The motion and deformation of a neutrally buoyant drop in a rectangular channel experiencing a pressure-driven flow at a low Reynolds number has been investigated both experimentally and numerically. A moving-frame boundary-integral algorithm was used to simulate the drop dynamics, with a focus on steady-state drop velocity and deformation. Results are presented for drops of varying undeformed diameters relative to channel height ($D/H$), drop-to-bulk viscosity ratio ($\lambda$), capillary number ($Ca$, ratio of deforming viscous forces to shape-preserving interfacial tension) and initial position in the channel in a parameter space larger than considered previously. The general trend shows that the drop steady-state velocity decreases with increasing drop diameter and viscosity ratio but increases with increasing $Ca$. An opposite trend is seen for drops with small viscosity ratio, however, where the steady-state velocity increases with increasing $D/H$ and can exceed the maximum background flow velocity. Experimental results verify theoretical predictions. A deformable drop with a size comparable to the channel height when placed off centre migrates towards the centreline and attains a steady state there. In general, a drop with a low viscosity ratio and high capillary number experiences faster cross-stream migration. With increasing aspect ratio, there is a competition between the effect of reduced wall interactions and lower maximum channel centreline velocity at fixed average velocity, with the former helping drops attain higher steady-state velocities at low aspect ratios, but the latter takes over at aspect ratios above approximately 1.5.

Crowdy et al. (2023 Phys. Rev. Fluids, vol. 8, 094201), recently showed that liquid suspended in the Cassie state over an asymmetrically spaced periodic array of alternating cold and hot ridges such that the menisci spanning the ridges are of unequal length will be pumped in the direction of the thermocapillary stress along the longer menisci. Their solution, applicable in the Stokes flow limit for a vanishingly small thermal Péclet number, provides the steady-state temperature and velocity fields in a semi-infinite domain above the superhydrophobic surface, including the uniform far-field velocity, i.e. pumping speed, the key engineering parameter. Here, a related problem in a finite domain is considered where, opposing the superhydrophobic surface, a flow of liquid through a microchannel is bounded by a horizontally mobile smooth wall of finite mass subjected to an external load. A key assumption underlying the analysis is that, on a unit area basis, the mass of the liquid is small compared with that of the wall. Thus, as shown, rather than the heat equation and the transient Stokes equations governing the temperature and flow fields, respectively, they are quasi-steady and, as a result, governed by the Laplace and Stokes equations, respectively. Under the further assumption that the ridge period is small compared with the height of the microchannel, these equations are resolved using matched asymptotic expansions which yield solutions with exponentially small asymptotic errors. Consequently, the transient problem of determining the velocity of the smooth wall is reduced to an ordinary differential equation. This approach is used to provide a theoretical demonstration of the conversion of thermal energy to mechanical work via the thermocapillary stresses along the menisci.

The transport of droplets in microfluidic channels is strongly dominated by interfacial properties, which makes it a relevant tool for understanding the mechanisms associated with the presence of more or less soluble surfactants. In this paper, we show that the mobility of an oil droplet pushed by an aqueous carrier phase in a Hele-Shaw cell qualitatively depends on the nature of the surfactants: the drop velocity is an increasing function of the drop radius for highly soluble surfactants, whereas it is a decreasing function for poorly soluble surfactants. These two different behaviours are experimentally observed by using two families of surfactant with a carbon chain of variable length. We first focus on the second regime, observed here for the first time, and we develop a model which takes into account the flux of surfactants on the whole droplet interface, assuming an incompressible surfactant monolayer. This model leads to a quantitative agreement with the experimental data, without any adjustable parameter. We then propose a model for a stress-free interface, i.e. for highly soluble surfactants. In these two limits, the models become independent on the physico-chemical properties of the surfactants, and should be valid for any surfactant complying with the incompressible or stress-free limit. As such, we provide a theoretical framework with two limits for all the experimental physico-chemical configurations, which constitute the bounds for the droplet mobility for intermediate surfactant solubility.

Analytical expressions are derived for the velocity field, and effective slip lengths, associated with pressure-driven longitudinal flow in a circular superhydrophobic pipe whose boundary is patterned with a general arrangement of longitudinal no-shear stripes not necessarily possessing any rotational symmetry. First, the flow in a superhydrophobic pipe with $M$ different no-shear stripes in general position is found for $M=1, 2, 3$. The method, which is based on use of so-called prime functions, is such that with these cases covered, generalisation to any $M \geqslant 1$ follows in a straightforward manner. It is shown how any one of these solutions can be generalised to solve for flow along superhydrophobic pipes where that pattern of $M$ menisci is repeated $N \geqslant 1$ times around the boundary in a rotational symmetric arrangement. The work provides an extension of the canonical pipe flow solution for an $N$-fold rotationally symmetric pattern of no-shear stripes due to Philip (Angew. Math. Phys., vol. 23, 1972, pp. 353–372). The novel solution method, and the solutions that it produces, have significance for a wide range of mixed boundary value problems involving Poisson’s equation arising in other applications.

We present numerical analysis of the lateral movement of a spherical capsule in the steady and pulsatile channel flow of a Newtonian fluid for a wide range of oscillatory frequencies. Each capsule membrane satisfying strain-hardening characteristics is simulated for different Reynolds numbers $Re$ and capillary numbers $Ca$. Our numerical results showed that capsules with high $Ca$ exhibit axial focusing at finite $Re$ similarly to the inertialess case. We observe that the speed of the axial focusing can be substantially accelerated by making the driving pressure gradient oscillate in time. We also confirm the existence of an optimal frequency that maximises the speed of axial focusing, which remains the same found in the absence of inertia. For relatively low $Ca$, however, the capsule exhibits off-centre focusing, resulting in various equilibrium radial positions depending on $Re$. Our numerical results further clarify the existence of a specific $Re$ for which the effect of the flow pulsation to the equilibrium radial position is maximum. The roles of channel size on the lateral movements of the capsule are also addressed. Throughout our analyses, we have quantified the radial position of the capsule in a tube based on an empirical expression. Given that the speed of inertial focusing can be controlled by the oscillatory frequency, the results obtained here can be used for label-free cell alignment/sorting/separation techniques, e.g. for circulating tumour cells in cancer patients or precious haematopoietic cells such as colony-forming cells.

In a quiescent medium, chemically active particles propel themselves by emitting or absorbing solutes, creating concentration gradients that induce a slip at the particle surface. This self-propulsion occurs when solute advection overcomes diffusion. However, an imposed flow field can alter these dynamics. This study explores the propulsion characteristics and the related rheological consequences of chemically active particles in an imposed uniaxial extensional flow analytically and numerically. An asymptotic solution is obtained for weak imposed flow relative to self-induced diffusiophoretic slip. Meanwhile, finite element simulations are carried out over a wide range of imposed flow strength and Péclet number. The results reveal that the interplay between solute advection, imposed flow and diffusiophoretic slip significantly affects particle propulsion and suspension rheology. While solute advection and diffusiophoretic slip tend to create asymmetric solute distributions, promoting self-propulsion, imposed extensional flow promotes symmetric distributions, hindering self-propulsion. This not only delays the start of self-propulsion but also results in an early transition from a propulsion state to a stationary state characterised by an abrupt halt at relatively lower Péclet number compared to a quiescent medium. Post the abrupt halt, a stirring effect induced by particle activity and imposed extensional flow results in an increased magnitude of stresslet, thus a sudden change in the effective viscosity of the active suspension. The effect of imposed extensional flow on active particle dynamics and suspension rheology can be described succinctly by categorising the overall dynamics into three separate regimes, determined by the Péclet number and the intensity of the extensional flow.

The demand for separating and analysing rare target cells is increasing dramatically for vital applications such as cancer treatment and cell-based therapies. However, there remains a grand challenge for high-throughput and label-free segregation of lesion cells with similar sizes. Cancer cells with different invasiveness usually manifest distinct deformability. In this work, we employ a hydrogel microparticle system with similar sizes but varied stiffness to mimic cancer cells and examine in situ their deformation and focusing under microfluidic flow. We first demonstrate the similar focusing behaviour of hydrogel microparticles and cancer cells in confined flow that is dominated by deformability-induced lateral migration. The deformation, orientation and focusing position of hydrogel microparticles in microfluidic flow under different Reynolds numbers are then systematically observed and measured using a high-speed camera. Linear correlations of the Taylor deformation and tilt angle of hydrogel microparticles with the capillary number are revealed, consistent with theoretical predictions. Detailed analysis of the dependence of particle focusing on the flow rate and particle stiffness enables us to identify a linear scaling between the equilibrium focusing position and the major axis of the deformed microparticles, which is uniquely determined by the capillary number. Our findings provide insights into the focusing and dynamics of soft beads, such as cells and hydrogel microparticles, under confined flow, and pave the way for applications including the separation and identification of circulating tumour cells, drug delivery and controlled drug release.

Manipulation of small-scale particles across streamlines is the elementary task of microfluidic devices. Many such devices operate at very low Reynolds numbers and deflect particles using arrays of obstacles, but a systematic quantification of relevant hydrodynamic effects has been lacking. Here, we explore an alternative approach, rigorously modelling the displacement of force-free spherical particles in vortical Stokes flows under hydrodynamic particle–wall interaction. Certain Moffatt-like eddy geometries with broken symmetry allow for systematic deflection of particles across streamlines, leading to particle accumulation at either Faxen field fixed points or limit cycles. Moreover, particles can be forced onto trajectories approaching channel walls exponentially closely, making possible quantitative predictions of particle capture (sticking) by short-range forces. This rich, particle-size-dependent behaviour suggests the versatile use of inertia-less flow in devices with a long particle residence time for concentration, sorting or filtering.

The transport of particles in elastoviscoplastic (EVP) fluids is of significant interest across various industrial and scientific domains. However, the physical mechanisms underlying the various particle distribution patterns observed in experimental studies remain inadequately understood in the current literature. To bridge this gap, we perform interface-resolved direct numerical simulations to study the collective dynamics of spherical particles suspended in a pressure-driven EVP duct flow. In particular, we investigate the effects of solid volume fraction, yield stress, inertia, elasticity, shear-thinning viscosity, and secondary flows on particle migration and formation of plug regions in the suspending fluid. Various cross-streamline migration patterns are observed depending on the rheological parameters of the carrier fluid. In EVP fluids with constant plastic viscosity, particles aggregate into a large cluster at the duct centre. Conversely, EVP fluids with shear-thinning plastic viscosity induce particle migration towards the duct walls, leading to formation of particle trains at the corners. Notably, we observe significant secondary flows ($O(10^{-2})$ compared to the mean velocity) in shear-thinning EVP suspensions, arising from the interplay of elasticity, shear-thinning viscosity and particle presence, which further enhances corner-ward particle migration. We elucidate the physical mechanism by which yield stress augments the first normal stress difference, thereby significantly amplifying elastic effects. Furthermore, through a comprehensive analysis of various EVP suspensions, we identify critical thresholds for elasticity and yield stress necessary to achieve particle focusing at the duct corners.

A numerical study supplemented with theoretical analysis is made, to analyse the electrophoresis of highly charged soft particles in electrolytes with trivalent counterions. The electrokinetic model is devised under the continuum hypothesis, which incorporates the ion–ion electrostatic correlations, hydrodynamic steric interactions of finite sized ions and ion–solvent interactions. The governing equations for ion transport and electric field are derived from the volumetric free energy of the system, which includes the first-order correction for the non-local electrostatic correlations of interacting ions, excess electrochemical potential due to finite ion size as well as the Born energy difference of ions due to dielectric permittivity variation. The electrolyte viscosity is considered to be a function of the local volume fraction of finite-sized ions, which causes the diffusivity of ions to vary locally. The occurrence of mobility reversal of a soft particle having the same polarity of its core and soft shell charge and formation of a coion-dominated zone in the soft layer is elaborated through this study. This can explain the mechanisms for the attraction between like-charged soft particles, as seen in the condensation of DNAs. The impact of ion–ion correlations and ion–solvent interactions of finite-sized ions are analysed by comparing them with the results based on the standard model. At a higher range of the core charge density, the ion–ion correlations induce a condensed layer of counterions on the outer surface of the core, which draws coions in the electric double layer, leading to an inversion in polarity of the charge density and mobility reversal. The dielectric decrement and ion steric interactions create a saturation in ion distribution and hence, modify the condensed layer of counterions. The enhanced fixed charge density of the polyelectrolyte layer diminishes the ion correlations due to the stronger screening effects and prevents the formation of a coion dominated zone in the Debye layer. The impact of the counterion size and the mixture of monovalent and trivalent counterions on mobility is analysed.

Particle-laden flow through conduits is ubiquitous in both natural and industrial systems. In such flows, particles often migrate across the main fluid stream, resulting in non-uniform spatial distribution owing to particle–fluid and particle–particle interactions. The most relevant lateral particle migration mechanism by particle–fluid interaction is the Segré–Silberberg effect, which is induced by the inertial forces exerted on a particle, as the flow rate increases. However, methods to suppress it have not been suggested yet. Here, we demonstrate that adding a small amount of polymer to the particle-suspending solvent effectively suppresses the Segré–Silberberg effect in a square channel. To accurately determine the position of the particles within the channel cross-sections, we devised a dual-view imaging system applicable to microfluidic systems. Our analyses show that the Segré–Silberberg effect is effectively suppressed in a square microchannel due to the balance between the inertial and elastic forces at an optimal polymer concentration while maintaining nearly constant shear viscosity.

When one fluid is injected into a confined geometry such as a porous medium filled with another immiscible fluid, even at an extremely low injection speed, rapid filling of several pore spaces accompanied by retraction of multiple fluid–fluid interfaces can be observed. Such processes with fast liquid redistribution within the solid structure, called Haines jumps, are ubiquitous in many multiphase flow systems, which can impact fluid trapping, energy dissipation and hysteretic saturation in various engineering applications. Inspired by this mechanism, here, we propose a dual-channel structure to realise controlled Haines jumps during fluid displacement processes. Via theoretical analysis and numerical simulations, we show that the dynamics of fluid interfaces during Haines jumps can be quantitatively correlated with the driving capillary pressure and dissipating viscous stress, which enables simultaneous determination of the fluid viscosity and interfacial tension in the dual-channel multiphase system.

This study explores the efficacy of a novel microfluidic device in isolating rheotactic sperm and assesses their advantages compared with other motile sperm. Two microfluidic devices were used in this study: the microfluidic device we designed to separate sperm based on rheotaxis and a simple passive microfluidic device. We compared the results with the density gradient centrifugation technique. Sperm attributes including concentration, morphology, viability and motility were assessed using related procedures. Statistical analyses were conducted using one-way analysis of variance. Results showed differences in sperm concentration, motility, morphology and vitality using different sperm separation techniques. The sperms separated using our microfluidic device demonstrated the highest motilities, normal morphology percentages and higher sperm vitality but significantly lower sperm concentrations. These findings suggest the potential of our microfluidic design in enhancing sperm quality. Our findings are in agreement with previous research, emphasizing the capability of microfluidics in enhancing sperm quality. Specifically, our designed microfluidic device exhibited exceptional efficacy in isolating highly motile sperm, a critical factor for successful fertilization.

Understanding interfacial instability in a coflow system has relevance in the effective manipulation of small objects in microfluidic applications. We experimentally elucidate interfacial instability in stratified coflow systems of Newtonian and viscoelastic fluid streams in microfluidic confinements. By performing a linear stability analysis, we derive equations that describe the complex wave speed and the dispersion relationship between wavenumber and angular frequency, thus categorizing the behaviour of the systems into two main regimes: stable (with a flat interface) and unstable (with either a wavy interface or droplet formation). We characterize the regimes in terms of the capillary numbers of the phases in a comprehensive regime plot. We decipher the dependence of interfacial instability on fluidic parameters by decoupling the physics into viscous and elastic components. Remarkably, our findings reveal that elastic stratification can both stabilize and destabilize the flow, depending on the fluid and flow parameters. We also examine droplet formation, which is important for microfluidic applications. Our findings suggest that adjusting the viscous and elastic properties of the fluids can control the transition between wavy and droplet-forming unstable regimes. Our investigation uncovers the physics behind the instability involved in interfacial flows of Newtonian and viscoelastic fluids in general, and the unexplored behaviour of interfacial waves in stratified liquid systems. The present study can lead to a better understanding of the manipulation of small objects and production of droplets in microfluidic coflow systems.

Recognising that surfactants can impede the drag reduction resulting from superhydrophobic surfaces (SHS), we investigate the impact of spatio–temporal fluctuations in surfactant concentration on the drag-reduction properties of SHS. We model the unsteady transport of soluble surfactant in a channel flow bounded by two SHS. The flow is laminar, pressure driven and the SHS are periodic in the streamwise and spanwise directions. We assume that the channel length is much longer than the streamwise period, the streamwise period is much longer than the channel height and spanwise period, and bulk diffusion is sufficiently strong for cross-channel concentration gradients to be small. By combining long-wave and homogenisation theories, we derive an unsteady advection–diffusion equation for surfactant-flux transport over the length of the channel, which is coupled to a quasi-steady advection–diffusion equation for surfactant transport over individual plastrons. As diffusion over the length of the channel is typically small, the surfactant flux is governed by a nonlinear wave equation. In the fundamental case of the transport of a bolus of surfactant, we predict its propagation speed and describe its nonlinear evolution via interaction with the SHS. The propagation speed can fall below the average streamwise velocity as the surfactant adsorbs and rigidifies the plastrons. Smaller concentrations of surfactant are advected faster than larger ones, so that wave-steepening effects can lead to shock formation in the surfactant-flux distribution. Our asymptotic results reveal how unsteady surfactant transport can affect the spatio–temporal evolution of the slip velocity, drag reduction and effective slip length in SHS channels.

Most of the existing theories on electrophoresis are based on the consideration of a weak applied electric field and ions as point charges, which create a mean electric potential and neglect ion–solvent interactions. These theories cannot demonstrate the dependence of electrophoretic mobility on the applied electric field (nonlinear electrophoresis), reversal in mobility with increasing ion concentration and/or surface charge density or counterion saturation in the electric double layer. In this study we consider a modified electrokinetic model to analyse nonlinear electrophoresis by taking into account the finite ion size effects and ion–ion electrostatic correlations. In this approach, the mean-field-based model is extended to capture the many-body phenomena by considering the non-local electrostatic contribution in the ion free energy functional and the ion–ion hydrodynamic steric interactions are incorporated through the volume exclusion effect in the electrochemical potential. The viscosity of the medium is considered to vary with the local ionic volume fraction. Stronger correlations for multivalent counterions create ion layering, charge density oscillation and mobility reversal. Such phenomena are captured by the present continuum model. The ion crowding attenuates the growth of the electrophoretic mobility with the electric field. At a higher range of the imposed electric field, the ion concentration in the electric double layer enhances, which modifies both the overscreening and ion crowding processes.

We investigate theoretically the steady incompressible viscoelastic flow in a rigid axisymmetric cylindrical pipe with varying cross-section. We use the Oldroyd-B viscoelastic constitutive equation to model the fluid viscoelasticity. First, we derive exact general formulae: for the total average pressure-drop as a function of the wall shear rate and the viscoelastic axial normal extra-stress; for the viscoelastic extra-stress tensor and the Trouton ratio as functions of the fluid velocity on the axis of symmetry; and for the viscoelastic extra-stress tensor along the wall in terms of the shear rate at the wall. Then we exploit the classic lubrication approximation, valid for small values of the square of the aspect ratio of the pipe, to simplify the original governing equations. The final equations are solved analytically using a regular perturbation scheme in terms of the Deborah number, De, up to eighth order in De. For a hyperbolically shaped pipe, we reveal that the reduced pressure-drop and the Trouton ratio can be recast in terms of a modified Deborah number, Dem, and the polymer viscosity ratio, η, only. Furthermore, we enhance the convergence and accuracy of the eighth-order solutions by deriving transformed analytical formulae using Padé diagonal approximants. The results show the decrease of the pressure drop and the enhancement of the Trouton ratio with increasing Dem and/or increasing η. Comparison of the transformed solutions with numerical simulations of the lubrication equations using pseudospectral methods shows excellent agreement between the results, even for high values of Dem and all values of η, revealing the robustness, validity and efficiency of the theoretical methods and techniques developed in this work. Last, it is shown that the exact solution for the Trouton ratio gives a well-defined and finite solution for any value of Dem and reveals the reason for the failure of the corresponding high-order perturbation series for Dem > 1/2.

Understanding the effect of intricate surface wettability conditions on microswimmers is crucial for precisely navigating them across narrow microcirculatory networks. Here, we adopt the spherical squirmer model and Navier slip condition to delineate the microswimmer locomotion under a Poiseuille flow in a slit microchannel. Through a combined analytical–numerical approach utilizing bispherical coordinates and the superposition technique, we resolve the slip-modulated simultaneous hydrodynamic interaction with substrate boundaries. Phase portraits reveal that slip significantly alters propulsion mechanisms, destabilizing centreline stable oscillations of pullers beyond a threshold slip length. Superhydrophobic surfaces suppress near-wall rheotaxis states but preserve centreline focusing, facilitating slip-assisted directed transport without surface accumulation. Under strong background flows, subcritical Hopf bifurcation emerges for pullers at a critical slip length, transitioning dynamics from coexisting stable and unstable states to purely unstable behaviour. Contrastingly, for pushers, slip causes a transition from unstable to either stable or fixed-amplitude oscillations. Increased slip length reduces hydrodynamic repulsion on pullers from the walls by enhancing rotational velocity near the walls, whereas it counteracts the torque that causes unstable oscillations of pushers. Three-dimensional analysis of the trajectories reveals the significant role of the out-of-plane orientation of the microswimmer in its transitions between different swimming states. The presented regime maps offer parametric combinations for specific motion behaviours, guiding the development of smart microfluidic drug delivery systems and preventing biofilm deposition in biomedical devices.

The elasto-inertial focusing and rotating characteristics of spheroids in a square channel flow of Oldroyd-B viscoelastic fluids are studied by the direct forcing/fictitious domain method. The rotational behaviours, changes in the equilibrium positions and travel distances are explored to analyse the mechanisms of spheroid migration in viscoelastic fluids. Within the present simulated parameters (1 ≤ Re ≤ 100, 0 ≤ Wi ≤ 2, 0.4 ≤ α ≤3), the results show that there are four kinds of equilibrium positions and six (five) kinds of rotational behaviours for the elasto-inertial migration of prolate (oblate) spheroids. We are the first to identify a new rotational mode for the migration of prolate spheroids. Only when the particles are initially located at a corner and wall bisector, some special initial orientations of the spheroids have an impact on the final equilibrium position and rotational mode. In other general initial positions, the initial orientation of the spheroid has a negligible effect. A higher Weissenberg number means the faster the particles migrate to the equilibrium position. The spheroid gradually changes from the corner (CO), channel centreline (CC), diagonal line (DL) and cross-section midline (CSM) equilibrium positions as the elastic number decreases, depending on the aspect ratio, initial orientation and rotational behaviour of the particles and the elastic number of the fluid. When the elastic number is less than the critical value, the types of rotational modes of the spheroids are reduced. By controlling the elastic number near the critical value, spheroids with different aspect ratios can be efficiently separated.

We numerically study the influence of a soluble surfactant on the microjetting mode of the liquid–liquid flow focusing configuration. The surfactant adsorbs on the interface next to the feeding capillary and accumulates in front of the emitted jet, significantly lowering the surface tension there. The resulting Marangoni stress substantially alters the balance of the tangential stresses at the interface but does not modify the interface velocity. The global stability analysis at the minimum flow rate stability limit shows that the Marangoni stress collaborates with soluto-capillarity to stabilize the microjetting mode. Our analysis unveils the noticeable effect of the Marangoni stress associated with the surface tension perturbation. Surfactant diffusion and desorption hardly affect the stability limit. Transient numerical simulations show how subcritical and supercritical base flows respond to a spatially localized initial perturbation. Our parametric study indicates that the minimum flow rate ratio depends on the adsorption constant and the surfactant concentration through the product of these two variables. The surfactant stabilizing effect increases with the outer stream flow rate. We show that surfactants not only stabilize the microemulsion resulting from the jet breakup in hydrodynamic focusing, but also allow for the reduction of droplet size. Our findings advance the fundamental understanding of the complex role of surfactants in tip streaming via hydrodynamic focusing. In particular, our results contradict the common assumption that adding surfactant favours tip streaming simply because it reduces the meniscus tip surface tension.