Fingering instabilities that develop at the advancing front of viscous gravity currents originate from the competition between gravitationally driven amplification of the front and capillary regularisation. Although the large-scale evolution of gravity currents is accurately captured by similarity solutions, the physical mechanisms underlying wavelength selection are not yet fully understood. In this study, a reduced front-tracking framework is formulated by coupling lubrication theory, self-similar spreading laws and a localised description of front dynamics. The approach is first applied to the classical experiments of Huppert (Nature, vol. 300, 1982b, pp. 427–429), involving currents propagating down inclined planes, and is subsequently generalised to gravity currents spreading over conical surfaces. New experiments conducted in conical geometry with Newtonian fluids that span a broad range of physical properties validate the predicted self-similar spreading behaviour and yield quantitative measurements of the emergent fingering wavelength. The results demonstrate that wavelength selection arises from the interplay between the global spreading dynamics and local processes at the front. In particular, the conical geometry modifies the instability by inducing a radial expansion of the front, which diverts a portion of the incoming flux into the azimuthal direction and thereby alters the balance between destabilising and stabilising mechanisms. The proposed framework offers a physically transparent interpretation of wavelength selection and establishes a direct connection between similarity solutions, front dynamics and pattern formation in viscous gravity currents. A distinctive feature of the model is that the selected wavelength is not purely a local property of the advancing front, but depends on the cumulative amplification history of the perturbations during current propagation.

Flow reorganisation and intermittent cessations of the large-scale convection (LSC) are investigated in turbulent liquid-metal Rayleigh–Bénard convection at a Prandtl number $ \textit{Pr} \thickapprox 0.03$. Flow measurements are performed at Rayleigh numbers of $\textit {Ra} = 9.33 \times 10^6, 5.31 \times 10^7$ and $6.02 \times 10^8$ in a cylindrical sample of aspect ratio $\varGamma$ = 0.5 filled with the ternary alloy GaInSn. The LSC structure is characterised using two complementary diagnostics: the classical multi-thermoprobe method based on sidewall temperature measurements and contactless inductive flow tomography (CIFT), which reconstructs the three-dimensional velocity field and enables a proper orthogonal decomposition of the dominant flow modes. While both diagnostics provide consistent descriptions of well-organised circulation states, they increasingly diverge during strongly three-dimensional flow reorganisation as they probe different physical quantities. The volumetric CIFT reconstructions reveal that cessations are associated with a temporary loss of global flow coherence accompanied by a redistribution of kinetic energy from the dominant single-roll and double-roll circulation modes towards higher-order structures. In contrast, localised flow reorganisations are identified as events representing transitions between competing circulation modes without complete loss of global coherence. The transition pathways are highly stochastic and show no unique modal sequence. The results provide the first structurally resolved experimental characterisation of the dynamics of flow cessations in turbulent liquid-metal convection and demonstrate the role of flow mode transitions in triggering large-scale intermittency at very low Prandtl numbers.

What do the ocean surface and a swaying flag have in common? Both are deformable surfaces exhibiting chaotic motion when exposed to turbulent flows. Whether such motion is primarily driven by flow turbulence or by nonlinear dynamics intrinsic to the surface remains debated. Surface waves can interact nonlinearly and transfer energy across scales through the cascade of wave turbulence, a behaviour observed at interfaces between otherwise quiescent fluids and in controlled laboratory experiments. They can as well induce turbulent motions in the neighbouring fluids (wave-induced-turbulence), provided the local Reynolds number is large enough. Realistic environments, however, are more complex and typically involve the simultaneous presence of wave turbulence and wave-induced turbulence with turbulence-induced waves, the dynamic relevance of which remains unclear. Here we develop a theoretical framework describing the response of a deformable surface to pressure fluctuations generated by a turbulent flow, and validate it using numerical simulations of the air–water interface in quasi-realistic conditions, complemented by simulations of a deformable rubber layer. Our linear theory, which excludes nonlinear wave–wave interactions, predicts distinct dynamical regimes depending on whether intrinsic surface dynamics emerges or whether the interface is enslaved by flow turbulence. Remarkably, although our fully resolved and nonlinear simulations do not inhibit the onset of wave turbulence, interface spectra measured in the configurations studied here are controlled at leading order by turbulent pressure forcing and linear surface response, in strong agreement with our theoretical assumptions. Our predictions appear compatible with aerial surveys of the ocean surface, highlighting the need for further, joint measurements of the pressure, elevation and velocity at the surface to distinguish among wave turbulence and turbulence-induced-waves.

Direct numerical simulation is employed to investigate the transport of inertial particles in turbulent channel flows over spanwise rod-roughened walls. The study examines the effects of roughness geometry by varying the streamwise spacing-to-height ratio $w/k$ from 1 to 199. Results indicate that rod roughness elements significantly modulate near-wall turbulent structures and drag coefficients, as both form and total drag reach a maximum value at approximately $w/k=9$. For the dispersed phase, the presence of rough walls suppresses the near-wall accumulation of inertial particles typically observed in smooth-wall turbulence. This reduction in preferential concentration is attributed to the modification of near-wall flow structures, which weakens the correlation between particle and fluid motions. Specifically, the roughness elements disrupt canonical turbulent coherent structures and generate additional roughness-induced flow structures. Unlike the direct collision rebounding effect observed in curved roughness geometries, this decorrelation diminishes the effectiveness of flow structures in trapping particles within the near-wall region and serves as the primary mechanism for reduced near-wall accumulation. Furthermore, a more uniform particle distribution in the wall-normal direction leads to increased streamwise bulk velocity and enhanced transport efficiency, which vary depending on particle inertia and the streamwise spacing-to-height ratio. These findings underscore the critical role of wall roughness geometry in modulating particle spatial distribution and streamwise transport in wall turbulence.

We propose a Navier–Stokes-driven analysis of mean and fluctuating wall shear stresses (WSS) applied to fully developed turbulent channel flow data from direct numerical simulations at friction Reynolds numbers up to $ \textit{Re}_\tau \approx 2000$. Starting from the streamwise momentum equation, we derive exact integral equations that relate the square plane-average and the square fluctuating WSS to wall-normal integrals of terms combining shear with acceleration, shear with pressure gradient, and shear with viscous diffusion. The square plane-average WSS can be well approximated by the product of plane-average shear and plane-average acceleration integrated over the buffer layer with corrections from the mean pressure gradient, which diminish as the reciprocal of the Reynolds number. The square fluctuating WSS is similarly well approximated by the shear-acceleration and shear-pressure-gradient covariances integrated over the buffer layer, but the latter increases in magnitude with Reynolds number and is therefore not negligible. The acceleration fluctuations around the plane-average acceleration consist of a local Eulerian fluctuating acceleration, an advective acceleration and a term that gives rise to a turbulence production contribution to the shear-acceleration covariance. By Taylor’s frozen turbulence hypothesis, the Eulerian acceleration and the streamwise mean advection part of the advective acceleration cancel each other. The shear-acceleration covariance is characterised by a near-wall peak that results from turbulence production and, more specifically, sweeps.

We investigate the enhancement of turbulent heat transport by phase-change bi-phasic particles in a cylindrical Rayleigh–Bénard convection cell. The system comprises deionised water and a small volume fraction ($\phi = 0.86\,\%$) of HFE-7000, a secondary fluid with a low boiling point ($T_{cr} = 42\,^{\circ }\mathrm{C}$). By fixing the top-plate temperature at $T_t = 10\,^{\circ }\mathrm{C}$ and varying the bottom-plate temperature $T_b$, we characterise the global Nusselt number $\textit{Nu}$ as a function of the superheat $\Delta T_{\textit{sup}}\equiv T_b - T_{cr}$. While the formation of bi-phasic particles initially catalyses heat transport, we report the discovery of a saturation plateau in the enhancement of heat transport for superheats exceeding $2\,^{\circ }\mathrm{C}$, a phenomenon not observed in previous higher-Rayleigh-number studies. Flow visualisation demonstrates that the bi-phasic particles are restricted to the upper subcooled region of the cell, effectively partitioning the flow into a two-layer system. An interface is identified separating a lower single-phase liquid layer from an upper layer containing vigorously moving bi-phasic particles. We find that as the superheat increases, the vertical extent of the upper layer (the effective particle travelling distance) decreases. This geometric constriction leads to a reduction in the local heat transport efficiency of the upper region, which is offset by an increase in the efficiency of the lower layer.

Effective slip boundary conditions have proven effective in mechanistic interpretation and predictive modelling over micro-textured drag-reducing surfaces at moderate Reynolds numbers, but their accuracy deteriorates as the texture size and flow rate increase. Torque measurements and direct numerical simulations of Taylor–Couette turbulence over patterned slip surfaces show that this breakdown occurs in a large-texture, high-turbulence regime in which near-wall inertial convection is no longer negligible. Decomposition of angular-momentum transport reveals a near-wall modulated layer where convection is dominated by surface-induced secondary vortices, in contrast to the outer region where turbulent mixing and Taylor rolls govern the transport. Spectral kinetic-energy analysis further shows that spanwise heterogeneity generates wall modes at the texture wavenumber, which undergo self-interaction within the modulated layer and couple to the outer large-scale circulation through cascade-like transfer in the outer region. Motivated by these observations, the slip-only boundary model is extended by introducing an effective near-wall turbulent viscosity, $\nu _{t,\textit{eff}}$, to represent the convective correction. The resulting model links the texture parameters (texture width $w^+$ and solid fraction $\phi _s$) to the effective coefficients (wall slip $U_s$ and $\nu _{t,\textit{eff}}$), and thereby to the skin-friction coefficient $C_f$. This model captures the observed transition in effective slip scaling from a viscous-dominated regime ${\sim}(w^+)^{3/4}$ to an inertial-affected regime ${\sim}(w^+)^{1/4}$, thus providing a predictive effective-boundary framework in turbulence over patterned slip surfaces.

The mechanisms underlying the generation and evolution of Rayleigh–Taylor instability (RTI) in rarefied gases are an important issue, particularly in many emerging problems where rarefaction effects become significant. Using a graphics processing unit parallelised direct simulation Monte Carlo approach, we simulate RTI across a wide range of Knudsen numbers ($Kn$), and find that instability growth is progressively inhibited as $Kn$ increases, with complete suppression above a critical value $Kn \approx 0.011$. Molecular-scale analysis reveals that rarefaction enhances super-diffusive transport of the molecules, extending the ballistic regime, and favouring the interfacial mixing. This microscale behaviour macroscopically manifests as a rapid decay of the density gradient and the effective Atwood number. To determine whether instability emerges in rarefied conditions, we introduce a stability criterion based on the competition between a diffusive time scale ($\tau _D$) and an instability time scale ($\tau _I$). The transition to instability occurs only when $\tau _I \lt \tau _D$, a condition that fails for $Kn \gtrsim 0.011$. The effect of domain height is interpreted through its influence on the Froude number ($Fr$). Small domains (large $Fr$) hinder the transition from linear to nonlinear growth, while for larger domains, instability suppression is mainly determined by the threshold Knudsen number. The critical threshold $Kn \approx 0.011$ remains consistent for sufficiently small $Fr$. These findings provide a multiscale perspective on flow instability in rarefied gases, and establish a time scale competition framework that quantitatively explains how rarefaction governs the onset of RTI.

Simulations are conducted on an elliptical-nosed, square-backed cylinder in the presence of a moving ground plane. Open-loop active flow control is applied to the body in the form of zero-net-mass-flux combined injection and suction near the top trailing edge. Both steady and periodic control are used, and their effects on the aerodynamic forces on the body and wake flow are examined. Steady control is shown to deflect the separating shear layer and reduce the wake width, subsequently reducing the drag. The effect of periodic control is shown to be sensitive to the control frequency–different frequencies can either enhance or inhibit the formation of large-scale coherent structures in the wake. Notably, all five actuation cases tested resulted in a drag reduction over the natural flow case except when the frequency matched the wake shedding frequency. A drag reduction of ${\sim}25$ % occurred for the highest control frequency tested ($\textit{St}= \textit{fH} /U_\infty=5$, with $f$ the forcing frequency, $H$ the body height, and $U_\infty$ the free-stream velocity) for a non-dimensional momentum coefficient of 0.006.

In this study, direct numerical simulations are performed to investigate inertial particle transport in supersonic turbulent flow in a straight-to-bent pipe. Three particle populations with viscous Stokes numbers (St+) of 0.9, 4 and 16 are considered. It is revealed that under the combined influence of wall confinement, flow compressibility and centrifugal forces, distinct turbulent structures emerge in different pipe regions, leading to inertia-dependent particle dynamics. In the straight section, particle streaks and near-wall accumulation are more significant for St+ = 4 particles. When entering the curved section, Görtler vortices along the outer bend enhance the streak formation and the accumulation of St+ = 16 particles. Further downstream, expansion waves suppress Görtler vortices while Dean vortices become dominant, transporting particles along the sidewalls towards the inner bend, thereby reducing outer-wall collisions and accumulation. Particles tend to escape the confinement of Dean vortices and accumulate on the sidewalls. A larger Stokes number results in an earlier detachment, accompanied by higher collision frequency, particle accumulation and slip velocity. In contrast, small inertia particles follow the vortical motions more closely, migrating into the inner low-speed region and resulting in a higher local particle concentration. These different behaviours stem primarily from differences in the centripetal force imparted by the flow in the streamwise and circumferential directions. The strongest centrifugal effect occurs in the areas on both lateral sides of the pipeline. These elucidations offer new physical insights for modelling and predicting particle-laden turbulent flows in curved geometries.

The intrinsic effect of shear-thinning on successive instabilities in narrow-gap Taylor–Couette flow is investigated using direct numerical simulation (DNS) with axially periodic boundary conditions, thereby eliminating endwall effects. The Carreau model is employed with three shear-thinning indices ($n = 0.52$, $0.62$, $0.68$), matching the experimental parameters of Cagney and Balabani (Phys. Fluids, 2019, vol. 31, p. 053102). Shear-thinning exerts opposing influences on the two instabilities. For the primary transition from circular Couette flow to Taylor vortex flow, enhanced shear-thinning (decreasing $n$) is destabilising, with the critical Reynolds number $\textit{Re}_{c,1}$ decreasing from $101$ to $94$. In contrast, the secondary transition to wavy vortex flow is strongly stabilised, with $\textit{Re}_{c,2}$ increasing from $115$ to $142$ over the same range of $n$. This opposing behaviour arises from a competition between two consequences of radial viscosity variation: the reduced viscosity near the inner wall promotes centrifugal disturbances (destabilising), while the elevated viscosity in the bulk gap damps inter-vortex oscillations (stabilising). Comparison with the non-monotonic experimental measurements of Cagney and Balabani (Phys. Fluids, 2019, vol. 31, p. 053102) shows that the observed discrepancies originate from endwall-induced wavelength constraints and vortex drift rather than intrinsic rheological effects. These results establish the intrinsic bifurcation structure of shear-thinning Taylor–Couette flow in the narrow-gap configuration and clarify the distinct roles of rheology and confinement in successive flow transitions.

Bioinspired propulsion demonstrates broad potential for engineering applications. In this study, the hydrodynamic behaviour of an undulating hydrofoil operating at a fixed submergence depth near the free surface is numerically investigated using the smoothed particle hydrodynamics method. The body-wave wavelength $\lambda$ is introduced to characterise the spatial pattern of the prescribed undulatory motion of the hydrofoil, and the effects of submergence depth and wavelength on propulsion performance are systematically analysed together with flow field structures. The results show that cruising speed decreases monotonically with decreasing submergence depth, whereas propulsion efficiency exhibits a non-monotonic trend, indicating an optimal submergence depth for maximum efficiency. Propulsion efficiency decreases monotonically with increasing wavelength, whereas, at shallow submergence depths, an optimal wavelength exists that maximises the cruising speed. When operating close to the free surface, the hydrofoil experiences an attractive force towards the surface; therefore, a negative angle of attack is required to maintain a fixed depth. Free-surface wave generation intensifies with decreasing submergence depth and increasing wavelength; however, under mild wave-breaking conditions, the cruising efficiency of near-surface hydrofoils may still exceed that achieved in deep-water operation. In addition, a scaling relationship between cruising speed, submergence depth and wavelength is obtained through curve fitting and can be used to predict cruising speed. These results provide useful guidance for the optimisation and control of undulating hydrofoils operating near the free surface.

The onset and control of unsteadiness in an idealised porous medium are investigated using global stability and sensitivity analysis. The flow is modelled by a periodic representative elementary volume composed of tandem-arranged circular cylinders, and the Hopf bifurcation is characterised over a broad porosity range ($0.45 \leqslant \phi \leqslant {0.95}$). The neutral stability curve exhibits an approximately linear dependence on $\phi$, indicating that porosity serves as an effective predictor for the transition threshold. In the vicinity of instability onset, several eigenmode families emerge, clarifying the organisation of the competing branches. Energy budget analysis further shows that perturbation production is dominated by the lift-up mechanism. In addition, adjoint-based sensitivity analysis near criticality identifies where a localised steady forcing produces the largest drift of the leading global mode. The result highlights the throat of the main passage as the region of strongest receptivity, offering a clear target and an efficient tool for designing flow-control strategies.

Three-dimensional experiments are presented on a school of three pitching hydrofoils. Two side-by-side leader foils maintain the same relative positions while the location of a third follower foil is varied. Force and flow measurements detail the mechanisms leading to collective thrust and efficiency 58 % and 24 % higher than isolated foils, respectively. In traditional drafting, a follower is positioned in the wake of an upstream object. In wakes with a net momentum deficit, drafting reduces drag by lowering oncoming flow speed. By contrast, wakes from oscillatory swimmers feature strong momentum surplus regions, which increases drag by increasing oncoming flow. Despite that, our results show that the best performance benefits occur for compact schools where the follower is directly in the vortex wake of a leader, whereas regions of reduced mean flow do not improve performance. The performance benefits are driven by vortex–body interactions that increase the thrust and efficiency of the follower and by body-to-body upstream interactions that reduce the power of the leaders. There is an optimal spatial phase to maximise the thrust and efficiency of the follower that depends upon the actual wake wavelength rather than the estimated wavelength used in previous literature. Moreover, wake breakdown, and its associated elimination of vortex–body performance benefits, is not observed within at least three chord lengths downstream of the leaders. Lastly, cross-stream stability measurements of the downstream foil indicate that compact, high-performance formations may require active control strategies in order to maintain their organisation and maximise the hydrodynamic benefits of schooling.

We investigate head-on droplet–bubble collisions in a quiescent, surfactant-free viscous liquid using three-dimensional volume-of-fluid simulations. By systematically varying the bubble-to-droplet radius ratio ($0.5\le \varDelta \le 2$) and Weber number ($3\le \textit{We}\le 270$) at small Ohnesorge number ($ \textit{Oh}\lt 1$), we identify the mechanisms governing interfacial deformation, attachment, jet formation and vortex evolution during an isolated collision. Four distinct collision regimes emerge, ranging from oscillatory attachment to jet-mediated penetration and coupled toroidal deformations. A transition from weak- to inertia-dominated dynamics occurs at a critical Weber number of $ \textit{We}\approx 23$, nearly independent of $\varDelta$, whereas the onset of complete jet penetration depends strongly on the size ratio. Comparison with the unsteady Oseen solution for an impulsively accelerated rigid sphere establishes a hydrodynamic baseline that isolates the effects of interface mobility and deformation, revealing the progressive emergence of toroidal vortices, interfacial oscillations and jetting. An energy analysis demonstrates that jet formation is controlled by the conversion of impact kinetic energy into surface energy, yielding a characteristic jet velocity comparable to the impact speed, while viscous dissipation suppresses jetting above a critical Ohnesorge number scaling as $ \textit{Oh}_c\sim We^{-1/2}$. These results provide a unified physical framework linking inertia, capillarity and vortex dynamics in elementary droplet–bubble interactions, offering predictive insight into multiphase processes such as flotation, aeration, emulsification and liquid–liquid separation.

Air lubrication regimes were studied using simultaneous drag force measurements and multi-plane imaging to characterise the regimes and identify the governing mechanisms of drag reduction. Bubbly, transitional and air layer regimes are identified over a large range of free-stream velocities ($U_{\infty }$), air flow rates ($Q_{\textit{air}}$) and Froude-depth numbers ($\textit{Fr}_d$). For the lowest $U_{\infty }$, drag reduction lags significantly behind the non-wetted area coverage in all cases and no simple correlation exists. Within the bubbly regime, a drag increase is found for low $U_{\infty }$ with large, slow-moving bubbles forming a single layer over the plate height. For higher velocities, bubbles become smaller and disperse vertically, while the drag starts decreasing. For higher $Q_{\textit{air}}$, irrespective of $U_{\infty }$, air patches start to form (transitional regime) and drag monotonically decreases, with the onset of the air layer regime at 60 % drag reduction. A new scaling of the associated critical $Q_{\textit{air}}$ is proposed, combining the air exit velocity, the liquid velocity close to the air layer and $\textit{Fr}_d$. For a further increase of $Q_{\textit{air}}$ and a low $U_{\infty }$, a thicker and smoother air layer is formed with even lower drag; for higher $U_{\infty }$, marginal differences are observed. The air layer morphology is significantly altered, however, depending on $\textit{Fr}_d$: for $\textit{Fr}_d\gt 0.7$, it is unbounded, extending beyond the current test section length, and for subcritical conditions (deep water regime, $\textit{Fr}_d\lt 0.61$) a closure is formed and the air layer transitions to a cavity of a specific length.

We report results from experiments and direct numerical simulation (DNS) of the temperature and velocity fields in horizontal convection (HC) over a range of Rayleigh numbers $10^{8} \leqslant Ra \leqslant 5\times 10^{12}$ and Prandtl numbers $4.3 \leqslant Pr \leqslant 7$ in a rectangular cell of aspect ratio $\varGamma \equiv L:W:H = 10:1:1$ ($L$, $W$ and $H$ denote the length, width and height of the cell, respectively). The DNS considers four combinations of velocity and thermal boundary conditions (BCs), spanning free-slip and no-slip velocity constraints together with isothermal and adiabatic thermal conditions. In the experiments, the upper boundary consists of an open water–air interface, which approximates a slippery surface. The DNS results reveal systematic differences in the thermal boundary-layer thickness, vertical temperature gradients and overall flow organisation across the four BC configurations. Building on these findings, we develop a theoretical model based on a prescribed streamfunction that satisfies the relevant boundary constraints. From this formulation, general analytical expressions for the vertical mean temperature profile are derived for each configuration, capturing the interplay among advection, diffusion and the imposed BCs. The model successfully reproduces self-similar features of the temperature profiles observed in the DNS and agrees with available experimental data. This framework thus provides a unified description of vertical temperature and velocity structure in HC and clarifies how velocity and thermal BCs jointly regulate boundary-layer formation and heat transport efficiency. The findings also offer insight into analogous geophysical processes, including urban near-surface heating and oceanic stratification.

We examine the onset of wave breaking in highly viscous gravity-driven vertical falling liquid films in a quiescent atmosphere via multiphase direct numerical simulations. Both Newtonian and Herschel–Bulkley liquids are considered. Analysis of the force balance acting on wave humps shows that the transition from travelling waves to breaking states is governed primarily by the competition between gravitational and capillary forces. The transition is primarily characterised by a criterion based on the Eötvös number (quantifying gravitational to capillary forces), which we identify through a comprehensive parameter study involving over 500 simulations, where we vary the Reynolds number, the plasticity number and the power-law index. We provide a regression model for $\textit{Eo}_{\textit{crit}}$ in terms of these parameters, as well as for the corresponding wetting rate. The parameter study indicates that breaking generally occurs for large film thicknesses and high viscosities, yielding large Eötvös numbers while remaining in the wavy–laminar flow regime. Close to the onset of breaking, we observe overhanging travelling wave solutions, where the wave front deforms into a steady overhang, and this state generally precedes wave breaking irrespective of the investigated rheologies. Our results establish a concise criterion for predicting wave breaking in falling films involving both Newtonian and complex fluids representative of different industrial processes, and have implications for heat and mass transfer, bubble entrainment and liquid residence time.

When jet noise is cast in the form of a linear input–output problem, the cross-spectral-density (CSD) matrix of the sound field is forced by the CSD of nonlinear interactions via the resolvent operator. The forcing CSD is difficult to compute or measure, but its projection onto the resolvent input space can be identified from the CSD of the sound field. We performed two-point acoustic measurements in isothermal turbulent jets over a broad range of subsonic Mach numbers. The CSD matrices are used to obtain an empirical model of the forcing-projection matrix. We investigate the parametric dependence of this low-rank forcing-projection matrix on Mach number, frequency and azimuthal mode, and we propose an empirical model that captures downstream radiation with 2 dB precision for subsonic jets.

In this work, we compute the hydrodynamic force and the first and second moments of force acting on a translating spherical droplet immersed in a uniform flow using the reciprocal theorem. We consider the low but finite Reynolds number regime, $\textit{Re} = a U \rho _{\!f} / \mu _{\!f}$, and the dilute limit of small droplet volume fraction $\phi$. Here, $U$ denotes the magnitude of the relative velocity between the phases, $a$ the droplet radius and $\rho _{\!f}$ and $\mu _{\!f}$ the density and viscosity of the continuous phase, respectively. We show that the $O(\textit{Re})$ inertial corrections to the first and second moments of force scale as $O(\rho _{\!f} \phi U^2)$ and $O(a\rho _{\!f}\phi U^2)$, respectively. Equivalently, in dimensionless form, these corrections scale as $O(\phi \textit{Re})$ and $O(({a}/{R})\phi \textit{Re})$, where $R$ denotes the macroscopic length scale. Moreover, the ensemble average of the drag force and the higher-order force moments over the distribution of droplet velocities introduces additional contributions proportional to the velocity variance of the dispersed phase, both in the interphase momentum exchange and in the effective stress of the continuous phase. As a consequence, in dilute emulsions of buoyant droplets, the effective stress depends quadratically on the relative velocity between the phases, on the velocity variance of the dispersed phase and on the spatial gradients of these quantities. Finally, we consider a steady-state pipe flow to examine how the force moments derived in this study affect the behaviour of the two-phase flow in this particular configuration.