Dynamics of spheroidal particle migration within the elasto-inertial square duct flow of Giesekus viscoelastic fluids were studied by using the direct forcing/fictitious domain method. The results show rich migration behaviours, a spheroidal particle gradually transitions from the corner (CO), channel centreline (CC), inertial rotational (IR), diagonal line and cross-section midline equilibrium positions with a decrease in the elastic number, depending on the initial particle position, initial particle orientation and fluid elasticity. From the effect of secondary flow, the IR equilibrium position is reported when the fluid inertia is relatively strong. Six (five) kinds of rotational behaviours are observed for the elasto-inertial migration of prolate (oblate) spheroids. Moreover, the critical elastic number is determined for the migration of spheroidal particles in Giesekus fluids. Near the critical elastic number, oblate and prolate spheroids can simultaneously maintain the CC, CO and IR equilibrium positions, and the initial orientation of particles affects their final rotational modes and equilibrium positions. Through comprehensive analysis, empirical formulas governing the ability of oblate and prolate spheroids to maintain the CC equilibrium position are proposed as $\textit{Wi} = 0.055\,\textit{Re}{-0.1}$ and Wi = 0.045 Re−0.35 when n = 0.5, 0.01 ≤ Wi ≤ 1. Due to the different directions of the pressure forces acting on the particles and the forces from the first normal stress difference and the second normal stress difference, the equilibrium position in Giesekus fluids is rapidly increased by increasing the secondary flow at higher elastic numbers, which is contrary to the phenomenon observed in the Oldroyd-B fluid.

The acoustically excited vibrations of a micrometric object in a viscous liquid induce a net fluid flow known as microstreaming. This phenomenon can be harnessed for a variety of microscale applications, including particle transport, fluid mixing and the propulsion of micro-swimmers. Acoustic propulsion holds significant promise for in vivo manipulation due to its inherent biocompatibility and remote actuation capability, eliminating the need for an onboard energy source. However, designing steerable swimmers powered by vibrating tails requires a detailed understanding of the relationship between the input acoustic signal and the resulting streaming flow. In this paper, we characterise experimentally and model the microstreaming generated by a vertically standing micro-cantilever attached to a vibrating plate, as a function of the excitation frequency. Significant streaming is observed only at specific frequencies corresponding to the vibration modes of the support, which both translate and bend the cantilever. Computations based on a two-dimensional semi-analytical model enable quantitative predictions of the in-plane streaming flow structure and velocity magnitude, using as input the cantilever’s vibration profile, fully characterised by laser Doppler vibrometry. In particular, comparison between experiments and simulations allows us to rationalise the frequency-dependent emergence of dipolar, circular and elliptical streaming patterns, which are respectively induced by rectilinear, circular and elliptical translations of the cantilever. This analysis also explains the prevalence of elliptical streaming structures observed in our system. Beyond advancing our fundamental understanding of streaming generated by vibrating slender bodies, these results highlight the potential for frequency-based control of micro-swimmers through predictable, mode-specific flow responses.

The classical problem of steady rarefied gas flow past an infinitely thin circular disk is revisited, with particular emphasis on the gas behaviour near the disk edge. The uniform flow is assumed to be perpendicular to the disk surface. An integral equation for the velocity distribution function, derived from the linearised Bhatnagar–Gross–Krook model of the Boltzmann equation and subject to diffuse reflection boundary conditions, is solved numerically. The numerical method fully accounts for the discontinuity in the velocity distribution function that arises due to the presence of the edge. It is found that a kinetic boundary layer forms near the disk edge, extending over several mean free paths, and that its magnitude scales as $\textit{Kn}^{1/2}$ as the Knudsen number $\textit{Kn}$ (defined with respect to the disk radius) tends to zero. A thermal polarisation effect, previously studied for spherical geometries, is also observed in the disk case, with a more pronounced manifestation near the edge that exhibits the same $\textit{Kn}^{1/2}$ scaling. The drag force acting on the disk is computed over a wide range of Knudsen numbers and shows good agreement with existing results for a hard-sphere gas and in the near-free-molecular regime.

We study flows generated within a two-dimensional corner by the chemical activity of the confining boundaries. Catalytic reactions at the surfaces induce diffusio-osmotic motion of the viscous fluid throughout the domain. The presence of chemically active sectors can give rise to steady eddies reminiscent of classical Moffatt vortices, which are mechanically induced in similar confined geometries. In our approach, an exact analytical solution of the diffusion problem in a wedge geometry is derived and coupled to the diffusio-osmotic slip-velocity formulation, yielding the stream function of associated Stokes flow. In selected limiting cases, simple closed-form expressions provide clear physical insight into the underlying mechanisms. Our results open new perspectives for the design of microscale mixing strategies in dead-end pores and cornered microfluidic channels, and offer benchmarks for numerical simulations of confined (diffusio-)osmotic systems.

The modal and non-modal stability of laminar flow in a rectangular microchannel is investigated by incorporating the effects of Coriolis forces due to rotation, cross-sectional aspect ratio and superhydrophobic wall slip. The full Navier–Stokes equations are linearised into modified Orr–Sommerfeld and Squire equations, which are then formulated as an eigenvalue problem using small disturbances of the Tollmien–Schlichting type. These equations are subsequently solved by the spectral collocation method. The transition to instability in rotating microchannel flows, influenced by aspect ratio and slip conditions, is analysed through eigenvalue spectra and neutral stability curves. For non-modal analysis, we express the solution in matrix exponential form and then, using the singular value decomposition method, calculate the maximum energy growth. The study reveals that the flow becomes unstable in the presence of rotation at a critical Reynolds number of $ Re_c \approx 40$ for a low aspect ratio and $ Re_c \approx 50.4$ for a high aspect ratio. We find that instability is more pronounced in spanwise-rotating flows at higher aspect ratios compared with those at lower aspect ratios. Rotation induces disturbances from both walls along the spanwise direction, forming secondary flow structures near the centreline. Furthermore, we examine the influence of anisotropic slip by separately considering streamwise and spanwise slip as limiting cases. The numerical results demonstrate that while streamwise slip has a stabilising effect on rotating flows at small scales, a sufficiently large spanwise slip length can trigger instability at Reynolds numbers lower than those observed in the no-slip case. Rotation has the potential to enhance non-modal transient energy growth, while streamwise slip can effectively suppress this instability. These findings suggest that the onset of instability and transient energy growth can be effectively regulated by adjusting the aspect ratio and spanwise slip of the channel walls.

We consider the flow of a viscous fluid through a two-dimensional symmetric cross-slot geometry with sharp corners. The problem is analysed using the unified transform method in the complex plane, providing a quasi-analytical solution that can be used to compute all the physical quantities of interest. This study is a novel application of this method to a complicated geometry featuring multiple sharp corner singularities and multiple inlets and outlets. Our approach offers the advantage of resolving unbounded domains, as well as providing quantities of interest, such as the velocity and stress profiles, and the Couette pressure correction, from the solution of low-order linear systems. Our results agree well with the existing literature, which has largely used truncated bounded geometries with rounded or curved corners.

We present theoretical models for flow and diffusion in microfluidic polygonal mixers of arbitrary shapes. Combining work based on Boussinesq coordinates with modern methods for the calculation of the Schwarz–Christoffel transform, we present an integrated method that yields analytical solutions for both flow and concentration profiles everywhere in microfluidic mixers with arbitrary numbers of inlets. We illustrate how the problem can be reduced to a sequence of conformal maps to a known domain, where the advection–diffusion problem can be readily solved, and map back the solution to the geometry of interest. We use the method to model a number of previously published microfluidic mixer geometries, used in lipid nanoparticle synthesis, among others. The method is also applicable to other problems described by planar transport equations in polygonal domains, for instance, in groundwater flows or electrokinetics.

For the first time, an analytical solution has been derived for Stokes flow through a conical diffuser under the condition of partial slip. Recurrent relations are obtained that allow determination of the velocity, pressure and stream function for a certain slip length λ. The solution is analysed in the first order of decomposition with respect to a small dimensionless parameter ${\lambda }/{r}$. It is shown that the sliding of the liquid over the surface of the cone leads to a vorticity of the flow. At zero slip length, we obtain the well-known solution to the problem of a diffuser with a no-slip boundary condition corresponding to strictly radial streamlines. To solve that problem, we use an alternative form of the general solution of the linearised, stationary, axisymmetric Navier–Stokes equations for an incompressible fluid in spherical coordinates. A previously published solution to this problem, dating back to the paper by Sampson (1891 Phil. Trans. R. Soc. A, vol. 182, pp. 449–518), is given in terms of a stream function that leads to formulae that are difficult to apply in practice. By contrast, the new general solution is derived in the vector potential representation and is simpler to apply.

Inspired by small intestine motility, we investigate the flow induced by a propagating pendular wave along the walls of a channel lined with rigid, villi-like microstructures. The villi undergo harmonic axial oscillations with a phase lag relative to their neighbours, generating travelling patterns of intervillous contraction. Using two-dimensional lattice Boltzmann simulations, we resolve the flow within the villi zone and the lumen, sampling small to moderate Womersley numbers. We uncover a mixing boundary layer (MBL) just above the villi, composed of semi-vortical structures that travel with the imposed wave. In the lumen, an axial steady flow emerges, surprisingly oriented opposite to the wave propagation direction, contrary to canonical peristaltic flows. We attribute this flow reversal to the non-reciprocal trajectories of fluid trapped between adjacent villi and derive a geometric scaling law that captures its magnitude in the Stokes regime. The MBL thickness is found to depend solely on the wave kinematics given by intervillous phase lag in the low-inertia limit. Above a critical threshold, oscillatory inertia induces dynamic confinement, limiting the radial extent of the MBL and leading to non-monotonic behaviour of the axial steady flux. We further develop an effective boundary condition at the villus tips, incorporating both steady and oscillatory components across relevant spatial scales. This framework enables coarse-grained simulations of intestinal flows without resolving individual villi. Our results shed light on the interplay among active microstructure, pendular wave and finite inertia in biological flows, and suggests new avenues for flow control in biomimetic and microfluidic systems.

Surfactants are usually added in droplet-based systems to stabilise them. When their concentration exceeds the critical micelle concentration (CMC), they self-assemble into micelles, which act as reservoirs regulating the availability of monomers in the continuous phase, thereby promoting interfacial remobilisation. The monomers get adsorbed onto a drop’s interface to alter its surface tension, and thus, governs how the drop moves within the suspending phase. Indeed, fine tuning droplet trajectories remain crucial in many classical as well as modern applications. Yet, the role of soluble surfactants in modulating droplet movement, especially at high concentrations, hitherto remains poorly understood. To address this, here we investigate the motion and cross-stream migration of a non-deforming drop in an unbounded Poiseuille flow, in the presence of bulk-soluble surfactants at concentrations above the CMC. We build a mixed semi-analytical-cum-numerical framework using spherical harmonics to determine the ensuing velocity and concentration fields. Our results suggest that the drop migrates towards the flow centreline, the extent of which depends on the interplay between the bulk concentration and the sensitivity of the interfacial tension to the surfactant molecules. This propensity for migration plateaus in the presence of micelles, although changing their specific properties seems to have relatively little impact. We further establish that adsorption–desorption between the interface and the bulk tends to suppress migration, while a relatively stronger coupling between bulk and interfacial transport facilitates the same. These findings highlight the crucial role of micelles in droplet motion, with implications in microfluidic control strategies and surfactant-driven flow manipulation.

The inertial migration of neutrally buoyant spherical particles in viscoelastic fluids flowing through square channels is experimentally and numerically studied. In the experiments, using dilute aqueous solutions of polymers with various concentrations that have nearly constant viscosities, we measured the distribution of suspended particles in downstream cross-sections for the Reynolds number ($\textit{Re}$) up to 100 and the elasticity number ($El$) up to 0.07. There are several focusing patterns of the particles, such as four-point focusing near the centre of the channel faces on the midlines for low $\textit{El}$ and/or high $\textit{Re}$, four-point focusing on the diagonals for medium $\textit{El}$, single-point focusing at the channel centre for relatively high $\textit{El}$ and low $\textit{Re}$, and five-point focusing near the four corners and the channel centre for high $\textit{El}$ and very low $\textit{Re}$. Among these focusing patterns, various types of particle distributions suggesting the presence of a new equilibrium position located between the midline and the diagonal, and multistable states of different equilibrium positions were observed. In general, as $\textit{El}$ increases from 0 at a constant $\textit{Re}$, the particle focusing positions shift from the midline to the diagonal in the azimuthal direction first, and then inward in the radial direction to the channel centre. These focusing patterns and their transitions were numerically well reproduced based on a FENE-P model with measured values of viscosity and relaxation time. Using the numerical results, the experimentally observed focusing patterns of particles are elucidated in terms of the fluid elasticity-induced lift and the wall-induced elastic lift.

Deformable microchannels emulate a key characteristic of soft biological systems and flexible engineering devices: the flow-induced deformation of the conduit due to slow viscous flow within. Elucidating the two-way coupling between oscillatory flow and deformation of a three-dimensional (3-D) rectangular channel is crucial for designing lab-on-a-chip and organ-on-a-chip microsystems and eventually understanding flow–structure instabilities that can enhance mixing and transport. To this end, we determine the axial variations of the primary flow, pressure and deformation for Newtonian fluids in the canonical geometry of a slender (long) and shallow (wide) 3-D rectangular channel with a deformable top wall under the assumption of weak compliance and without restriction on the oscillation frequency (i.e. on the Womersley number). Unlike rigid conduits, the pressure distribution is not linear with the axial coordinate. To validate this prediction, we design a polydimethylsiloxane-based experimental platform with a speaker-based flow-generation apparatus and a pressure acquisition system with multiple ports along the axial length of the channel. The experimental measurements show good agreement with the predicted pressure profiles across a wide range of the key dimensionless quantities: the Womersley number, the compliance number and the elastoviscous number. Finally, we explore how the nonlinear flow–deformation coupling leads to self-induced streaming (rectification of the oscillatory flow). Following Zhang and Rallabandi (J. Fluid Mech., vol. 996, 2024, p. A16), we develop a theory for the cycle-averaged pressure based on the primary problem’s solution, and we validate the predictions for the axial distribution of the streaming pressure against the experimental measurements.

Microfluidic paper-based analytical devices (${\unicode{x03BC}}$PADs) have gained considerable attention due to their ability to transport fluids without external pumps. Fluid motion in ${\unicode{x03BC}}$PADs is driven by capillary forces through the network of pores within paper substrates. However, the inherently low flow speeds resulting from the small pore sizes in paper often limit the performance of ${\unicode{x03BC}}$PADs. Recent studies have introduced multilayered ${\unicode{x03BC}}$PADs composed of stacked paper sheets, which enable significantly faster fluid transport through inter-layer channels. In this study, we present a combined theoretical and experimental investigation of water imbibition dynamics through channels formed by multiple paper layers. Upon contact with water, the paper layers absorb water and undergo swelling, altering channel geometry and consequently affecting flow dynamics. We develop a mathematical model that extends the classical Washburn equation to incorporate the effects of water absorption and swelling. The model predictions show excellent agreement with experimental observations of water flow through multilayered paper channels. The results elucidate how water absorption and swelling influence capillary imbibition, and suggest potential strategies for regulating flow rates in multilayered ${\unicode{x03BC}}$PADs.

Steady flow at low Reynolds (Re) number through a planar channel with converging or diverging width is investigated in this study. Along the primary direction of flow, the small dimension of the channel cross-section remains constant while the sidewalls bounding the larger dimension are oriented at a constant angle. Due in part to ease of manufacturing, parallel-plate geometries such as this have found widespread use in microfluidic devices for mixing, heat exchange, flow control and flow patterning at small length scales. Previous analytical solutions for flows of this nature have required the converging or diverging aspect of the channel to be gradual. In this work, we derive a matched asymptotic solution, validated against numerical modelling results, that is valid for any sidewall angle, without requiring the channel width to vary gradually. To accomplish this, a cylindrical coordinate system defined by the angle of convergence between the channel sidewalls is considered. From the mathematical form of the composite expansion, a delineation between two secondary flow components emerges naturally. The results of this work show how one of these two components, originating from viscous shear near the channel sidewalls, corresponds to convective mixing, whereas the other component impresses the sidewall geometry on streamlines in the outer flow.

The inertial migration of hydrogel particles suspended in a Newtonian fluid flowing through a square channel is studied both experimentally and numerically. Experimental results demonstrate significant differences in the focusing positions of the deformable and rigid particles, highlighting the role of particle deformability in inertial migration. At low Reynolds numbers (${Re}$), hydrogel particles migrate towards the centre of the channel cross-section, whereas the rigid spheres exhibit negligible lateral motion. At finite ${Re}$, they focus at four points along the diagonals in the downstream cross-section, in contrast to the rigid particles which focus near the centre of the channel face at similar ${Re}$. Numerical simulations using viscous hyperelastic particles as a model for hydrogel particles reproduced the experimental results for the particle distribution with an appropriate Young’s modulus of the hyperelastic particles. Further numerical simulations over a broader range of ${Re}$ and the capillary number ($Ca$) reveal various focusing patterns of the particles in the channel cross-section. The phase transitions between them are discussed in terms of the inertial lift and the lift due to particle deformation, which would act in the direction towards lower shear. The stability of the channel centre is analysed using an asymptotic expansion approach to the migration force at low ${Re}$ and $Ca$. The theoretical analysis predicts the critical condition for the transition, which is consistent with the direct numerical simulation. These experimental, numerical and theoretical results contribute to a deeper understanding of inertial migration of deformable particles.

This paper theoretically introduces a new architecture for pumping leaky-dielectric fluids. For two such fluids layered in a channel, the mechanism utilises Maxwell stresses on fluid interfaces (referred to as menisci) induced by a periodic array of electrode pairs inserted between the two fluids and separated by the menisci. The electrode pairs are asymmetrically spaced and held at different potentials, generating an electric field with variation along the menisci. To induce surface charge accumulation, an electric field (and thus current flow) is also imposed in the direction normal to the menisci, using flat upper and lower electrodes, one in each fluid. The existence of both normal and tangential electric fields gives rise to Maxwell stresses on each meniscus, driving the flow in opposite directions on adjacent menisci. If the two menisci are the same length, then a vortex array is generated that results in no net flow; however, if the spacing is asymmetric, then the longer meniscus dominates, causing a net pumping in one direction. The pumping direction can be controlled by the (four) potentials of the electrodes, and the electrical properties of the two fluids. In the analysis, an asymptotic approximation is made that the interfacial electrode period is small compared to the fluid layer thicknesses, which reduces the analytical difficulty to an inner region close to the menisci. Closed-form solutions are presented for the potentials, velocity field and resulting pumping speed, for which maximum values are estimated, with reference to the electrical power required and feasibility.

Cilia exist ubiquitously in nature, and they are very effective in generating flow in a low Reynolds number environment. Inspired by nature, various artificial cilia have been invented for microfluidic applications, and a nature-mimicking tilted conical motion was often used for flow generation due to its simplicity and effectiveness. However, the current theoretical model for predicting the net flow rate generated by the tilted conical motion fails when the cilia are in close confinement, i.e. when the tips of the cilia are close to the ceiling of their channel or chamber, which is, in reality, the most practical way to enhance flow rate generation. Moreover, numerical simulations are very expensive for optimisation of such designs. In this study, we derive a new theoretical model, taking into account the tilting and opening angles of the cone, the height of the chamber and the length of the cilia. The results differ significantly from when the ceiling is not considered, and counter-intuitively in some cases the flow can even reverse. These unexpected results have important implications for artificial cilium design and applications. We validate the model with both numerical simulations and experiments using magnetic artificial cilia, and show that the flow optimisation based on tilted conical cilium motion can now be performed accurately in a realistic and practical manner. This study not only offers a simple tool for optimising designs of artificial cilium-based systems for microfluidic applications, but it also provides fresh insights for understanding natural cilium-driven flows.

This paper presents a theoretical investigation of vortex modes in acoustofluidic cylindrical resonators with rigid boundaries and viscous fluids. By solving the Helmholtz equation for linear pressure, incorporating boundary conditions that account for no-slip surfaces and vortex and non-vortex excitation at the base, we analyse both single- and dual-eigenfunction modes near system resonance. The results demonstrate that single-vortex modes generate spin angular momentum exclusively along the axial direction, while dual modes introduce a transverse spin component due to the nonlinear interaction between axial and transverse ultrasonic waves, even in the absence of vortex excitation. We find that nonlinear acoustic fields, including energy density, radiation force potential and spin, scale with the square of the shear wave number, defined as the ratio of the cavity radius to the thickness of the viscous boundary layer. Theoretical predictions align closely with finite element simulations based on a model for an acoustofluidic cavity with adiabatic and rigid walls. These findings hold particular significance for acoustofluidic systems, offering potential applications in the precise control of cells and microparticles.

Flows of particles through bottlenecks are ubiquitous in nature and industry, involving both dry granular materials and suspensions. However, difficulties in precisely controlling particle properties in conventional set-ups hinder the full understanding of these flows in confined geometries. Here, we present a microfluidic model set-up to investigate the flow of dense suspensions in a two-dimensional hopper channel. Particles with controlled properties such as shape and deformability are in situ fabricated with a photolithographic projection method and compacted at the channel constriction using a Quake valve. The set-up is characterised by examining the flow of a dense suspension of hard, monodisperse disks through constrictions of varying widths. We demonstrate that the microfluidic hopper discharges particles at a constant rate under both imposed pressure and flow rate. The discharge of particles under imposed flow rate follows a Beverloo-like scaling, while it varies nonlinearly with particle size under imposed pressure. Additionally, we show that the statistics of clog formation in our microfluidic hopper follow the same stochastic laws as reported in other systems. Finally, we show how the versatility of our microfluidic model system can be used to investigate the outflow and clogging of suspensions of more complex particles.

Previous studies claimed that the non-monotonic effects of wettability came mainly from the heterogeneity of geometries or flow conditions on multiphase displacements in porous media. For macroscopic homogeneous porous media, without permeability contrast or obvious preferential flow pathways, most pore-scale evidence showed a monotonic trend of the wettability effect. However, this work reports transitions from monotonic to non-monotonic wettability effects when the dimension of the model system rises from two-dimensional (2-D) to three-dimensional (3-D), validated by both the network modelling and the microfluidic experiments. The mechanisms linking the pore-scale events to macroscopic displacement patterns have been analysed through direct simulations. For 2-D porous media, the monotonic effect of wettability comes from the consistent transition pattern for the full range of capillary numbers $Ca$, where the capillary fingering mode transitions to the compact displacement mode as the contact angle $\theta$ decreases. Yet, it is indicated that the 3-D porous geometries, even though homogeneous without permeability contrast or obvious preferential flow pathways, introduce a different $Ca$–$\theta$ phase diagram with new pore-scale events, such as the coupling of capillary fingering with snap-off during strong drainage, and frequent snap-off events during strong imbibition. These events depend strongly on geometric confinements and capillary numbers, leading to the non-monotonicity of wettability effects. Our findings provide new insights into the multiphase displacement dependent on wettability in various natural porous media and offer design principles for engineering artificial porous media to achieve desired immiscible displacement behaviours.