1. Introduction
Rayleigh–Taylor (R–T) instability (Rayleigh Reference Rayleigh1883; Taylor Reference Taylor1950), which occurs when a lighter fluid accelerates a heavier one and forms asymmetric structures that lead to turbulence, is a major obstacle to achieving energy gain in inertial confinement fusion (ICF) (Betti & Hurricane Reference Betti and Hurricane2016; Zhou, Sadler & Hurricane Reference Zhou, Sadler and Hurricane2025). Saffman–Taylor (S–T) instability (Saffman & Taylor Reference Saffman and Taylor1958), which occurs when a less viscous fluid displaces a more viscous one in a confined space, is key to designing effective injection strategies for enhanced oil recovery (Stokes et al. Reference Stokes, Weitz, Gollub, Dougherty, Robbins, Chaikin and Lindsay1986; Li et al. Reference Li, Yang, Pahlavan, Zhang, Hu and Chen2024). In ICF, the deuterium–tritium inner layer is gaseous, while the outer fuel layer can be solid or liquid. This creates multiphase interfaces with significant density and viscosity differences where hydrodynamic instabilities, particularly at liquid–gas interfaces, become prominent.
The coupling of R–T and S–T instabilities is observed when a gas displaces a high-viscosity liquid in a multiphase system. There is an ongoing controversy as to the effect of viscosity contrast on liquid–gas R–T instability and of density contrast on S–T instability, which has left the coupled interfacial dynamics poorly understood. Crucially, whether viscosity contrast merely dampens the R–T instability growth or actively triggers a new instability pathway through the S–T mechanism remains unclear. This ambiguity defines the problem of Rayleigh–Saffman–Taylor (R–S–T) instability: What is the dominant physical mechanism governing the instability when both density and viscosity contrasts are significant, and how does it dictate the transition to nonlinearity? Can an idealised experimental configuration be designed to precisely study the R–S–T instability?
Answering this requires dissecting the competing roles of normal-stress-driven R–T instability, viscous-stress-driven S–T instability and the restoring force of surface tension in a confined geometry where all effects are coupled. A radial Hele-Shaw cell consists of two parallel plates separated by a narrow space, providing an ideal platform for this study and allowing precise measurement (Morrow, Cuttle & MacMinn Reference Morrow, Cuttle and MacMinn2023; Liang & Luo Reference Liang and Luo2025).
Theoretically, Mikaelian (Reference Mikaelian2005) derived the linear solution for cylindrical R–T instability, where the interface amplitude
$a$
follows
\begin{align} \ddot {a}+2\frac {\dot {R}}{R}\dot {a}-(nA_{{d}}-1)\frac {\ddot {R}}{R}a=0, \end{align}
with
$n$
as the azimuthal wavenumber,
$R$
as the interface radius and
$A_{d}$
(
${=}(\rho _{{h}}-\rho _{{l}})/ (\rho _{{h}}+\rho _{{l}})$
, where
$\rho _{{h}}$
and
$\rho _{{l}}$
are the densities of heavier and lighter fluids, respectively) characterising density contrast. Moreover, Wilson (Reference Wilson1975) derived the linear solution for cylindrical S–T instability considering surface tension:
\begin{align} \dot {a}-\left [(nA_{\mu }-1)\frac {\dot {R}}{R}-\frac {h^2}{12}\frac {\sigma }{R^3}\frac {n(n^2-1)}{\mu _h+\mu _l}\right ]a=0, \end{align}
with
$h$
as the space height of a Hele-Shaw cell,
$\sigma$
as the surface tension coefficient and
$A_{\mu }$
(
${=}(\mu _h-\mu _l)/(\mu _h+\mu _l)$
, where
$\mu _h$
and
$\mu _l$
are the viscosity coefficients of more and less viscous fluids, respectively) characterising viscosity contrast. Let us suppose that the interface is contracting, so that
$\dot {R} \lt 0$
. Then the interface is unstable even if
$A_{{d}}=0$
in (1.1) or
$A_{{\mu }}=0$
in (1.2), in other words, if only one fluid is present. This geometry-driven amplification of perturbations, known as the Bell–Plesset effect (Bell Reference Bell1951; Plesset Reference Plesset1954), also manifests in radial Hele-Shaw cells – making them a valid simplified system for studying spherical interfacial instabilities in ICF (Zhou Reference Zhou2024).
The R–T instability in a cylindrical geometry with time-varying acceleration exhibits fundamental differences from its planar counterpart under constant acceleration. From the geometric perspective, Yu & Livescu (Reference Yu and Livescu2008) demonstrated that acceleration direction strongly influences cylindrical R–T instability: compared with planar R–T instability, inward gravity (contraction) enhances growth, while outward gravity (expansion) suppresses it. Zhao et al. (Reference Zhao, Wang, Liu and Lu2020) extended the Layzer model to cylindrical geometry, developing a nonlinear model for two-dimensional single-mode R–T instability. Their analysis revealed that unlike planar R–T instability where the bubble velocity
$v_{{b}}$
approaches a constant, in cylindrical geometry the value of
$v_{{b}}/\sqrt {a+R}$
asymptotically tends to a constant. From the acceleration perspective, recent systematic studies have employed group theory (Abarzhi et al. Reference Abarzhi, Bhowmick, Naveh, Pandian, Swisher, Stellingwerf and Arnett2019; Hill et al. Reference Hill, Bhowmick, Ilyin and Abarzhi2019; Abarzhi & Williams Reference Abarzhi and Williams2020; Chan et al. Reference Chan, Jain, Hwang, Naveh and Abarzhi2023) to investigate R–T instability under time- and space-varying acceleration. These works focus on power-law acceleration profiles
$t^n$
. It has been discovered that the instability character depends on the exponent
$n$
: R–T-instability-like behaviour occurs when
$n\gt -2$
, whereas behaviour resembling Richtmyer–Meshkov (R–M) instability (Richtmyer Reference Richtmyer1960; Meshkov Reference Meshkov1969) – a specialised form of R–T instability triggered by impulsive acceleration – emerges when
$n\lt -2$
. The transition between R–T and R–M instabilities occurs at
$n=-2$
(Hill et al. Reference Hill, Bhowmick, Ilyin and Abarzhi2019; Abarzhi & Williams Reference Abarzhi and Williams2020; Chan et al. Reference Chan, Jain, Hwang, Naveh and Abarzhi2023).
In this work, we aimed to determine whether density contrast or viscosity contrast serves as the primary destabilising mechanism governing interfacial instability in a confined geometry. Experiments on cylindrically divergent water–air and glycerin–air interfaces were compared. We developed a unified model that quantitatively evaluates each physical effect and derived various solutions for the instability. This work establishes a new paradigm for manipulating hydrodynamic instabilities under high density and viscosity contrasts.
2. Experimental method
We employed two transparent parallel acrylic sheets to construct a radial Hele-Shaw cell (gap height
$h = 3.0$
mm) to study the instability growth (see figure 1
a). Generating a well-defined liquid–gas interface with well-controlled initial conditions remains a challenge. A key innovation in this study lies in the use of hydrophobic-coated acrylic rings embedded within microchannels on the cell surfaces (marked in red), which effectively increase the liquid–air interface contact angle (typically
$\gt 90^{\circ }$
) and shape the initial inner liquid–gas interface into a well-defined single-mode perturbation described by
$\eta = R_0 + a_0 \cos (n\theta )$
, at an initial radius
$R_0 = 25.0$
mm (see figure 1
b). Table 1 summarises the experimental parameters. The outer liquid–gas interface at
$R_L = 100.0$
mm is similarly constrained. The region between the inner and outer rings is not coated with hydrophobic material. The Hele-Shaw cell is filled with either water or glycerin between the inner and outer hydrophobic rings, and the instability is driven by impulsively injecting air at a controlled rate of 25 l min–1 through a central port. The initial amplitude-to-wavelength ratio (
$na_0/2\mathrm{\pi }R_0$
) is fixed at 0.076 (
${\lt } 0.1$
) to ensure initial linear evolution.
Schematic of the experimental set-up (a), initial interface configuration (b) and sketches of velocity profiles of uniform distribution (c) and Poiseuille flow (d) in the Hele-Shaw cell.
Initial interface parameters, where
$n$
and
$a_0$
denote the azimuthal wavenumber and initial amplitude of the liquid–gas interface, respectively;
$A_{{d}}$
and
$A_{{\mu }}$
denote the density contrast and viscosity contrast, respectively.
A high-speed camera captured the flow with backscatter illumination. The ambient pressure and temperature are 101.3 kPa and
$293.5\pm 0.5$
K, respectively. The surface tension coefficients of water–air and glycerin–air interfaces are 67.5 and 60.89 mN m, respectively. The densities of air, water and glycerin are 1.2, 997.0 and 1240.0 kg m
$^{-3}$
, respectively. The viscosities of air, water and glycerin are 1.8
$\times$
10
$^{-2}$
, 1.0 and 887 mPa s, respectively. Large density contrast
$A_{{d}}$
and viscosity contrast
$A_{{\mu }}$
are achieved on the two types of liquid–gas interfaces, as listed in table 1.
The experimentally measured radius of a water–air interface is well described by a cubic fit:
$R = R_0 + 0.118t + 2.797t^2 - 11.643t^3$
, while that of a glycerin–air interface follows
$R = R_0 + 0.061t + 1.264t^2 - 4.597t^3$
. The liquid–air flow constitutes an impulsive Poiseuille-type outward radial flow within the Hele-Shaw cell (Saffman & Taylor Reference Saffman and Taylor1958), which typically achieves full development after the characteristic viscous diffusion time scale across the cell gap,
$t_p \sim h^2/\nu$
(Carles et al. Reference Carles, Huang, Carbone and Rosenblatt2006; Wang, Xue & Han Reference Wang, Xue and Han2021), with
$\nu$
(
${=}(\mu _h+\mu _l)/(\rho _{{h}}+\rho _{{l}})$
) as the weighted viscosity coefficient. For the water–air interface (
$t_p\sim 8.85$
s, significantly longer than the experimental duration of 120 ms), the velocity profile perpendicular to the flow plane exhibits a uniform distribution (see figure 1
c), and the flow remains within an inertia-dominated regime governed by R–T instability. In contrast, for the glycerin–air interface (
$t_p\sim 12.6$
ms), viscous effects dominate early in the process. Here, the perpendicular velocity profile adopts a Poiseuille-type distribution (see figure 1
d), leading to the coupling of R–T instability with S–T mechanisms and resulting in a combined R–S–T instability.
To minimise boundary-layer effects, a gap height of 3.0 mm was used in this study – larger than the typical gap (
${\sim} 1$
mm) employed in many classic Hele-Shaw experiments. Although the Bond number is of order unity (
$1.30$
for water–air and
$1.79$
for glycerin–air), the vertical Péclet number (which compares gravity-driven vertical advection with diffusion) is small (
${\ll}1$
) for both fluid pairs. Furthermore, the experimental duration (120 ms) is considerably shorter than the characteristic time required for gravity to establish vertical stratification. The interface contour remains thin relative to the perturbation amplitude and wavelength (see figure 3), indicating limited three-dimensional effects induced by gravity. Consequently, the three-dimensional gravity-driven phenomena reported by Petitjeans & Maxworthy (Reference Petitjeans and Maxworthy1996) do not arise in our experiments.
The Weber number (We) and Reynolds number (Re) of a liquid–gas interface are calculated using
\begin{align} \textit{We}=\frac {(\rho _{{h}}+\rho _{{l}})R}{n\sigma }A_{{d}}^2\dot {R}^2,\quad {Re}=\frac {\dot {R}R}{n\nu }, \end{align}
and shown in figure 2. The Weber number is slightly higher for the water–air interface than for glycerin–air due to its faster expansion rate. Our measured We values (
$2{-}25$
) lie within the transitional regime (
$\textit{We} \lt 80$
), where surface tension exerts a significant influence on interface evolution, consistent with earlier droplet deformation studies (Hsiang & Faeth Reference Hsiang and Faeth1992). In contrast, the Reynolds number differs markedly between systems: the water–air interface exhibits Re (
${\gt } 1000$
) far exceeding the critical value of 256, confirming operation in the inviscid regime (Walchli & Thornber Reference Walchli and Thornber2017), whereas the glycerin–air interface yields very low Re (
$1.0{-}3.5$
), indicating strongly viscosity-dominated dynamics. This stark contrast arises from the three-order-of-magnitude difference in viscosity between water and glycerin.
Time-varying (a) Weber number (We) and (b) Reynolds number (Re) of water–air and glycerin–air interfaces.
3. Results and discussion
Figures 3(a) and 3(b) show the evolution of radially divergent water–air and glycerin–air interfaces under different wavenumber conditions, respectively. During expansion, instability grows faster at lower wavenumbers on water–air interfaces but at higher wavenumbers on glycerin–air interfaces. Water–air interfaces maintain symmetry between bubbles (lighter fluid penetrating heavier fluid) and spikes (heavier fluid penetrating lighter fluid) (e.g. water-n6 at
$t=80$
ms), indicating weak nonlinearity. Differently, glycerin–air interfaces develop pronounced asymmetry – spikes narrow and elongate while bubbles widen – signalling strong nonlinearity driven by viscosity contrast (e.g. glycerin-n9 at
$t=120$
ms).
Evolution of radially divergent water–air interfaces (a) and glycerin–air interfaces (b).
Through image processing, we extracted the
$r{-}\theta$
coordinates of interface contours and applied fast Fourier transform to quantify the temporal evolution of the amplitudes of the first three modes (see figure 4). Amplitudes are normalised as
$a_n/a_0$
(where
$a_n$
is the
$n$
th-mode amplitude) and time scaled by
$t/t^*$
(
$t^*= R_0/\dot {R}_l$
, with
$\dot {R}_l$
being the linearly fitted interface velocity during the final 60 ms). For water–air interfaces, the instability growth is dominated by the first-order (fundamental) mode (
$a_1$
), with higher-order modes remaining negligible. Unlike the planar geometry under constant acceleration, the liquid–air interface in the present study expands under decaying acceleration, leading to a slower progression of the R–T instability. For glycerin–air interfaces, we observed a pronounced negative second-order mode (
$a_2$
) concurrent with a vigorous growth of the third-order mode (
$a_3$
). This specific modal coupling is a hallmark of S–T instability-driven dynamics: the second-order mode enhances bubble–spike asymmetry, while the rapid growth of the third-order mode directly triggers an early transition to nonlinearity, promoting fingering bifurcation through surface tension destabilising the advancing finger tip (Miranda & Widom Reference Miranda and Widom1998). Remarkably, at
$n=9$
, the first-order mode amplitude for the glycerin–air interface is about six times greater than that for the water–air interface. This unequivocally demonstrates that viscosity contrast, through the S–T mechanism, is the primary driver of destabilisation in this regime.
Time-varying amplitudes of the first-order modes (red symbols), second-order modes (blue symbols) and third-order modes (green symbols) of water–air interfaces (a) and glycerin–air interfaces (b).
As Homsy (Reference Homsy1987) outlines, the classical S–T instability in radial source flow is governed by a dimensionless growth rate of the form (equation (10) in Homsy (Reference Homsy1987))
\begin{align} \gamma =A_{\mu }n-1-{\textit{Ca}'}n(n^2-1)(A_{\mu }+1)/2, \end{align}
where
${\textit{Ca}'}$
is a modified capillary number (
${=}12\mu _h\dot {R}(R/h)^2/\sigma$
). For high viscosity contrast (
$A_{\mu }\approx 1$
), this theory predicts that all modes with (equation (11) in Homsy (Reference Homsy1987))
\begin{align} n\leq \frac {1}{2} \big[1+(1+4{\textit{Ca}'})^{(1/2)} \big] \end{align}
are linearly unstable. In our experiments, as the interface expands, the range of
$n$
for the water–air interface (
$A_{\mu }\approx 1$
,
${\textit{Ca}'}\sim O$
(
$10^0{-}10^1$
)) is 1.8–4.9, while that for the glycerin–air interface (
$A_{\mu }\approx 1$
,
${\textit{Ca}'}\sim O$
(
$10^2{-}10^3$
)) is 27.7–77.3. The wavenumbers examined in this study (
$n=3$
, 6 and 9) fall within the glycerin–air range but outside the water–air range (except for the water-n3 case), indicating that instability in the water–air interface is driven primarily by density contrast rather than viscosity contrast, whereas the glycerin–air instability is indeed dominated by viscosity contrast. The S–T theory proposed by Homsy (Reference Homsy1987) – although it accurately predicts the destabilising effect of viscosity contrast – is based on a constant interface velocity and does not account for time-dependent acceleration. Consequently, an interpretation based solely on S–T instability is insufficient for our system.
To quantitatively predict the R–S–T instability growth, a new linear model, that goes beyond a straightforward superposition of existing solutions for R–T instability (1.1) and S–T instability (1.2), is required. First, the flow field is assumed to be irrotational and the viscous component is included only in the normal stress balance at the interface. Second, we assumed Poiseuille flow has been fully developed in the Hele-Shaw cell with a gap
$h$
(see figure 1
d). Considering the large aspect ratio
$R/h$
, the viscous drag is mainly created by the velocity gradients in the thin
$z$
direction (Logvinov Reference Logvinov2016; Wang et al. Reference Wang, Xue and Han2021).
The velocity potential for the lighter fluid (
$\phi _l$
) and heavier fluid (
$\phi _h$
) satisfies the kinematic continuity condition for normal velocity components across the interface:
\begin{align} -\left(\frac {\partial \phi _l}{\partial r}\right)_{\!\eta }=-\left(\frac {\partial \phi _h}{\partial r}\right)_{\!\eta }=\dot {R}+\dot {a}\cos (n\theta ). \end{align}
Then, we can obtain the perturbed velocity potential for each side of the interface as
\begin{eqnarray} \begin{aligned} &\phi _l=-R\dot {R}\ln r-\frac {1}{n}(\dot {a}R+a\dot {R})\frac {r^n}{R^n}\cos (n\theta ),\\ &\phi _h=-R\dot {R}\ln r+\frac {1}{n}(\dot {a}R+a\dot {R})\frac {R^n}{r^n}\cos (n\theta ). \end{aligned} \end{eqnarray}
The expressions for the pressure in the lighter fluid (
$p_l$
) and heavier fluid (
$p_h$
) can be evaluated using the Bernoulli integral (Wang et al. Reference Wang, Xue and Han2021):
\begin{eqnarray} \begin{aligned} & p_l=F_l+\rho _{{l}}\left [\left(\frac {\partial \phi _l}{\partial t}\right)_{\eta }-\frac {3}{5}\left(\frac {\partial \phi _l}{\partial r}\right)_{\eta }^2\right ]+\frac {12\mu _l}{h^2}(\phi _l)_{\eta },\\ &p_h=F_h+\rho _{{h}}\left [\left(\frac {\partial \phi _h}{\partial t}\right)_{\eta }-\frac {3}{5}\left(\frac {\partial \phi _h}{\partial r}\right)_{\eta }^2\right ]+\frac {12\mu _h}{h^2}(\phi _h)_{\eta }, \end{aligned} \end{eqnarray}
where
$F_l$
and
$F_h$
are integration constants. The terms
$({\partial \phi }/{\partial r})_{\eta }^2$
are of second order and are therefore to be neglected. At the interface, normal stress continuity including surface tension and viscous component yields
\begin{align} -p_l+2\mu _l\left (\frac {\partial u_{l}}{\partial r}\right )_{\eta }+\sigma \kappa =-p_h+2\mu _h\left (\frac {\partial u_{h}}{\partial r}\right )_{\eta } \!, \end{align}
with interface curvature
$\kappa$
approximated as
\begin{align} \kappa =\frac {\left|\dfrac {{\rm d}^2\eta }{{\rm d}s^2} \right|}{\left[1+\left( {\rm d}\eta /{\rm d}s\right)^2\right]^{3/2}}\approx \frac {1}{R}-\frac {a(n^2-1)\cos (n\theta )}{R^2}, \end{align}
where
$s$
is the arc length parameter. Collecting terms proportional to
$\cos (n\theta )$
in (3.6) gives the governing equation for radial R–S–T instability:
\begin{align} \ddot {a} & +\frac {2\dot {R}}{R}\dot {a}-(nA_d-1)\frac {\ddot {R}}{R}a + \left [\frac {\sigma (n^2-1)n}{(\rho _{{h}}+\rho _{{l}})R^3}\right ]a \nonumber \\& +\frac {2n\nu }{R^2}\left [(n+A_{{\mu }})\dot {a}+\frac {\dot {R}}{R}(n-A_{{\mu }})a\right ]+Z\frac {12\nu }{h^2}\left [\dot {a}-\frac {\dot {R}}{R}(nA_\mu -1)a\right ]=0, \end{align}
where
$Z$
is a dimensionless factor for adjusting the contribution of S–T instability. This linear model incorporates the governing physics: inertial R–T instability (line 1), surface tension oscillation (line 2, term 1), viscous damping (line 2, term 2) and the crucial S–T destabilisation driven by viscosity contrast (line 2, term 3).
(a) Time-varying experimental interfacial amplitudes in cases water-n9 (triangle symbols), glycerin-n9 (circle symbols), glycerin-n6 (diamond symbols) and glycerin-n3 (square symbols), where the solid, dotted, dashed and dash-dot lines (colour-matched to the symbols) represent the theoretical predictions from (3.8) with
$Z=0$
, (1.2) and (3.8) with
$Z=1$
, and (3.8) with
$Z=0.2$
, respectively. (b) Contributions of surface tension (solid lines), viscous damping (dash-dot lines) and S–T destabilisation (dashed lines) to the R–T instability.
Figure 5(a) compares theoretical predictions and measured experimental amplitudes. The pure R–T instability model ((3.8) with
$Z=0$
, solid line) agrees well for water–air interfaces while fails for glycerin–air interfaces due to a lack of S–T destabilisation. The classic S–T instability model ((1.2), dotted line) grossly overpredicts growth of glycerin–air interfaces. Our full R–S–T instability model ((3.8) with
$Z=1$
, dashed line) captures the trend but still overestimates the growth of glycerin–air interfaces, particularly after the onset of nonlinearity (
$t/t^* \gt 0.1$
). This indicates that the linear S–T instability term overestimates the destabilisation effect as fingers develop. We found that introducing a phenomenological damping factor
$Z = 0.2$
to the S–T instability term (dash-dot line) yields remarkable agreement with the data. This factor physically represents the saturation of the instability due to early nonlinearity induced by high viscosity contrast. This reduction could reflect several effects that emerge during finger development: (i) increased curvature at the spike tips enhances local surface tension, counteracting further destabilisation; (ii) transverse flow between developing fingers suppresses instability growth, analogous to bubble competition in R–T dynamics; and (iii) the rapid transition into the nonlinear regime violates the exponential-growth assumption of linear theory. Therefore,
$Z$
is not merely a fitting parameter, but rather a phenomenological representation of the early nonlinear saturation that modulates the linear S–T destabilisation term in (3.8).
We quantified the influence of each physical mechanism – surface tension, viscous damping and S–T destabilisation – by computing the amplitude difference
$a_{{diff}}$
between the full linear model and a version that selectively neglects the corresponding term. The results reveal three key features: all three effects intensify with wavenumber
$n$
, highlighting the strongly mode-dependent nature of the instability; surface tension provides the weakest influence; and, most notably, the destabilising role of the S–T mechanism nearly balances the stabilisation from viscous damping at
$n=6$
, with S–T instability dominating at lower wavenumbers (
$n\lt 6$
) and viscous damping prevailing at higher ones (
$n\gt 6$
). However, it should be noted that at higher wavenumbers (
$n\gg 9$
) or lower Weber numbers (
$\textit{We}\ll 1$
), surface tension can become a dominant stabilising mechanism.
For cylindrical R–S–T instability, the interfacial growth follows an exponential behaviour as given by (3.8), where the growth rate
$\gamma$
satisfies
\begin{align} \gamma ^2+\xi \gamma -\zeta =0, \end{align}
with
\begin{align} \xi =\frac {2n\nu }{R^2}(n+A_{{\mu }})+\frac {12\nu }{h^2} \end{align}
and
\begin{align} \zeta =\frac {(nA_d-1)\ddot {R}}{R}-\frac {\sigma (n^2-1)n}{(\rho _{{h}}+\rho _{{l}})R^3}-\frac {2n\dot {R}}{R^3}(n-A_{{\mu }})\nu +Z\frac {12\nu }{h^2}\frac {\dot {R}}{R}(nA_\mu -1). \end{align}
The growth rate
$\gamma$
is then expressed as
\begin{align} \gamma =-\frac {\xi }{2}+\sqrt {\frac {\xi ^2}{4}+\zeta }. \end{align}
In the context of radial R–S–T instability within a Hele-Shaw cell, the value of
$\zeta$
determines three regimes of instability: unstable solution (
$\zeta \gt 0$
), freeze-out solution (
$\zeta =0$
) and oscillatory solution (
$\zeta \lt 0$
). Considering high density and viscosity ratios (
$A_{{d}}=1$
,
$A_{{\mu }}=1$
and
$\rho _{{l}}=0$
), the most unstable azimuthal wavenumber
$n_u$
is derived from
$\dot {\zeta }=0$
and given by
\begin{align} n_u=\frac {\rho _{{h}}}{6\sigma }\left (-4\dot {R}\nu +\sqrt {16\dot {R}^2\nu ^2+\frac {12\sigma }{\rho _{{h}}}\left (\ddot {R}R^2+2\dot {R}\nu +\frac {\sigma }{\rho _{{h}}}+Z\frac {12\nu }{h^2}\dot {R}R^2\right )}\right ) \!. \end{align}
Furthermore, in ICF experiments, suppressing hydrodynamic instability via the freeze-out condition (
$\zeta = 0$
) is essential to reduce mixing between the hot fuel and the cold pusher, thereby enhancing nuclear ignition efficiency. The freeze-out wavenumber
$n_{\kern-1.5pt f}$
, obtained by solving
$\zeta = 0$
, is
\begin{align} n_{\kern-1.5pt f}=-\frac {\rho _{{h}}}{2\sigma }\left (2\dot {R}\nu +\frac {\sigma }{\rho _{{h}}}\right )+\frac {\rho _{{h}}}{2\sigma }\sqrt {\left (2\dot {R}\nu +\frac {\sigma }{\rho _{{h}}}\right )^2+\frac {4\sigma }{\rho _{{h}}}\left (R^2\ddot {R}+Z\frac {12\dot {R}R^2\nu }{h^2}\right )}.\end{align}
The time-varying most unstable wavenumber
$n_u$
, freeze-out wavenumber
$n_{\kern-1.5pt f}$
and the radius acceleration
$\ddot {R}$
obtained from cubic fitting are presented in figure 6. Acceleration
$\ddot {R}$
decreases linearly, leading to non-monotonic growth in both
$n_u$
and
$n_{\kern-1.5pt f}$
. Initially,
$n_u$
is 7; it then increases with time, reaching a maximum value of 18 at
$t/t^*=0.89$
. Similarly,
$n_{\kern-1.5pt f}$
starts at 12 and increases to a maximum value of 31 at
$t/t^*=0.89$
.
Time-dependent most unstable wavenumbers
$n_u$
(red) and freeze-out wavenumbers
$n_{\kern-1.5pt f}$
(blue) for glycerin–air interfaces. The inset displays the temporal evolution of the radius acceleration of the interfaces.
These results indicate that the growth behaviour of the cylindrical R–S–T instability evolves over time due to changes in the radius histories (
$R$
,
$\dot {R}$
and
$\ddot {R}$
). As a result, a mode that is initially stable (e.g.
$n=15$
) may become unstable during interface expansion and perturbation growth – for instance,
$t/t^*=0.1$
– and may subsequently restabilise, repeating this cyclic pattern. Likewise, a mode that is initially unstable (e.g.
$n=9$
) can become the most unstable mode at certain times (e.g.
$t/t^*=0.1$
), then lose that status, in a recurring manner. Overall, this work offers new insights into the manipulation of hydrodynamic instabilities through surface tension and viscosity, demonstrating that the critical parameters
$n_u$
and
$n_{\kern-1.5pt f}$
can be used to rapidly enhance or suppress instability growth.
It is instructive to contrast the present findings with our earlier work on water–air interfaces (Liang & Luo Reference Liang and Luo2025), where surface tension dominated the interfacial dynamics, leading to oscillatory or freeze-out behaviours at higher wavenumbers. In that regime, viscosity contrast was negligible, and instability growth was primarily governed by the balance between inertial acceleration and capillary restoring forces. In the current study, however, the large viscosity contrast in glycerin–air interfaces activates the S–T mechanism, which overcomes both viscous damping and surface tension stabilisation, driving accelerated nonlinearity and early fingering. This comparison underscores that interfacial evolution in radially expanding geometries is highly sensitive to fluid properties: when viscosity contrast is small, surface tension controls stability; when viscosity contrast is large, the S–T mechanism becomes the dominant driver of destabilisation.
4. Conclusions
This study decouples the interplay between inertial-driven R–T and stress-driven S–T instabilities in the evolution of interfaces under normal and viscous stresses. Using a radial hydrophobic ring technique, we conducted experiments on cylindrically divergent hydrodynamic instabilities at water–air and glycerin–air interfaces. Contrary to expectations, the higher-viscosity glycerin–air interface exhibited significantly greater instability than the water–air interface. Contrary to classical R–T instability, we demonstrated that viscosity contrast can be a powerful driver of instability through the S–T mechanism, overwhelming its traditional damping role. This results in accelerated nonlinear evolution and earlier onset of fingering.
Our unified R–S–T instability model, which accounts for early nonlinear saturation due to high viscosity contrast, successfully describes the observed dynamics. The effects of surface tension, viscous damping and S–T destabilisation intensify with wavenumber
$n$
, highlighting the strongly mode-dependent nature of the instability. For the first time, we quantitatively demonstrated that surface tension has the weakest influence on instability development – most strikingly, the S–T destabilisation nearly balances viscous damping at an azimuthal wavenumber of 6. Moreover, we established criteria for the most unstable mode and the freeze-out mode, providing new insights into controlling hydrodynamic instabilities through surface tension and viscosity. This work clarifies the mechanisms of interfacial instability in systems where both normal and viscous stresses play essential roles.
Funding
This work was supported by the Natural Science Foundation of China (12572256, 12388101, 12441203 and 24FAA00998). Y.L. acknowledges partial support from Tamkeen under NYU Abu Dhabi Research Institute grant CG002.
Declaration of interests
The authors report no conflict of interest.
