The flow of liquids through paper is challenging to model due to the complexity and disordered layout of the fibre matrix. The expanding use and capability of microfluidic paper-based analytical devices ($\unicode[STIX]{x1D707}$PADs) and their requirement for precision has increased the need to accurately predict the flow of liquids through paper. Many studies have developed models and revealed some of the physical mechanisms responsible for the flow behaviour, but we still lack a complete understanding, particularly in relation to how the fluid fills the various voids with a wide range of shapes and sizes in the fibre matrix of paper. In the featured article, Chang et al. (J. Fluid Mech., vol. 845, 2018, 36–50) used a combined experimental and theoretical approach to uncover the importance of the liquid filling the intra-fibre pores, showed that this results in deviation from the flow behaviour predicted by the Lucas–Washburn equation and developed a model which accounts for this effect.

Certain geological features have been interpreted as evidence of channelized magma flow in the mantle, which is a compacting porous medium. Aharonov et al. (J. Geophys. Res., vol. 100 (B10), 1995, pp. 20433–20450) developed a simple model of reactive porous flow and numerically analysed its instability to channels. The instability relies on magma advection against a chemical solubility gradient and the porosity-dependent permeability of the porous host rock. We extend the previous analysis by systematically mapping out the parameter space. Crucially, we augment numerical solutions with asymptotic analysis to better understand the physical controls on the instability. We derive scalings for the critical conditions of the instability and analyse the associated bifurcation structure. We also determine scalings for the wavelengths and growth rates of the channel structures that emerge. We obtain quantitative theories for and a physical understanding of, first, how advection or diffusion over the reactive time scale sets the horizontal length scale of channels and, second, the role of viscous compaction of the host rock, which also affects the vertical extent of channelized flow. These scalings allow us to derive estimates of the dimensions of emergent channels that are consistent with the geologic record.

Diffusiophoresis describes the motion of colloids in an electrolyte or non-electrolyte solution where there is a concentration gradient. While most of the studies of diffusiophoresis focus on the motion of solid particles, soft objects such as drops and bubbles are also known to experience diffusiophoresis. Here, we investigate the diffusiophoresis of charged drops in an electrolyte solution both analytically and experimentally. The drop is assumed to remain spherical. An analytical solution of the diffusiophoretic velocity of drops is obtained by perturbation methods. We find that the flow inside the drop is driven by the tangential electric stress at the interface and it directly influences the diffusiophoretic speed of the drop. Using charged oil droplets, we measure the drop speed under solute concentration gradients and find good agreement with the analytical solution. Our findings have potential applications for oil recovery and drug delivery.

Drop flow in rectangular microchannels has been utilized extensively in microfluidics. However, the pressure-gradient versus flow-rate relation is still not well understood. We study the motion of a long drop in a rectangular microchannel in the limit the capillary number $Ca\rightarrow 0$ ($Ca=\unicode[STIX]{x1D707}U/\unicode[STIX]{x1D70E}$, where $U$ is the constant drop velocity, $\unicode[STIX]{x1D707}$ is the viscosity of the carrier liquid and $\unicode[STIX]{x1D70E}$ is the interfacial tension). In this limit, the moving drop looks like the static drop and has two end caps connected by a long column, which is surrounded by thin films on the microchannel wall and by menisci along the microchannel corners. Integral axial force balances on the drop fluid and on the carrier liquid surrounding the drop relate the carrier-liquid pressure gradient to the drop-fluid pressure gradient and the contact-line drag. The contact-line drag is argued to be the same as that for a long bubble (which has been determined by Wong et al. (J. Fluid Mech., vol. 292, 1995b, pp. 95–110)) if the viscosity ratio $\unicode[STIX]{x1D706}\ll Ca^{-1/3}$ and $\unicode[STIX]{x1D706}\ll L$, where $\unicode[STIX]{x1D706}=\bar{\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D707}$, $\bar{\unicode[STIX]{x1D707}}$ is the drop viscosity and $L~(\gg 1)$ is the dimensionless drop length. Thus, the force balances yield one equation relating the two pressure gradients. The two pressure gradients also drive unidirectional flows in the drop and in the corner channels along the long middle column. These coupled flows are solved by a finite-element method to yield another equation relating the two pressure gradients. From the two equations, we determine the pressure gradients and thus the unidirectional velocity fields inside and outside the drop for $\unicode[STIX]{x1D706}=0$–100 and various microchannel aspect ratios. We find that in the limit $LCa^{1/3}\rightarrow 0$, the contact-line drag dominates and the carrier liquid bypasses the drop through the corner channels alongside the drop. For $LCa^{1/3}\gg 1$, the contact-line drag is negligible and the corner fluid is stationary. Thus, the drop moves as a leaky piston. We extend our model to a train of long drops, and compare our model predictions with published experiments.

The dynamics of bubbles inertially collapsing in water near solid objects have been the subject of numerous studies in the context of cavitation erosion. While non-spherical bubble collapse, re-entrant jet dynamics and emitted shock waves have received significant interest, less is known about the temperatures thereby produced and their possible connection to damage. In this article, we use highly resolved numerical simulations of a single bubble inertially collapsing near a rigid surface to measure the temperatures produced in the fluid and estimate those in the solid, as well as to identify the responsible mechanisms. In particular, we find that elevated temperatures along the wall can be produced by one of two mechanisms, depending on the initial stand-off distance of the bubble from the wall and the driving pressure: for bubbles initially far from the wall, the shock generated by the bubble collapse is the source of the high temperature, while bubbles starting initially closer migrate towards the wall and eventually come into contact with it. A scaling is introduced to describe the maximum fluid temperature along the wall as a function of the initial stand-off distance and driving pressure. To predict the temperature of the solid, we develop a semianalytical heat transfer model, which supports recent experimental observations that elevated temperatures achieved during collapse could play a role in cavitation damage to soft heat-sensitive materials.

This paper presents a two-dimensional (2-D) cavity-by-cavity description of a convective instability near a lined wall with low dissipation due to the coupling of hydrodynamic modes with resonance of the wall. For a liner consisting of an array of deep cavities periodically placed along a duct containing a mean shear flow, the acoustic and hydrodynamic disturbances are described by the linearized Euler equations. The Bloch modes and the scattering matrix of periodic cells are used to examine the instability over the liner. The unstable Bloch mode is due to the coupling of a hydrodynamic mode in the shear flow with the cavity resonance. It is demonstrated that even when all the transverse modes are stable in the duct–cavity system, i.e. when the Kelvin–Helmholtz instability of the shear flow over the cavities does not occur, such an instability over the liner can still exist. The unstable Bloch wave, excited by the incident sound wave at the upstream part of the liner, convectively grows along the liner, and regenerates sound near the downstream edge of the liner with a sound level higher than the incident sound level. It is shown that a homogenized approach, where the wall effect is described by a homogeneous impedance, can also explain the unstable behaviour above the liner. It reveals that a small wall resistance and a small and positive reactance are two necessary conditions for such an instability.

In this investigation, we use a simple model of the dynamics of an inviscid vortex sheet given by the Birkhoff–Rott equation to obtain fundamental insights about the potential for stabilization of shear layers using feedback control. As actuation, we consider two arrays of point sinks/sources located a certain distance above and below the vortex sheet and subject to the constraint that their mass fluxes separately add up to zero. First, we demonstrate using analytical computations that the Birkhoff–Rott equation linearized around the flat-sheet configuration is in fact controllable when the number of actuator pairs is sufficiently large relative to the number of discrete degrees of freedom present in the system, a result valid for generic actuator locations. Next, we design a state-based linear-quadratic regulator stabilization strategy, where the key difficulty is the numerical solution of the Riccati equation in the presence of severe ill-conditioning resulting from the properties of the Birkhoff–Rott equation and the chosen form of actuation, an issue that is overcome by performing computations with a suitably increased arithmetic precision. Analysis of the linear closed-loop system reveals exponential decay of the perturbation energy and the corresponding actuation energy in all cases. Computations performed for the nonlinear closed-loop system demonstrate that initial perturbations of non-negligible amplitude can be effectively stabilized when a sufficient number of actuators is used. We also thoroughly analyse the sensitivity of the closed-loop stabilization strategies to the variation of a number of key parameters. Subject to the known limitations of inviscid vortex models, our findings indicate that, in principle, it may be possible to stabilize shear layers for relatively large initial perturbations, provided that the actuation has sufficiently many degrees of freedom.

The pinch-off of a capillary thread is studied at large Ohnesorge number for non-Brownian, neutrally buoyant, mono-disperse, rigid, spherical particles suspended in a Newtonian liquid with viscosity $\unicode[STIX]{x1D702}_{0}$ and surface tension $\unicode[STIX]{x1D70E}$. Reproducible pinch-off dynamics is obtained by letting a drop coalesce with a bath. The bridge shape and time evolution of the neck diameter, $h_{\mathit{min}}$, are studied for varied particle size $d$, volume fraction $\unicode[STIX]{x1D719}$ and liquid contact angle $\unicode[STIX]{x1D703}$. Two successive regimes are identified: (i) a first effective-viscous-fluid regime which only depends upon $\unicode[STIX]{x1D719}$ and (ii) a subsequent discrete regime, depending both on $d$ and $\unicode[STIX]{x1D719}$, in which the thinning localises at the neck and accelerates continuously. In the first regime, the suspension behaves as an effective viscous fluid and the dynamics is solely characterised by the effective viscosity of the suspension, $\unicode[STIX]{x1D702}_{e}\sim -\unicode[STIX]{x1D70E}/{\dot{h}}_{\mathit{min}}$, which agrees closely with the steady shear viscosity measured in a conventional rheometer and diverges as $(\unicode[STIX]{x1D719}_{c}-\unicode[STIX]{x1D719})^{-2}$ at the same critical particle volume fraction, $\unicode[STIX]{x1D719}_{c}$. For $\unicode[STIX]{x1D719}\gtrsim 35\,\%$, the thinning rate is found to increase by a factor of order one when the flow becomes purely extensional, suggesting non-Newtonian effects. The discrete regime is observed from a transition neck diameter, $h_{\mathit{min}}\equiv h^{\ast }\sim d\,(\unicode[STIX]{x1D719}_{c}-\unicode[STIX]{x1D719})^{-1/3}$, down to $h_{\mathit{min}}\approx d$, where the thinning rate recovers the value obtained for the pure interstitial fluid, $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D702}_{0}$, and lasts $t^{\ast }\sim \unicode[STIX]{x1D702}_{e}h^{\ast }/\unicode[STIX]{x1D70E}$.

This study is motivated by the need to develop stochastic parameterisations for representing the effects of mesoscale oceanic eddies in non-eddy-resolving and eddy-permitting ocean circulation models. A necessary logical step on the way to such parameterisations is the understanding of flow responses to spatially stationary and localised, time-dependent ‘plunger’ forcings intended to represent transient eddy flux divergences. Specifically, this study develops an understanding of the plunger-induced convergence of potential vorticity (PV) fluxes using the linearised single-layer shallow-water model. Time-periodic solutions are obtained and the ‘footprint’, defined as the time-mean, quasi-linear PV flux convergence, quantifies the cumulative PV redistribution induced by the plunger. Using the footprint, the equivalent eddy flux (EEF) is defined such that it succinctly quantifies the extent of the PV redistribution, and its dependencies on the forcing latitude and the background flow are examined in detail. For a uniform background flow the EEF is positive for all forcing latitudes, corresponding to net-poleward PV flux convergence, as expected by current theory of $\unicode[STIX]{x1D6FD}$-plane Rossby waves. The EEF also has a robust dependence on the direction and magnitude of a uniform background flow, which is a useful quality for the EEF to provide a basis for a parameterisation of eddy PV fluxes. We also examine the PV redistribution due to forcing on top of a Gaussian jet background flow and find that forcing proximity to the jet core is the primary factor in determining whether the jet is sharpened or broadened.

This paper presents a numerical investigation of oscillatory flow around a circular cylinder that is placed in proximity to a plane boundary that is parallel to the cylinder axis. The onset and development of the Honji instability are studied over a range of Stokes numbers ($\unicode[STIX]{x1D6FD}$) and gap-to-diameter ratios ($e/D$) at a fixed Keulegan–Carpenter number ($KC=2$). Four flow regimes are identified in the ($e/D,\unicode[STIX]{x1D6FD}$)-plane: (I) featureless two-dimensional flow, (II) stable Honji vortex, (III) unstable Honji vortex and (IV) chaotic flow. As $e/D$ increases from $-0.5$ (embedment) to $1$, the critical Stokes number $\unicode[STIX]{x1D6FD}_{cr}$ for the onset of the Honji instability follows two side-by-side convex functions, peaking at the connection point of $e/D=0.125$ and reaching troughs at $e/D=0$ and 0.375. The Honji instability is always initiated on the gap side of the cylinder surface for $0.375\leqslant e/D\leqslant 2$ and occurs only on the top side for $-0.5\leqslant e/D<0.125$. The location for the initiation of the Honji instability switches from the gap side to the top side of the cylinder surface for $0.125<e/D<0.375$. No Honji instability is observed at $e/D=0.125$, where the flow three-dimensionality is developed through a different flow mechanism. Consistently, the three-dimensional kinetic energy of the flow, which represents a measure of the strength of flow three-dimensionality, varies with $e/D$ in a trend opposite to that of $\unicode[STIX]{x1D6FD}_{cr}$. Three physical mechanisms are identified as being responsible for the observed variation trend of $\unicode[STIX]{x1D6FD}_{cr}$ with $e/D$ and for various flow phenomena, which are the blockage effect induced by the geometry setting, the existence of the Stokes layer on the plane boundary and the favourable pressure gradient in the flow direction over the gap between the cylinder and the plane surface.

Direct numerical simulations of turbulent suspension flows are carried out with the force-coupling method in plane Couette and pressure-driven channel configurations. Dilute to moderately concentrated suspensions of neutrally buoyant finite-size spherical particles are considered when the Reynolds number is slightly above the laminar–turbulent transition. Tests performed with synthetic streaks, in both turbulent channel and Couette flows, show clearly that particles trigger the instability in channel flow whereas the plane Couette flow becomes laminar. Moreover, we have shown that particles have a pronounced impact on pressure-driven flow through a detailed temporal and spatial analysis whereas they have no significant impact on the plane Couette flow configuration. The substantial difference between the two flow configurations is related to the spatial preferential distribution of particles in the large-scale rolls (inactive motion) in Couette flow, whereas they are accumulated in the ejection (active motion) regions in pressure-driven flow. Through investigation of particle modification in two distinct flow configurations, we were able to show the specific response of turbulent structures and the modulation of the fundamental mechanisms composing the regeneration cycle in the buffer layer of the near-wall turbulence. Especially for pressure-driven flow, the particles enhance the lift-up and let it act continuously whereas the particles do not significantly alter the streak breakdown process. The reinforcement of the streamwise vortices is attributed to the vorticity stretching term by the wavy streaks. The smaller and more numerous wavy streaks enhance the vorticity stretching and consequently strengthen the circulation of large-scale streamwise vortices in suspension flow.

We construct a model to investigate the interfacial stability of film boiling, and discover that instability of very thin vapour films and subsequent large interface superheating is only possible if thermocapillary instabilities are present. The model concerns horizontal saturated film boiling, and includes novel features such as non-equilibrium evaporation based on kinetic theory, thermocapillary and vapour thrust stresses and van der Waals interactions. From linear stability analysis applied to this model, we are led to suggest that vapour film collapse depends on a balance between thermocapillary instabilities and vapour thrust stabilization. This yields a purely theoretical prediction of the Leidenfrost temperature. Given that the evaporation coefficient is in the range 0.7–1.0, this model is consistent with the average Leidenfrost temperature of every fluid for which data could be found. With an evaporation coefficient of 0.85, the model can predict the Leidenfrost point within 10 % error for every fluid, including cryogens and liquid metals where existing models and correlations fail.

The statistics of the deformation and breakup of neutrally buoyant sub-Kolmogorov ellipsoidal drops is investigated via Lagrangian simulations of homogeneous isotropic turbulence. The mean lifetime of a drop is also studied as a function of the initial drop size and the capillary number. A vector model of a drop previously introduced by Olbricht et al. (J. Non-Newtonian Fluid Mech., vol. 10, 1982, pp. 291–318) is used to predict the behaviour of the above quantities analytically.

We present a numerical study of non-colloidal spherical and rigid particles suspended in Newtonian, shear thinning and shear thickening fluids employing an immersed boundary method. We consider a linear Couette configuration to explore a wide range of solid volume fractions ($0.1\leqslant \unicode[STIX]{x1D6F7}\leqslant 0.4$) and particle Reynolds numbers ($0.1\leqslant Re_{p}\leqslant 10$). We report the distribution of solid and fluid phase velocity and solid volume fraction and show that close to the boundaries inertial effects result in a significant slip velocity between the solid and fluid phase. The local solid volume fraction profiles indicate particle layering close to the walls, which increases with the nominal $\unicode[STIX]{x1D6F7}$. This feature is associated with the confinement effects. We calculate the probability density function of local strain rates and compare the latter’s mean value with the values estimated from the homogenisation theory of Chateau et al. (J. Rheol., vol. 52, 2008, pp. 489–506), indicating a reasonable agreement in the Stokesian regime. Both the mean value and standard deviation of the local strain rates increase primarily with the solid volume fraction and secondarily with the $Re_{p}$. The wide spectrum of the local shear rate and its dependency on $\unicode[STIX]{x1D6F7}$ and $Re_{p}$ point to the deficiencies of the mean value of the local shear rates in estimating the rheology of these non-colloidal complex suspensions. Finally, we show that in the presence of inertia, the effective viscosity of these non-colloidal suspensions deviates from that of Stokesian suspensions. We discuss how inertia affects the microstructure and provide a scaling argument to give a closure for the suspension shear stress for both Newtonian and power-law suspending fluids. The stress closure is valid for moderate particle Reynolds numbers, $O(Re_{p})\sim 10$.

As a first step towards understanding the distribution of swimming micro-organisms in flowing shallow water containing vegetation, we formulate a continuum model for dilute suspensions in horizontal shear flow, with a maximum Reynolds number of 100, past a single, rigid, vertical, circular cylinder that extends from a flat horizontal bed and penetrates the free water surface. A numerical platform was developed to solve this problem, in four stages: first, a scheme for computation of the flow field; second, a solver for the Fokker–Planck equation governing the probability distribution of the swimming direction of gyrotactic cells under the combined action of gravity, ambient vorticity and rotational diffusion; third, the construction of a database for the mean swimming velocity and the translational diffusivity tensor as functions of the three vorticity components, using parameters appropriate for the swimming alga, Chlamydomonas nivalis; fourth, a solver for the three-dimensional concentration distribution of the gyrotactic micro-organisms. Upstream of the cylinder, the cells are confined to a vertical strip of width equal to the cylinder diameter, which enables us to visualise mixing in the wake. The flow downstream of the cylinder is divided into three zones: parallel vortex shedding in the top zone near the water surface, oblique vortex shedding in the middle zone and quasi-steady flow in the bottom zone. Secondary (vertical) flow occurs just upstream and downstream of the cylinder. Frequency spectra of the velocity components in the wake of the cylinder show two dominant frequencies of vortex shedding, in the parallel- and oblique-shedding zones respectively, together with a low frequency, equal to the difference between those two frequencies, that corresponds to a beating modulation. The concentration distribution is calculated for both active particles and passive, non-swimming, particles for comparison. The concentration distribution is very similar for both active and passive particles, except near the top surface, where upswimming causes the concentration of active particles to reach values greater than in the upstream strip, and in a thin boundary layer on the downstream surface of the cylinder, where a high concentration of active particles occurs as a result of radial swimming.

An important attraction of saline formations for CO2 storage is that their high salinity renders their associated brine unlikely to be identified as a potential water resource in the future. However, high salinity can lead to dissolved salt precipitating around injection wells, resulting in loss of injectivity and well deterioration. Earlier numerical simulations have revealed that salt precipitation becomes more problematic at lower injection rates. This article presents a new similarity solution, which is used to study the relationship between capillary pressure and salt precipitation around CO2 injection wells in saline formations. Mathematical analysis reveals that the process is strongly controlled by a dimensionless capillary number, which represents the ratio of the CO2 injection rate to the product of the CO2 mobility and air-entry pressure of the porous medium. Low injection rates lead to low capillary numbers, which in turn are found to lead to large volume fractions of precipitated salt around the injection well. For one example studied, reducing the CO2 injection rate by 94 % led to a tenfold increase in the volume fraction of precipitated salt around the injection well.

Miscible liquids often come into contact with one another in natural and technological situations, commonly as a drop of one liquid present in a second, miscible liquid. The shape of a liquid droplet present in a miscible environment evolves spontaneously in time, in a distinctly different fashion than drops present in immiscible environments, which have been reported previously. We consider drops of two classical types, pendant and sessile, in building upon our prior work with miscible systems. Here we present experimental findings of the shape evolution of pendant drops along with an expanded study of the spreading of sessile drops in miscible environments. We develop scalings considering the diffusion of mass to group volumetric data of the evolving pendant drops and the diffusion of momentum to group leading-edge radial data of the spreading sessile drops. These treatments are effective in obtaining single responses for the measurements of each type of droplet, where the volume of a pendant drop diminishes exponentially in time and the leading-edge radius of a sessile drop grows following a power law of $t^{1/2}$ at long times. A complementary numerical approach to compute the concentration and velocity fields of these systems using a simplified set of governing equations is paired with our experimental findings.

The aim of this paper is to develop a stability theory and a numerical model for three density-stratified electrically conductive liquid layers. Using regular perturbation methods to reduce the full three-dimensional problem to the shallow layer model, the coupled wave and electric current equations are derived. The problem set-up allows for weakly nonlinear velocity field action and an arbitrary vertical magnetic field. Further linearisation of the coupled equations is used for the linear stability analysis in the case of a uniform vertical magnetic field. New analytical stability criteria accounting for the viscous damping are derived for particular cases of practical interest and compared to the numerical solutions for a variety of materials used in batteries. These new criteria are equally applicable to the aluminium electrolysis cell magnetohydrodynamic (MHD) stability estimates.

Simulations of cylindrical clouds of concentrated, neutrally buoyant, suspended particles are used to investigate the dispersion of the particles in an oscillating Couette flow. In experiments by Metzger & Butler (Phys. Fluids, vol. 24 (2), 2012, 021703) with spherical clouds of non-Brownian particles, the clouds are shown to elongate at volume fraction $\unicode[STIX]{x1D719}=0.4$ but form ‘galaxies’ where the cloud rotates as a single body with extended arms when $\unicode[STIX]{x1D719}>0.4$ and the ratio of the cloud radius to particle radius, $R/a$, is sufficiently large. The simulations, which use the force coupling method, are completed for $\unicode[STIX]{x1D719}=0.4$ and $\unicode[STIX]{x1D719}=0.55$, with $R/a$ between $5$ and $20$. The cloud shape for $\unicode[STIX]{x1D719}=0.4$ is shown to be reversible at low strain amplitude, and extend in the streamwise direction along the centre of the cloud at moderate strain amplitude. For higher strain amplitude the clouds extend near the channel walls to form a parallelogram. The results demonstrate that the particle contact force determines the transition between these states and plays a large role in the irreversibility of the parallelograms. Rotating galaxies form at $\unicode[STIX]{x1D719}=0.55$ with $R/a\geqslant 15$, and are characterized by a particle-induced flow in the wall-normal direction.

A direct numerical simulation (DNS) of a plane wall jet is performed at a Reynolds number of $Re_{j}=7500$. The streamwise length of the domain is long enough to achieve self-similarity for the mean flow and the Reynolds shear stress. This is the highest Reynolds number wall jet DNS for a large domain achieved to date. The high resolution simulation reveals the unsteady flow field in great detail and shows the transition process in the outer shear layer and inner boundary layer. Mean flow parameters of maximum velocity decay, wall shear stress, friction coefficient and jet spreading rate are consistent with several other studies reported in the literature. Mean flow, Reynolds normal and shear stress profiles are presented with various scalings, revealing the self-similar behaviour of the wall jet. The Reynolds normal stresses do not show complete similarity for the given Reynolds number and domain length. Previously published inner layer budgets based on LES are inaccurate and those that have been measured are only available in the outer layer. The current DNS provides fully balanced, explicitly calculated budgets for the turbulence kinetic energy, Reynolds normal stresses and Reynolds shear stress in both the inner and outer layers. The budgets are scaled with inner and outer variables. The inner-scaled budgets in the near wall region show great similarity with turbulent boundary layers. The only remarkable difference is for the turbulent diffusion in the wall-normal Reynolds stress and Reynolds shear stress budgets. The outer layer interacts with the inner layer through turbulent diffusion and the excess energy from the wall-normal direction is transferred to the spanwise direction.