We investigate through numerical simulations the hydrodynamic interactions between two rigid spherical particles suspended on the axis of a cylindrical tube filled with an elastoviscoplastic fluid subjected to pressure-driven flow. The simulations are performed by the finite-element method with the arbitrary Lagrangian–Eulerian formulation. We carry out a parametric analysis to examine the impact of the yield stress and relaxation time of the fluid and of particle confinement on the dynamics of the system. We identify master curves of the particle relative velocity as a function of the inter-particle distance. When the yield stress of the suspending phase is much lower than the viscous stress, those curves highlight short-range attractive interactions and long-range repulsive interactions between particles, with the latter specifically promoting their alignment. As the yield stress increases, the attractive interaction is replaced by stasis at short distance, characterised by a vanishing relative velocity and the formation of an unyielded region that connects the two spheres, where the fluid behaves like a viscoelastic solid. Additionally, the combined effects of plasticity and elasticity enhance the repulsion between the particles, promoting their ordering. Also, increasing the confinement of the particles enhances repulsion, thus allowing us to achieve ordering within shorter lengths in the flow direction. Reducing shear thinning amplifies peak relative velocities and expands the attractive region due to increased viscoelastic stresses and stress gradients. While a stable equilibrium may appear at larger separations, its impact is limited by low relative velocities.

In recent years, various unique properties of microswimmer suspensions have been revealed. Some microswimmers are deformable; however, the influence of the swimmer’s deformability has been overlooked. The present study examined the impact of soft microswimmers’ membrane deformations in a mono-dispersed dense suspension on microstructure formation. Due to the small size of the microswimmers, the flow field is described by the Stokes equation. The soft microswimmer was modelled as a capsule with a two-dimensional hyperelastic membrane enclosing a Newtonian fluid that is driven by propulsion torques distributed slightly above the membrane surface. Changes to the torque distribution caused the soft swimmer to exhibit different swimming modes as a pusher or puller. Similar to rigid squirmers, soft swimmers displayed self-organised local clusters in the suspension. Membrane deformation changed the mutual interference among swimmers in the cluster, bringing the interactions closer together than those of rigid squirmers. Especially among soft pushers, rotational diffusion due to hydrodynamic interference was reduced and the swimming trajectory became relatively straight. As a result, polar order was less likely to form, especially in regions of high $Ca$. On the other hand, pullers showed strong interactions due to retraction flow and an increase in mean membrane tension. For pushers (pullers), the rear (side) interaction produced the greatest change in tension. These findings are expected to be useful for effort to understand the propulsion mechanisms of medical and industrial soft microrobots, as well as the biological responses of microorganisms induced by mechanical stimuli.

At constant pressure, a mixture of water parcels with equal density but differing salinity and temperature will be denser than the parent water parcels. This is known as cabbeling and is a consequence of the nonlinear equation of state for seawater density. With a source of turbulent vertical mixing, cabbeling has the potential to trigger and drive convection in gravitationally stable water columns and there is observational evidence that this process shapes the thermohaline structure of high-latitude oceans. However, the evolution and maintenance of turbulent mixing due to cabbeling has not been fully explored. Here, we use turbulence-resolving direct numerical simulations to investigate cabbeling’s impact on vertical mixing and pathways of energy in closed systems. We find that cabbeling can sustain convection in an initially gravitationally stable two-layer configuration where relatively cold/fresh water sits atop warm/salty water. We show the mixture of the cold/fresh and warm/salty water is constrained by a density maximum and that cabbeling enhances mixing rates by four orders of magnitude. Cabbeling’s effect is amplified as the static stability limit is approached, leading to convection being sustained for longer. We find that available potential energy, which is classically thought to only decrease with mixing, can increase with mixing due to cabbeling’s densification of the mixed water. Our direct numerical dimulations support the notion that cabbeling could be a source of enhanced ocean mixing and that conventional definitions of energetic pathways may need to be reconsidered to take into account densification under mixing.

We investigate the dynamics of close-contact melting (CCM) on ‘gas-trapped’ hydrophobic surfaces, with specific focus on the effects of geometrical confinement and the liquid–air meniscus below the liquid film. By employing dual-series and perturbation methods under the assumption of small meniscus deflections, we obtain numerical solutions for the effective slip lengths associated with velocity $\lambda$ and temperature $\lambda _t$ fields, across various values of aspect ratio $\Lambda$ (defined as the ratio of the film thickness $h$ to the structure’s periodic length $l$) and gas–liquid fraction $\phi$. Asymptotic solutions of $\lambda$ and $\lambda _t$ for $\Lambda \ll 1$ and $\Lambda \gg 1$ are derived and summarised for different surface structures, interface shapes and $\Lambda$, which reveal a different trend of $\lambda$ for $\Lambda \ll 1$ and depending on the presence of a meniscus. In the context of constant-pressure CCM, our results indicate that longitudinal grooves can enhance heat transfer under the effects of confinement and a meniscus when $\Lambda \lesssim 0.1$ and $\phi \lt 1 - 0.5^{2/3} \approx 0.37$. For gravity-driven CCM, the parameters of $l$ and $\phi$ determine whether the melting rate is enhanced, reduced or nearly unaffected. We construct a phase diagram based on the parameter matrix $(\log _{10} l, \phi )$ to delineate these three regimes. Lastly, we derive two asymptotic solutions for predicting the variation in time of the unmelted solid height.

Head-on collisions between elliptic vortex rings (EVRs) and walls were studied experimentally using planar laser-induced fluorescence visualisations and time-resolved particle image velocimetry. Aspect ratios of $AR=2$ and 4 EVRs at a Reynolds number of $Re=4000$ were used. Collision locations were based on four key axis-switching stages of freely translating EVRs, which would shed light upon how axis-switching behaviour and aspect ratio variations affect the collision outcomes. Results show that non-uniform circumferential induced velocities in both colliding EVRs produce different behaviours along major and minor planes, where vortex-stretching/compression and hence circumferential flows play key roles in the vortex dynamics. Non-uniform formations of secondary/tertiary EVRs also lead to varied entanglements around the primary EVRs. As such, secondary vortex rings form vortex loops that may congregate along the collision axis, depending on the exact collision location. Vortex-core trajectories show the net primary/secondary vortex-core movements result from a balance between EVR diameter expansion due to collision and EVR segment motions associated with the axis-switching stage at the point of collision. Confinement effects are also observed to dominate over aspect ratio effects when the collision occurs closest to the orifice. While increasing the aspect ratio leads to different vortex-stretching/compression behaviour and more varied vortex-core trajectories due to the greater non-uniform induced velocities, they could still be understood by the preceding interpretations. Finally, three-dimensional vortex flows are reconstructed based on the experimental results to further explain the flow mechanisms.

In gas evolving electrolysis, bubbles grow at electrodes due to a diffusive influx from oversaturation generated locally in the electrolyte by the electrode reaction. When considering electrodes of micrometre size resembling catalytic islands, direct numerical simulations show that bubbles may approach dynamic equilibrium states at which they neither grow nor shrink. These are found in under- and saturated bulk electrolytes during both pinning and expanding wetting regimes of the bubbles. The equilibrium is based on the balance of local influx near the bubble foot and global outflux. To identify the parameter regions of bubble growth, dissolution and dynamic equilibrium by analytical means, we extend the solution of Zhang & Lohse (2023) J. Fluid Mech. 975, R3, by taking into account modified gas fluxes across the bubble interface, that result from a non-uniform distribution of dissolved gas. The Damköhler numbers at equilibrium are found to range from small to intermediate values. Unlike pinned nano-bubbles studied earlier, for micrometre-sized bubbles the Laplace pressure plays only a minor role. With respect to the stability of the dynamic equilibrium states, we extend the methodology of Lohse & Zhang (2015a) Phys. Rev. E 91 (3), 031003(R), by additionally taking into account the electrode reaction. Under contact line pinning, the equilibrium states are found to be stable for flat nano-bubbles and for micro-bubbles in general. For unpinned bubbles, the equilibrium states are always stable. Finally, we draw conclusions on how to possibly enhance the efficiency of electrolysis.

The attached-eddy model (AEM) predicts that the mean streamwise velocity and streamwise velocity variance profiles follow a logarithmic shape, while the vertical velocity variance remains invariant with height in the overlap region of high Reynolds number wall-bounded turbulent flows. Moreover, the AEM coefficients are presumed to attain asymptotically constant values at very high Reynolds numbers. Here, the AEM predictions are examined using sonic anemometer measurements in the near-neutral atmospheric surface layer, with a focus on the logarithmic behaviour of the streamwise velocity variance. Utilizing an extensive 210-day dataset collected from a 62 m meteorological tower located in the Eastern Snake River Plain, Idaho, USA, the inertial sublayer is first identified by analysing the measured momentum flux and mean velocity profiles. The logarithmic behaviour of the streamwise velocity variance and the associated ‘$-1$’ scaling of the streamwise velocity energy spectra are then investigated. The findings indicate that the Townsend–Perry coefficient ($A_1$) is influenced by mild non-stationarity that manifests itself as a Reynolds number dependence. After excluding non-stationary runs, and requiring the bulk Reynolds number defined using the atmospheric boundary layer height to be larger than $4 \times 10^{7}$, the inferred $A_1$ converges to values ranging between 1 and 1.25, consistent with laboratory experiments. Furthermore, nine benchmark cases selected through a restrictive quality control reveal a close relation between the ‘$-1$’ scaling in the streamwise velocity energy spectrum and the logarithmic behaviour of streamwise velocity variance. However, additional data are required to determine whether the plateau value of the pre-multiplied streamwise velocity energy spectrum is identical to $A_1$.

Opposition control (OC) is a reactive flow-control approach that mitigates the near-wall fluctuations by imposing blowing and suction at the wall, being opposite to the off-wall observations. We carried out high-resolution large-eddy simulations to investigate the effects of OC on turbulent boundary layers (TBLs) over a wing at a chord-based Reynolds number (${Re}_c$) of $200 \ 000$. Two cases were considered: flow over the suction sides of the NACA0012 wing section at an angle of attack of $0^{\circ }$, and the NACA4412 wing section at an angle of attack of $5^{\circ }$. These cases represent TBLs subjected to mild and strong non-uniform adverse pressure gradients (APGs), respectively. First, we assessed the control effects on the streamwise development of TBLs and the achieved drag reduction. Our findings indicate that the performance of OC in terms of friction-drag reduction significantly diminishes as the APG intensifies. Analysis of turbulence statistics subsequently reveals that this is directly linked to the intensified wall-normal convection caused by the strong APG: it energizes the control intensity to overload the limitation that guarantees drag reduction. The formation of the so-called virtual wall that reflects the mitigation of wall-normal momentum transport is also implicitly affected by the pressure gradient. Control and pressure-gradient effects are clearly apparent in the anisotropy invariant maps, which also highlight the relevance of the virtual wall. Finally, spectral analyses indicate that the wall-normal transport of small-scale structures to the outer region due to the APG has a detrimental impact on the performance of OC. Uniform blowing and body-force damping were also examined to understand the differences between the various control schemes. Despite the distinct performance of friction-drag reduction, the effects of uniform blowing are akin to those induced by a stronger APG, while the effects of body-force damping exhibit similarities to those of OC in terms of the streamwise development of the TBL although there are differences in the turbulent statistics. To authors’ best knowledge, the present study stands as the first in-depth analysis of the effects of OC applied to TBL subjected to non-uniform APGs with complex geometries.

We analyse the small-scale characteristics, such as enstrophy, total strain and normality/non-normality, in the three-dimensional, separated flow around a NACA 0018 wing using direct numerical simulations. The angle of attack is $10^\circ$ and the Reynolds number (based on the chord length) is $Re_c=5000$. The role of non-normality is investigated by performing Schur decomposition of the velocity gradient tensor. We also apply the Schur decomposition to derive new expressions for the production of enstrophy and total strain arising from the mean flow inhomogeneity. We focus on two sections of the flow, across the recirculating zone and along the transitioning shear layer, and compare our results with homogeneous isotropic turbulence (HIT). Within the recirculating region, the non-normality index is approximately 0 (and close to the HIT value), indicating almost equal normal and non-normal contributions. However, in the separating layer non-normal effects strongly dominate, especially in the region of kinetic energy growth. Only in the decay region do the values of the non-normality index gradually approximate HIT values. The production of enstrophy due to vortex stretching is dominated by the mixed (interaction) term, where normal strain stretches non-normal vorticity. The same component also dominates the strain self-amplification term. The contributions of different QR regions to the production terms are also examined. Production due to mean strain rate is triggered upstream compared with production due to fluctuating strain fields.

We consider the drawing of a hollow Newtonian fibre with temperature-dependent viscosity. The drawing is affected by surface tension, inertia, hole pressurisation and externally applied cooling. We apply long-wavelength techniques to determine the steady states and examine their stability. In the presence of surface tension but with no cooling or internal hole pressure, we show the counter-intuitive result that the hole radius at the outlet of the device is a non-monotonic function of the hole radius at the inlet. We also show that if the internal hole is pressurised and the hole size at the inlet is sufficiently large, then the exit temperature of the fibre is a non-monotonic function of the applied cooling rate. We have found a number of surprising mechanisms related to how the various physical effects influence the stability of drawing. For the isothermal case, we show that increasing the internal hole pressure has a destabilising effect for non-zero surface tension while the stability is completely independent of the internal hole pressure for zero surface tension. We further show that there is a complicated interplay between internal hole pressure, external cooling and surface tension in determining the stability and that it is possible that increasing the hole size at the inlet can act to destabilise, then stabilise and finally destabilise the flow. We discuss the mechanisms that determine the counter-intuitive steady-state behaviour and stability.

The critical points of vorticity in a two-dimensional viscous flow are essential for identifying coherent structures in the vorticity field. Their bifurcations as time progresses can be associated with the creation, destruction or merging of vortices, and we analyse these processes using the equation of motion for these points. The equation decomposes the velocity of a critical point into advection with the fluid and a drift proportional to viscosity. Conditions for the drift to be small or vanish are derived, and the analysis is extended to cover bifurcations. We analyse the dynamics of vorticity extrema in numerical simulations of merging of two identical vortices at Reynolds numbers ranging from 5 to 1500 in the light of the theory. We show that different phases of the merging process can be identified on the basis of the balance between advection and drift of the critical points, and identify two types of merging, one for low and one for high values of the Reynolds number. In addition to local maxima of positive vorticity and minima of negative vorticity, which can be considered centres of vortices, minima of positive vorticity and maxima of negative vorticity can also exist. We find that such anti-vortices occur in the merging process at high Reynolds numbers, and discuss their dynamics.

The Edgerton crown is an iconic manifestation of drop impact splashing, with its prominent cylindrical edge decorated with detaching droplets. Herein, we identify the formation of an intriguing double-crown, when a high-viscosity drop impacts on a shallow pool of a lower-viscosity immiscible liquid. High-speed imaging shows that after the initial fine horizontal ejecta sheet, the first inner crown emerges vertically from the film liquid. This is followed by the second crown which forms near the outer base of the first crown, as the tip of the horizontally spreading viscous drop approaches the outer free surface. Axisymmetric numerical simulations, using the volume-of-fluid method with adaptive grid refinement, show that the flow squeezed out between the viscous drop and the solid surface, generates two counter-rotating vortex rings, which travel radially outwards together and drive out the second crown through the free surface. The bottom vortex emerges from the separated boundary layer at the solid wall, while the top one detaches from the underside of the viscous drop. We map out the narrow parameter regime, where this ephemeral structure emerges, in terms of viscosity ratio, impact velocity and film thickness.

We employ a novel computational modelling framework to perform high-fidelity direct numerical simulations of aero-structural interactions in bat-inspired membrane wings. The wing of a bat consists of an elastic membrane supported by a highly articulated skeleton, enabling localised control over wing movement and deformation during flight. By modelling these complex deformations, along with realistic wing movements and interactions with the surrounding airflow, we expect to gain new insights into the performance of these unique wings. Our model achieves a high degree of realism by incorporating experimental measurements of the skeleton’s joint movements to guide the fluid–structure interaction simulations. The simulations reveal that different segments of the wing undergo distinct aeroelastic deformations, impacting the flow dynamics and aerodynamic loads. Specifically, the simulations show significant variations in the effectiveness of the wing in generating lift, drag and thrust forces across different segments and regions of the wing. We employ a force partitioning method to analyse the causality of pressure loads over the wing, demonstrating that vortex-induced pressure forces are dominant while added-mass contributions to aerodynamic loads are minimal. This approach also elucidates the role of various flow structures in shaping pressure distributions. Finally, we compare the fully articulated, flexible bat wing with equivalent stiff wings derived from the same kinematics, demonstrating the critical impact of wing articulation and deformation on aerodynamic efficiency.

In this paper, we discuss the transport of sediment and the formation of bedforms in turbulent river flows, under flow conditions typical of flooding events. Through the implementation of an immersed boundary method, a wall model and a morphological model, we were able to simulate complex and mobile geometries under high Reynolds numbers at an affordable computational cost. In particular, we examined the evolution of bedforms on a loose sediment bed under turbulent flow conditions, using input parameters obtained from laboratory measurements. Over time, the bedforms become more three-dimensional and irregular in shape, leading to changes in the shear layer, crest angle and separation patterns. The bedforms continue to evolve until a quasi-steady equilibrium is reached. Our simulations highlight the crucial role played by the small-scale bedforms, which significantly affect the flow dynamics: an increase in the total drag is observed, related to the form drag generated by the local recirculation and the increased size of the large-scale recirculation bubble. Furthermore, a stronger turbulent activity ensues from the shear layers forming on the crests of the small-scale bedforms. Finally, a wider shedding angle of the shear layer is caused by the irregular crest line.

Motivated by microfluidic applications, we investigate drag reduction in laminar pressure-driven flows in channels with streamwise-periodic superhydrophobic surfaces (SHSs) contaminated with soluble surfactant. We develop a model in the long-wave and weak-diffusion limit, where the streamwise SHS period is large compared with the channel height and the Péclet number is large. Using asymptotic and numerical techniques, we determine the influence of surfactant on drag reduction in terms of the relative strength of advection, diffusion, Marangoni effects and bulk–surface exchange. In scenarios with strong exchange, the drag reduction exhibits a complex dependence on the thickness of the bulk-concentration boundary layer and surfactant strength. Strong Marangoni effects immobilise the interface through a linear surfactant distribution, whereas weak Marangoni effects yield a quasi-stagnant cap. The quasi-stagnant cap has an intricate structure with an upstream slip region, followed by intermediate inner regions and a quasi-stagnant region that is mediated by weak bulk diffusion. The quasi-stagnant region differs from the immobile region of a classical stagnant cap, observed for instance in surfactant-laden air bubbles in water, by displaying weak slip. As exchange weakens, the bulk and interface decouple: the surfactant distribution is linear when the surfactant is strong, whilst it forms a classical stagnant cap when the surfactant is weak. The asymptotic solutions offer closed-form predictions of drag reduction across much of the parameter space, providing practical utility and enhancing understanding of surfactant dynamics in flows over SHSs.

The settling velocity of frozen hydrometeors in the atmospheric surface layer depends on their inertial and drag properties, and on the intensity of ambient turbulence. Thin, solid and perforated circular disks have been investigated through high-speed imaging, under laboratory conditions, to reproduce the settling of snow plates and dendrites in quiescent and turbulent flows. Different perforations made it possible to test the parameterisation of the fall speed in quiescent air, based on the geometric description of the solidity of the disk cross-sectional area. Interestingly, different falling styles, ranging from stable horizontal to fluttering and tumbling, were observed to depend significantly on the perforation geometry, which resulted in the stabilisation of the particle rotation and in a modulation of the drag coefficient. Ambient turbulence is observed to primarily induce cross-flow drag on the disks settling in the nonlinear regime, causing a reduction of the settling velocity in all cases investigated. Turbulence also manifests a secondary effect on the disk rotational dynamics, in particular when tumbling and stable falling styles co-exist, based on the phase space defined by the Reynolds number $Re$ and the inertia ratio $I^*$. The interaction between ambient turbulence, particle anisotropy and permeability and the likelihood of tumbling is inferred to be the main reason for the observed settling velocity variability of snow dendrites in nature.

Planar entropy waves are commonly assumed for predicting indirect combustion noise. However, the non-planar and turbulent nature of flows found in most practical combustors challenges this assumption. In the present paper, we examine the indirect noise generated by non-planar and turbulent entropy fields through subsonic nozzles. Firstly, we introduce a new transfer function framework that accounts for the contribution of non-planar Fourier modes of the entropy field to the indirect noise spectra. When applied to a turbulent flow field, this method demonstrates a significant improvement in spectral predictions compared with a conventional approach that only considers the planar mode. Secondly, simulations show that non-planar Fourier modes become significant above a threshold frequency $f_{thresh}$, found in the mid- to high-frequency range. This contribution of non-planar modes is explained by two-dimensional shear effects that distort the entropy waves. A scaling relation that uses residence times along streamlines is developed for $f_{thresh}$, showing good agreement with simulation results. Finally, we show that the indirect noise from non-planar entropy modes found in aviation combustors can be significant at frequencies below 1 kHz, which might be relevant in situations of thermo-acoustic instabilities coupled to indirect noise.

In aerodynamic and hydrodynamic devices and locomotive organisms, passive appendages have practical purposes such as drag reduction and flow control. Although these appendages also affect the dynamics of freely falling bodies, underlying principles of their functions remain elusive. We investigate experimentally the dynamics of a falling sphere with a filament appended on its rear side by varying the ratio of filament length to sphere diameter ($l/D=0{-}3.0$) and sphere-to-fluid density ratio ($\rho _s/\rho _f= 1.06{-}1.36$), and maintaining a similar dimensionless moment of inertia ($I^* \approx 0.96$). At the Reynolds number of $O(10^3)$, a sphere without any filament exhibits vertical descent. However, the falling of the sphere with a filament is accompanied by periodic horizontal displacements, and the degree of zigzag motion is maximised under specific filament length. The filament induces periodic rotation of the sphere by shifting the centre of mass of the entire model and through the hydrodynamic interaction of the filament with the surrounding fluid. The rotation of the sphere increases the drag force acting on the model, reducing tangential velocity along the trajectory by 14 % compared to a plain sphere. Furthermore, the sphere rotation enhances the lift force normal to the trajectory, extending trajectory length by 5 %. These combined effects improve falling time over a certain vertical distance by 20 % compared to the plain sphere. With increasing sphere density, the effects of the filament on the falling dynamics weaken, because the offset distance between the centre of mass of the model and the geometric centre of the sphere becomes smaller.

Experiments are conducted over smooth and rough walls to explore the influence of pressure-gradient histories on skin friction and mean flow of turbulent boundary layers. Different pressure-gradient histories are imposed on the boundary layer through an aerofoil mounted in the free stream. Hot-wire measurements are taken at different free-stream velocities downstream of the aerofoil where the flow has locally recovered to zero pressure gradient but retains the history effects. Direct skin friction measurements are also made using oil film interferometry for smooth walls and a floating-element drag balance for rough walls. The friction Reynolds number, $Re_\tau$, varies between $3000$ and $27\,000$, depending both on the surface conditions and the free-stream velocity ensuring sufficient scale separation. Results align with previous findings, showing that adverse pressure gradients just upstream of the measurement location increase wake strength and reduce the local skin friction while favourable pressure gradients suppress the wake and increase skin friction. The roughness length scale, $y_0$, remains constant across different pressure-gradient histories for rough wall boundary layers. Inspired by previous works, a new correlation is proposed to infer skin friction based on the mean flow. The difference in skin friction by matching the turbulence profiles and flow structure between an arbitrary pressure-gradient history and zero pressure-gradient condition can be predicted using only the local wake strength parameter ($\Pi$), and the variations in wake strength for different histories are related to a weighted integral of the pressure-gradient history normalised by local quantities. This allows us to develop a general correlation that can be used to infer skin friction for turbulent boundary layers experiencing arbitrary pressure-gradient histories.

We investigate the deformation, dynamics and rheology of a single and a suspension of elastic capsules in inertial shear flow using high-fidelity particle-resolved simulations. For a single capsule in the shear flow, we elucidate the interplay of flow inertia and viscosity ratio, revealing the mechanism behind the stretching of capsule surface during tank-treading motion and the sign changes in normal stress differences with increasing inertia. When examining capsule suspensions, we thoroughly discuss the impact of volume fraction on average deformation, diffusion and rheology. Notably, we observe the formation of bridge structures due to hydrodynamic interactions, which enhance the inhomogeneity of the microstructure and alter the surface stress distribution within the suspension. We identify a critical Reynolds number range that marks the transition of capsule diffusion from non-inertial to inertial regimes. Furthermore, we reveal close connections between the behaviour of individual capsules and dense suspensions, particularly regarding capsule deformation and dynamics. Additionally, we propose multiple new empirical correlations for predicting the deformation factor of a single capsule and the relative viscosity of the suspension. These findings provide valuable insights into the complex behaviour of elastic capsules in inertial flows, informing the design of more accurate and efficient inertial microfluidic systems.