Spatial linear instability analysis is employed to investigate the instability of a viscoelastic liquid jet in a co-flowing gas stream. The theoretical model incorporates a non-uniform axial base profile represented by a hyperbolic tangent, capturing the shear layer. The Oldroyd-B model discretised with Chebyshev polynomials is employed, and energy budget analysis is used to interpret underlying mechanisms. At low Weber numbers, the jet evolves axisymmetrically and the instability is governed by interfacial gas-pressure fluctuations; as the Weber number increases, the growing inertia drives a transition of the predominant mode from axisymmetric to helical. At weak elasticity, the instability is also primarily governed by gas-pressure fluctuations. As elasticity increases, the predominant mode transitions from axisymmetric to helical. This transition is accompanied by a migration of disturbance structures from the interface toward the jet interior and an enhanced coupling between velocity perturbation and the basic flow. These trends reveal a new predominant instability mechanism – the elasticity-enhanced shear-driven instability – which is distinct from capillary or Kelvin–Helmholtz instabilities in Newtonian jets. A $\textit{We}$–$El$ phase diagram delineates the boundary between predominant modes and experimental results obtained in a flow-focusing configuration validate the theoretical predictions. Compared with temporal stability results, the spatial framework – by directly resolving the convective downstream amplification of disturbances – achieves quantitative agreement with experiments and highlights the superiority of spatial instability analysis in capturing the dynamics of strongly convective, non-parallel jet flows. These findings provide mechanistic insight into viscoelastic jet instabilities and offer guidance for applications involving droplet and fibre formation in co-flow systems.

The vortex-induced vibration of multiple spring-mounted bodies free to move in the orthogonal direction of the flow is investigated. In a first step, we derive a linear arbitrary Lagrangian–Eulerian method to solve the fluid–structure linear problem as well as a forced problem where a harmonic motion of the bodies is imposed. We then propose a low computational-cost impedance-based criterion to predict the instability thresholds. A global stability analysis of the fluid–structure system is then performed for a tandem of cylinders and the instability thresholds obtained are found to be in perfect agreement with the predictions of the impedance-based criterion. An extensive parametric study is then performed for a tandem of cylinders and the effects of mass, damping and spacing between the bodies are investigated. Finally we also apply the impedance-based method to a three-body system to show its validity to a higher number of bodies.

The stability of the interface in a core–annular flow (CAF) of two immiscible Newtonian fluids with contrasting densities has been investigated, emphasising the role of strong circumferential rotation for the first time. The aim of the investigation is to give insight into the physical mechanisms underlying interfacial disruption. We examine the combined effects of gravity, interfacial tension, axial and azimuthal shear stresses, and centrifugal force on interface stability. The Rayleigh–Taylor instability, induced by gravity, appears as a spiral mode with a azimuthal wavenumber of one. As gravitational effects decrease, the most unstable mode number increases sharply before decreasing with increasing rotation. This non-monotonic behaviour is attributed to the interplay between azimuthal shear and centrifugal acceleration. We demonstrate that this velocity ratio fundamentally governs the onset of spiral modes by varying the ratio of the axial velocities of the core and annular fluids. Higher Reynolds numbers in the annular phase promote the emergence of higher-order spiral modes concomitant with amplified azimuthal shear at the interface. In a parametric study of the gap between the core and pipe wall, we identified a suppressive effect of reduced annular thickness on the growth of higher azimuthal wavenumbers. An energy budget analysis further delineated distinct mechanisms underpinning each instability regime and clarified transitions between them. These findings extend our understanding of interfacial stability in swirling CAFs and provide a predictive framework to control spiral-mode selection.

A joint experimental–computational investigation was conducted to examine the aerodynamic behaviour of a partially closed cavity model in Mach-6 flow. The model, consisting of a flat plate with a rectangular cavity and a forward-facing hinged door, resulted in a strong 500 Hz fluctuation with a 7.5$^\circ$ door and 25 mm cavity depth. The experiments revealed a recirculation bubble present upstream of the cavity region. The fluctuations, detected by surface pressure sensors on the upper surface, upstream cavity wall and cavity floor, were caused by oscillations of the separation bubble along the streamwise axis. Notably, this phenomenon is not explained by established empirical models for cavity flows, such as the Rossiter mechanism or closed-box acoustic resonance. To further elucidate the flow physics, detached eddy simulations (DESs) of the flow were conducted, providing a detailed understanding of the complex flow phenomena. The DES results complemented the experimental data, offering insights into the unsteady flow behaviour and the mechanisms driving the pressure fluctuations. Additional experiments and simulations were conducted for other door angles to simulate different stages of opening. The strong pressure fluctuations at approximately 500 Hz were only experimentally observed for door angles between 5.0$^\circ$ and 7.5$^\circ$ but were absent at much smaller and larger angles. Additionally, several cavity depths were tested, which demonstrated that a shallower cavity delayed the onset of fluctuations until a higher free-stream Reynolds number was reached. The combination of experimental and numerical results provides valuable initial data on the aerodynamic performance of a hypersonic forward-facing door over a cavity.

This paper examines two-dimensional liquid curtains ejected from a narrow horizontal outlet at an angle to the vertical. Curtains are characterised by the Froude number ${\textit{Fr}}=U/ ( gH ) ^{1/2}$, Reynolds number ${\textit{Re}}=UH/\nu$ and Weber number ${\textit{We}}=\rho U^{2}H/\sigma$, where $U$ is the ejection velocity, $g$ the gravity, $H$ the outlet’s half-width, $\nu$ the kinematic viscosity and $\sigma$ the surface tension. It is assumed that ${\textit{Fr}}\gg 1$ (so that the radius of the curtain’s curvature due to gravity exceeds $H$), ${\textit{Re}}\ll 1$ (viscosity is strong) and ${\textit{We}}\sim 1$ (surface tension is on par with inertia). It is shown that steady oblique curtains exist only subject to a constraint of the form ${\textit{We}}\gt f({\textit{Fr}}^{2}{\textit{Re}})$, which is more restrictive than the previously known constraint ${\textit{We}}\gt 1$. Thus, sufficiently strong viscosity and/or surface tension eliminate the steady regime and make the curtain evolve – typically, rotate around the outlet, eventually producing the teapot effect.

Upon radial liquid sheet expansion, a bounding rim forms, with a thickness and stability governed, in part, by the liquid influx from the unsteady connected sheet. We examine how the thickness and fragmentation of such a radially expanding rim change upon its severance from its sheet, absent of liquid influx. To do so, we design an experiment enabling the study of rims pre- and post-severance by vaporising the thin neck connecting the rim. No vaporisation occurs of the bulk rim itself. We confirm that the severed rim follows a ballistic motion, with a radial velocity inherited from the sheet at severance time. We identify that the severed rim undergoes fragmentation in two types of junctions: the base of inherited, pre-severance, ligaments and the junction between nascent rim corrugations, with no significant distinction between the two associated time scales. The number of ligaments and fragments formed is captured well by the theoretical prediction of rim corrugation and ligament wavenumbers established for unsteady expanding sheets upon droplet impact on surfaces of comparable size to the droplet. Our findings are robust to changes in impacting laser energy and initial droplet size. Finally, we report and analyse the re-formation of the rim on the expanding sheet and propose a prediction for its characteristic corrugation time scale. Our findings highlight the fundamental mechanisms governing interfacial destabilisation of connected fluid-fed expanding rims that become severed, thereby clarifying destabilisation of freely radially expanding toroidal fluid structures absent of fluid influx.

This study investigates the low-frequency dynamics of a turbulent separation bubble (TSB) forming over a backward-facing ramp, with a focus on large-scale coherent structures associated with the so-called `breathing motion’. Using time-resolved particle image velocimetry (PIV) in both streamwise and spanwise planes, we examine the role of sidewall confinement, an aspect largely overlooked in previous research. Spectral proper orthogonal decomposition (SPOD) of the streamwise velocity field reveals a dominant low-rank mode at low Strouhal numbers ($St \lt 0.05$), consistent with prior observations of TSB breathing. Strikingly, the spanwise-oriented PIV data uncover a previously unreported standing-wave pattern, characterised by discrete spanwise wavenumbers and nodal/antinodal structures, suggesting the presence of spanwise resonance. To explain these observations, we construct a resolvent-based model that imposes free-slip conditions at the sidewall locations by superposing left- and right-travelling three-dimensional modes. The model accurately reproduces the spanwise structure and frequency content of the measured SPOD modes, demonstrating that sidewall reflections lead to the formation of standing-wave-like patterns. Global stability analysis reveals a zero-frequency eigenmode originating from a centrifugal instability, giving rise to the observed low-frequency breathing. Downstream, the associated coherent structures are further amplified through non-modal lift-up mechanisms. Our findings highlight the critical influence of spanwise boundary conditions on the selection and structure of low-frequency modes in TSBs. This has direct implications for both experimental and numerical studies relying on spanwise-periodic boundary conditions and offers a low-order framework for predicting sidewall-induced modal dynamics in separated flows.

The linear stability of nanofluid boundary-layer flow over a flat plate is investigated using a two-phase formulation that incorporates the Brinkman (1952 J. Chem. Phys., vol. 20, pp. 571–581) model for viscosity along with Brownian motion (BM) and thermophoresis (TP), building upon the earlier work of Buongiorno (2006 J. Heat Transfer, vol. 128, pp. 240–250). Solutions to the steady boundary-layer equations reveal a thin nanoparticle concentration layer near the plate surface, with a characteristic thickness of $O({\textit{Re}}^{-1/2}{\textit{Sc}}^{-1/3})$, for a Reynolds number ${\textit{Re}}$ and Schmidt number ${\textit{Sc}}$. When BM and TP are neglected, the governing equations reduce to the standard Blasius formulation for a single-phase fluid, and the nanoparticle concentration layer disappears, resulting in a uniform concentration across the boundary layer. Neutral stability curves and critical conditions for the onset of the Tollmien–Schlichting (TS) wave are computed for a range of nanoparticle materials and volume concentrations. Results indicate that while the effects of BM and TP are negligible, the impact of nanoparticle density is significant. Denser nanoparticles, such as silver and copper, destabilise the TS wave, whereas lighter nanoparticles, like aluminium and silicon, establish a small stabilising effect. Additionally, the viscosity model plays a crucial role, with alternative formulations leading to different stability behaviour. Finally, a high Reynolds number asymptotic analysis is undertaken for the lower branch of the neutral stability curve.

The coupling between Rayleigh–Taylor (R–T) and Saffman–Taylor (S–T) instabilities, when a gas displaces a high-viscosity liquid, remains challenging to elucidate due to the unclear roles of density and viscosity contrasts. Counterintuitively, our radial Hele-Shaw cell experiments revealed that viscosity contrast – typically considered a damping factor – serves as the primary driver of instability. We observed that the glycerin–air interface, despite its higher viscosity, exhibits significantly greater instability than the water–air interface. This anomalous behaviour arises from the S–T mechanism, which accelerates the onset of nonlinearity and induces an early transition to fingering. We applied a unified model to decouple the competing influences of surface tension oscillation and viscous damping on R–T instability and the S–T destabilisation. Moreover, we proposed criteria for either mostly enhancing or completely freezing the instability. These findings offer valuable insights into manipulating hydrodynamic instabilities in contracting/expanding geometries through surface tension and viscosity.

We present a combined experimental and theoretical exploration of three-layer, horizontal core–annular pipe flow, in which two fluids are separated by a deformable elastic solid. In the experiments, an elastic solid created by an in situ chemical reaction maintains the separation of the core and annular fluids. Corrugations of the elastic interface are observed, and stable pipelining, where the elastic shell created separating the two fluids remains intact, is successfully demonstrated even when the core fluid is buoyant. The theoretical model combines lubrication theory for the fluids with standard shell theory for the elastic solid. The model is used to predict the buckling states resulting from radial compression of the shell, and to explore the sedimentation of a buoyant core. The self-sculpting of the shell by buckling cannot by itself generate hydrodynamic lift owing to symmetry in the direction of flow. Instead, we demonstrate that hydrodynamic lift can be achieved by other elastohydrodynamic effects, when that symmetry becomes broken during the bending of the shell.

This paper investigates the nonlinear dynamics of horizontal shear instability in an incompressible, stratified and rotating fluid in the non-traditional $f$-plane, i.e. with the full Coriolis acceleration, using direct numerical simulations. The study is restricted to two-dimensional horizontal perturbations. It is therefore independent of the vertical (traditional) Coriolis parameter. However, the flow has three velocity components due to the horizontal (non-traditional) Coriolis parameter. Three different scenarios of nonlinear evolution of the shear instability are identified, depending on the non-dimensional Brunt–Väisälä frequency $N$ and the non-dimensional non-traditional Coriolis parameter $\tilde {f}$ (non-dimensionalised by the maximum shear), in the range $\tilde {f}\lt N$ for fixed Reynolds and Schmidt numbers $ \textit{Re}=2000$, $ \textit{Sc}=1$. When the stratification is strong $N\gg 1$, the shear instability generates stable Kelvin–Helmholtz billows like in the traditional limit $\tilde {f}=0$. Furthermore, when $N\gg 1$, the governing equations for any $\tilde {f}$ can be transformed into those for $\tilde {f}=0$. This enables us to directly predict the characteristics of the flow depending on $\tilde {f}$ and $N$. When $N$ is around unity and $\tilde {f}$ is above a threshold, the primary Kelvin–Helmholtz vortex is destabilised by secondary instabilities but it remains coherent. For weaker stratification, $N\leqslant 0.5$ and $\tilde {f}$ large enough, secondary instabilities develop vigorously and destroy the primary vortex into small-scales turbulence. Concomitantly, the enstrophy rises to high values by stretching/tilting as in fully three-dimensional flows. A local analysis of the flow prior to the onset of secondary instabilities reveals that the Fjørtoft necessary condition for instability is satisfied, suggesting that they correspond to shear instabilities.

We investigate a novel Marangoni-induced instability that arises exclusively in diffuse fluid interfaces, that is absent in classical sharp-interface models. Using a validated phase-field Navier–Stokes–Allen–Cahn framework, we linearise the governing equations to analyse the onset and development of interfacial instability driven by solute-induced surface tension gradients. A critical interfacial thickness scaling inversely with the Marangoni number, $\delta _{\textit{cr}} \sim \textit{Ma}^{-1}$, emerges from the balance between advective and diffusive transport. Unlike sharp-interface scenarios where matched viscosity and diffusivity stabilise the interface, finite thickness induces asymmetric solute distributions and tangential velocity shifts that destabilise the system. We identify universal power-law scalings of velocity and concentration offsets with a modified Marangoni number $\textit{Ma}_\delta$, independent of capillary number and interfacial mobility. A critical crossover at $ \textit{Ma}_\delta \approx 590$ distinguishes diffusion-dominated stabilisation from advection-driven destabilisation. These findings highlight the importance of diffuse-interface effects in multiphase flows, with implications for miscible fluids, soft matter, and microfluidics where interfacial thickness and coupled transport phenomena are non-negligible.

We investigate the convective stability of a thin, infinite fluid layer with a rectangular cross-section, subject to imposed heat fluxes at the top and bottom and fixed temperature along the vertical sides. The instability threshold depends on the Prandtl number as well as the normalized flux difference ($f$) and decreases with the aspect ratio ($\epsilon$), following a $\epsilon f^{-1}$ power law. Using a three-dimensional (3-D) initial value and two-dimensional eigenvalue calculations, we identify a dominant 3-D mode characterized by two transverse standing waves attached to the domain edges. We characterize the dominant mode’s frequency and transverse wavenumber as functions of the Rayleigh number and aspect ratio. An analytical asymptotic solution for the base state in the bulk is obtained, valid over most of the domain and increasingly accurate for lower aspect ratios. A local stability analysis, based on the analytical base state, reveals oscillatory transverse instabilities consistent with the global instability characteristics. The source term for this most unstable mode appears to be interactions between vertical shear and horizontal temperature gradients.

In deep learning, interval neural networks are used to quantify the uncertainty of a pre-trained neural network. Suppose we are given a computational problem $P$ and a pre-trained neural network $\Phi _P$ that aims to solve $P$. An interval neural network is then a pair of neural networks $(\underline {\phi }, \overline {\phi })$, with the property that $\underline {\phi }(y) \leq \Phi _P(y) \leq \overline {\phi }(y)$ for all inputs $y$, where the inequalities are meant componentwise. $(\underline {\phi }, \overline {\phi })$ are specifically trained to quantify the uncertainty of $\Phi _P$, in the sense that the size of the interval $[\underline {\phi }(y),\overline {\phi }(y)]$ quantifies the uncertainty of the prediction $\Phi _P(y)$. In this paper, we investigate the phenomenon when algorithms cannot compute interval neural networks in the setting of inverse problems. We show that in the typical setting of a linear inverse problem, the problem of constructing an optimal pair of interval neural networks is non-computable, even with the assumption that the pre-trained neural network $\Phi _P$ is an optimal solution. In other words, there exist classes of training sets $\Omega$, such that there is no algorithm, even randomised (with probability $p \geq 1/2$), that computes an optimal pair of interval neural networks for each training set ${\mathcal{T}} \in \Omega$. This phenomenon happens even when we are given a pre-trained neural network $\Phi _{{\mathcal{T}}}$ that is optimal for $\mathcal{T}$. This phenomenon is intimately linked to instability in deep learning.

We study the stability of a steady Eckart streaming jet flowing in a closed cylindrical cavity. This configuration is a generic representation of industrial processes where driving flows in a cavity by means of acoustic forcing offers a contactless way of stirring or controlling flows. Successfully doing so, however, requires sufficient insight into the topology induced by the acoustic beam. This, in turn, raises the more fundamental question of whether the basic jet topology is stable and, when it is not, of the alternative states that end up being acoustically forced. To answer these questions, we consider a flow forced by an axisymmetric diffracting beam of attenuated sound waves emitted by a plane circular transducer at one cavity end. At the opposite end, the jet impingement drives recirculating structures spanning nearly the entire cavity radius. We rely on linear stability analysis (LSA) together with three-dimensional nonlinear simulations to identify the flow destabilisation mechanisms and to determine the bifurcation criticalities. We show that flow destabilisation is closely related to the impingement-driven recirculating structures, and that the ratio $C_R$ between the cavity and the maximum beam radii plays a key role on the flow stability. In total, we identified four mode types destabilising the flow. For $4 \leqslant C_R \leqslant 6$, a non-oscillatory perturbation rooted in the jet impingement triggers a supercritical bifurcation. For $C_R = 3$, the flow destabilises through a subcritical non-oscillatory bifurcation and we explain the topological change of the unstable perturbation by analysing its critical points. Further reducing $C_R$ increases the shear within the flow and gradually moves the instability origin to the shear layer between impingement-induced vortices: for $C_R = 2$, an unstable travelling wave grows out of a subcritical bifurcation, which becomes supercritical for $C_R=1$. For each geometry, the nonlinear three-dimensional (3-D) simulations confirm both the topology and the growth rate of the unstable perturbation returned by LSA. This study offers fundamental insight into the stability of acoustically driven flows in general, but also opens possible pathways to either induce turbulence acoustically or to avoid it in realistic configurations.

The effects of Reynolds number across ${\textit{Re}}=1000$, $2500$, $5000$ and $10\,000$ on separated flow over a two-dimensional NACA0012 airfoil at an angle of attack of $\alpha =14^\circ$ are investigated through biglobal resolvent analysis. We identify modal structures and energy amplifications over a range of frequencies, spanwise wavenumbers, and values of the discount parameter, providing insights across various time scales. Using temporal discounting, we find that the shear-layer dynamics dominates over short time horizons, while the wake dynamics becomes the primary amplification mechanism over long time horizons. Spanwise effects also appear over long time horizons, sustained by low frequencies. The low-frequency and high-wavenumber structures are found to be dominated by elliptic mechanisms within the recirculation region. At a fixed angle of attack and across the Reynolds numbers, the response modes shift from wake-dominated structures at low frequencies to shear-layer-dominated structures at higher frequencies. The frequency at which the dominant mechanism changes is independent of the Reynolds number. Comparisons at a different angle of attack ($\alpha =9^\circ$) show that the transition from wake to shear-layer dynamics with increasing frequency only occurs if the unsteady flow is three-dimensional. We also study the dominant frequencies associated with wake and shear-layer dynamics across the angles of attack and Reynolds numbers, and confirm characteristic scaling laws from the literature.

Direct numerical simulations of a uniform flow past a fixed spherical droplet are performed to determine the parameter range within which the axisymmetric flow becomes unstable. The problem is governed by three dimensionless parameters: the drop-to-fluid dynamic viscosity ratio, $\mu ^\ast$, and the external and internal Reynolds numbers, ${\textit{Re}}^e$ and ${\textit{Re}}^i$, which are defined using the kinematic viscosities of the external and internal fluids, respectively. The present study confirms the existence of a regime at low-to-moderate viscosity ratio where the axisymmetric flow breaks down due to an internal flow instability. In the initial stages of this bifurcation, the external flow remains axisymmetric, while the asymmetry is generated and grows only inside the droplet. As the disturbance propagates outward, the entire flow first transits to a biplanar-symmetric flow, characterised by two pairs of counter-rotating streamwise vortices in the wake. A detailed examination of the flow field reveals that the vorticity on the internal side of the droplet interface is driving the flow instability. Specifically, the bifurcation sets in once the maximum internal vorticity exceeds a critical value that decreases with increasing ${\textit{Re}}^i$. For sufficiently large ${\textit{Re}}^i$, internal flow bifurcation may occur at viscosity ratios of $\mu ^\ast = {\mathcal{O}}(10)$, an order of magnitude higher than previously reported values. Finally, we demonstrate that the internal flow bifurcation in the configuration of a fixed droplet in a uniform fluid stream is closely related to the first path instability experienced by a buoyant, deformable droplet of low-to-moderate $\mu ^\ast$ freely rising in a stagnant liquid.