The evolution of settling fine particle clouds in transition or rarefied flow regimes is a fundamental yet insufficiently understood problem in fluid mechanics. Here, we address this challenge numerically using a kinematic model, and approximate the hydrodynamic interaction between particles by superposing velocity disturbances from rarefied gas flows past individual particles. The effect of electrostatic interactions among charged particles is also studied. As an application, we simulate the sedimentation of small dust clouds under Martian conditions, focusing on the 10$\,\unicode{x03BC}$m diameter fraction of ‘settled dust’. Our results show that under Martian conditions, dust clouds develop elongated tails during sedimentation, with up to 25 % of particles leaking from the bulk over a 10 minute period. Unlike Earth-based scenarios, the clouds do not break apart owing to the weaker hydrodynamic interactions in Mars’ thin atmosphere. By examining the interplay between hydrodynamic and electrostatic interactions, which influence particle leakage in opposite ways, we demonstrate that larger dust clouds are also likely to evolve with sustained tail formation. Fully suppressing particle leakage would require particle charges well above $10^4e$, levels unlikely to occur under typical Martian conditions. New analytical expressions are derived for the cloud settling velocity and tail evolution, providing theoretical insights and a foundation for future studies on particle dynamics in transition/rarefied environments.

This work experimentally explores the alignment of the vorticity vector and the strain-rate tensor eigenvectors at locations of extreme upscale and downscale energy transfer. We show that the turbulent von Kármán flow displays vorticity–strain alignment behaviour across a large range of Reynolds numbers, which is very similar to previous studies on homogeneous, isotropic turbulence. We observe that this behaviour is amplified for the largest downscale energy transfer events, which tend to be associated with sheet-like geometries. These events are also shown to have characteristics previously associated with high flow field nonlinearity and singularities. In contrast, the largest upscale energy transfer events display very different structures which showcase a strong preference for vortex compression. Notably, in both cases we find that these trends are strengthened as the probed scales approach the Kolmogorov scale. We then show further evidence for the argument that strain self-amplification is the most salient feature in characterising the cascade direction. Finally, we identify possible invariant behaviour for the largest energy transfer events, even at scales near the Kolmogorov scale.

Droplet impacts with rough surfaces described by Fourier series are investigated assuming gas cushioning is negligible. For impacts leading to a contiguous contact patch, a mixed boundary value problem for the displacement potential is formulated by extending models of inertially dominated droplet impacts with a flat plate. For large times after impact, the contact line evolution for impacts with periodic rough substrates is found to tend to the contact line evolution obtained for a droplet impact with a flat plate vertically positioned at the average height of the rough substrate. For symmetric impacts with even substrate geometries represented by Fourier cosine series, the contact line evolution is given by a Schlömilch series in which the coefficients are related to the coefficients of the corresponding Fourier series. A method for determining whether secondary impacts occur for particular geometries is described and regime diagrams, which show the boundary of the region of substrate parameters associated with single contiguous impacts, are obtained. The loads associated with droplet impacts with periodic rough substrates are calculated and compared with the loads associated with impacts with a flat plate. As the height of the roughness increases, the load associated with an impact with a rough substrate may initially differ significantly from the flat-plate case, although the load on a flat plate is recovered in the limit of large time. The implications of the results for more general droplet impacts with roughness are discussed from both a theoretical and experimental standpoint.

Optimal transitional mechanisms are analysed for an incompressible shear layer developing over a short, pressure gradient-induced laminar separation bubble (LSB) with peak reversed flow of 2 %. Although the bubble remains globally stable, the shear layer destabilises due to the amplification of external time- and spanwise-periodic disturbances. Using linear resolvent analysis, we demonstrate that the pressure gradient modifies boundary layer receptivity, shifting from Tollmien–Schlichting (T-S) waves and streaks in a zero-pressure-gradient environment to Kelvin–Helmholtz (K-H) and centrifugal instabilities in the presence of the LSB. To characterise the nonlinear evolution of these disturbances, we employ the harmonic-balanced Navier–Stokes (N-S) framework, solving the N-S equations in spectral space with a finite number of Fourier harmonics. Additionally, adjoint optimisation is incorporated to identify forcing disturbances that maximise the mean skin friction drag, conveniently chosen as the cost function for the optimisation problem since it is commonly observed to increase in the transitional stage. Compared with attached boundary layers, this transition scenario exhibits both similarities and differences. While oblique T-S instability is replaced by oblique K-H instability, both induce streamwise rotational forcing through the quadratic nonlinearity of the N-S equations. However, in separated boundary layers, centrifugal instability first generates strong streamwise vortices due to multiple centrifugal resolvent modes, which then develop into streaks via lift-up. Finally, we show that the progressive distortion and disintegration of K-H rollers, driven by streamwise vortices, lead to the breakdown of large coherent structures.

We use direct numerical simulations to examine the onset of stratified turbulence triggered by the zigzag instability recently identified in columnar Taylor–Green vortices (Guo et al. 2024, J. Fluid Mech., vol. 997, A34) and its role in layer formation within the flow. The study focuses on Froude numbers $0.125 \leqslant \textit{Fr} \leqslant 2.0$ and Reynolds numbers ${\textit{Re}}$ ranging from 800 to 3200. The breakdown of the freely evolving vortex array is driven by local density overturns, combining shear and convective mechanisms initiated by the primary zigzag instability. Our results show a linear relationship between the peak buoyancy Reynolds number ${{\textit{Re}}}_b^{\star }$, driven by the zigzag instability, and ${\textit{Re}}\, {\textit{Fr}}^2$. When the flow does not exhibit local shear or convective instability, the value of ${{\textit{Re}}}_b^{\star }$ falls below unity. Both density and momentum layers arise from the zigzag instability: horizontal velocity layers are strong and persistent, while density layers are weaker and more transient. The vertical scale of the mean shear layers increases with ${\textit{Fr}}$ for ${\textit{Fr}} \leqslant 1$, shows weak dependence on ${\textit{Re}}$, and agrees well with the length scale associated with the fastest-growing linear mode of the zigzag instability. Further analysis in the sorted buoyancy coordinate highlights the role of density overturns caused by the zigzag instability in forming buoyancy layers during the transition to turbulence.

We develop an optimal resolvent-based estimator and controller to predict and attenuate unsteady vortex-shedding fluctuations in the laminar wake of a NACA 0012 airfoil at an angle of attack of 6.5°, chord-based Reynolds number of 5000 and Mach number of 0.3. The resolvent-based estimation and control framework offers several advantages over standard methods. Under equivalent assumptions, the resolvent-based estimator and controller reproduce the Kalman filter and LQG controller, respectively, but at substantially lower computational cost using either an operator-based or data-driven implementation. Unlike these methods, the resolvent-based approach can naturally accommodate forcing terms (nonlinear terms from Navier–Stokes) with coloured-in-time statistics, significantly improving estimation accuracy and control efficacy. Causality is optimally enforced using a Wiener–Hopf formalism. We integrate these tools into a high-performance-computing-ready compressible flow solver and demonstrate their effectiveness for estimating and controlling velocity fluctuations in the wake of the airfoil immersed in clean and noisy free streams, the latter of which prevents the flow from falling into a periodic limit cycle. Using four shear–stress sensors on the surface of the airfoil, the resolvent-based estimator predicts a series of downstream targets with approximately $3\,\%$ and $30\,\%$ error for the clean and noisy free stream conditions, respectively. For the latter case, using four actuators on the airfoil surface, the resolvent-based controller reduces the turbulent kinetic energy in the wake by $98\,\%$.

The presence of trapped air on a solid surface can alter the direction of the liquid jets induced by cavitation bubbles, which prevents or reduces erosion. In this study, we numerically investigate mutual interaction between air trapped in a pocket on a wall and a nearby bubble in water, as well as the resultant hydrodynamic loading. Both the depth and radius of the cylindrical pocket are similar to the maximum bubble radius. The pressure imposed on the inner wall of the air pocket is assessed for various values of the air pocket size and the stand-off parameter. The deformation of the air pocket and the bubble is analysed in each of three sequential stages. During the bubble expansion stage, a shock wave reflects at the water–air interface of the pocket, and the wall inside the compressed pocket is protected from the shock wave. As the bubble jet induced during bubble contraction tends to move away from the air pocket, other liquid jets formed at the water–air interface, namely central and lateral pocket jets, can directly collide with the inner wall of the pocket after the bubble collapses. These collisions exert significant pressure on the wall under certain conditions. The formation of the central pocket jet originates from the strong fluctuation of the water–air interface by the expanding and contracting bubble. The development of the lateral pocket is related to changes in the potential energy of the air under its second contraction.

Discontinuous shear-thickening (DST) fluids exhibit unique instability properties in a wide range of flow conditions. We present numerical simulations of a scalar model for DST fluids in a planar simple shear using the smoothed particle hydrodynamics approach. The model reproduces the spatially homogeneous instability mechanism based on the competition between the inertial and microstructural time scales, with good congruence to the theoretical predictions. Spatial inhomogeneities arising from a stress-splitting instability are rationalised within the context of local components of the microstructure evolution. Using this effect, the addition of non-locality in the model is found to produce an alternative mechanism of temporal instabilities, driven by the inhomogeneous pattern formation. The reported arrangement of the microstructure is generally in agreement with the experimental data on gradient pattern formation in DST. Simulations in a parameter space representative of realistic DST materials resulted in aperiodic oscillations in measured shear rate and stress, driven by formation of gap-spanning frictional structures.

A new arbitrary Lagrangian–Eulerian (ALE) formulation for Navier–Stokes flow on self-evolving surfaces is presented. It is based on a general curvilinear surface parameterisation that describes the motion of the ALE frame. Its in-plane part becomes fully arbitrary, while its out-of-plane part follows the material motion of the surface. This allows for the description of flows on deforming surfaces using only surface meshes. The unknown fields are the fluid density or pressure, the fluid velocity and the surface motion, where the latter two share the same normal velocity. The corresponding field equations are the continuity equation or area-incompressibility constraint, the surface Navier–Stokes equations and suitable surface mesh equations. Particularly advantageous are mesh equations based on membrane elasticity. The presentation focuses on the coupled set of strong and weak form equations, and presents several manufactured steady and transient solutions. These solutions are used together with numerical simulations to illustrate and discuss the properties of the proposed new ALE formulation. They also serve as basis for the development and verification of corresponding computational methods. The new formulation allows for a detailed study of fluidic membranes such as soap films, capillary menisci and lipid bilayers.

This study explores the Faraday instability as a mechanism to enhance heat transfer in two-phase systems by exciting interfacial waves through resonance. The approach is particularly applicable to reduced-gravity environments where buoyancy-driven convection is ineffective. A reduced-order model, based on a weighted residual integral boundary layer method, is used to predict interfacial dynamics and heat flux under vertical oscillations with a stabilising thermal gradient. The model employs long-wave and one-way coupling approximations to simplify the governing equations. Linear stability theory informs the oscillation parameters for subsequent nonlinear simulations, which are then qualitatively compared against experiments conducted under Earth’s gravity. Experimental results show up to a 4.5-fold enhancement in heat transfer over pure conduction. Key findings include: (i) reduced gravity lowers interfacial stability, promoting mixing and heat transfer; and (ii) oscillation-induced instability significantly improves heat transport under Earth’s gravity. Theoretical predictions qualitatively validate experimental trends in wavelength-dependent enhancement of heat transfer. Quantitative discrepancies between model and experiment are rationalised by model assumptions, such as neglecting higher-order inertial terms, idealised boundary conditions, and simplified interface dynamics. These limitations lead to underprediction of interface deflection and heat flux. Nevertheless, the study underscores the value of Faraday instability as a means to boost heat transfer in reduced gravity, with implications for thermal management in space applications.

Underwater capillary tubes fill rapidly with the surrounding liquid. Capillary and hydrostatic pressures push the liquid into the tube, causing the air to exit as bubbles at the other end. We study the natural filling process of a vertical capillary tube immersed in water during several bubble formation events. A theoretical model is proposed that captures the dynamics of the meniscus inside the capillary tube as it fills with water. We find good agreement with the experimental data that describe this special case of spontaneous flow using a dynamic contact angle model based on molecular kinetic theory.

The ‘vorticity transport’ theory by G. I. Taylor states that, in two-dimensional (2-D) turbulent flows, it is not the momentum of the eddies which is conserved from one step of their random walk to the other (the so-called Reynolds–Prandtl analogy), but their vorticity, implying that the conservation laws for the time-averaged profiles for the velocity $u$ and concentration of a passive scalar $c$ must be different. This theory predicts that, across a 2-D wake or a jet, both fields (scaled by their maximal value) are exactly related to each other by $u=c^2.$ We reexamine critically this problem on hand of several experiments with plane and round turbulent jets seeded with high and low diffusing scalars, and conclude that the microscopic equations for $u$ and $c$ are identical, but that the differences between the $u$- and $c$-fields is a genuine mixing problem, sensitive to the dimensionality of the flow and to the intrinsic diffusivity of the scalar $D$, through the Schmidt number ($Sc=\nu /D$) dependence of the flow coarsening scale. We observe that $u=c^{\beta }$ with $\beta =2$ in plane jets irrespective of $Sc$, $\beta =3/2$ in round jets at $Sc=1$ and $\beta =1$ in round jets for $Sc\to \infty$. We explain why, because measurements dating back to the 1930s–40s were all made for heat transport in air ($Sc\approx 1$), agreement with Taylor‘s vision was only coincidental. The experiments and the new representation proposed here are strictly at odds with Reynolds’ analogy, although essentially an adaptation of it to eddies transporting momentum and mass, but liable to exchange mass with a smooth reservoir along their Brownian path.

Hydrodynamic density functional theory (DFT) is applied to analyse dynamic contact angles of droplets to assess its predictive capability regarding wetting phenomena at the microscopic scale and to evaluate its feasibility for multiscale modelling. Hydrodynamic DFT incorporates the influence of fluid–fluid and solid–fluid interfaces into a hydrodynamic theory by including a thermodynamic model based on classical DFT for the chemical potential of inhomogeneous fluids. It simplifies to the isothermal Navier–Stokes equations far away from interfaces, thus connecting microscopic molecular modelling and continuum fluid dynamics. In this work, we use a Helmholtz energy functional based on the perturbed-chain statistical associating fluid theory (PC-SAFT) and the viscosity is obtained from generalised entropy scaling, a one-parameter model which takes microscopic information of the fluid and solid phase into account. Deterministic (noise-free) density and velocity profiles reveal wetting phenomena including different advancing and receding contact angles, the transition from equilibrium to steady state and the rolling motion of droplets. Compared with a viscosity model based on bulk values, generalised entropy scaling provides more accurate results, which stresses the importance of including microscopic information in the local viscosity model. Hydrodynamic DFT is transferable as it captures the influence of different external forces, wetting strengths and (molecular) solid roughness. For all results, good quantitative agreement with non-equilibrium molecular dynamics simulations is found, which emphasises that hydrodynamic DFT is able to predict wetting phenomena at the microscopic scale.

The constant temperature and constant heat flux thermal boundary conditions, both developing distinct flow patterns, represent limiting cases of ideally conducting and insulating plates in Rayleigh–Bénard convection flows, respectively. This study bridges the gap in between, using a conjugate heat transfer (CHT) set-up and studying finite thermal diffusivity ratios $\kappa _s \! / \! \kappa _f$ to better represent real-life conditions in experiments. A three-dimensional Rayleigh–Bénard convection configuration including two fluid-confining plates is studied via direct numerical simulations given a Prandtl number ${Pr}=1$. The fluid layer of height $H$ and horizontal extension $L$ obeys no-slip boundary conditions at the two solid–fluid interfaces and an aspect ratio of ${\Gamma }=L/H=30$ while the relative thickness of each plate is ${\Gamma _s}=H_s/H=15$. The entire domain is laterally periodic. Here, different $\kappa _s \! / \! \kappa _f$ are investigated for moderate Rayleigh numbers $Ra=\left \{ 10^4, 10^5 \right \}$. We observe a gradual shift of the size of the characteristic flow patterns and their induced heat and mass transfer as $\kappa _s \! / \! \kappa _f$ is varied, suggesting a relation between the recently studied turbulent superstructures and supergranules for constant temperature and constant heat flux boundary conditions, respectively. Performing a linear stability analysis for this CHT configuration confirms these observations theoretically while extending previous studies by investigating the impact of a varying solid plate thickness $\Gamma _s$. Moreover, we study the impact of $\kappa _s \! / \! \kappa _f$ on both the thermal and viscous boundary layers. Given the prevalence of finite $\kappa _s \! / \! \kappa _f$ in nature, this work is a starting point to extend our understanding of pattern formation in geo- and astrophysical convection flows.

We consider laminar forced convection in a shrouded longitudinal-fin heat sink (LFHS) with tip clearance, as described by the pioneering study of (Sparrow, Baliga & Patankar 1978 J. Heat Trans. 100). The base of the LFHS is isothermal but the fins, while thin, are not isothermal, i.e. the conjugate heat transfer problem is of interest. Whereas Sparrow et al. numerically solved the fully developed flow and thermal problems for a range of geometries and fin conductivities, we consider the physically realistic asymptotic limit where the fins are closely spaced, i.e. the spacing is small relative to their height and the clearance above them. The flow problem in this limit was considered by (Miyoshi et al. 2024, J. Fluid Mech. 991, A2), and we consider the corresponding thermal problem. Using matched asymptotic expansions, we find explicit solutions for the temperature field (in both the fluid and fins) and conjugate Nusselt numbers (local and average). The structure of the asymptotic solutions provides further insight into the results of Sparrow et al.: the flow is highest in the gap above the fins, hence heat transfer predominantly occurs close to the fin tips. The new formulas are compared with numerical solutions and are found to be accurate for practical LFHSs. Significantly, existing analytical results for ducts are for boundaries that are either wholly isothermal, wholly isoflux or with one of these conditions on each wall. Consequently, this study provides the first analytical results for conjugate Nusselt numbers for flow through ducts.

The breakup dynamics of viscous liquid bridges on solid surfaces is studied experimentally. It is found that the dynamics bears similarities to the breakup of free liquid bridges in the viscous regime. Nevertheless, the dynamics is significantly influenced by the wettability of the solid substrate. Therefore, it is essential to take into account the interaction between the solid and the liquid, especially at the three-phase contact line. It is shown that when the breakup velocity is low and the solid surface is hydrophobic, the dominant channel of energy dissipation is likely due to thermally activated jumping of molecules, as described by the molecular kinetic theory. Nevertheless, the viscous dissipation in the bulk due to axial flow along the bridge can be of importance for long bridges. In view of this, a scaling relation for the time dependence of the minimum width of the liquid bridge is derived. For high viscosities, the scaling relation captures the time evolution of the minimum width very well. Furthermore, it is found that external geometrical constraints alter the dynamic behaviour of low and high viscosity liquid bridges in a different fashion. This discrepancy is explained by considering the dominant forces in each regime. Lastly, the morphology of the satellite droplets deposited on the surface is qualitatively compared with that of free liquid bridges.

Fourier analysis is the standard tool of choice for quantifying the distribution of kinetic energy amongst the eddies in a turbulent flow. The resulting spectral energy-density function is the well-known energy spectrum. And yet, because eddies are distinct from waves, alternative approaches to finding energy-density functions have long been sought. Townsend (1976) outlined a promising approach to finding a spatial energy-density function, $V\!(r)$, where $r$ is the eddy size. Notably, this approach led to two distinct and mutually inconsistent formulations of $V\!(r)$ in homogeneous, isotropic turbulence. We revisit Townsend’s proposal and derive the corresponding three-dimensional $V\!(r)$ as well as introduce its one-dimensional variants (which, to our knowledge, have not been explicitly discussed before). By training our focus on the associated dimensionality of the function, we resolve the discrepancies between the previous formulations. Additionally, we generalise our analysis to include anisotropic flows. Finally, by means of concrete examples, we illustrate how one-dimensional spatial energy-density functions are useful for analysing empirical data. Some notable findings include new insights into the $k_1^{-1}$ scaling (where $k_1$ is the streamwise wavenumber) and a possible resolution of the enigmatic sizes of organised motions at large scales.

This study uses the diffusion analogy (Miyake, Sci. Rep., 5R-6, 1965, Univ. of Washington, Seattle, USA) to predict the full growth behaviour of internal boundary layers (IBLs) induced by a roughness change for neutrally – and especially stably – stratified boundary layers with finite thickness. The physics of the diffusion analogy shows that the streamwise variation of the IBL thickness is dictated by $\sigma _w/U$ at the interface, where $\sigma _w$ and $U$ represent wall-normal Reynolds stress and mean streamwise velocity, respectively. The existing variants of the model, summarised by Savelyev & Taylor (2005, Boundary-Layer Meteorol., vol. 115, pp. 1–25), are tailored to IBLs confined within the constant shear stress layer. To extend the applicability of the model to the outer region, we investigate the relation between $\sigma _w/U$ and $U/U_\infty$ in the outer region across varying stratification, where $U_\infty$ is the free-stream velocity. Our analysis reveals that wind tunnel data from a number of facilities collapse onto a master curve when $\sigma _w/U$ is premultiplied by a height-independent parameter, which is a function of the ratio of Monin–Obukhov length to the boundary layer thickness. The scaled $\sigma _w/U$ decreases inversely with $U/U_\infty$ in the surface layer, transitioning to a linear decrease as $U/U_\infty$ increases. The new model, which integrates these findings, along with the effects of streamline displacement and acceleration, captures the complete characteristics of IBLs as they develop within turbulent boundary layers of finite thickness.

The Taylor–Melcher leaky dielectric (LD) model is often used to study the physics of electrosprays operating in the cone-jet mode. Despite its success, there are electrospraying conditions in which the ion concentration fields must be retained, which requires an electrokinetic model. This article reproduces cone-jets with two electrokinetic formulations: the standard Poisson–Nernst–Planck (PNP) equations, and a modified electrokinetic (MEK) model that accounts for overscreening and overcrowding of electrolytes, which is important in fluids with high electrical conductivities such as ionic liquids (Kilic et al. 2007 Phys. Rev. E vol. 75, no. 2, 021502, 021503; Bazant et al. 2011 Phys. Rev. Lett. vol. 106, no. 4, 46102). In the case of liquids with low electrical conductivities, it is observed that the LD and PNP models agree under certain limiting conditions, but they are less restrictive than previously proposed (Baygents & Saville 1990 AIP Conf. Proc. vol. 197, 7–17; Schnitzer & Yariv 2015 Fluid Mech. vol. 773, 1–33); the effects of dissimilar ion diffusivities are also investigated. In the case of liquids with high electrical conductivities, in particular ionic liquids, overscreening and overcrowding effects are important, resulting in significant differences between the solutions of the PNP, MEK and LD models. In particular, the electrokinetic models yield increased dissipation and self-heating, leading to higher temperature variations and currents, in agreement with measurements. Furthermore, the MEK formulation describes the ion concentration fields with higher fidelity than the PNP equations.