This study explores the application of a wall-attached ferrofluid film to decrease skin-friction drag in turbulent channel flow. We conduct experiments using water as a working fluid in a turbulent channel flow set-up, where one wall is coated with a ferrofluid layer held in place by external permanent magnets. Depending on the flow conditions, the interface between the two fluids is observed to form unstable travelling waves. While ferrofluid coating has been previously employed in laminar and moderately turbulent flows (Reynolds number $Re<4000$) to reduce drag by creating a slip condition at the fluid interface, its effectiveness in fully developed turbulent conditions, particularly when the interface exhibits instability, remains uncertain. Our primary objective is to assess the effectiveness of ferrofluid coating in reducing turbulent drag with particular focus on scenarios when the ferrofluid layer forms unstable waves. To achieve this, we measure flow velocity using two-dimensional particle tracking velocimetry (2D-PTV), and the interface contour between the fluids is determined using an interface tracking algorithm. Our results reveal the significant potential of ferrofluid coating for drag reduction, even in scenarios where the interface between the surrounding fluid and ferrofluid exhibits instability, with observed drag reduction rates up to 95 %. In particular, waves with an amplitude significantly smaller than a viscous length scale positively contribute to drag reduction, while larger waves are detrimental, because of induced turbulent fluctuations. However, for the latter case, slip outcompetes the extra turbulence so that drag is still reduced.

Magnetic soap films under the forcing of an inhomogeneous magnetic field are governed by a wide range of interconnected physics. The study of magnetic soap films requires the development of comprehensive models to support experimental observations. In this study, the thin film approximation is applied to the Navier–Stokes equations to derive a model for the film thickness of magnetic soap films that incorporates the effects of interfacial mobility, surfactant transport and magnetite nanoparticle (NP) transport. This derived model consists of a coupled system of equations for the film thickness, interfacial velocity, interfacial surfactant concentration and magnetite NP concentration. Simulations are performed for both soap films and magnetic soap films by solving the system of equations using the Galerkin finite element method, and results are compared with experiments. Simulation results highlight that interfacial flows can dominate the rate of film thinning and that accounting for the dependence of the magnetisation on the local magnetite NP concentration can influence the predicted speed of magnetically driven flows. Furthermore, simulation results demonstrate that the model is able to predict marginal regeneration in qualitative agreement with the experiments for soap films; the model also predicts the same flow pattern as seen in the experiments for magnetic soap films. Overall, this study advances the state of soap film and magnetic soap film modelling and will contribute to acquiring control over the drainage and stability of magnetic soap films in the long term.

A Lagrangian approach to both hydrostatic non-dispersive in the short-wave range and non-hydrostatic dispersive rotating shallow-water magnetohydrodynamics is developed, and used to analyse weakly and fully nonlinear waves described by the model. Hyperbolic structure in the non-dispersive case is displayed and Riemann invariants are constructed. Characteristic equations are used to establish criteria of breaking and formation of shocks by magneto-gravity waves, and conditions of the appearance of contact discontinuities in Alfvén waves. As in the case of non-magnetic rotating shallow water, rotation cannot prevent breaking. The Lagrangian equations of the model are reduced to a single partial differential ‘master’ equation, which is used to analyse the propagation of weakly nonlinear waves of both families, with or without weak rotation, and with or without weak short-wave dispersion. Corresponding modulation equations are constructed and their main properties sketched. The same master equation is used to obtain fully nonlinear finite-amplitude wave solutions in particular cases of no short-wave dispersion or no rotation.

We examined the influences of slip parameters on the velocity and thermal characteristics of a ferrofluid film of fixed thickness. The flow is generated on a rough and inclined whirling surface that is positioned in an external magnetic (dipole) field. The similarity transformation reduces the model equations (continuity, momentum, energy and concentration), and the solution of the normalized coupled ordinary differential equations is carried out through the finite element process. The influences of slip effects, Brownian motion, thermophoresis and a heat source on the velocity (radial, tangential and axial), gravity (drainage, induced), temperature profile and concentration profile are determined. The tangential flow and temperature are both decreased by an increase in the velocity slip parameter, whereas drainage, induced, radial and axial flows are increased. Enlarging the thermal slip parameter decreases the temperature. Improving slip parameters (velocity and thermal) also improves the concentration profile. Both Nusselt and Sherwood numbers are found to improve on improving the velocity slip parameter, while they decrease on decreasing the thermal slip parameter. The results and insights from this work could be applied to a wide range of medicinal fields, such as targeted medication therapy and delivery, tissue engineering, etc. as well as different industrial processes including coating, lubrication, heat transfer, etc.

Flow generation by colloidal motors activated by external stimuli is an important issue for active matter physics and several nanotechnological or biomedical applications. For instance, flow recirculation generated by rotating magnetic self-assemblies allows effective ‘pumping’ of a thrombolytic drug towards a blood clot along a blocked vessel. However, the physics of the flow generation in this case remains still poorly explored. This study is focused on the generation of a recirculation flow of a magnetic colloid (aqueous suspension of iron oxide nanoparticles with partially screened electrostatic repulsion) within a closed microfluidic channel via application of an external rotating magnetic field. The colloid undergoes reversible phase separation manifested through the appearance of micron-sized elongated aggregates. They synchronously rotate with the magnetic field and can generate macroscopic flows only in the presence of gradients of the aggregate concentration across the channel induced by superposition of a weak magnetic field gradient to the homogeneous rotating field. We achieve recirculation flows with a characteristic speed ${\sim} 5{-}8\;{\rm \mu}\textrm{m}\;{\textrm{s}^{ - \textrm{1}}}$ at low magnetic field amplitude and frequency (${H_0} \approx 3{-}10\;\textrm{kA}\;{\textrm{m}^{ - 1}}$, ${f = 5{-}15\ \textrm{Hz}}$) at low nanoparticle volume fraction ${\varphi _p} = (1.6{-}3.2) \times {10^{ - 3}}$. The concentration and velocity profiles have been assessed experimentally through particle tracking and particle image velocimetry, and have also been computed using the hydrodynamic diffusion approach coupled with the momentum balance equation with a magnetic torque term. The model correctly reproduces the shape of the experimental concentration and velocity fields and explains complex behaviours of the average recirculation speed as a function of governing parameters (${H_0}$, f, ${\varphi _p}$, channel size).

We consider the dynamics of a set of reduced equations describing the evolution of a magnetised, rotating stably stratified fluid layer, atop a stagnant dense, perfectly conducting layer. We consider two closely related models. In the first, the layer has, above it, relatively light fluid where the magnetic pressure is much larger than the gas pressure, and the magnetic field is largely force-free. In the second model, the magnetic field is constrained to lie within the dynamical layer by the implementation of a model diffusion operator for the magnetic field. The model derivation proceeds by assuming that the horizontal velocity and the horizontal magnetic field are independent of the vertical coordinate, whilst the vertical components in the layer have a linear dependence on height. The full system comprises evolution equations for the magnetic field, horizontal velocity and height field together with a linear elliptic equation for the vertically integrated non-hydrostatic pressure. In the magneto-hydrostatic limit, these equations simplify to equations of shallow-water type. Numerical solutions for both models are provided for the fiducial case of a Gaussian vortex interacting with a magnetic field. The solutions are shown to differ negligibly. We investigate how the interaction of the vortex changes in response to the magnetic Reynolds number ${Rm}$, the Rossby deformation radius $L_D$, and a Coriolis buoyancy frequency ratio $f/N$ measuring the significance of non-hydrostatic effects. The magneto-hydrostatic limit corresponds to $f/N\to 0$.

This study presents the results of a numerical simulation of two horizontal flows of miscible magnetic and non-magnetic fluids at low Reynolds numbers in a vertical uniform magnetic field. The problem is solved by taking into account the dependence of the viscosity and magnetization of the fluid on the concentration of the magnetic phase, and the dependence of the magnetic field on the concentration. Four flow modes are found: the diffusion mixing mode with a flat diffusion front, the wave mode and two different plug flow modes. In the first of them, the growing wave instability forms the plugs, whereas in the second, the growing magnetostatic instability does. A combination of dimensionless criteria is found that determines the transition from one mode to another. The dependences of the phase velocity of the waves on the diffusion front and the period of the oscillations of the front near the point of the confluence of the two flows on dimensionless criteria are found.

This paper studies the instability of two-dimensional magnetohydrodynamic systems on a sphere using analytical methods. The underlying flow consists of a zonal differential rotation and a toroidal magnetic field is present. Semicircle rules that prescribe the possible domain of the wave velocity in the complex plane for general flow and field profiles are derived. The paper then sets out an analytical study of the ‘clamshell instability’, which features field lines on the two hemispheres tilting in opposite directions (Cally, Sol. Phys., vol. 199, 2001, pp. 231–249). An asymptotic solution for the instability problem is derived for the limit of weak shear of the zonal flow, via the method of matched asymptotic expansions. It is shown that when the zonal flow is solid body rotation, there exists a neutral mode that tilts the magnetic field lines, referred to as the ‘tilting mode’. A weak shear of the zonal flow excites the critical layer of the tilting mode, which reverses the tilting direction to form the clamshell pattern and induces the instability. The asymptotic solution provides insights into properties of the instability for a range of flow and field profiles. A remarkable feature is that the magnetic field affects the instability only through its local behaviour in the critical layer.

We study swirling electrovortex flows in a cylinder filled with GaInSn metal using axisymmetric and large-scale three-dimensional numerical simulations. In our set-up electrical currents enter and exit the cell symmetrically through wires and the result is a von Kármán-like flow. Three inductionless and an inductive flow regimes are identified. Scaling laws for the magnitude of the velocity in each of these regimes are obtained both numerically and explained theoretically. We study how the aspect ratio of the cell affects the flow and how symmetrically wired cells are different from asymmetrical wired cells. We vary the radius of the connecting wires and propose a simple model that captures how the flow's intensity varies with the wire radius.

The influence of a magnetic field on the pattern of stationary waves formed on the surface of a magnetic fluid (ferrofluid) when an obstacle moves has been studied both theoretically and experimentally. It is found that a vertical magnetic field narrows the cone of stationary waves and increases their amplitude. In the wake region, the peaks of the Rosensweig instability appear in a magnetic field that is smaller than the critical field that determines this instability occurrence. A horizontal magnetic field parallel to the obstacle velocity expands the cone of waves but reduces their amplitude up to the suppression of stationary waves. A horizontal field perpendicular to the obstacle velocity also expands the cone of waves and stabilizes their amplitude.

We report the breakup dynamics of a magnetically active (ferrofluid) droplet in a T-shaped Lab on a Chip (LOC) device under the modulation of a non-uniform magnetic field. We adhere to high-speed imaging modalities for the experimental quantification of the droplet splitting phenomenon, while the underlying phenomenon is supported by the numerical results in a qualitative manner as well. On reaching the T-junction divergence, the droplet engulfs the intersection fully and eventually deforms into the dumbbell-shaped form, making its bulges move towards the branches of the junction. We observe that the asymmetric distribution of the magnetic force lines, acting over the T-junction divergence, induces an accelerating motion to the left of the moving bulge (since the magnet is placed adjacent to the left branch). We show that the non-uniform force field gradient allows the formation of a hump-like structure inside the left moving bulge, which triggers the onset of augmented convection in its flow field. We reveal that this augmented internal convection developed in the left moving volume/bulge, on becoming coupled to the various involved time scales of the flow field, leads to the asymmetric splitting of the droplet into two sister droplets. Our analysis establishes that, at the critical strength of the applied forcing, as realized by the critical magnetic Bond number, the flow time scale becomes minimum at the left branch of the channel, leading to the formation of larger sized sister droplets therein. Inferences of the present analysis, which demonstrates a plausible means of independently controlling the size of the sister droplet by manoeuvring the applied force field gradient, will provide a potential solution for rapid droplet splitting, which typically finds significant importance in point-of-care diagnostics.

Investigated first is the linear stability of a Newtonian ferrofluid centred on a rigid wire, surrounded by another ferrofluid with a different magnetic susceptibility. An electric current runs through the wire, generating an azimuthal magnetic field that produces a magnetic stress at the interface of the fluids. Three-dimensional disturbances to the system are considered, and the linearised Navier–Stokes equations are solved analytically in terms of an implicit expression for the growth rate of the disturbance. The growth rate is found numerically for arbitrary Reynolds number, and given explicitly in the inviscid and Stokes regimes. Investigated next is a ferrofluid whose magnetic susceptibility varies radially, centred on a rigid wire, subject to a non-uniform azimuthal field. It is proven that if the gradient of the susceptibility is positive anywhere in the fluid, then the system is linearly unstable. Moreover, it is proven that applying an axial field can stabilise disturbances for both continuous and discontinuous susceptibilities.

We performed a linear stability analysis of a Newtonian ferrofluid cylinder surrounded by a Newtonian non-magnetic fluid in an azimuthal magnetic field. A wire is used at the centre of the ferrofluid cylinder to create this magnetic field. Isothermal conditions are considered and gravity is ignored. An axisymmetric perturbation is imposed at the interface between the two fluids and a dispersion relation is obtained allowing us to predict whether the flow is stable or unstable with respect to this perturbation. This relation is dependent on the Ohnesorge number of the ferrofluid, the dynamic viscosity ratio, the density ratio, the magnetic Bond number, the relative magnetic permeability and the dimensionless wire radius. Solutions to this dispersion relation are compared with experimental data from Arkhipenko et al. (Fluid Dyn., vol. 15, issue 4, 1981, pp. 477–481) and, more recently, Bourdin et al. (Phys. Rev. Lett., vol. 104, issue 9, 2010, 094502). A better agreement than the inviscid theory and the theory that only takes into account the viscosity of the ferrofluid is shown with the data of Arkhipenko et al. (Fluid Dyn., vol. 15, issue 4, 1981, pp. 477–481) and those of Bourdin et al. (Phys. Rev. Lett., vol. 104, issue 9, 2010, 094502) for small wavenumbers.

The present study deals with the effect of micromagnetorotation (MMR) on a micropolar Poiseuille flow in the presence of a uniform magnetic field. Micromagnetorotation is associated with the impact of magnetization on magnetohydrodynamic (MHD) micropolar flows. Previously, magnetization was assumed to be parallel to the applied magnetic field and thus, its influence on the flow was ignored. This assumption is incorrect in the case of micropolar fluids, because their anisotropy affects magnetization. Here, the velocity and microrotation fields, as well as the skin friction coefficient are examined analytically by using a new MHD micropolar fluid theory that includes a constitutive equation for magnetization. Results reveale that MMR has a strong braking effect both on velocity and microrotation. Flow deceleration is found to be up to 16 %, while an increase in the skin friction coefficient is also observed. Moreover, the stability of the MHD micropolar flow is studied by introducing a modified version of the Orr–Sommerfeld equation, which incorporates MMR. The eigenvalue problem is solved with the use of the open-source Chebfun library. It is found that the MMR has a strong stabilizing effect on the MHD micropolar flow. Thus, the MMR is proved to be a mechanism similar to the Lorentz force, which dissipates additional magnetic energy to the flow via microrotation. In summary, the important effect of MMR, neglected by researchers so far, should be considered for industrial and bioengineering applications that involve micropolar fluids and magnetic fields.

We report experimental investigations on the mixing of a ferrofluid droplet with a non-magnetic miscible fluid in the presence of a time-dependent magnetic field on an open surface microfluidic platform. The bright-field visualization technique, in combination with micro-particle image velocimetry analysis, is carried out to explore the internal hydrodynamics of the ferrofluid droplet. Also, using the laser-induced fluorescence technique, we quantify the mass transfer occurring between the two droplets, which in effect, determines the underlying mixing performance under the modulation of the frequency of the applied magnetic field. We show that the magnetic nanoparticles exhibit complex spatio-temporal movements inside the ferrofluid droplet domain in a transient magnetic forcing environment, which, in turn, promotes the mixing efficiency in the convective mixing regime. Our analysis establishes that the movement of magnetic nanoparticles in the presence of the time-periodic field strengthens the flow instability, which initiates an augmented mixing in the present scenario. By performing numerical simulations, we also review the onset of instability phenomena, mainly stemming from the susceptibility mismatch between the magnetic and non-magnetic fluids. Inferences of the present analysis, which focuses on the simple, wireless, robust and low-cost open surface micromixing mechanism, will provide a potential solution for rapid droplet mixing without requiring a pH level or ion concentration dependency of the fluids.

We investigate the stability of a Newtonian ferrofluid cylinder under a steady magnetic field. Linear stability analysis is performed by imposing a small-amplitude axisymmetric disturbance to the basic state. Admissible magnetic field shapes are found and dispersion relations are obtained for two general shapes: axial and non-axial. For the latter, a wire is needed to avoid a singularity at the cylinder centre. Solutions are plotted for two particular shapes previously discussed in the literature for an inviscid ferrofluid: axial and azimuthal. The applicability of the developed theory to cylinder and jet experiments using a solenoid and a wire carrying a current to create the axial and azimuthal magnetic fields, respectively, is studied. For both cases, results show that the magnetic field has a stabilising effect, as already evidenced. In addition, by solving the equation for the cutoff wavenumber, we show that, unlike the azimuthal case, the axial case cannot be stable for all wavenumbers. Taking into account the ferrofluid viscosity better explains the experimental results on ferrofluid cylinders in an azimuthal magnetic field. By choosing a specific admissible magnetic fields shape, the range of wavenumbers in which the cylinder is unstable can be controlled. A radial magnetic field is one example studied here without providing any details on the means to create the field. This latter case reveals the potential of magnetic fields to control the drop size during the ferrofluid jet breakup. These results may be of interest for applications in the printing and medical fields.

The behaviour of the diffusion front of a magnetic fluid in contact with a miscible non-magnetic fluid in a normal magnetic field is studied. It was found that the magnetic field is a cause of the diffusion front bending and its movement accompanied by intense advective flows. These flows lead to the fast growth of the wavy shape of the diffusion front and formation of the peaks. This phenomenon is studied both numerically and experimentally. The reasons for the instability of the diffusion front in a magnetic field are discussed. The influence of the parameters of the problem (Schmidt number, magnetic Rayleigh number, magnetic field, the thickness of the layer, diffusion front width, etc.) on the instability parameters is studied both numerically and experimentally. It is shown that the studied instability differs from Rosensweig's instability.

We present an Euler–Lagrange approach for simulating magneto-Archimedes separation of almost neutrally buoyant spherical particles in the flow of a paramagnetic liquid, which is of direct relevance for separating different types of plastic by magnetic density separation. A four-way coupled point-particle method is employed where all relevant interactions between an external magnetic field, a magnetic fluid and discrete immersed particles are taken into account. Particle–particle interaction is modelled by a hard-sphere collision model which takes the interstitial fluid effects into account. First, the motion of rigid spherical particles in a paramagnetic liquid is studied in single- and two-particle systems. We find good agreements between our numerical results and experiments performed in a paramagnetic liquid exposed to a non-homogeneous magnetic field, also in the case of two colliding particles. Next, we investigate the magneto-Archimedes separation of particles with different mass densities in many-particle systems interacting with the fluid. Our results reveal that history effects and interparticle interactions significantly influence the levitation dynamics of particles and have a detrimental impact on the separation performance. We also investigate the effect of particle size and initial distribution on the separation performance. Results show that a reduction in the particle size from 4 to 2 mm leads to a $40\,\%$ increase in the separation time. Moreover, preseparation of particles into two groups of light and heavy particles decreases the separation time by $33\,\%$. The presented method is shown to be a robust and efficient computational framework for the investigation of particle-laden flows of magnetically responsive fluids.

Finite-amplitude hydromagnetic Rossby waves in the magnetostrophic regime are studied. We consider the slow mode, which travels in the opposite direction to the hydrodynamic or fast mode, in the presence of a toroidal magnetic field and zonal flow by means of quasi-geostrophic models for thick spherical shells. The weakly nonlinear long waves are derived asymptotically using a reductive perturbation method. The problem at the first order is found to obey a second-order ordinary differential equation, leading to a hypergeometric equation for a Malkus field and a confluent Heun equation for an electrical wire field, and is non-singular when the wave speed approaches the mean flow. Investigating its neutral non-singular eigensolutions for different basic states, we find the evolution is described by the Korteweg–de Vries equation. This implies that the nonlinear slow wave forms solitons and solitary waves. These may take the form of a coherent eddy, such as a single anticyclone. We speculate on the relation of the anticyclone to the asymmetric gyre seen in the Earth's fluid core, and in state-of-the-art dynamo direct numerical simulations.

We measure the Kelvin–Helmholtz instability in between a layer of a diamagnetic fluid flowing in a channel and a layer of ferrofluid resting on top. When the diamagnetic fluid exceeds a critical flow velocity the interface in between both fluids becomes unstable and waves develop. It has been predicted by Sutyrin & Taktarov (J. Appl. Math. Mech., vol. 39, 1975, pp. 520–524) that a homogeneous magnetic field, oriented horizontally, stabilizes the liquid interface. To test this prediction we apply in a closed flow channel a local periodic perturbation of the interface by magnetic or mechanic means. From the measured growth and decay rates of the interface undulations we determine the critical flow velocity for various driving frequencies and applied magnetic fields. In this way we confirm quantitatively the stabilizing effect of the horizontal field. Moreover we measure the dispersion relation of the interfacial waves.