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.

The linear stability of miscible displacement for radial source flow at infinite Péclet number in a Hele-Shaw cell is calculated theoretically. The axisymmetric self-similar flow is shown to be unstable to viscous fingering if the viscosity ratio $m$ between ambient and injected fluids exceeds $3/2$, and to be stable if $m\lt {3/2}$. If $1\lt m\lt {3/2}$, then small disturbances decay at rates between $t^{-3/4}$ and $t^{-1}$ (the exact range depending on $m$) relative to the $t^{1/2}$ radius of the axisymmetric base-state similarity solution; if $m\lt 1$, then they decay faster than $t^{-1}$. Asymptotic analysis confirms these results and gives physical insight into various features of the numerically determined relationship between the growth rate and the azimuthal wavenumber and viscosity ratio.

The Hele-Shaw–Cahn–Hilliard model, coupled with phase separation, is numerically simulated to demonstrate the formation of anomalous fingering patterns in a radial displacement of a partially miscible binary-fluid system. The composition of injected fluid is set to be less viscous than the displaced fluid and within the spinodal or metastable phase-separated region, in which the second derivative of the free energy is negative or positive, respectively. Because of phase separation, concentration evolves non-monotonically between the injected and displaced fluids. The simulations reveal four areas of the concentration distribution between the fluids: the inner core; the low-concentration grooves/high-concentration ridges; the isolated fluid fragments or droplets; the mixing zone. The grooves/ridges and the fragments/droplets, which are the unique features of phase separation, form in the spinodal and metastable regions. Four typical types of patterns are categorized: core separation (CS); fingering separation (FS); separation fingering (SF); lollipop fingering, in the order of the dominance of phase separation, respectively. For the patterns of CS and FS, isolated fluid fragments or droplets around the inner core are the main features. Fingering formation is better maintained with droplets in the SF pattern if the phase separation is relatively weaker than viscous fingering (VF). Even continuous fingers are well preserved in the case of dominant VF; phase separation results in lollipop-shaped fingers. The evolving trend of the patterns is in line with the experiments. These patterns are summarized in a pattern diagram, mainly by the magnitude of the second derivative of the free energy profile.

We undertake an experimental investigation into the instabilities that emerge when a shear-thinning fluid intrudes a less viscous Newtonian fluid axisymmetrically in a lubricated Hele-Shaw cell. Pre-formed lubrication layers of Newtonian fluid that separate the shear-thinning fluid from the cell walls are incorporated into the experimental design. Provided the lubrication layers remain effective at reducing shear stress, so that extensional stresses dominate the flow of the intruding fluid, the instabilities evolve to form branch-like structures, which exhibit fracturing or tearing behaviour at their troughs. Thicker lubrication layers enable the branches to propagate radially outwards, whilst thinner, less effective ones hinder their development and progression. In the absence of lubrication layers, the shear-thinning fluid spreads radially and remains axisymmetric. For lubricated flows, we show that the number of branches is dependent primarily on the strain rate at the radial distance where they first emerge, and that the number of branches decreases with increasing strain rate.

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.

When a less-viscous solution of a reactant $A$ displaces a more-viscous solution of another reactant $B$, a fast bimolecular $A + B \rightarrow C$ reaction decreasing locally the viscosity can influence the viscous fingering (VF) instability taking place between the two miscible solutions. We show both experimentally and numerically that, for monotonic viscosity profiles, this decrease in viscosity has opposite effects on the fingering pattern depending on the injection flow rate. For high flow rates, the reaction enhances the shielding effect, creating VF patterns with a lower surface density, i.e. thinner fingers covering a smaller area. In contrast, for lower flow rates, the reaction stabilises the VF dynamics, i.e. delays the instability and gives a less-deformed displacement, reaching in some cases an almost-stable displacement. Nonlinear simulations of reactive VF show that these opposite effects at low or high flow rates can only be reproduced if the diffusivity of $A$ is larger than that of $B$, which favours a larger production of the product $C$ and, hence, a larger viscosity decrease. The analysis of one-dimensional viscosity profiles reconstructed on the basis of a one-dimensional reaction–diffusion–advection model confirms that the VF stabilisation at low Péclet number and in the presence of differential diffusion of reactants originates from an optimum reaction-driven decrease in the gradient of the monotonic viscosity profile.

The impact of a chemical reaction, $A+B \rightarrow C$, on the stability of a miscible radial displacement in a porous medium is established. Our study involves a comprehensive analysis employing both linear stability analysis and nonlinear simulations. Through linear stability analysis, the onset of instability for monotonic as well as non-monotonic viscosity profiles corresponding to the same end-point viscosity are discussed and compared. We establish a $(R_b,R_c)$ phase plane for a wide range of Damköhler number ($Da$) and Péclet number ($Pe$) into stable and unstable regions. Here, $R_b=\ln (\mu _B/ \mu _A)$ and $R_c=\ln (\mu _C/ \mu _A)$ and $\mu _{i}$ is the viscosity of fluid $i$ $\in \{A$, $B$, $C$}. The stable zone in the $(R_b, R_c)$ phase plane contracts with increased $Da$ and $Pe$ but never vanishes. It exists even for $Da \rightarrow \infty$. Interestingly, we obtain a $Da$ independent stable region in the neighbourhood of $R_c=R_b$ where no transition occurs in stability despite changes in reaction rate. The study allows us to acquire knowledge about the transition of the stability for varying $Da, Pe$ and different reactions classified using $R_b, R_c$.

We study the effect of geometrical confinement on thermal convection by laboratory experiments and direct numerical simulations using Hele-Shaw geometries (typically the gap-to-height aspect ratio $0.12$) for the Prandtl number $Pr \geq 40$ and the Rayleigh number $Ra \leq 6 \times 10^7$. Under such strong unidirectional confinement, the convective flows are forced to squeeze within the narrow gap and exhibit unique spatiotemporal signatures, which contrast those in unconfined systems. With the increase of $Ra$, we identify that the system experiences five convective regimes that can be classified from two aspects, time dependency and flow dimensionality: (I) quasi-two-dimensional (quasi-2-D) steady flow; (II) quasi-2-D flow with oscillatory corner rolls; (III) three-dimensional (3-D) flow with oscillatory corner rolls; (IV) 3-D steady flow; and (V) 3-D time-dependent motion of plumes around sidewalls. Notably, unsteadiness does not emerge globally, but is localised near the sidewalls as oscillatory corner rolls, resulting in the regime transitions happening in a quasi-steady manner. We confirm that these regime transitions show less dependence on both $Pr$ and the other (wider) horizontal scale of the geometry. Moreover, we find that a recently proposed criterion ‘degree of confinement’ (Noto et al., Proc. Natl Acad. Sci. USA, vol. 121, issue 28, 2024, e2403699121) successfully explains the emergence of 3-D structures, expanding its applicable range to smaller $Ra$. This study deepens the comprehension of the thermal convection emerging in tight geometries, impacting across disciplines, such as Earth and planetary science, and thermal engineering.

In this paper, we investigate a sink-driven three-layer flow in a radial Hele-Shaw cell. The three fluids are of different viscosities, with one fluid occupying an annulus-like domain, forming two interfaces with the other two fluids. Using a boundary integral method and a semi-implicit time stepping scheme, we alleviate the numerical stiffness in updating the interfaces and achieve spectral accuracy in space. The interaction between the two interfaces introduces novel dynamics leading to rich pattern formation phenomena, manifested by two typical events: either one of the two interfaces reaches the sink faster than the other (forming cusp-like morphology), or they come very close to each other (suggesting a possibility of interface merging). In particular, the inner interface can be wrapped by the other to have both scenarios. We find that multiple parameters contribute to the dynamics, including the width of the annular region, the location of the sink, and the mobilities of the fluids.

The flow of a nematic liquid crystal in a Hele-Shaw cell with an electrically controlled viscous obstruction is investigated using both a theoretical model and physical experiments. The viscous obstruction is created by temporarily electrically altering the viscosity of the nematic in a region of the cell across which an electric field is applied. The theoretical model is validated experimentally for a circular cylindrical obstruction, demonstrating user-controlled flow manipulation of an anisotropic liquid within a heterogeneous single-phase microfluidic device.

We explored the instability dynamics of the viscous fingering interaction in dual displacement fronts by varying the viscosity configuration. Four regimes of rear-dominated fingering, front-dominated fingering, dual fingering and stable were identified. By using the breakthrough time, which refers to the breakup of the dual displacement fronts, the instability dynamics were modelled, and a regime map was developed. These serve as a tool for effectively harnessing the dual displacement fronts for various applications, such as hydrogeology, petroleum, chemical processes and microfluidics.

Consider the motion of a thin layer of electrically conducting fluid, between two closely spaced parallel plates, in a classical Hele-Shaw geometry. Furthermore, let the system be immersed in a uniform external magnetic field (normal to the plates) and let electrical current be driven between conducting probes immersed in the fluid layer. In the present paper, we analyse the ensuing fluid flow at low Hartmann numbers. Physically, the system is particularly interesting because it allows for circulation in the flow, which is not possible in the standard pressure-driven Hele-Shaw cell. We first elucidate the mechanism of flow generation both physically and mathematically. After formulating the problem using complex variables, we present mathematical solutions for a class of canonical multiply connected geometries in terms of the prime function framework developed by Crowdy (Solving Problems in Multiply Connected Domains, SIAM, 2020). We then demonstrate how recently developed fast numerical methods may be applied to accurately determine the flow field in arbitrary geometries.

When a viscous fluid partially fills a Hele-Shaw channel, and is pushed by a pressure difference, the fluid interface is unstable due to the Saffman–Taylor instability. We consider the evolution of a fluid region of finite extent, bounded between two interfaces, in the limit that the interfaces are close, that is, when the fluid region is a thin liquid filament separating two gases of different pressure. In this limit, we derive a second-order ‘thin-filament’ model that describes the normal velocity of the filament centreline, and evolution of the filament thickness, as functions of the thickness, centreline curvature and their derivatives. We show that the second-order terms in this model, that include the effect of transverse flow along the filament, are necessary to regularise the instability. Numerical simulation of the thin-filament model is shown to be in accordance with level-set computations of the complete two-interface model. Solutions ultimately evolve to form a bubble of rapidly increasing radius and decreasing thickness.

We study self-similar viscous fingering for the case of divergent flow within a wedge-shaped Hele-Shaw cell. Previous authors have conjectured the existence of a countably infinite number of selected solutions, each distinguished by a different value of the relative finger angle. Interestingly, the associated solution branches have been posited to merge and disappear in pairs as the surface tension decreases. For the first time, we demonstrate how the selection mechanism can be derived based on exponential asymptotics. Asymptotic predictions of the finger-to-wedge angle are additionally given for different sized wedges and surface-tension values. The merging of solution branches is explained; this feature is qualitatively different to the case of classic Saffman–Taylor viscous fingering in a parallel channel configuration. Moreover, because the asymptotic framework does not highly depend on specifics of the wedge geometry, the proposed theory for branch merging in our self-similar problem likely relates much more widely to tip-splitting instabilities in time-dependent flows in circular and other geometries, where the viscous fingers destabilise and divide in two.

Existing theoretical analyses of Faraday waves in Hele-Shaw cells rely on the Darcy approximation and assume a parabolic flow profile in the narrow direction. However, Darcy's model is known to be inaccurate when convective or unsteady inertial effects are important. In this work, we propose a gap-averaged Floquet theory accounting for inertial effects induced by the unsteady terms in the Navier–Stokes equations, a scenario that corresponds to a pulsatile flow where the fluid motion reduces to a two-dimensional oscillating Poiseuille flow, similarly to the Womersley flow in arteries. When gap-averaging the linearised Navier–Stokes equation, this results in a modified damping coefficient, which is a function of the ratio between the Stokes boundary layer thickness and the cell's gap, and whose complex value depends on the frequency of the wave response specific to each unstable parametric region. We first revisit the standard case of horizontally infinite rectangular Hele-Shaw cells by also accounting for a dynamic contact angle model. A comparison with existing experiments shows the predictive improvement brought by the present theory and points out how the standard gap-averaged model often underestimates the Faraday threshold. The analysis is then extended to the less conventional case of thin annuli. A series of dedicated experiments for this configuration highlights how Darcy's thin-gap approximation overlooks a frequency detuning that is essential to correctly predict the locations of the Faraday tongues in the frequency–amplitude parameter plane. These findings are well rationalised and captured by the present model.

We present results for Newtonian laminar miscible displacement flows in a narrow, vertical, eccentric annulus, obtained from experiments carried out in a scaled laboratory set-up. These are compared with two computational models introduced in Part 1, Zhang & Frigaard (J. Fluid Mech., vol. 947, 2022, A732). Quite close matches have been found for different approaches, regarding the overall evolving displacement process: front shape, dispersion levels and front velocities. Standardized criteria have been established to identify whether a flow has steady/unsteady and dispersive/non-dispersive displacement front, applicable to both concentric and eccentric annuli. Three characteristic flows have been observed, unsteady and dispersive, dispersive with steady front, and non-dispersive with steady front. Through both qualitative and quantitative studies, the importance of the buoyancy number has been established: it both restrains dispersion on the annular gap scale and induces secondary flows in the azimuthal direction.

The coupled effects of thermodynamic and hydrodynamic instabilities are studied during viscous fingering (VF). We introduced a modified Cahn–Hilliard phase-field model in conjunction with the Korteweg force in the classical VF model and derived consistent governing equations. The free energy of the partially miscible system is described using a modified Flory–Huggins model, which allows us to investigate the temporal evolution of spatial inhomogeneities. The mass flux in the Cahn–Hilliard equations is modified according to modern diffusion theory. The governing equations have been solved through an in-house model implementation using the COMSOL multiphysics software. We successfully demonstrated the transition from the finger-like structures to the droplet formation during spinodal decomposition as demonstrated experimentally in the literature. Our results are also in agreement with earlier numerical results obtained using a classical Landau type mixing energy. We further systematically studied the effects of the Margules parameter (interaction parameter) and the gradient parameter, which is associated to the thermodynamic length scale and capillary number on the VF. Aysmmetric features of the binary mixture are also investigated showing a stronger thermodynamic effect on the system with increasing phase separation and, hence, droplet formation.

In this work we investigate the effect of vertical confinement and inertia on the flow past thin ellipses in a Hele-Shaw cell (with centre line velocity $U_c$ and height 2$h$) with different aspect ratios for symmetrical flows and at an angle of attack, using asymptotic methods and numerical simulations. A Stokes region is identified at the ellipse vertices which results in flow different to flow past bluff bodies. Comparison with asymptotic analysis indicates close agreement over the ‘flat’ portion of the ellipse, for $\delta =(b/a)=0.05$, where $a$ and $b$ are the semi-major and -minor ellipse axes, respectively. Two flow conditions are investigated for ellipses at an angle of attack of 10$^\circ$ for a fixed $\delta =0.05$. Firstly, for $\varLambda =(U_ca/\nu )(h/a)^2 \ll 1$, the effect of increasing the vertical confinement of the Hele-Shaw cell results in the rear stagnation point (RSP) moving from close to the potential-flow prediction when $\epsilon =h/a$ is very small to the two-dimensional Stokes-flow prediction when $\epsilon$ is large. Secondly, for a fixed $\epsilon \ll 1$, when inertia is increased past $\varLambda ={O}(\epsilon )$ the RSP moves towards the trailing edge and is located there for $\varLambda ={O}(1)$. Under these conditions an attached exponentially decaying shear layer or ‘viscous tail’ is formed. A modified Bernoulli equation of the depth-averaged flow, together with the Kutta–Joukowski theorem is used to predict the drag and lift coefficients on the ellipse, which include a linear and a nonlinear contribution, corresponding to a Hele-Shaw and circulation component, respectively. Close agreement is found up to $\varLambda ={O}(1)$.

We experimentally study front propagation in a vortex lattice providing closed steady cellular flows and no mean flow. To this end, we trigger an autocatalytic reaction in a solution stirred by magnetohydrodynamic flows in a Hele-Shaw cell. We evidence a scale-invariant regime below some flow magnitude and a scale-dependent regime above, the scales referring here to the vortex scale and the front thickness. The transition between these regimes corresponds to a unitary Damköhler number $Da$: $Da=1$. The enhancement of the mean front velocity with the flow magnitude nicely agrees with the literature on numerical simulations and theoretical analyses in the scale-invariant regime $Da>1$, but displays noticeable discrepancies in the scale-dependent one $Da<1$. This shows that the transition between regimes is qualitatively sharp but quantitatively smooth.

Active emulsions can spontaneously form self-propelled droplets or phoretic micropumps. However, it remains unclear how these active systems interact with their self-generated chemical fields, which can lead to emergent chemodynamic phenomena and multistable interfacial flows. Here, we simultaneously measure the flow and chemical concentration fields using dual-channel fluorescence microscopy for active micropumps, i.e. immobilised oil droplets that dynamically solubilise in a supramicellar aqueous surfactant solution. With increasing droplet radius, we observe (i) a migration of vortices from the posterior to the anterior, analogous to a transition from pusher- to puller-type swimmers, (ii) a bistability between dipolar and quadrupolar flows and, eventually, (iii) a transition to multipolar modes. We also investigate the long-time dynamics. Together, our observations suggest that a local build-up of chemical products leads to a saturation of the surface, which controls the propulsion mechanism. These multistable dynamics can be explained by the competing time scales of slow micellar diffusion governing the chemical buildup and faster molecular diffusion powering the underlying transport mechanism. Our results are directly relevant to phoretic micropumps, but also shed light on the interfacial activity dynamics of self-propelled droplets and other active emulsion systems.