Elastoviscoplastic (EVP) fluids, characterised by the coexistence of elastic, viscous and yield-stress properties, play a central role in diverse applications, including drug delivery, 3D printing and hydraulic fracturing. These fluids often transport non-spherical particles whose migration dynamics strongly influences flow behaviour. In this work, we employ interface-resolved direct numerical simulations to investigate the migration and orientation dynamics of finite-size spheroidal particles suspended in EVP duct flows across a wide range of governing parameters. Our results show that the equilibrium position and orientation of the particles are influenced significantly by both their aspect ratio and the carrier fluid rheology. In Saramito fluids, spheroidal particles migrate towards the duct centre and align along the duct diagonals in the presence of inertia. At sufficiently high elasticity, they penetrate the central plug and reach the duct core, irrespective of their initial position or shape. At lower elasticities, where larger plug regions persist, interactions with the plug alter the angular dynamics of the particles, leading to unsteady, quasi-periodic tumbling and spinning motions. In contrast, in Saramito–Giesekus fluids, the interplay between inertial forces, shear-thinning plastic viscosity and yield stress drives particles towards the duct corners, aligning them perpendicular to the duct diagonals. In semi-dilute suspensions, flattened particles maintain a greater distance from the walls, whereas their spherical counterparts tend to cluster directly at the corners. These findings reveal complex migration and orientation behaviours unique to EVP media and suggest new opportunities for geometry-based particle separation in microfluidic applications.

We present an experimental and theoretical investigation of steady Taylor cone-jetting of highly viscous liquids at the minimum flow rate required for steady jetting. To achieve a steady cone-jet, the viscous liquid is flown through a conducting needle under the action of a strong electric field acting between the needle and a flat collector plate. Experiments reveal that the minimum flow rate and the corresponding jet diameter depend on both the needle diameter and the electrical conductivity of the liquid. Subsequently, we used the experimental measurements to formulate correlations among the minimum flow rate, the needle diameter and the physical properties of the liquid, including the electrical conductivity. To elucidate the underlying physics and experimental observations, we performed an order-of-magnitude analysis. The scaling analysis reveals that in the cone region, the viscous and interfacial tension forces are of comparable magnitude, while in the current transfer region, the viscous and electrostatic suction forces are the dominant resisting and driving forces, respectively. Subsequently, we theoretically derived the scaling relations for the minimum flow rate and the corresponding jet diameter by considering the balance of forces at the cone-tip and in the current transfer region. The empirical and theoretical scaling laws for the minimum flow rate and the corresponding jet diameter agree well for highly viscous liquids with electrical conductivity spanning over four orders of magnitude. Lastly, we present the limits that describe the regime for which the minimum flow rate depends on the needle diameter, and the derived scaling laws are applicable.

Systematic deflection of microparticles off of initial streamlines is a fundamental task in microfluidics, aiming at applications including sorting, accumulation or capture of the transported particles. In a large class of set-ups, including deterministic lateral displacement and porous media filtering, particles in non-inertial (Stokes) flows are deflected by an array of obstacles. We show that net deflection of force-free particles passing an obstacle in Stokes flow is possible solely by hydrodynamic interactions if the flow and obstacle geometry break fore–aft symmetries. The net deflection is maximal for certain initial conditions and we analytically describe its scaling with particle size, obstacle shape and flow geometry, confirmed by direct trajectory simulations. For realistic parameters, separation by particle size is comparable to what is found assuming contact (roughness) interactions. Our approach also makes systematic predictions on when short-range attractive forces lead to particle capture or sticking. In separating hydrodynamic effects on particle motion strictly from contact interactions, we provide novel, rigorous guidelines for elementary microfluidic particle manipulation and filtering.

Over 125 years ago, Henry Selby Hele-Shaw realised that the depth-averaged flow in thin-gap geometries can be closely approximated by two-dimensional (2-D) potential flow, in a surprising marriage between the theories of viscous-dominated and inviscid flows. Hele-Shaw approximation allows visualisation of potential flows over 2-D aerofoils and also undergirds important discoveries in the dynamics of interfacial instabilities and convection, yet it has found little use in modelling flows in microfluidic devices, although these devices often have thin-gap geometries. Here, we derive a Hele-Shaw approximation for the flow in the kinds of thin-gap geometries created within microfluidic devices. Using the method of weighted residuals, we reinterpret the Hele-Shaw approximation as the leading term of an orthogonal polynomial expansion that can be systematically extended to higher-order corrections. The resulting leading-order equation coincides with the previously derived 2-D approximations, but our derivation is shorter and more direct. By extending the expansion beyond leading order, we obtain a new reduced model that captures non-parabolic gapwise velocity profiles and out-of-plane flow effects. We provide substantial numerical evidence showing that approximate equations can successfully model real microfluidic and inertial-microfluidic device geometries. By reducing three-dimensional flows to 2-D models, our validated model will allow for accelerated device modelling and design.

Evaporation and condensation have been reported to be able to considerably alter the pinch-off profile of nano liquid threads. However, it is still not well understood how evaporation and condensation will impact the instability of nano liquid threads. In this article, we propose a modified stochastic lubrication equation (MSLE) that incorporates both thermal fluctuations and evaporation–condensation. We conduct stability analysis based on the MSLE and show that the curvature-dependent evaporation tends to enhance the growth of perturbations of small wavenumber but impede those with large wavenumber. For realistic fluids, the effect is actually very small. However, in a supersaturated vapour environment, condensation is dominant. Stability analysis and molecular dynamics simulations both show that the growth of the perturbations is considerably impeded by condensation. The spectrum curves of the perturbations shift to smaller wavenumber and lower magnitude. The effect of condensation becomes very significant on short nano liquid threads whose lengths are around the critical length in Rayleigh’s classic theory. Although condensation is usually a slow process, it can completely alter the stability of short liquid threads, rendering an originally unstable thread stable. Our results open up a new avenue to control the instability of nano liquid threads through environmental vapour pressure.

We investigated the influence of the slip velocity on particle migration in viscoelastic microchannel flows using a hybrid computational approach that coupled the lattice Boltzmann method with coarse-grained molecular dynamics. Our results demonstrate that the slip velocity changes lateral migration mechanisms by affecting the balance of inertial and elastic lift forces. In Newtonian fluids, forward slip drives particles toward the channel walls due to dominant inertial lift, while backward slip promotes migration toward the channel centreline. In viscoelastic fluids, however, slip-induced elastic lift forces arising from asymmetric polymer deformation around particles exceed inertial effects by an order of magnitude. This leads to a complete reversal of migration behaviour. We established that elastic lift scales linearly with the slip velocity and the block ratio, consistent with theoretical predictions, while polymer chain length influences elastic lift through a power-law dependence ($F_{e,s}^*\sim M^{1.66}$). These findings reveal that viscoelasticity-mediated slip effects provide a robust mechanism for particle manipulation in complex fluids. By connecting the microscopic polymer dynamics to macroscopic transport phenomena, our work offers new design principles for particle sorting and focusing applications in microfluidic systems.

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.