The rheology of bubble suspensions is critical for the prediction and control of bubbly flows in various industrial processes. It is well known that bubble suspensions exhibit shear-thinning behaviour due to bubble shape deformation under pure shear, but the shear rheological response to dilatation under time-varying pressure remains unexplored. Here, we propose a constitutive equation for dilute bubble suspensions that accounts for both shear and dilatational effects, demonstrating that bubble compressibility can significantly influence the shear viscosity under time-varying pressures. Under constant-rate pressure variations, compression leads to a progressive reduction in viscosity, whereas decompression induces an increase. This peculiar compression-thinning behaviour arises microscopically from the fact that a shrinking bubble surface effectively weakens the flow resistance of the surrounding liquid. Under oscillatory pressure, the amplitude of the dilatation-induced viscosity grows with frequency, and the phase shift of the shear stress response transitions from $-\pi /2$ to $-\pi$. Notably, viscosity troughs emerge during oscillation cycles, leading to transient flow resistance lower than that in pure shear, indicating a potential route for drag reduction when combined with shear-thinning. These findings highlight bubble compressibility as a controllable factor for tuning bubbly flow rheology, with potential for enhanced flow efficiency in practical applications.

We perform axisymmetric numerical simulations to investigate the coalescence dynamics of a liquid drop in a deep liquid pool. This study aims to generalise the mechanisms of partial coalescence across a range of drop shapes, elucidate the underlying mechanism of neck oscillations, and examine the roles of inertial, viscous and gravitational forces, quantified by the Weber, Ohnesorge and Bond numbers, in governing the coalescence behaviour. A phase diagram is constructed to delineate the boundaries between partial and complete coalescence regimes based on these dimensionless parameters. Our analysis of the height-to-neck ratio shows that, upon contact with the pool, the primary drop forms an upward liquid column that ultimately pinches off due to inwardly directed horizontal momentum. Additionally, the study suggests that as the dimensionless numbers increase, the effect of the vertical collapse rate plays a significant role in the outcome of the coalescence process. Notably, the Rayleigh–Plateau instability is found to be insignificant in driving partial coalescence within the explored parameter space. We identified a transition regime between partial and complete coalescence, characterised by multiple neck oscillations that delay the pinch-off of secondary droplets. The formation of secondary droplets is most prominent for prolate drops, followed by spherical and oblate drops of comparable volume. Furthermore, we observe that the tendency to form multiple droplets from elongated liquid columns diminishes with an increase in the impact velocity of the primary drop.

The dispersion phenomenon of mass and heat transport in oscillatory flows has wide applications in environmental, physiological and microfluidic flows. The method of concentration moments is a powerful theoretical framework for analysing transport characteristics and is well developed for steady flows: general solution expressions of moments have been profoundly derived by Barton (J. Fluid Mech. 126, 1983, 205–218). However, it was thought that these expressions could not be directly applied to unsteady flows. Prior studies needed to re-solve the governing equations of moments from scratch, encountering the complication induced by the time-periodic velocity, leaving higher-order statistics like skewness and kurtosis analytically intractable except for specific cases. This work proposes a novel approach based on a two-time-variable extension to tackle these challenges. By introducing an auxiliary time variable, referred to as oscillation time to characterise the inherent oscillation in the dispersion due to the oscillating flow, the transport problem is extended to a two-time-variable system with a ‘steady’ flow term. This enables the direct use of Barton’s expressions and thus avoids the prior complication. This approach not only offers an intuitive physical perspective for the influence of the velocity oscillation, but also clarifies the solution structure of concentration moments. As a preliminary verification, we examine the transport problem in an oscillatory Couette flow. The analytical solution agrees well with the numerical result by Brownian dynamics simulations. The effects of the point-source release and the phase shift of velocity on the transport characteristics are investigated. By extending the classic steady-flow solution to the time-dependent flows, this work provides a versatile framework for transient dispersion analysis, enhancing predictions in oscillatory transport problems.

Impact pressure temporal and spatial profiles caused by a droplet on a surface can assist in understanding of microscale erosion mechanisms. We derive analytical solutions of pressure profiles and impact force time-series on a surface for early and intermediate impact times, up to dimensionless time $0.27$, using unsteady potential flow. The solutions reproduce the ‘ring pattern’ of surface pressure at early impact and the temporal evolution to a shift to, at intermediate time, the centred maximum. This is due to the emerging dominance of the steady component of the Bernoulli equation. The analytical solutions use a model of an unsteady disk in infinite liquid to induce flows similar to those within impacting drops. We identify a separation point close to the expanding edge of the impacted droplet, which interprets the Wagner condition for droplet impact and extends the validity of the analytical wet radius from time $\textit {O}(10^{-2})$ in the literature to beyond $\textit {O}(1)$. This separation point resolves singularity issues at the wet radius and solutions address the droplet impact problem. The theoretical predictions agree with high-fidelity numerical calculations of the impact pressure at the surface centre, middle and at the radius of the ring pressure, and with the impact force for more than four time decades. The theoretical predictions remain at least qualitatively correct in dimensionless time for more than $\textit {O}(1)$. The analytical solutions are closed and explicit functions of space, time, wet radius expanding velocity and incidence angle, and provide ready estimation of droplet impact loadings for erosion problems.

Tangential interactions between particles play a central role in suspension rheology. We show that surface roughness significantly enhances the strength of hydrodynamic interactions between closely separated particles in relative sliding motion. Using numerical solutions of the lubrication equation, we show that tangential forces due to sliding motion between rough spheres scale inversely with the separation distance, as opposed to the weaker logarithmic scaling for smooth spheres. A fully analytic theory identifies these features as the consequence of asperity-scale squeeze flows, quantitatively recovering the numerical results. These singular hydrodynamic forces are associated with similarly singular torques. The need to resolve the hydrodynamic singularity couples the particles’ rotation to their translation, and forces them to roll without slip, recovering a kinematic constraint that is central to understanding dense suspension rheology. Despite their purely hydrodynamic origin and occurring without contact, these features resemble several aspects of rolling and sliding contact friction.

The present study investigates the mass transfer of a dilute species from a dispersed bubbly phase to a carrier liquid phase using interface-resolved simulations and proposes a phenomenological model for its transient dynamics. To this end, we individually vary several input parameters, i.e. the species diffusivity (the Schmidt number, ${\textit{Sc}}$), the void fraction $(\alpha )$ and the bubble size (which affects the Galilei number ${\textit{Ga}}$ and Bond number ${\textit{Bo}}$) – while maintaining conditions representative of air bubbles in water. For the parametric range – ${\textit{Sc}}$ = $1,5$ and 10; $\alpha$ = 0.5 %, 1.9 % and 3.6 %; and three initial bubble diameters $d_0$ = $0.63\, \text{mm}$ $(\text{i.e.}\ Ga$ = $1.75$ and ${\textit{Bo}}$ = $0.0125)$, $1.2\, \text{mm}$ (${\textit{Ga}}$ = $4.6$ and ${\textit{Bo}}$ = 0.045) and $1.58\, \text{mm}$ (${\textit{Ga}}$ = $7$ and ${\textit{Bo}}$ = $0.07921$) – we show that increasing diffusivity and decreasing bubble size decreases scalar advection as compared with diffusion and hence the time needed for the species concentration to saturate in the carrier phase, which can be well represented by a Péclet number $Pe = {\sqrt {gr_0} r_0}/{D_c}$ based on the bubble rising speed, its radius and the species diffusivity. We also document the increase of the mass transfer rate with the void fraction when the carrier-phase scalar concentration is low, driven by a larger interfacial area and enhanced velocity fluctuations. When the species-rich bubble wakes start to interact, however, the transfer rate decreases, which occurs earlier at higher values of $\alpha$. The proposed phenomenological model agrees closely with simulations, capturing a self-similar temporal evolution of the mean carrier-phase concentration. By rescaling the non-dimensional time $\tau$ as ${Sc^{{2}/{3}}Ga}/{\alpha ^{0.45}}$, the numerical results collapse onto a master curve.

Travelling wave control is a promising technique for reducing turbulent skin friction by suppressing turbulence in boundary layers. This study presents experimental observations of the streamwise evolution of turbulence over a travelling wavy wall. Particle image velocimetry measurements were performed at multiple downstream locations. The travelling wave was generated by oscillating a rubber sheet to attain properties that are known to achieve drag reduction. The results reveal a two-stage process where the drag-reduction mechanism qualitatively changes in the streamwise direction. In the upstream region (up to approximately two wavelengths from the start of control), turbulent fluctuations are rapidly reduced, however, being limited to the near-wall region. Further downstream, the suppression effect diffuses in the wall-normal direction, leading to a modification on the edge of the boundary layer. The diffusion process of the turbulence-suppression effect is consistently interpreted within the framework of an internal boundary layer, whose development follows a power law. Two-point correlation analysis indicates that the wave crests initially disrupt the near-wall streaky structures, and subsequently reorganise into a characteristic state with a shorter streamwise coherence length downstream. While fully developed states have been studied previously, this work presents the streamwise development of turbulence suppression within a finite length, informing the design of practical drag-reduction devices.

We present direct numerical simulations of planar intrusions from a constant source into a linearly stratified ambient fluid for Reynolds numbers between $200$ and $5000$, inlet widths $W\geqslant 2.2 \sqrt {Q/N}$ and ambient layer thicknesses $H_a$ between $W$ and $20\sqrt {Q/N}$, where $Q$ is the supply rate (area per unit time) and $N$ is the buoyancy frequency. Across this broad parameter space, the intrusions form a universal self-similar shape with a constant thickness of approximately $2.2\sqrt {Q/N}$ at the source tapering towards a tip that propagates at a constant speed of approximately $0.7\sqrt {NQ}$. This broad-scale structure does not change regardless of whether the intrusions are subcritical or supercritical relative to internal waves. The perturbations to the ambient resulting from the intrusive flow appear as a near-universal uplift/depression of isopycnals immediately above/below the intrusion, upstream blocking ahead of the intrusion and, for subcritical intrusions, columnar disturbances. A moderate-amplitude wave train is also formed on the surface of subcritical intrusions. This appears at approximately the mid-length of the intrusion, with the waves propagating towards the tip. We also compare our results with the solution of the Mei shallow-water model. The comparison is poor and we rederive the model, carefully examining the underlying assumptions against the simulation data. The only assumption that is violated is that the ambient density is unperturbed. We present extensions to the model allowing for a density perturbation based on (i) simple data fits and (ii) a solution to the Dubreil-Jacotin–Long equation for shallow ambients. These significantly improve the predictive ability of the model for this geometry.

Turbulent line fountains are often operated in environments with a lateral density difference which are integral to engineering applications such as air curtains and bubble screens. While the dynamics of fountains in a uniform ambient (UA) has been widely studied, the underlying flow physics of their interaction with a lateral density stratification remain poorly understood. This study addresses this knowledge gap through a synergistic investigation combining high-fidelity large-eddy simulations and time-resolved particle image velocimetry measurements. We quantify the trajectory and spreading characteristics, flow statistics and entrainment dynamics of line fountains for a wide range of lateral density stratification and compare them with the canonical case of a UA. Our results reveal a new scaling law for the fountain’s trajectory $x_{_{cl}}$, which follows a noticeably steep power law ($x_{_{cl}} \propto z^2$) that is fundamentally different from jets in a unidirectional cross-flow. This is due to the bidirectional non-uniform lateral forcing and an adverse pressure gradient generated at the impingement surface. These mechanisms also oppose the fountain motion and result in a reduction in their centreline velocity compared with the UA cases. The spatial development of fountains is also affected by the lateral density stratification as it spreads more on the buoyant side exhibiting an asymmetry in the local half-width. The mean velocity statistics was shown to be symmetric if appropriate length scales are chosen for normalization on the buoyant and non-buoyant side. However, the turbulence profiles exhibit an asymmetric behaviour. A quantitative analysis of the fountain’s unsteadiness from experiments reveals that the lateral stratification actively energizes the natural, low-frequency flapping mode. The quantification of entrainment coefficient confirms that the fountains in laterally stratified ambient entrains up to 20 % more fluid than in a UA. These findings provide a deeper understanding of the structural and dynamical aspects of line fountains in complex, stratified environments.

Understanding fluid-elastic instabilities in slender bodies is crucial for predicting and controlling flow-induced vibration (FIV) in engineering and biological systems. The FIV of a prolate spheroid with an aspect ratio of $\epsilon = 3$, a mass ratio of $m^* = 3$ and a damping ratio of $\zeta = 0$, elastically mounted in a uniform flow at ${\textit{Re}} = 600$, are investigated using direct numerical simulations and a reduced-order model (ROM). As reduced velocity $U_r$ increases, five vibration states emerge: quasi-steady (QS), periodic (PM), large-amplitude chaotic (LAC), quasi-periodic (QP) and small-amplitude chaotic (SAC) modes. These mode regimes form two categories of response branches, namely synchronised branches (SB) and desynchronised branches (DB). In SB, three synchronisation mechanisms are identified, i.e. conventional lock-in (CLI), secondary-component lock-in (SCLI) and superharmonic lock-in (SHLI), corresponding to PM, LAC and QP modes, respectively. In contrast, DB comprises two types. The flow-dominated desynchronised (FDD) branch corresponds to QS mode, where flow instability dominates while structural vibrations remain weak. The dual-mode competition desynchronised (DMCD) branch corresponds to the SAC mode, where fluid and structural instabilities coexist but fail to synchronise. Analysis of wake dynamics identifies spanwise, transverse and high-frequency spanwise shedding patterns that are closely correlated with vibration regimes. The overlap of the response branches produces three distinct hysteresis zones, emphasising the sensitivity of spheroidal FIV to initial conditions and its inherently path-dependent behaviour. Dynamic mode decomposition (DMD) and an ERA-based ROM, which together resolve mode-specific spatial structures and frequency evolution, show that the vibration dynamics is governed by a persistently unstable wake mode (WM) and a structural mode (SM) whose stability alternates across branches. This clarifies the resulting sequence of vibration modes and provides insight into how different branches compete and transition.

Recent experiments by Daneshi & Frigaard (J. Fluid Mech., vol. 957, 2023, p. A16) examined the response of an initially spherical, stationary bubble in an elastic yield-stress material to stepwise ambient-pressure variations under two protocols. In the first protocol, pressure decreases and the bubble swells, elongates, and mobilises. In the second protocol, pressure decreases then increases; the bubble stays stationary, but its volume shows hysteresis between the two phases. This hysteresis was attributed to elastic non-recoverable strain. In the present study, we numerically investigate these two protocols, accounting for elasticity, residual stresses and nonlinear viscoelastic deformation before yielding, using the Saramito–Herschel–Bulkley model. In the first protocol, assuming constant bubble mass yields clear deviations from experiments in (a) the bubble radius evolution, (b) the pressure–volume product at different pressures and (c) the bubble mobilisation. These deviations are resolved by including mass transfer of gas from the surrounding material, which increases bubble mass. This is caused by the pressure reduction, which decreases the gas concentration at the bubble interface below the ambient value, generating a mass influx. In the second protocol, we demonstrate that hysteresis can be predicted only when mass transfer is included. Finally, we propose a simplified model to predict the bubble dynamics during either pressure protocol, which can also be used to extract the mass-transfer properties of gas–fluid systems in yield stress materials.

We study the stability of the bubble rising in the presence of a soluble surfactant numerically and experimentally. For the range of surfactant concentrations considered, the Marangoni stress almost immobilises the interface. However, the non-zero surface velocity is crucial to understanding the surfactant behaviour. Global linear stability analysis predicts the transition to an oblique path above the threshold of the Galilei number (the bubble radius). This transition is followed by the coexistence of stationary and oscillatory instabilities as the Galilei number increases. These predictions agree with the experimental observations without any fitting parameters. We evaluate the bubble deformation, hydrostatic pressure variation and perturbed viscous stress. The perturbation of the velocity field causes a destabilising vortex in the rear of the bubble, while the perturbed viscous stress produces a torque opposing this vortex. We found that the torque significantly decreases above the critical Galilei number, which may constitute the origin of instability. The linear stability analysis and the experiments were conducted for Surfynol, which can be regarded as a fast (fast-kinetics) surfactant. Our experiments show the considerable differences between the rising of bubbles in the presence of a fast and a non-fast surfactant.

Inspired by recent experiments demonstrating that vibrating elastic sheets can function as seemingly contactless suction cups, we investigate the elastohydrodynamic hovering of a thin elastic sheet vibrating near a wall. Previous theoretical work suggests that the hovering height results from a balance between the active forcing that triggers the vibrations, the bending stresses associated with the sheet’s deformation, the viscous lubrication flow between the sheet and the wall, and the sheet’s weight. Here, we extend this analysis beyond the asymptotic regime of weak forcing and explore the regime of strong forcing through numerical simulations. We identify the scalings for the equilibrium hovering height and the maximum load that can be supported. We further quantify the influence of fluid inertia and compressibility: both effects are found to introduce repulsive contributions to the net force on the sheet, which can significantly reduce its adhesive strength. Beyond providing insights into soft contactless grippers and swimming near surfaces, our analysis is relevant to the elastohydrodynamics of squeeze films and near-field acoustic levitation.

For regular reflection (RR) and Mach reflection (MR), the critical parameter of the trailing-edge height ($H_{R,min }$), at which the reflected shock grazes the trailing edge, is the critical condition for stable and unstable reflection. A proof of the statement that $H_{R,min }$ for MR is larger than $H_{R,min }$ for RR, within some region in the dual-solution domain, is important for confirming the existence of a dual-solution stability gap, within which RR is stable while MR is unstable. This proof is accomplished here by transitivity, with the intermediate value corresponding to the minimum height of the Mach stem. By establishing a bridge between the evaluation of $H_{R,min }$ for MR and that of the linear coefficients for Mach stem height variation with the trailing-edge height, we overcome the difficulty of quantifying $H_{R,min }$ exactly, and show that the difference between $H_{R,min }$ for MR and $H_{R,min }$ for RR is significant, meaning that there is a large enough dual-solution stability gap. The confirmation of this gap has further impact on shock transition, suggesting a new transition scenario: stable to unstable dynamic transition, i.e., within the dual-solution stability gap, a stable RR can undergo a dynamic transition to an unstable MR state (unstart flow) under suitable disturbance of the flow parameters. This dynamic transition is demonstrated here numerically. The time history of dynamic transitions displays (i) direct transitions from RR to MR to unstart flow, with complex flow structures such as hybrid MR–type VI shock interference and double MR–MR reflections, and (ii) inverted transitions, in which RR first shifts to MR and then returns back to RR.

We present a semi-analytic investigation of the resolvent operator, and its associated forcing and response modes for quasi-one-dimensional shock-laden flows. Using a Green’s function approach, we derive resolvent solutions for isentropic (subsonic and supersonic) and transonic flows with shocks in converging–diverging nozzles of arbitrary geometry. Our analysis demonstrates that shock-induced heightened sensitivity in the resolvent across flow discontinuities leads to significant discrepancies between numerically computed and the analytical input and output modes if shock effects are not properly accounted for. In particular, we find that the resolvent operator exhibits singular behaviour at the shock location. Specifically, the inviscid (where the shock is treated purely as a flow discontinuity) and viscous analytical leading resolvent modes do not converge as the viscosity parameter $\mu \rightarrow 0$, which affects the accuracy of flow control and stability analyses that rely on resolvent-based methods. Furthermore, the derived solutions serve as benchmarks for verifying numerical schemes designed to compute adjoint and resolvent modes in shock-laden flows, ensuring that they capture the correct physical behaviour in the presence of shocks.

Over 125 years ago, Henry Selby Hele-Shaw realised that the depth-averaged flow in thin-gap geometries can be closely approximated by two-dimensional (2-D) potential flow, in a surprising marriage between the theories of viscous-dominated and inviscid flows. Hele-Shaw approximation allows visualisation of potential flows over 2-D aerofoils and also undergirds important discoveries in the dynamics of interfacial instabilities and convection, yet it has found little use in modelling flows in microfluidic devices, although these devices often have thin-gap geometries. Here, we derive a Hele-Shaw approximation for the flow in the kinds of thin-gap geometries created within microfluidic devices. Using the method of weighted residuals, we reinterpret the Hele-Shaw approximation as the leading term of an orthogonal polynomial expansion that can be systematically extended to higher-order corrections. The resulting leading-order equation coincides with the previously derived 2-D approximations, but our derivation is shorter and more direct. By extending the expansion beyond leading order, we obtain a new reduced model that captures non-parabolic gapwise velocity profiles and out-of-plane flow effects. We provide substantial numerical evidence showing that approximate equations can successfully model real microfluidic and inertial-microfluidic device geometries. By reducing three-dimensional flows to 2-D models, our validated model will allow for accelerated device modelling and design.