We develop a general framework to describe the cubically nonlinear interaction of a degenerate quartet of deep-water gravity waves in one or two spatial dimensions. Starting from the discretised Zakharov equation, and thus without restriction on spectral bandwidth, we derive a planar Hamiltonian system in terms of the dynamic phase and a modal amplitude. This is characterised by two free parameters: the wave action and the mode separation between the carrier and the sidebands. For unidirectional waves, the mode separation serves as a bifurcation parameter, which allows us to fully classify the dynamics. Centres of our system correspond to non-trivial, steady-state nearly resonant degenerate quartets. The existence of saddle-points is connected to the instability of uniform and bichromatic wave trains, generalising the classical picture of the Benjamin–Feir instability. Moreover, heteroclinic orbits are found to correspond to discrete, three-mode breather solutions, including an analogue of the famed Akhmediev breather solution of the nonlinear Schrödinger equation.

A general mixed kinetic-diffusion boundary condition is formulated to account for the out-of-equilibrium kinetics in the Knudsen layer. The mixed boundary condition is used to investigate the problem of quasi-steady evaporation of a droplet in an infinite domain containing inert gases. The widely adopted local thermodynamic equilibrium assumption is found to be the limiting case of infinitely large kinetic Péclet number ${{Pe}_k}$, and it introduces significant error for ${{Pe}_k} \leqslant O(10)$, which corresponds to a typical droplet radius $a$ of a few micrometres or smaller. When compared with experimental data, solutions based on the mixed boundary condition, which take into account the temperature jump across the Knudsen layer, better predict the time evolution of $a$ than the classical $D^2$-law (i.e. $a^2 \propto t$, where $t$ denotes time). In the slow evaporation limit, an analytical solution is obtained by linearising the full formulation about the equilibrium condition, which shows that the $D^2$-law can be recovered only in the large ${{Pe}_k}$ limit. For small ${{Pe}_k}$, where the process is dominated by kinetics, a linear relation, i.e. $a \propto t$, emerges. When the gas phase density approaches the liquid density (e.g. at high-pressure or low-temperature conditions), the increase in the chemical potential of the liquid phase due to the presence of inert gases needs to be accounted for when formulating the mixed boundary condition, an effect largely ignored in the literature so far.

The purpose of the paper is to identify Mach-number effects on pressure fluctuations $p'$ in compressible turbulent plane channel flow. We use data from a specifically constructed $(Re_{\tau ^\star },\bar {M}_{{CL}_x})$-matrix direct numerical simulation (DNS) database, with systematic variation of the centreline streamwise Mach number $0.32\leqslant \bar {M}_{{CL}_x}\leqslant 2.49$ and of the HCB (Huang et al., J. Fluid Mech., vol. 305, 1995, pp. 185–218) friction Reynolds number $66\leqslant Re_{\tau ^\star }\lessapprox 1000$. Strong $\bar {M}_{{CL}_x}$ effects (enhanced by the increasingly cold-wall condition) appear for $\bar {M}_{{CL}_x}\gtrapprox 2$, for all $Re_{\tau ^\star }$, very close to the wall ($y^\star \lessapprox 15$). Compared with incompressible flow at the same $Re_{\tau ^\star }$, the wall root-mean-square $[p'_{rms}]^+_w$ (in wall-units, i.e. scaled by the average wall shear stress $\bar {\tau }_w$) strongly increases with $\bar {M}_{{CL}_x}$. In contrast, the peak level across the channel, $[p'_{rms}]^+_{PEAK}$, slightly decreases with increasing $\bar {M}_{{CL}_x}$. In order to study the near-wall coherent structures we introduce a new wall-distance-independent non-local system of units, based for all $y$ on wall friction and the extreme values of density and dynamic viscosity, namely, for cold walls $\{\bar {\tau }_w,\min _y\bar {\rho },\max _y\bar {\mu }\}$. The average spanwise distance between streaks, scaled by this length-unit, is nearly independent of $\bar {M}_{{CL}_x}$ at constant $Re_{\tau ^\star }$. Using the in-plane (parallel to the wall) Laplacian $\nabla ^2_{xz}p'$ we find that the $(+/-)\text {-}p'$ wave-packet-like structures appearing inside the low-speed streaks ($y^\star \lessapprox 15$) with increasing $\bar {M}_{{CL}_x}\gtrapprox 2$ are part of a more complex wave system with spanwise extent over several streaks, whose spatial density decreases rapidly with decreasing $\bar {M}_{{CL}_x}$ or increasing $y^\star$. These $p'$ wave packets appear to be collocated with strong $(+/-)$-$v'$ events and could be responsible for compensating towards 0 the negative incompressible-flow correlation coefficient $c_{p'v'}$, with increasing $\bar {M}_{{CL}_x}$ very near the wall.

Non-spherical particles exhibit peculiar behaviour in non-Newtonian flows. In this paper, we numerically investigate the dynamics of a neutrally buoyant prolate spheroid immersed in viscoelastic shear flows at finite Reynolds numbers by means of the immersed boundary method. Our results show that the period of particle rotation changes monotonically with the solvent viscosity ratio but non-monotonically with the mobility factor. Furthermore, we find five rotation modes of the spheroid under the effects of fluid inertia and fluid rheology in the present flow configuration. With weak fluid inertia, the particle rotation rate is remarkably reduced by fluid elasticity, which also induces asymmetric rotational behaviour. While the particle tends to tumble in the shear plane with weak fluid elasticity and moderate fluid inertia. However, as the fluid elasticity increases, the particle rotates with a newly observed mode, named the asymmetric-kayaking mode, which is classified by two additional critical elastic numbers that differ from the earlier studies on Stokesian viscoelastic shear flows. The present findings imply the importance of fluid inertia and fluid elasticity on the particle dynamics and could be potentially used to control the particle orientations in viscoelastic fluid flows.

The scattering of three-dimensional inertia-gravity waves by a turbulent geostrophic flow leads to the redistribution of their action through what is approximately a diffusion process in wavevector space. The corresponding diffusivity tensor was obtained by Kafiabad et al. (J. Fluid Mech., vol. 869, 2019, R7) under the assumption of a time-independent geostrophic flow. We relax this assumption to examine how the weak diffusion of wave action across constant-frequency cones that results from the slow time dependence of the geostrophic flow affects the distribution of wave energy. We find that the stationary wave-energy spectrum that arises from a single-frequency wave forcing is localised within a thin boundary layer around the constant-frequency cone, with a thickness controlled by the acceleration spectrum of the geostrophic flow. We obtain an explicit analytic formula for the wave-energy spectrum which shows good agreement with the results of a high-resolution simulation of the Boussinesq equations.

The surface wave instability (SWI) of thermocapillary migration is examined by linear stability analysis for a droplet on a unidirectional heated plane. Both a Newtonian fluid and an Oldroyd-B fluid are considered. The droplet, flattened by gravity, is susceptible to two kinds of instabilities: convective instability (CI), which is independent of surface deformation; and SWI, which occurs only when the Galileo number and the surface-tension number are not too large. The wavenumber of the latter is much smaller than that of the former, while the reverse is true for the wave speed. SWI is found at different Prandtl numbers (Pr), while its mode includes streamwise and oblique waves. Energy analysis suggests that the energy of the long-wave mode comes from the shear stress induced by the surface deformation, the energy source for the mode with finite wavelength is the work done by Marangoni forces, while the energy from the basic flow is only important in some cases at small Pr. For the Oldroyd-B fluid, a small elasticity slightly changes the critical Marangoni number of SWI, while larger elasticity changes the preferred mode from SWI to CI. The instability mechanism is discussed and comparisons are made with experimental results.

In the classical irreversible thermodynamics (CIT) framework, the Navier–Stokes–Fourier constitutive equations are obtained so as to satisfy the entropy inequality, by and large assuming that the entropy flux is equal to the heat flux over the temperature. This article is focused on the derivation of second-order constitutive equations for polyatomic gases; it takes the basis of CIT, but most importantly, allows up to quadratic nonlinearities in the entropy flux. Mathematical similarities between the proposed model and the classic Stokes–Laplace equations are exploited so as to construct analytic/semi-analytic solutions for the slow rarefied gas flow over different shapes. A set of second-order boundary conditions are formulated such that the model's prediction for the drag force is in excellent agreement with the experimental data over the whole range of Knudsen numbers. We have also computed the normal shock structure in nitrogen for Mach ${Ma} \lesssim 4$. A very good agreement was observed with the kinetic theory, as well as with the experimental data.

The rebound of droplets impacting a deep fluid bath is studied both experimentally and theoretically. Millimetric drops are generated using a piezoelectric droplet-on-demand generator and normally impact a bath of the same fluid. Measurements of the droplet trajectory and other rebound metrics are compared directly with the predictions of a linear quasipotential model, as well as fully resolved direct numerical simulations of the unsteady Navier–Stokes equations. Both models resolve the time-dependent bath and droplet shapes in addition to the droplet trajectory. In the quasipotential model, the droplet and bath shape are decomposed using orthogonal function decompositions leading to two sets of coupled damped linear harmonic oscillator equations solved using an implicit numerical method. The underdamped dynamics of the drop are directly coupled to the response of the bath through a single-point kinematic match condition which we demonstrate to be an effective and efficient model in our parameter regime of interest. Starting from the inertio-capillary limit in which both gravitational and viscous effects are negligible, increases in gravity or viscosity lead to a decrease in the coefficient of restitution and an increase in the contact time. The inertio-capillary limit defines an upper bound on the possible coefficient of restitution for droplet–bath impact, depending only on the Weber number. The quasipotential model is able to rationalize historical experimental measurements for the coefficient of restitution, first presented by Jayaratne & Mason (Proc. R. Soc. Lond. A, vol. 280, issue 1383, 1964, pp. 545–565).

When a liquid drop falls on a solid substrate, the air layer between them delays the occurrence of liquid–solid contact. For impacts on smooth substrates, the air film can even prevent wetting, allowing the drop to bounce off with dynamics identical to that observed for impacts on superamphiphobic materials. In this paper, we investigate similar bouncing phenomena, occurring on viscous liquid films, that mimic atomically smooth substrates, with the goal to probe their effective repellency. We elucidate the mechanisms associated with the bouncing to non-bouncing (floating) transition using experiments, simulations, and a minimal model that predicts the main characteristics of drop impact, the contact time and the coefficient of restitution. In the case of highly viscous or very thin films, the impact dynamics is not affected by the presence of the viscous film. Within this substrate-independent limit, bouncing is suppressed once the drop viscosity exceeds a critical value, as on superamphiphobic substrates. For thicker or less viscous films, both the drop and film properties influence the rebound dynamics and conspire to inhibit bouncing above a critical film thickness. This substrate-dependent regime also admits a limit, for low-viscosity drops, in which the film properties alone determine the limits of repellency.

Non-wetting substrates allow impacting liquid drops to spread, recoil and take-off, provided they are not too heavy (Biance et al., J. Fluid Mech., vol. 554, 2006, pp. 47–66) or too viscous (Jha et al., Soft Matt., vol. 16, no. 31, 2020, pp. 7270–7273). In this article, using direct numerical simulations with the volume of fluid method, we investigate how viscous stresses and gravity oppose capillarity to inhibit drop rebound. Close to the bouncing to non-bouncing transition, we evidence that the initial spreading stage can be decoupled from the later retraction and take-off, allowing us to understand the rebound as a process converting the surface energy of the spread liquid into kinetic energy. Drawing an analogy with coalescence-induced jumping, we propose a criterion for the transition from the bouncing to the non-bouncing regime, namely by the condition ${Oh}_{c} + {Bo}_{c} \sim 1$, where ${Oh}_{c}$ and ${Bo}_{c}$ are the Ohnesorge number and Bond number at the transition, respectively. This criterion is in excellent agreement with the numerical results. We also elucidate the mechanisms of bouncing inhibition in the heavy and viscous drop limiting regimes by calculating the energy budgets and relating them to the drop's shape and internal flow.

This work deals with the investigation and modelling of wall pressure fluctuations induced by a supersonic jet over a tangential flat plate. The analysis is performed at several nozzle pressure ratios around the nozzle design Mach number, including slightly over-expanded and under-expanded conditions, and for different radial positions of the rigid plate. Pitot measurements and flow visualizations through the background oriented schlieren technique provided a general overview of the aerodynamic interactions between the jet flow and the plate at the different regimes and configurations. Wall pressure fluctuations were measured using a couple of piezoelectric pressure transducers flush mounted over the plate surface. The spectral analysis has been carried out to clarify the effect of the plate position on the single and multivariate wall pressure statistics, including the screech tone amplitude. The experimental dataset is used to assess and validate a surrogate model based on artificial neural networks. Sound pressure levels and coherence functions are modelled by means of a single fully connected network, built on the basis of a recently implemented fully deterministic topology optimization algorithm. The metamodel uncertainty is also quantified using the spatial correlation function. It is shown that the flow behaviour as well as the screech and broadband noise signatures are significantly influenced by the presence of the plate, and the effects on spectral quantities are correctly reproduced by the proposed data-driven model that provides predictions in agreement with the available data.

The spontaneous emergence of structure is a ubiquitous process observed in fluid and plasma turbulence. These structures typically manifest as flows which remain coherent over a range of spatial and temporal scales, resisting statistically homogeneous description. This work conducts a computational and theoretical study of coherence in turbulent flows in the stochastically forced barotropic $\beta$-plane quasi-geostrophic equations. These equations serve as a prototypical two-dimensional model for turbulent flows in Jovian atmospheres, and can also be extended to study flows in magnetically confined fusion plasmas. First, analysis of direct numerical simulations demonstrates that a significant fraction of the flow energy is organized into coherent large-scale Rossby wave eigenmodes, comparable with the total energy in the zonal flows. A characterization is given for Rossby wave eigenmodes as nearly integrable perturbations to zonal flow Lagrangian trajectories, linking finite-dimensional deterministic Hamiltonian chaos in the plane to a laminar-to-turbulent flow transition. Poincaré section analysis reveals that Lagrangian flows induced by the zonal flows plus large-scale waves exhibit localized chaotic regions bounded by invariant tori, manifesting as Rossby wave breaking in the absence of critical layers. It is argued that the surviving invariant tori organize the large-scale flows into a single temporally and zonally varying laminar flow, suggesting a form of self-organization and wave stability that can account for the resilience of the observed large-amplitude Rossby waves.

There has been significant recent interest in the study of water waves coupled with non-zero vorticity. We derive analytical approximations for the exponentially small free-surface waves generated in two dimensions by one or several submerged point vortices when driven at low Froude numbers. The vortices are fixed in place, and a boundary-integral formulation in the arclength along the surface allows the study of nonlinear waves and strong point vortices. We demonstrate that, for a single point vortex, techniques in exponential asymptotics prescribe the formation of waves in connection with the presence of Stokes lines originating from the vortex. When multiple point vortices are placed within the fluid, trapped waves may occur, which are confined to lie between the vortices. We also demonstrate that, for the two-vortex problem, the phenomenon of trapped waves occurs for a countably infinite set of values of the Froude number. This work will form a basis for other asymptotic investigations of wave–structure interactions where vorticity plays a key role in the formation of surface waves.

The Lighthill–Panton and Lyman–Huggins interpretations of vorticity dynamics are extended to the dynamics of enstrophy. There exist two competing definitions of the vorticity current tensor, which describes the flow rate of vorticity in the fluid interior, and the corresponding boundary vorticity flux, which represents the local vorticity creation rate on a boundary. It is demonstrated that each definition of the vorticity current tensor leads to a consistent set of definitions for the enstrophy current, boundary enstrophy flux and the enstrophy dissipation term. This leads to two alternative interpretations of vorticity and enstrophy dynamics: the Lighthill–Panton and Lyman–Huggins interpretations. Although the kinematic evolution of the vorticity and enstrophy fields are the same under each set of definitions, the dynamical interpretation of the motion generally differs. For example, we consider the Stokes flow over a rotating sphere, and find that the flow approaches a steady state where, under the Lyman–Huggins interpretation, there is no enstrophy creation or dissipation. Under the Lighthill–Panton interpretation, however, the steady-state flow features a balance between the continuous generation and subsequent dissipation of enstrophy. Moreover, the Lyman–Huggins interpretation has previously been shown to offer several benefits in understanding the dynamics of vorticity, and therefore it is beneficial to extend this interpretation to the dynamics of enstrophy. For example, the Lyman–Huggins interpretation allows the creation of vorticity, and therefore enstrophy, to be interpreted as an inviscid process, due to the relative acceleration between the fluid and the boundary.

Synthetic jets have received extensive attention due to their superior mixing property. However, its mechanisms have not been investigated from the perspective of the turbulent/non-turbulent interface (TNTI). To shed new light on this issue, the entrainment and TNTI properties of a synthetic jet are experimentally investigated and compared with a continuous jet at $Re_j = 3150$. The fuzzy clustering method is applied to select an appropriate vorticity threshold to detect the TNTI. Statistically, it is revealed that the entrainment coefficients of the two jets significantly differ in the near field, while they become almost identical in the far field. Instead of the vortex ring, the ‘breakdown of the vortex ring’ enhances the entrainment in the present synthetic jet. Instantaneously, the TNTI more violently fluctuates in the near field of the synthetic jet, which leads to a larger fluctuation in the TNTI radial position, a higher fractal dimension and enhanced local entrainment. Moreover, the transition of the probability density function of the TNTI orientation from the unimodal distribution in the near field to the bimodal distribution in the far field is found in both jets. The multi-scale analysis reveals a new mechanism for the bimodal distribution caused by the TNTI-thickness-scale structures.

A droplet charged above the Rayleigh limit is unstable. In the resulting dynamical process, referred to as a Coulomb explosion, smaller droplets with higher charge-to-mass ratios are ejected, reducing the charge of the parent droplet below the Rayleigh limit. Furthermore, if the droplet is sufficiently small, the electric field on its surface can promote ion field emission. Ion emission can lower the charge of a spherical droplet below its Rayleigh limit, keeping it stable, or reduce the charge of a deforming droplet, changing its dynamics and potentially preventing the Coulomb explosion. This article develops a continuum phase field electrohydrodynamic model to study the interplay between Coulomb explosions and ion emission, using charged nanodroplets of the ionic liquid 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide (EMI-Im) as a case study. In small droplets (diameter $D \lesssim 20$ nm for EMI-Im), the electric field is strong enough to emit ions in the early phase of the droplet's evolution, suppressing the Coulomb explosion. For $20 \lesssim D \lesssim 45$ nm, the electric field on the EMI-Im droplet may not promote significant ion emission; however, as the unstable droplet becomes ellipsoidal, ions are emitted from its vertices, ultimately suppressing the Coulomb explosion while shedding $20\unicode{x2013}40\,\%$ of the initial charge. For larger EMI-Im droplets, $45 \lesssim D \lesssim 100$ nm, the evolution typical of a Coulomb explosion is observed, accompanied by ion emission which is however insufficient to prevent the Coulomb explosion. Ion emission and the smaller progeny droplets account for $24\,\%$ and $16\,\%$ of the initial charge, respectively.

Marangoni spreading on thin films is widely observed in nature and applied in industry. It has serious implications for airway drug delivery, especially in surfactant displacement therapy. This paper reports the results of experimental investigations of a surfactant-laden droplet spreading on films made of more viscous Newtonian fluids as well as on films made of viscoelastic fluids. The experiments used particle seeding, the transmission-speckle method and particle tracking velocimetry (PTV) to determine the deformation of the film–droplet interface and to measure velocity fields. Radially aligned patterns were observed on Newtonian films. Similar patterns, but with much smaller wavenumber, were observed on viscoelastic films in combination with rapid azimuthal variations of the film thickness. The Saffman–Taylor instability at the film–droplet interface explains the formation of patterns on a more viscous Newtonian film, and their onset requires exceeding the critical capillary number. The pattern formation on viscoelastic films is correlated with an instability at the film–droplet–air contact line when the liquid is expelled radially by the spreading droplet. PTV revealed azimuthal variations of the velocity field in the vicinity of the contact line. The observed contact line instability is different from previously reported fingering instabilities of Newtonian thin films. A simple scaling law accounting for the Marangoni-stress-induced elastic shear deformation is proposed to describe the flow field in the patterns formed in the viscoelastic films.

This study presents the spreading of a single filament of a yield stress (viscoplastic) fluid extruded onto a pre-wetted solid surface. The filaments spread laterally under surface tension forces until they reach a final equilibrium shape when the yield stress dominates. We use a simple experimental set-up to print the filaments on a moving surface and measure their final width using optical coherence tomography. Additionally, we present a scaling law for the final width and determine the corresponding prefactor using asymptotic analysis. We then analyse the level of agreement between the theory and experiments, and discuss the possible origins of discrepancies. The process studied here has applications in extrusion-based thermoplastic and bio-three-dimensional printing.

We numerically study the impact of a droplet on superhydrophobic flexible plates, aiming to understand how the flexible substrate influences the maximum spreading of the droplet. Compared with the rigid case, the vertical movement of the flexible substrate due to droplet impact reduces the maximum spreading. Besides, the average acceleration $a$ during droplet spreading changes significantly. Arising from energy conservation, we rescale the acceleration $a$ for cases with different bending stiffness $K_B$ and mass ratio $M_r$. Moreover, through theoretical analysis, we propose a scaling for the droplet's maximum spreading diameter ratio $\beta _{max}$. In the scaling, based on the derived $a$, an effective Weber number $We_m$ is well defined, which accounts for the substrate properties without any adjustable parameters. In the ($\beta _{max}, We_m$) plane, the two-dimensional numerical results of different $K_B$, $M_r$ and rigid cases all collapse into a single curve, as do the experimental and three-dimensional (3-D) results. In particular, the collapsed 3-D data can be well represented by the universal rescaling of $\beta _{max}$ proposed by Lee et al. (J. Fluid Mech., vol. 786, 2016, R4). Furthermore, an a posteriori energy analysis confirms the validation of our a priori scaling law.

Fluid interfacial instability induced by gravity or external acceleration, known as the Rayleigh–Taylor instability, plays an important role in both scientific research and industrial application. How to control this instability is challenging. Researchers have been actively exploring the suppression method of applying electric fields parallel to dielectric fluid interfaces. The instability is characterized by the penetration of fingers at the interface. The velocities at the finger tips are the most important quantities since they characterize how fast the penetration occurs. The dynamics of the fingers is nonlinear. We present a nonlinear perturbation procedure for determining the amplitude and velocity of fingers at a Rayleigh–Taylor unstable interface between two incompressible, inviscid, immiscible and perfectly dielectric fluids in the presence of a horizontal electric field in two dimensions. The analytic formulas are displayed explicitly up to the third order of the initial disturbance. The comparison with the data from numerical simulations based on the vortex sheet method shows the theoretical formulas can capture well the nonlinear behaviour of the fingers. It is known that the interplay between the electric field and the fluids can lead to the suppression of interfacial instability. We further analyse the electrical force along the interface and show how this force leads to the instability suppression in our setting. It has been reported numerically in the literature that switching the electric permittivities of the fluids leads to quantitative differences in finger velocity. We show theoretically this phenomenon can only be explained by the nonlinear behaviour of the system.