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.

Viscous fingering or Saffman–Taylor instability shows fingering interfacial patterns when a more mobile fluid displaces a less mobile one in porous media. The effective interfacial tension (EIT) is like capillary force, acting at the miscible interface on time scales shorter than interface relaxation. It has been numerically reported so far that the fingers are wider with EIT compared with those without EIT. A recent experiment observed finger widening with increasing flow rate in a miscible system with EIT, which is not observed in classical immiscible and miscible systems. In this study, we have numerically investigated the effect of $Pe$ (which corresponds to the injection flow rates in the experiment) on miscible viscous fingering with/without EIT. We have shown that the fingers are monotonically thinner with an increase in $Pe$ in the system without EIT, while finger widening with increasing $Pe$ is observed in the system with EIT. Furthermore, we have also examined a one-dimensional underlying concentration profile and EIT profile by using a one-dimensional diffusion–convection model because EIT is proportional to the squared concentration gradient. We have then shown that the concentration gradient is steeper and, thus, the EIT is larger as $Pe$ is larger. Therefore, this is the first numerical study that can theoretically verify finger widening with increasing flow rate, which occurs only in a miscible system with EIT to the extent of our targeting EIT values, and explain the mechanism by one-dimensional analysis.

Supercritical ${\rm CO}_2$ injection and dissolution into deep brine aquifers allow its sequestration within geological formations. After injection, ${\rm CO}_{2}$ gas phase is buoyancy-driven over the denser aqueous brine, reaching an apparent gravitational stable distribution. However, ${\rm CO}_2$ dissolution in brine propels convection since the mixture is even denser than the underlying brine. This process still needs to be characterised comprehensively. Here, we investigate the irreversible mixing of dissolved ${\rm CO}_2$ in brine through laboratory-scale numerical experiments utilising the Hele-Shaw model (Letelier et al., J. Fluid Mech., vol. 864, 2019, pp. 746–767) and a fully miscible two-fluid system. In this scenario, mixing the less dense fluid – mimicking ${\rm CO}_{2}$ gas phase – with the heavier fluid – representing aqueous brine – catalyses cabbeling-powered convection. Our numerical simulations recover the laboratory results in porous media by Neufeld et al. (Geophys. Res. Lett., vol. 37, issue 22, 2010, L22404) and may explain the scaling law obtained by Backhaus et al. (Phys. Rev. Lett., vol. 106, issue 10, 2011, 104501) in Hele-Shaw cells. More remarkably, we show that the mass flux between the two analogue fluids, characterised by the Sherwood number $ {{Sh}}$, obeys the universal scaling law $ {{Sh}}\sim {{Ra}}\, \vartheta _{scalar}$, with $ {{Ra}}$ the Rayleigh number and $\vartheta _{scalar}$ the mean scalar dissipation rate. This paper sheds light on the fluid dynamics and solubility trapping in geological carbon sequestration.

Understanding the hysteretic behaviour in fluid–fluid displacement processes in porous media is critical in many engineering applications. In this work, we study the quasi-static immiscible displacement process in two-dimensional porous media during cyclic injections in the context of carbon geosequestration. The role of wettability on the residual trapping of CO$_2$ is investigated numerically using an extended interface tracking algorithm. Despite that higher CO$_2$ saturation can be achieved in CO$_2$-wet porous media after the first CO$_2$ injection, the majority of CO$_2$ is found to be unstable and can be mobilised during subsequent water injection processes. An improvement in the residual trapping of CO$_2$ is observed as the number of injection cycles increases, which is associated with the dispersion of continuous CO$_2$ ganglia into numerous smaller blobs. Compared with either water-wet or CO$_2$-wet porous media, it is found that less CO$_2$ is trapped within the neutral-wet ones at equilibrium state after a sufficient number of injection cycles. The hysteretic behaviour of saturation between water/CO$_2$ injection cycles is found to follow an exponential decay, which eventually reaches a finite value. This process corresponds to the shift of the mobile region during displacement from typical capillary fingering to a less ramified regime, which ultimately converges towards main flow channels. This work highlights the hysteretic behaviour during cyclic injections, providing insights on the wettability impacts on multiphase flow in porous media, which is of great importance in applications such as carbon geosequestration and geological hydrogen storage.

Driving oscillatory flow around an obstacle generates, due to inertial rectification, a steady ‘streaming’ flow that is useful in a host of microfluidic applications. While theory has focused largely on two-dimensional flows, streaming in many practical microfluidic devices is three-dimensional due to confinement. We develop a three-dimensional streaming theory around an obstacle in a microchannel with a Hele-Shaw-like geometry, where one dimension (depth) is much shorter than the other two dimensions. Utilizing inertial lubrication theory, we demonstrate that the time-averaged streaming flow has a three-dimensional structure. Notably, the flow reverses direction across the depth of the channel, which is a feature not observed in less confined streaming set-ups. This feature is confirmed by our experiments of streaming around a cylinder sandwiched in a microchannel. Our theory also predicts that the streaming velocity decays as the inverse cube of the distance from the cylinder, faster than that expected from previous two-dimensional approaches. We verify this velocity decay quantitatively using particle tracking measurements from experiments of streaming around cylinders with different aspect ratios at different driving frequencies.

We consider the propagation of inviscid bubbles in a Hele-Shaw cell under a uniform background flow. We focus on the distinguished limit in which the hydrodynamic pressure gradient due to the external flow balances viscous drag effects due to thin liquid films between the bubbles and the cell walls (Bretherton, J. Fluid Mech., vol. 10, issue 2, 1961, pp. 166–188), with the ratio between these two effects measured by a single dimensionless parameter that we label $\delta$. In this regime, we find that each bubble remains approximately circular, and its propagation velocity is determined by a net force balance. The analytical solution for the problem of an isolated bubble in an infinite Hele-Shaw cell is found to agree well with experimental data in the literature. In particular, we find that the bubble may move faster or slower than the background fluid speed, depending on whether $\delta >1$ or $\delta <1$, or precisely with the background flow if $\delta =1$. When the model is generalised to include the effects of multiple bubbles and boundaries in the Hele-Shaw cell, we still find that the sign of $\delta -1$ causes striking changes in the qualitative behaviour. For a train of three or more bubbles moving along a Hele-Shaw channel, we observe longitudinal waves that propagate forwards or backwards along the bubble train, depending on whether $\delta >1$ or $\delta <1$, resembling a Hele-Shaw Newton's cradle.

Laminar miscible displacement flows in narrow, vertical, eccentric annuli show a wide range of interesting physical phenomena, e.g. static layers and channels, dispersive spikes/fronts and various instabilities. These are of relevance to the primary cementing of wells, where gas leakage and greenhouse gas emissions can result from ineffective cementing. The current popular way to model this process is via a Hele-Shaw approach, reducing the Navier–Stokes equations via scaling arguments and then averaging across the annular gap: the two-dimensional (2-D) gap averaged (2DGA) model. This leads to a reduced model that is computationally efficient and represents some important features of cementing flows, but is also deficient in capturing the effects of dispersion. While three-dimensional (3-D) simulations are able to capture dispersion and most other physical phenomena, they suffer from excessive computational times due to the extreme aspect ratios of cementing geometries and non-Newtonian properties of the fluids. Here we extend the 2DGA approach by modelling the high Péclet number limit of miscible duct displacement flows, to take into account the leading-order effects of dispersion. Our modified dispersive 2DGA model is well able to approximate dispersive effects that are observed by gap-averaging the results of 3-D computations, including geometric effects of eccentricity and instabilities. This represents a marked improvement over the 2DGA approach, while keeping the computational advantages of a 2-D model.

A numerical insight into the radial displacement of two viscously stable reactants undergoing an infinitely fast chemical reaction is obtained. This work broadens the numerical knowledge about the interaction between chemical reaction and hydrodynamic instability. A suitable transformation is utilised to deal with an infinite dimensionless parameter and reduce computational cost. Viscous fingering instability originates when the product has a different viscosity than the reactants. We calculate the onset time $t_{on}$ when the instability begins to appear for different reactants by varying the log mobility ratio $R_c$ and the Péclet number $Pe$. Based on $t_{on}$, we divide the time domain into stable and unstable zones with respect to instability. This helps to characterise reactants so as to have a flow with/without instability. For a fixed $Pe$ and $\vert R_c \vert$, it is found that $t_{on}$ is early for $R_c > 0$ in contrast to the result for rectilinear displacement. This results in a wider stable zone for $R_c < 0$. Interestingly, it is found that the stable zone can be made independent of $Pe$ by using a careful rescaling and we obtain the dependence of the onset time of instability on $R_c$ and $Pe$ as $t_{on} \propto (\vert R_c\vert Pe^{\beta _1}-\beta _2)^{\beta _3}$ where the constant of proportionality and $\beta _i$, $i=1,2,3$, depend upon the sign of $R_c$. In the unstable zone, we find that the length of the fingers depends upon the sign of $R_c$, which is not observed for any radial displacement of reactants undergoing a slow or moderately fast chemical reaction.

The rise of a single bubble confined between two vertical plates is investigated over a wide range of Reynolds numbers. In particular, we focus on the evolution of the bubble speed, aspect ratio and drag coefficient during the transition from the viscous to the inertial regime. For sufficiently large bubbles, a simple model based on power balance captures the transition for the bubble velocity and matches all the experimental data despite strong time variations of bubble aspect ratio at large Reynolds numbers. Surprisingly, bubbles in the viscous regime systematically exhibit an ellipse elongated along its direction of motion while bubbles in the inertia-dominated regime are always flattened perpendicularly to it.