A millimetric droplet may bounce and self-propel across the surface of a vertically vibrating liquid bath, guided by the slope of its accompanying Faraday wave field. The ‘walker’, consisting of a droplet dressed in a quasi-monochromatic wave form, is a spatially extended object that exhibits many phenomena previously thought exclusive to the quantum realm. While the walker dynamics can be remarkably complex, steady and periodic states arise in which the energy added by the bath vibration necessarily balances that dissipated by viscous effects. The system energetics may then be characterised in terms of the exchange between the bouncing droplet and its guiding or ‘pilot’ wave. We here characterise this energy exchange by means of a theoretical investigation into the dynamics of the pilot-wave system when time-averaged over one bouncing period. Specifically, we derive simple formulae characterising the dependence of the droplet’s gravitational potential energy and wave energy on the droplet speed. Doing so makes clear the partitioning between the gravitational, wave and kinetic energies of walking droplets in a number of steady, periodic and statistically steady dynamical states. We demonstrate that this partitioning depends exclusively on the ratio of the droplet speed to its speed limit.

Analytical expressions are derived for the velocity field, and effective slip lengths, associated with pressure-driven longitudinal flow in a circular superhydrophobic pipe whose boundary is patterned with a general arrangement of longitudinal no-shear stripes not necessarily possessing any rotational symmetry. First, the flow in a superhydrophobic pipe with $M$ different no-shear stripes in general position is found for $M=1, 2, 3$. The method, which is based on use of so-called prime functions, is such that with these cases covered, generalisation to any $M \geqslant 1$ follows in a straightforward manner. It is shown how any one of these solutions can be generalised to solve for flow along superhydrophobic pipes where that pattern of $M$ menisci is repeated $N \geqslant 1$ times around the boundary in a rotational symmetric arrangement. The work provides an extension of the canonical pipe flow solution for an $N$-fold rotationally symmetric pattern of no-shear stripes due to Philip (Angew. Math. Phys., vol. 23, 1972, pp. 353–372). The novel solution method, and the solutions that it produces, have significance for a wide range of mixed boundary value problems involving Poisson’s equation arising in other applications.

We analyse the steady viscoelastic fluid flow in slowly varying contracting channels of arbitrary shape and present a theory based on the lubrication approximation for calculating the flow rate–pressure drop relation at low and high Deborah ($De$) numbers. Unlike most prior theoretical studies leveraging the Oldroyd-B model, we describe the fluid viscoelasticity using a FENE-CR model and examine how the polymer chains’ finite extensibility impacts the pressure drop. We employ the low-Deborah-number lubrication analysis to provide analytical expressions for the pressure drop up to $O(De^4)$. We further consider the ultra-dilute limit and exploit a one-way coupling between the parabolic velocity and elastic stresses to calculate the pressure drop of the FENE-CR fluid for arbitrary values of the Deborah number. Such an approach allows us to elucidate elastic stress contributions governing the pressure drop variations and the effect of finite extensibility for all $De$. We validate our theoretical predictions with two-dimensional numerical simulations and find excellent agreement. We show that, at low Deborah numbers, the pressure drop of the FENE-CR fluid monotonically decreases with $De$, similar to the previous results for the Oldroyd-B and FENE-P fluids. However, at high Deborah numbers, in contrast to a linear decrease for the Oldroyd-B fluid, the pressure drop of the FENE-CR fluid exhibits a non-monotonic variation due to finite extensibility, first decreasing and then increasing with $De$. Nevertheless, even at sufficiently high Deborah numbers, the pressure drop of the FENE-CR fluid in the ultra-dilute and lubrication limits is lower than the corresponding Newtonian pressure drop.

We conducted a series of pore-scale numerical simulations on convective flow in porous media, with a fixed Schmidt number of 400 and a wide range of Rayleigh numbers. The porous media are modeled using regularly arranged square obstacles in a Rayleigh–Bénard (RB) system. As the Rayleigh number increases, the flow transitions from a Darcy-type regime to an RB-type regime, with the corresponding $Sh$–$Ra_D$ relationship shifting from sublinear scaling to the classical 0.3 scaling of RB convection. Here, $Sh$ and $Ra_D$ represent the Sherwood number and the Rayleigh–Darcy number, respectively. For different porosities, the transition begins at approximately $Ra_D = 4000$, at which point the characteristic horizontal scale of the flow field is comparable to the size of a single obstacle unit. When the thickness of the concentration boundary layer is less than approximately one-sixth of the pore spacing, the flow finally enters the RB regime. In the Darcy regime, the scaling exponent of $Sh$ and $Ra_D$ decreases as porosity increases. Based on the Grossman–Lohse theory (J. Fluid Mech. vol. 407, 2000, pp. 27–56; Phys. Rev. Lett. vol. 86, 2001, p. 3316), we provide an explanation for the scaling laws in each regime and highlight the significant impact of mechanical dispersion effects during the development of the plumes. Our findings provide some new insights into the validity range of the Darcy model.

The stability characteristics of a Mach $5.35$ boundary-layer flow over a flat plate with parametrised two-dimensional sinusoidal surface roughness are investigated. The investigation involves varying the roughness height from $10\,\%$ to $44\,\%$ of the boundary-layer thickness and exploring wavelengths ranging between $0.44$ and $3.56$ times the dominant second-mode wavelength in the region. The introduction of surface roughness leads to notable variations in the mean flow, resulting in separation behind the roughness elements and the propagation of local compression and expansion waves into the free stream. Stability investigations involved the utilisation of wave packet tracking in a linear disturbance simulation (LDS) framework and linear stability theory. The findings revealed significant effects on Mack modes including a reduction in frequency corresponding to maximum amplification with increased roughness height. Proper scaling of the dominant wavelength facilitates a collapse of the growth rate data. In contrast to the commonly reported stabilisation effects for roughness wavelengths significantly larger than the instability mode’s wavelength, the findings primarily revealed destabilisation compared with the smooth-wall case, except for cases with very small roughness wavelengths and large amplitudes approaching the threshold of being classified as porous media. The LDS findings depicted lobed wall pressure amplitude plots, indicating potential undiscovered instability mechanisms or differences compared with the smooth wall. A detailed stability analysis elucidates these LDS findings, establishing a connection between the lobed amplitude structures and substantial changes in local stability characteristics, along with the emergence of Mack’s first, second and third modes.

Parametric oscillations of an interface separating two fluid phases create nonlinear surface waves, called Faraday waves, which organise into simple patterns, such as squares and hexagons, as well as complex structures, such as double hexagonal and superlattice patterns. In this work, we study the influence of surfactant-induced Marangoni stresses on the formation and transition of Faraday-wave patterns. We use a control parameter, $B$, that assesses the relative importance of Marangoni stresses as compared with the surface-wave dynamics. Our results show that the threshold acceleration required to destabilise a surfactant-covered interface through vibration increases with increasing $B$. For a surfactant-free interface, a square-wave pattern is observed. As $B$ is incremented, we report transitions from squares to asymmetric squares, weakly wavy stripes and ultimately to ridges and hills. These hills are a consequence of the bidirectional Marangoni stresses at the neck of the ridges. The mechanisms underlying the pattern transitions and the formation of exotic ridges and hills are discussed.

Randomness is one of the most important characteristics of turbulence, but its origin remains an open question. By means of a ‘thought experiment’ via several clean numerical experiments based on the Navier–Stokes equations for two-dimensional turbulent Kolmogorov flow, we reveal a new phenomenon, which we call the ‘noise-expansion cascade’ whereby all micro-level noises/disturbances at different orders of magnitudes in the initial condition of Navier–Stokes equations enlarge consistently, say, one by one like an inverse cascade, to macro level. More importantly, each noise/disturbance input may greatly change the macro-level characteristics and statistics of the resulting turbulence, clearly indicating that micro-level noise/disturbance might have great influence on macro-level characteristics and statistics of turbulence. In addition, the noise-expansion cascade closely connects randomness of micro-level noise/disturbance and macro-level disorder of turbulence, thus revealing an origin of randomness of turbulence. This also highly suggests that unavoidable thermal fluctuations must be considered when simulating turbulence, even if such fluctuations are several orders of magnitudes smaller than other external environmental disturbances. We hope that the ‘noise-expansion cascade’, as a fundamental property of the Navier–Stokes equations, could greatly deepen our understandings about turbulence, and also be helpful for attacking the fourth millennium problem posed by the Clay Mathematics Institute in 2000.

This study explores an interesting fluid–structure interaction scenario: the flow past a flexible filament fixed at two ends. The dynamic performance of the filament under various inclination angles ($\theta$) was numerically investigated using the immersed boundary method. The motion of the filament in the $\theta$–$Lr$ space was categorised into three flapping modes and two stationary modes, where $Lr$ is the ratio of filament length to the distance between its two ends. The flow fields for each mode and their transitions were introduced. A more in-depth analysis was carried out for flapping at a large angle (FLA mode), which is widely present in the $\theta$–$Lr$ space. The maximum width $W$ of the time-averaged shape of the filament has been shown to strongly correlate with the flapping frequency. After non-dimensionalising based on $W$, the flapping frequency shows little variation across different $Lr$ and $\theta$. Moreover, two types of lift variation process were also identified. Finally, the total lift, drag and lift-to-drag ratio of the system were studied. Short filaments, such as those with $Lr\leqslant 1.5$, were shown to significantly increase lift and the lift-to-drag ratio over a wide range of $\theta$ compared with a rigid plate. Flow field analysis concluded that the increases in pressure difference on both sides of the filament, along with the upper part of the flexible filament having a normal direction closer to the $y$ direction, were the primary reasons for the increase in lift and lift-to-drag ratio. This study can provide some guidance for the potential applications of flexible structures.

We revisit the model problem of Squires & Brady (Phys. Fluids, vol. 17, 2005, 073101), where a Brownian probe is dragged through a dilute dispersion of Brownian bath particles. In this problem, the microrheology due to excluded-volume interactions is represented by an effective viscosity, with the nonlinearity in the driving force entering via the dependence of the viscosity increment (relative to the viscosity of a pure solvent) upon the deformation of the dispersion microstructure. Our interest is in the limit of large Péclet numbers, $ P{\kern-1pt}e\gg 1$, where the microstructural deformation adopts the form of a boundary layer about the upstream hemisphere of the probe. We show that the boundary-layer solution breaks down at the equator of the probe and identify a transition region about the equator, connecting the layer to a downstream wake. The microstructural deformation in this region is governed by a universal boundary-value problem in a semi-bounded two-dimensional domain. The equatorial region continues downstream as a transition layer, which separates the wake of the probe from the undisturbed ambient; in that layer, the microstructure is governed by a one-dimensional heat-like equation. Accounting for the combined contributions from the respective asymptotic provinces we find the approximation $ ({1}/{2})[1+ (\ln P{\kern-1pt}e + 1.046)/ P{\kern-1pt}e]$ for the ratio of the large-$ P{\kern-1pt}e$ viscosity increment to the corresponding linear-response increment. Our asymptotic approximation is in excellent agreement with the increment predicted by a finite-difference numerical calculation of the microstructure deformation, tailored to the large-$ P{\kern-1pt}e$ topology.

An optimal microswimmer with a given geometry has a surface velocity profile that minimises energy dissipation for a given swimming speed. An axisymmetric swimmer can be puller-, pusher- or neutral-type depending on the sign of the stresslet strength. We numerically investigate the type of optimal surface-driven active microswimmers using a minimum dissipation theorem for optimum microswimmers. We examine the hydrodynamic resistance and stresslet strength with nonlinear dependence on various deformation modes. Optimum microswimmers with fore-and-aft symmetry exhibit neutral-type behaviour. Asymmetrical geometries exhibit pusher-, puller- or neutral-type behaviour, depending on the dominant deformation mode and the nonlinear dependence of the stresslet for an optimum microswimmer on deformation mode and amplitude.

We present numerical analysis of the lateral movement of a spherical capsule in the steady and pulsatile channel flow of a Newtonian fluid for a wide range of oscillatory frequencies. Each capsule membrane satisfying strain-hardening characteristics is simulated for different Reynolds numbers $Re$ and capillary numbers $Ca$. Our numerical results showed that capsules with high $Ca$ exhibit axial focusing at finite $Re$ similarly to the inertialess case. We observe that the speed of the axial focusing can be substantially accelerated by making the driving pressure gradient oscillate in time. We also confirm the existence of an optimal frequency that maximises the speed of axial focusing, which remains the same found in the absence of inertia. For relatively low $Ca$, however, the capsule exhibits off-centre focusing, resulting in various equilibrium radial positions depending on $Re$. Our numerical results further clarify the existence of a specific $Re$ for which the effect of the flow pulsation to the equilibrium radial position is maximum. The roles of channel size on the lateral movements of the capsule are also addressed. Throughout our analyses, we have quantified the radial position of the capsule in a tube based on an empirical expression. Given that the speed of inertial focusing can be controlled by the oscillatory frequency, the results obtained here can be used for label-free cell alignment/sorting/separation techniques, e.g. for circulating tumour cells in cancer patients or precious haematopoietic cells such as colony-forming cells.

The interaction between acoustic and surface gravity waves is generally neglected in classical water-wave theory due to their distinct propagation speeds. However, nonlinear dynamics can facilitate energy exchange through resonant triad interactions. This study focuses on the resonant triad interaction involving two acoustic modes and a single gravity wave in water of finite and deep depths. Using the method of multiple scales, amplitude equations are derived to describe the spatio-temporal behaviour of the system. Energy transfer efficiency is shown to depend on water depth, with reduced transfer in deeper water and enhanced interaction in shallower regimes. Numerical simulations identify parameter ranges, including resonant gravity wavenumber, initial acoustic amplitude and wave packet width, where the gravity-wave amplitude is either amplified or reduced. These results provide insights into applications such as tsunami mitigation and energy harnessing.

Two-dimensional Euler flows, in the plane or on simple surfaces, possess a material invariant, namely the scalar vorticity normal to the surface. Consequently, flows with piecewise-uniform vorticity remain that way, and moreover evolve in a way which is entirely determined by the instantaneous shapes of the contours (interfaces) separating different regions of vorticity – this is known as ‘contour dynamics’. Unsteady vorticity contours or interfaces often grow in complexity (lengthen and fold), either as a result of vortex interactions (like mergers) or ‘filamentation’. In the latter, wave disturbances riding on a background, equilibrium contour shape appear to inevitably steepen and break, forming filaments, repeatedly– and perhaps endlessly. Here, we revisit the onset of filamentation. Building upon previous work and using a weakly nonlinear expansion to third order in wave amplitude, we derive a universal, parameter-free amplitude equation which applies (with a minor change) both to a straight interface and a circular patch in the plane, as well as circular vortex patches on the surface of a sphere. We show that this equation possesses a local, self-similar form describing the finite-time blow up of the wave slope (in a re-scaled long time proportional to the inverse square of the initial wave amplitude). We present numerical evidence for this self-similar blow-up solution, and for the conjecture that almost all initial conditions lead to finite-time blow up. In the full contour dynamics equations, this corresponds to the onset of filamentation.

The wave kinetic equation has become an important tool in different fields of physics. In particular, for surface gravity waves, it is the backbone of wave forecasting models. Its derivation is based on the Hamiltonian dynamics of surface gravity waves. Only at the end of the derivation are the non-conservative effects, such as forcing and dissipation, included as additional terms to the collision integral. In this paper, we present a first attempt to derive the wave kinetic equation when the dissipation/forcing is included in the deterministic dynamics. If, in the dynamical equations, the dissipation/forcing is one order of magnitude smaller than the nonlinear effect, then the classical wave action balance equation is obtained and the kinetic time scale corresponds to the dissipation/forcing time scale. However, if we assume that the nonlinearity and the dissipation/forcing act on the same dynamical time scale, we find that the dissipation/forcing dominates the dynamics and the resulting collision integral appears in a modified form, at a higher order.

We investigate the effects of thermal boundary conditions and Mach number on turbulence close to walls. In particular, we study the near-wall asymptotic behaviour for adiabatic and pseudo-adiabatic walls, and compare to the asymptotic behaviour recently found near isothermal cold walls (Baranwal et al. 2022. J. Fluid Mech. 933, A28). This is done by analysing a new large database of highly-resolved direct numerical simulations of turbulent channels with different wall thermal conditions and centreline Mach numbers. We observe that the asymptotic power-law behaviour of Reynolds stresses as well as heat fluxes does change with both centreline Mach number and thermal condition at the wall. Power-law exponents transition from their analytical expansion for solenoidal fields to those for non-solenoidal field as the Mach number is increased, though this transition is found to be dependent on the thermal boundary conditions. The correlation coefficients between velocity and temperature are also found to be affected by these factors. Consistent with recent proposals on universal behaviour of compressible turbulence, we find that dilatation at the wall is the key scaling parameter for these power-law exponents, providing a universal functional law that can provide a basis for general models of near-wall behaviour.

Debris flows are a growing natural hazard as a result of climate change and population density. To effectively assess this hazard, simulating field-scale debris flows at a reasonable computational cost is crucial. We enhance existing debris flow models by rigorously deriving a series of depth-averaged shallow models with varying complexities describing the behaviour of grain–fluid flows, considering granular mass dilatancy and pore fluid pressure feedback. The most complete model includes a mixture layer with an upper fluid layer, and solves for solid and fluid velocity in the mixture and for the upper fluid velocity. Simpler models are obtained by assuming velocity equality in the mixture or single-layer descriptions with a virtual thickness. Simulations in a uniform configuration mimicking submarine landslides and debris flows reveal that these models are extremely sensitive to the rheology, the permeability (grain diameter) and initial volume fraction, parameters that are hard to measure in the field. Notably, velocity equality assumptions in the mixture hold true only for low permeability (corresponding to grain diameter $d=10^{-3}$ m). The one-layer models’ results can strongly differ from those of the complete model, for example, the mass can stop much earlier. One-layer models, however, provide a rough estimate of two-layer models when permeability is low, initial volume fraction is far from critical and the upper fluid layer is very thin. Our work with uniform settings highlights the need of developing two-layer models accounting for dilatancy and for an upper layer made either of fluid or grains.

Turbulent flames in practical devices are subject to a superposition of broadband turbulence and narrowband harmonic flow oscillations. In such cases, flames have a superposition of space–time correlated wrinkles, superposed with broadband turbulent disturbances that interact nonlinearly. This paper extends our prior experimental work to characterise and quantify these flame dynamics. We extract ensemble-averaged flame edge and velocity by ensemble-averaging the instantaneous data at the same phase with respect to the forcing cycle. This paper shows that the ensemble-averaged spatio-temporal dynamics of the flame changes significantly with turbulence intensity. From a spatial viewpoint, the ensemble-averaged flame at weak turbulence intensities exhibits clear cusps and a large ratio between curvature in concave and convex regions. In contrast, at high turbulence intensities, the concave and convex parts of the ensemble-averaged flame are nearly symmetric. From a temporal viewpoint, increasing turbulence intensity monotonically suppresses higher harmonics of the forcing frequency that are manifestations of flame nonlinearities. Taken together, these both point to the interesting observation that the ensemble-averaged flame exhibits increasingly linear dynamics with increasing turbulence intensities, in contrast to its very strong nonlinear behaviours at weak turbulence intensities and juxtaposed with the increasingly nonlinear nature of its instantaneous dynamics with increasing turbulence intensity. In addition, prior studies have shown clear coherent modulation of turbulent flame speed correlated with coherent curvature modulation and that this relationship could be quantified via a ‘turbulent Markstein number’, $M_{T}$. We develop correlations for $M_{T}$ showing how it scales with turbulent and narrowband disturbance quantities, such as turbulent flame brush thickness and convective length scale.

Turbulent mixing in a supercritical CO$_2$ shear layer is examined using both experimental and numerical methods. Boundary conditions are selected to focus on the rarely studied near-critical regime, where thermophysical properties vary nonlinearly with respect to temperature and pressure. Experimental results are obtained via Raman spectroscopy and shadowgraphy, while numerical results are obtained via direct numerical simulation. The shear layer growth rate is found to be 0.2. Additionally, density profiles indicate a relaxation of density gradients between the mixed fluid and heavy fluid as the flow evolves downstream, which runs counter to existing supercritical shear layer data in the literature. The computational results identify significant anisotropy in the turbulence in the shear layer, which is discussed in terms of the development of regions of high density gradient magnitude. The Reynolds-averaged enstrophy budget at various streamwise locations indicates no significant dilatational or baroclinic contribution within the shear layer.

Thermal Marangoni effects play important roles in bubble dynamics such as bubbles generated by water electrolysis or boiling. As macroscopic bubbles often originate from nucleated nanobubbles, it is crucial to understand how thermocapillarity operates at the nanoscale. In this study, the motion of transient bulk gas nanobubbles in water driven by a vertical temperature gradient between two solid plates is investigated using molecular dynamics simulations and analytical theory. The simulation results show that due to the thermal Marangoni force, nanobubbles move towards the hot plate at a constant velocity, similar to the behaviour of macroscale gas bubbles. However, unlike macroscale gas bubbles whose thermal conductivity and viscosity are negligible compared to those of water, the thermal conductivity and viscosity of nanoscale gas bubbles are significantly increased due to their large gas density. The thermal resistance and the slip length are also found to matter at the liquid–gas interface, though they decrease with increasing gas densities. The previous thermocapillary theory for macroscale bubbles is extended by considering all these nanoscopic effects. Expressions of the Marangoni force and the drag force are derived. By balancing the Marangoni force and the drag force, the theoretical velocity of the nanobubble migration in a thermal gradient is obtained. When using the measured transport properties of liquid, gas, and their interfaces, the theoretically obtained velocity is consistent with the result of the molecular simulations. We find that the slip length is too small to have considerable effects on nanobubble motions in the current liquid–gas system.

This paper presents detailed analyses of the Reynolds stresses and their budgets in temporally evolving stratified wakes using direct numerical simulation. Ensemble averaging is employed to mitigate statistical errors in the data, and the results are presented as functions of both the transverse and vertical coordinates – at time instants across the near-wake, non-equilibrium, and quasi-two-dimensional regimes for wakes in weakly and strongly stratified environments. Key findings include the identification of dominant terms in the Reynolds stress transport equations and their spatial structures, the generation and destruction processes of the Reynolds stresses, and the energy transfer between the Reynolds stress and the mean flow. The study also clarifies the effects of the Reynolds number and the Froude number. Additionally, we assess the validity of the eddy-viscosity type models and some existing closures for the Reynolds stress model, highlighting the limitations of isotropy and return-to-isotropy hypotheses in stratified flows.