We study the dynamics of fracture deflation following hydraulic fracturing of an infinite elastic solid, with fluid removal from a narrow conduit at the centre. This process involves coupled lubricating flow and elastic deformation, now subject to appropriate descriptions of fluid removal through the conduit towards the ambient, driven by elastic stresses and extraction/suction. When the influence of material toughness is negligible, the dynamics is found to be governed by two dimensionless parameters that describe the relative influence of elasticity-driven backflow ($\Pi _c$) and ambient-pressure-driven backflow ($\Pi _e$), respectively. We also found that the fracture’s thickness eventually approaches zero at the centre, while the fracture evolves into a self-similar shape of the dipole type that conserves the dipole moment $M$. The fracture’s front continues to elongate according to $x_f \propto t^{1/9}$, while the total fluid volume within the fracture decreases according to $V \propto t^{-1/9}$. The model and solutions might find use in practical problems to estimate the rate of backflow and effective permeability of a fractured reservoir once pressure is released.

The aerodynamic deformation and breakup of wall-attached droplets in axisymmetric stagnation flow are investigated experimentally. A vertical shock tube is used to generate the shock wave accompanying the post-wave airflow, and the axisymmetric stagnation flow is formed through the impingement of an air stream on a solid wall. For the wall-attached droplets with initially hemispherical profile, four typical droplet deformation and breakup modes can be identified with the continuous increase of the droplet local Weber number, which are the vibrating mode, the compressing mode, the sheet thinning mode and the shear-induced entrainment mode. Quantitative analyses of droplet evolution dynamics are also conducted for the compressing mode and the sheet thinning mode, and the significant differences of air flow separation at the droplet lateral surface between these two modes are revealed. The potential flow model and the energy conservation model are further developed to predict the entire droplet deformation processes. The vibrating frequency and amplitude of droplets under the vibrating mode are predicted by a spring-mass model, and the surface perturbation wavelengths of droplets under the shear-induced entrainment mode are estimated based on the dispersion relation of Kelvin–Helmholtz instability. This work is proposed to give potential guidance for regulating the aerodynamic fragmentation of wall-attached droplets in practical engineering applications.

Supersonic impinging tones have been attracting significant interest because high-intensity discrete-frequency tones pose substantial risks to structural safety in applications such as rocket launch and recovery, and space vehicle attitude adjustment. However, various issues remain to be addressed regarding the jet oscillation and tone generation mechanism. In this study, a numerical simulation of the supersonic impinging jet with a nozzle pressure ratio of 4.03 and an impingement distance of 2.08 times the nozzle exit diameter is conducted. The results show good consistency with the reference data by other researchers. A phase-locked averaging analysis of 2960 flow field snapshots is employed to investigate jet structure oscillation dynamics and the tone generation mechanism. The phase-locked averaged images reveal that the pressure variation induced by Kelvin–Helmholtz vortices as they pass through the reflected shock results in the periodic motions of the reflected shock and Mach disk. The periodically oscillating Mach disk generates high-pressure fluid masses driving recirculation bubbles through a cyclic ‘compression–generation–merging’ oscillation. The streamline oscillation and sound-ray analyses reveal there are two distinct tone source regions: the impinging zone and the wall jet region. Consequently, it is proposed that vortex collapse in conjunction with wall jet oscillations coexist to generate the tone. According to the directivity, the tone emitted from the wall jet source region is believed to contribute to the feedback loop. These findings collectively contribute to an improved understanding of the jet plume oscillation and tone generation mechanisms of the supersonic impinging jet.

The dynamics of a stratified fluid in which the rotation vector is slanted at an angle with respect to the local vertical (determined by gravity) is considered for the case where the aspect ratio of the characteristic vertical scale of the motion D to the horizontal scale L is not small. In cases where the Rossby number of the flow is small the natural coordinate system is non-orthogonal and modifications to the dynamics are significant. Two regimes are examined in this paper. The first is the case in which the horizontal length scale of the motion, L, is sub-planetary where the quasi-geostrophic approximation is valid. The second is the case where the horizontal scale is commensurate with the planetary radius and so the dynamics must be formulated in spherical coordinates with imposing a full variation on the relevant components of rotation. In the quasi-geostrophic case the rotation axis replaces the direction of gravity as the axis along which the geostrophic flow varies in response to horizontal density gradients. The quasi-geostrophic potential vorticity equation is most naturally written in a non-orthogonal coordinate system with fundamental alterations in the dynamics. Examples such as the reformulation of the classical Eady problem are presented to illustrate the changes in the nature of the dynamics. For the second case where the horizontal scale is of the order of R, the planetary radius, more fundamental changes occur leading to more fundamental and difficult changes in the dynamical model.

We study nonlinear resonant triad interactions among flexural-gravity waves generated by a steadily moving load on a floating ice sheet. Of the many possible triad interactions involving at least one load-produced wave, we focus on the double-frequency case where the wavenumber of the leading wave is double that of the trailing wave. This case stands out because resonant interactions can occur with or without the presence of an ambient wave. Using multiple-scale perturbation analysis, we obtain amplitude evolution equations governing the leading-order, steady-state response. We complement the theoretical predictions with direct numerical simulations of the initial–boundary value problem using a high-order spectral method accurate to arbitrary order. Our results show that the double-frequency interaction can cause the trailing wave amplitude to decay with distance from the load, with its energy transferred to its second harmonic which radiates forwards to coherently interfere with the leading wave. Depending on the length and orientation of the load, the resonant interaction can in some cases cause the wave drag to become vanishingly small, or in other cases nearly double the maximum bending strain compared to the linear prediction. We also consider the effect of a small ambient wave that can initiate a resonant interaction in the leading wave field in addition to the trailing wave field interaction. This can result in a steady, localised wave packet containing two mutually trapped wave components, leading to vanishing wave drag. This work has potential implications for defining safe operating profiles for vehicles travelling on floating ice.

The thermocapillary flows generated by an inclined temperature gradient in and around a floating droplet are studied in the framework of the lubrication approximation. Numerical simulations of nonlinear flow regimes are fulfilled. It is shown that under the action of Marangoni stresses, a droplet typically moves as a whole. It is found that an inclined temperature gradient can lead to the excitation of periodic oscillations. With an increase of the inclination of the temperature gradient, temporally quasi-periodic oscillations have been obtained. In a definite region of parameters, an inclined temperature gradient can suppress oscillations, changing the droplet’s shape. The diagram of regimes in the plane of longitudinal and transverse Marangoni numbers has been constructed. Bistability has been found.

Spin coating is the process of generating a uniform coating film on a substrate by centrifugal forces during rotation. In the framework of lubrication theory, we investigate the axisymmetric film evolution and contact-line dynamics in spin coating on a partially wetting substrate. The contact-line singularity is regularized by imposing a Navier slip model. The interface morphology and the contact-line movement are obtained by numerical solution and asymptotic analysis of the lubrication equation. The results show that the evolution of the liquid film can be classified into two modes, depending on the rotational speed. At lower speeds, the film eventually reaches an equilibrium state, and we provide a theoretical description of how the equilibrium state can be approached through matched asymptotic expansions. At higher speeds, the film exhibits two or three distinct regions: a uniform thinning film in the central region, an annular ridge near the contact line, and a possible Landau–Levich–Derjaguin-type (LLD-type) film in between that has not been reported previously. In particular, the LLD-type film occurs only at speeds slightly higher than the critical value for the existence of the equilibrium state, and leads to the decoupling of the uniform film and the ridge. It is found that the evolution of the ridge can be well described by a two-dimensional quasi-steady analysis. As a result, the ridge volume approaches a constant and cannot be neglected to predict the variation of the contact-line radius. The long-time behaviours of the film thickness and the contact radius agree with derived asymptotic solutions.

Dean’s approximation for curved pipe flow, valid under loose coiling and high Reynolds numbers, is extended to study three-dimensional travelling waves. Two distinct types of solutions bifurcate from the Dean’s classic two-vortex solution. The first type arises through a supercritical bifurcation from inviscid linear instability, and the corresponding self-consistent asymptotic structure aligns with the vortex–wave interaction theory. The second type emerges from a subcritical bifurcation by curvature-induced instabilities and satisfies the boundary region equations. A connection to the zero-curvature limit was not found. However, by continuing from known self-sustained exact coherent structures in the straight pipe flow problem, another family of three-dimensional travelling waves can be shown to exist across all Dean numbers. The self-sustained solutions also possess the two high-Reynolds-number limits. While the vortex–wave interaction type of solutions can be computed at large Dean numbers, their branch remains unconnected to the Dean vortex solution branch.

Gravity-driven film flow in circular pipes with isolated topography was examined with fluorescence imaging for three flow rates, two angles of inclination, and four topography shapes. The time-averaged free surface response in the vicinity of the topography depended on flow rate, inclination angle and topography shape. For some flow conditions, the time-averaged free surface included a capillary ridge, and for a subset of those conditions, a series of capillary waves developed upstream with a spacing often approximated by half the capillary length. In contrast to film flow over planar topography, the capillary ridge often formed downstream of the topography, and for the lowest flow rate over rectangular step down topography, the free surface developed a steady overhang along the downstream face of the topography. Possible dynamic causes of the unique film flow behaviour in circular pipes are discussed. Transient free surface fluctuations were observed at half the magnitude reported in film flow over corrugated circular pipes, and local maxima in transient magnitude corresponded to axial locations of inflection points in the time-averaged free surface. Local maxima are related to low surface pressure regions that vary in location and amplitude. Rectangular step down topography generated an extra ridge of fluid that formed on top of the capillary ridge for flow conditions, resulting in a capillary ridge downstream of the step. The extra ridge varied in temporal duration and spatial extent, and finds no comparison in planar film flow. No evidence of periodic behaviour was detected in the transient film response.

Cilia perform various functions, including sensing, locomotion, generation of fluid flows and mass transport, serving to underpin a vast range of biological and ecological processes. However, analysis of the mass transport typically fails to resolve the near-field dynamics around individual cilia, and therefore overlooks the intricate role of power/recovery strokes of ciliary motion. Selvan et al. (2023, Phys. Rev. Fluids 8, 123103) observed that the flow field due to a point torque (i.e. a rotlet) accurately resolves both the near- and far-field characteristics of a single cilium’s flow in a semi-infinite domain. In this paper, we calculate the mass transport between a no-slip boundary and an adjacent fluid, as a model system for nutrient exchange with ciliated tissues. We develop a Langevin model in the presence of a point torque (i.e. a single cilium) to examine the nutrient flux from a localised surface source. This microscopic transport model is validated using a macroscopic continuum model, which directly solves the advection–diffusion equation. Our findings reveal that the flow induced by a point torque can enhance the particles’ transport, depending on their diffusivity and the magnitude of the point torque. Additionally, the average mass transport affected by a single cilium can be enhanced or diminished by the presence of an externally imposed linear shear flow, with a strong dependence on the alignment of the cilium. Taken together, this framework serves as a useful minimal model for examining the average nutrient exchange between ciliated tissues and fluid environments.

Two-dimensional gaseous detonations near critical propagation state were studied numerically in a channel with stoichiometric H$_2$/air and H$_2$/O$_2$ mixtures. Detonation waves exhibit a mode-locking effect (MLE) in a single-headed mode regime. Increasing the channel width alters the strength and propagation period of the single transverse wave. This leads to MLE failure and the occurrence of the single-dual-headed critical mode, featuring the emergence of a new transverse wave. For a stoichiometric H$_2$/air mixture, generation of the new transverse wave is due to interactions between the detonation front and the local explosion wave originating from interactions between the transverse wave and unreacted gas pocket downstream. Whereas, for a stoichiometric H$_2$/O$_2$ mixture, a transverse wave interacting with the wall produces Mach reflection bifurcation, causing MLE failure and generation of the new transverse wave. Our results show that all transverse waves manifest as strong transverse wave (STW) structures, with most belonging to the second kind, and an acoustic coupling exists between the typical second kind of STW structure and the acoustic wave in the induction zone behind the Chapman–Jouguet detonation front. A high-pressure region close to the STW structure plays a crucial role in exploring the transverse dynamics of this structure. Shock polars with rational assumptions are adopted to predict flow states in this region. The roles of pivotal factors in influencing the flow states and wave structure are clarified, and characteristic pressure values derived adequately represent the STW structure’s transverse dynamic behaviours. Lastly, the relationship between the kinematics and kinds of STW structures is unveiled.

The broad-band direct combustion noise is an important problem for industrial and domestic burners. The power spectral density (PSD) of this noise is related to the local spectral density of fluctuating heat release rate (HRR) ($\psi _{\dot {q}}$), which is challenging to measure but is readily available from large eddy simulations (LES) results. The behaviour of $\psi _{\dot {q}}$ for a wide range of thermochemical and turbulence conditions is investigated. Three burners are studied, namely a dual-swirl burner, a bluff-body burner and a jet in cross-flow burner, operating at atmospheric conditions with $\textrm {CH}_4$–air and $\textrm {H}_2$–air mixtures. In contrast to the classical $f^{-5/2}$ scaling, the far-field sound pressure level and volume-integrated HRR ($\psi _{\dot {Q}}$) spectra reveal a universal $f^{-5}$ scaling for high frequencies. This differing spectral decay rate for $\psi _{\dot {Q}}$ compared to the classical scaling is due to multi-regime combustion, related to either partial premixing or the local turbulence intensity. The dependence of $\psi _{\dot {q}}$ on the chosen spatial locations, flame configuration and its relation to velocity spectra are studied. A simple model for $\psi _{\dot {q}}$ involving the velocity spectra is found that compares well with LES results. The characteristic frequency involved in this model is related to the time scale of the coherent structures of the flow.

We examine the evaporation-induced coalescence of two droplets undergoing freezing by conducting numerical simulations employing the lubrication approximation. When two sessile drops undergo freezing in close vicinity over a substrate, they interact with each other through the gaseous phase and the simultaneous presence of evaporation/condensation. In an unsaturated environment, the evaporation flux over the two volatile sessile drops is asymmetric, with lower evaporation in the region between the two drops. This asymmetry in the evaporation flux generates an asymmetric curvature in each drop, which results in a capillary flow that drives the drops closer to each other, eventually leading to their coalescence. This capillary flow, driven by evaporation, competes with the upward movement of the freezing front, depending on the relative humidity in the surrounding environment. We found that higher relative humidity reduces the evaporative flux, delaying capillary flow and impeding coalescence by restricting contact line motion. For a constant relative humidity, the substrate temperature governs the coalescence phenomenon and the resulting condensation can accelerate this process. Interestingly, lower substrate temperatures are observed to facilitate faster propagation of the freezing front, which, in turn, restricts coalescence.

Manipulation of small-scale particles across streamlines is the elementary task of microfluidic devices. Many such devices operate at very low Reynolds numbers and deflect particles using arrays of obstacles, but a systematic quantification of relevant hydrodynamic effects has been lacking. Here, we explore an alternative approach, rigorously modelling the displacement of force-free spherical particles in vortical Stokes flows under hydrodynamic particle–wall interaction. Certain Moffatt-like eddy geometries with broken symmetry allow for systematic deflection of particles across streamlines, leading to particle accumulation at either Faxen field fixed points or limit cycles. Moreover, particles can be forced onto trajectories approaching channel walls exponentially closely, making possible quantitative predictions of particle capture (sticking) by short-range forces. This rich, particle-size-dependent behaviour suggests the versatile use of inertia-less flow in devices with a long particle residence time for concentration, sorting or filtering.

We perform direct numerical simulations of turbulent channel flows. Secondary motions are produced by applying a streamwise-homogeneous, spanwise-heterogeneous roughness pattern of spanwise period $\Lambda _s$ to the walls of the channel; their time evolution is observed. Notice that, owing to the geometry, the secondary motions are streamwise-invariant at any instant of time, so that no spatial development is seen. Once the secondary motions reach a statistically steady state, the roughness pattern is suddenly removed, so that the secondary motions decay. The time needed for the secondary motions to vanish is then measured; in doing so, we distinguish between the streamwise-momentum pathways and the cross-sectional circulatory motions that compose the secondary motions. Larger values of $\Lambda _s$ are generally associated with a longer time scale for the decay of the momentum pathways, although this might not hold true for $\Lambda _s/h\gt 4$ (where $h$ is the channel half-height). The value of such a time scale for the circulatory motions, instead, saturates for $\Lambda _s/h \geqslant 2$; this may be related to the observed spatial confinement of said circulatory motions. For specific values of $\Lambda _s$ ($2 \leqslant \Lambda _s/h \leqslant 4$), the volume-averaged energy associated with the momentum pathways undergoes an unexpected transient growth with respect to its value at the beginning of the decay. This might indicate that structures of such a specific size are able to self-sustain as postulated by Townsend (The Structure of Turbulent Shear Flow, 2nd edition, 1976, ch. 7.19); the evidence we gather in this respect is however inconclusive. Finally, the present data suggest that most of the energy of the momentum pathways is produced by the circulatory motions transporting the mean (spanwise-averaged) velocity.

Particle image velocimetry is used to study the control of swirl momentum, delivered through an orifice formed by a physically rotating tube of finite length, relevant to the evolution of vortex rings produced at a Reynolds number ${Re}\approx 1000$ based on the average discharge velocity, for swirl numbers ${S} \in [0, 1]$. Experiments without discharge, reinforced with complimentary numerical predictions, reveal the presence of an intriguing secondary flow pattern in the rotating tube, preventing attainment of a solid-body-like swirl distribution. Nevertheless, it is found that fully established rings produced in this way, following discharge once conditions in the tube have reached a steady state, exhibit similar characteristics to rings formed by an otherwise solid-body rotating initial condition as explored computationally by Ortega-Chavez et al. (2023, J. Fluid Mech. 967, A16). Namely, opposite-signed vorticity forms due to vortex tilting, which subsequently interacts with the ring, promoting vorticity cancellation and vortex ring breakdown. A key feature of the experimental work is that partially established vortex rings, produced before a steady-state rotating tube condition is reached, show unique characteristics. Their creation, a short time after the onset of tube rotation: (i) facilitates more efficient delivery of swirl momentum to the vortex core area; (ii) maintains a low level of swirl in the ring bubble’s central region which would otherwise promote the formation of opposite-signed vorticity and vortex breakdown.

We investigate the natural oscillations of sessile drops with a central trapped bubble on a plane using linear potential flow theory, considering both free and pinned contact lines. The system is governed by the contact angle $\alpha$ and the ratio $\tau$ of inner to outer contact line radii. For bubble-containing (BC) hemispherical drops with free contact lines (referred to as free BC semi-drops), the modes mirror half of those in concentric spherical BC drops due to plane symmetry. These modes are labelled ‘plus’ (with greater inner surface deformation) and ‘minus’ (with greater outer surface deformation). As $\tau \to 0$, minus modes converge to those of bubble-free drops. Results show that varying $\alpha$ from $90^\circ$ or pinning the contact line in free BC semi-drops alters the topology of spectral lines, turning original crossings of spectral lines between minus and plus modes into avoided crossings. This shift causes minus and plus modes to form spectral trends with avoided crossings, maintaining their original spectral shapes. In an avoided crossing, two coupled modes cannot be classified as plus or minus due to their comparable inner and outer surface deformations, resulting in mode beating when both are excited, as confirmed by our direct numerical simulations. This study on the impact of inner bubbles on the spectrum may help in predicting bubble size in opaque sessile drops.

Inertia–gravity waves are scattered by background flows as a result of Doppler shift by a non-uniform velocity. In the Wentzel–Kramers–Brillouin regime, the scattering process reduces to a diffusion in spectral space. Other inhomogeneities that the waves encounter, such as density variations, also cause scattering and spectral diffusion. We generalise the spectral diffusion equation to account for these inhomogeneities. We apply the result to a rotating shallow-water system, for which height inhomogeneities arise from velocity inhomogeneities through geostrophy, and to the Boussinesq system for which buoyancy inhomogeneities arise similarly. We compare the contributions that height and buoyancy variations make to the spectral diffusion with the contribution of the Doppler shift. In both systems, we find regimes where all contributions are significant. We support our findings with exact solutions of the diffusion equation and with ray tracing simulations in the shallow-water case.

We investigate fully developed turbulent flow in curved channels to explore the interaction between turbulence and curvature-driven coherent structures. By focusing on two cases of mild and strong curvature, we examine systematically the effects of the Reynolds number through a campaign of direct numerical simulations, spanning flow regimes from laminar up to the moderately high Reynolds number – based on bulk velocity and channel height – of $87\,000$. Our analysis highlights the influence of curvature on the friction coefficient, showing that flow transition is anticipated by concave curvature and delayed by convex curvature. In the case of mild curvature, a frictional drag reduction compared with plane channel flow is found in the transitional regime. Spectral analysis reveals that the near-wall turbulence regeneration cycle is maintained in mildly curved channels, while it is absent or severely inhibited on the convex wall of strongly curved channels. Streamwise large-scale structures resembling Dean vortices are found to be weakly dependent on the Reynolds number and strongly affected by curvature: increasing curvature shifts these vortices towards the outer wall and reduces their size and coherence, limiting their contribution to streamwise velocity fluctuations and momentum transport. In the case of strong curvature, spanwise large-scale structures are also detected. These structures are associated with large pressure fluctuations and the suppression of turbulent stresses near the convex wall, where a region with negative turbulence production is observed and characterised via quadrant analysis.