a_o$
and
$\sigma _i=\sigma _o$
conditions.
The predictions in figure 6 are derived from the linear theory under the quasi-static assumption and the additional simplifications listed above. While the theory itself has been validated against our experiments for the near-unity Atwood number regime, the specific quantitative trends for
$\epsilon _i$
and
$\epsilon _o$
versus
$\zeta$
in the limits
$g=0$
and
$g\rightarrow \infty$
have not been experimentally verified in the present study. Moreover, in actual ICF or supernova environments, compressibility, turbulence and nonlinear effects would modify the evolution; therefore, the present results should be interpreted as a qualitative trend rather than a quantitative prediction for those extreme conditions.
4. Conclusions
This study systematically examines the feedthrough between two successive cylindrical water–air interfaces in a cylindrically divergent (outward-expanding) geometry. Hydrophobic-coated rings are used to generate a water layer with controllable initial conditions. A linear model coupling the inner and outer perturbations is developed and validated against high-resolution experiments in the near-unity Atwood number regime.
The initial phase difference between the two interfaces profoundly alters the direction of feedthrough. In the initially in-phase (IP) case, the outer interface suppresses the inner perturbation before the outer interface reverses, but the inner interface promotes the outer perturbation afterwards. In the initially anti-phase (AP) case, the outer interface promotes the inner perturbation before reversal, but the inner interface suppresses the outer perturbation afterwards. The inner interface growth in the initially unperturbed outer interface (UP) case lies intermediate between the IP and AP cases. The agreement between theory and experiment demonstrates that the linear coupling model captures the essential physics of feedthrough in layered flows.
The derived feedthrough efficiency factors
$\epsilon _i$
and
$\epsilon _o$
show that the coupling strength depends on various physical parameters. Under zero acceleration (
$g=0$
, as in the present experiments), the feedthrough enhances instabilities on both interfaces. In contrast, under extremely high acceleration (
$g\rightarrow \infty$
, relevant to ICF), the feedthrough might suppress instabilities on both interfaces, with stronger suppression at lower wavenumbers. This suppression, if qualitatively applicable, might be relevant for ICF target design, as it may minimise fuel-pusher mixing and help to maintain temperature stratification. These results provide critical new insights into hydrodynamic instabilities in both radially convergent (e.g. ICF) and radially divergent (e.g. supernova) geometries.
Funding
This work was supported by the Natural Science Foundation of China (12572256, 12388101, 12441203 and 24FAA00998) and the Chinese Academy of Sciences Project for Young Scientists in Basic Research (grant no. YSBR-132).
Declaration of interests
The authors report no conflict of interest.
Appendix. Surface tension effects
To quantify the role of surface tension, which lies in the transitional Weber number regime (
${\textit{We}} = 2{-}5$
), we used our linear model to compute the interface amplitudes of both inner and outer interfaces in the three cases with surface tension set to zero, as shown in figure 7. Without surface tension, the amplitudes of the inner interfaces grow monotonically and do not exhibit the peak and subsequent decrease observed in the experiment. Surface tension suppresses the growth of the inner interface after
$\tau \approx 0.05$
. In contrast, under zero surface tension conditions, the dimensionless amplitude growth of the outer interface slightly decreases because of weaker feedthrough from the inner interface to the outer interface. These results demonstrate that surface tension plays a significant, non-negligible role in our experiments, especially in suppressing the inner interface instability.
Time-varying dimensionless amplitudes of the water–air interfaces predicted by the linear models. Solid and dashed lines denote predictions with and without surface tension, respectively.
n
Ri0
Ro0
ai0
ao0
Ri
Ro
Rei/o
Wei/o
ϵi
ϵo
ζ
n
Ai=1.0
Ao=−1.0
ai=ao
σi=σo
