Magnetic Bond Number Research Articles

Deformation of a ferrofluid droplet in simple shear flows under uniform magnetic fields

We numerically investigate the deformation and orientation of a ferrofluid droplet in a simple shear flow under a uniform magnetic field. The numerical simulation is based on the finite element method and couples the magnetic and flow fields. A level set method is used to model the dynamic motion of the droplet interface. Systematic numerical simulations are used to assess the effects of the direction and the strength of the magnetic field. Focusing on low Reynolds number flows (Re ≲ 0.02), the numerical results indicate that at a small capillary number (Ca ≈ 0.02), the magnetic field dominates over the shear flow above a certain magnetic bond number (Bom ≈ 3). The orientation of the droplet is aligned with the direction of the magnetic field, while the deformation of the droplet varies slightly when the direction of the magnetic field is varied. On the other hand, for large capillary numbers (Ca ≈ 0.23), the deformation and orientation of the droplet is influenced by both the shear flow and the magnetic field, except for a small magnetic bond number (Bom ≲ 0.2). In both the small and large capillary number cases, the droplet deformation is found to be maximum at α = 45° (the direction of magnetic field) and minimum at α = 135°. In addition, the effect of the magnetic field on the flow field inside and outside the droplet at different conditions is examined. We demonstrate active control of lateral migration of ferrofluid droplets in wall-bounded simple shear flows. The direction of the lateral migration depends on the orientation of the deformed droplets due to uniform magnetic fields at different directions.

Maximum spreading of a ferrofluid droplet under the effect of magnetic field

This study presents a theoretical and an experimental study of the effects of an applied external magnetic field on the maximum spreading of a ferrofluid droplet impacting on a solid substrate. Although many studies have explored the theoretical modeling of the droplet impact scenario, a theoretical model representing the impact of ferrofluid droplets of different magnetic characteristics, strongly affected by the magnetic field, is yet to be addressed. In this study, we developed a theoretical model based on the principle of the conservation of energy to predict the maximal deformation of both diamagnetic and paramagnetic ferrofluid droplets upon impact under the influence of the magnetic field. The physics behind the variation of maximum drop spread, as a function of Weber number (We), Reynolds number (Re), and magnetic Bond number (Bom) for 5–45, 150–400, and 150–3000, respectively, was studied. By validating the theoretical model with the experimental observations, we demonstrated that the proposed theoretical model could successfully predict experimental observations. Through theoretical analysis and extensive experimental investigations, a rational understanding was formulated which allowed us to comment on the effect of all the governing dimensionless numbers (We, Re, and Bom) on the maximum spreading of a ferrofluid droplet upon impact.

Numerical investigation on falling ferrofluid droplet under uniform magnetic field

In this study, ferrofluid droplet behavior in nonmagnetic viscous fluid under the uniform magnetic field is studied numerically by developing a robust two-dimensional hybrid lattice-Boltzmann/Finite-volume method. The lattice Boltzmann equation (LBE) with the magnetic force term is solved to update the flow field while the magnetic induction equation is solved using the finite volume method to calculate the magnetic field. To validate the flow field solution two test cases have been considered: the falling droplet and Laplace law. In order to validate the magnetic field, deformation of static drop under magnetic field is checked. The comparison of results between present study and previous researches shows a good agreement. The effects of the magnetic Bond number, susceptibility and magnetic field direction on deformation of the falling droplet are also investigated. The results show that the horizontal magnetic field accelerates the falling process of ferrofluid droplet while the vertical magnetic field slows down it by increasing the magnetic Bond number and susceptibility. Also, the increase in the magnetic Bond number and susceptibility enhances the magnitudes of magnetic force along the magnetic field.

Micro-magnetofluidics of ferrofluid droplet formation in a T-junction

The combination of microfluidics with magnetism has led to the emergence of a new scientific domain known as micro-magnetofluidics. The present study aims to investigate the interfacial dynamics of ferrofluid droplet formation under magnetic field in a microfluidic T-junction. A non-uniform magnetic field was constructed by placing a permanent magnet at one side of the T-junction. Three typical flow regimes including the slug, slug-dripping transition and dripping flow were observed. The external magnetic field promotes the transition from the slug-dripping transition flow to the slug flow, prolongs the generation cycle of ferrofluid droplets and hinders the ferrofluid droplet formation. As the magnetic flux density increases, both the increasing rate of the length for the thread tip and the expanding rate of the dispersed neck decrease, and the peak value of the minimum width of the dispersed neck increases. The ferrofluid droplet size increases with increasing the magnetic flux density while decreases with increasing the two-phase flow rate ratio and the capillary number. A scaling law is proposed to predict the size of ferrofluid droplets, by taking the two-phase flow rate ratio, capillary number and magnetic Bond number into account.

Numerical investigation on splitting of ferrofluid microdroplets in T-junctions using an asymmetric magnetic field with proposed correlation

The studies surrounding the concept of microdroplets have seen a dramatic increase in recent years. Microdroplets have applications in different fields such as chemical synthesis, biology, separation processes and micro-pumps. This study numerically investigates the effect of different parameters such as Capillary number, Length of droplets, and Magnetic Bond number on the splitting process of ferrofluid microdroplets in symmetric T-junctions using an asymmetric magnetic field. The use of said field that is applied asymmetrically to the T-junction center helps us control the splitting of ferrofluid microdroplets. During the process of numerical simulation, a magnetic field with various strengths from a dipole located at a constant distance from the center of the T-junction was applied. The main advantage of this design is its control over the splitting ratio of daughter droplets and reaching various microdroplet sizes in a T-junction by adjusting the magnetic field strength. The results showed that by increasing the strength of the magnetic field, the possibility of asymmetric splitting of microdroplets increases in a way that for high values of field strength, high splitting ratios can be reached. Also, by using the obtained results at various Magnetic Bond numbers and performing curve fitting, a correlation is derived that can be used to accurately predict the borderline between splitting and non-splitting zones of microdroplets flow in micro T-junctions.

Drop formation of ferrofluid at co-flowing microcahnnel under uniform magnetic field

A hybrid lattice-Boltzmann, with Shan–Chen model, and finite volume method is applied to simulate the effect of magnetic field on ferrofluid drop formation in Co-flowing configuration of non-magnetic viscous media, in microchannels. The lattice Boltzmann method is used to track the moving interface between the two immiscible phases, and to update the flow field by adding magnetic field force in lattice Boltzmann equations. Maxwell’s equations are solved by finite volume method to obtain the magnetic field. The accuracy of the present method is checked with previous numerical and experimental investigations. The comparison indicates that the present results are in good agreement with previous research in the literature. The effects of the flow rate ratio, magnetic Bond number, and susceptibility on the length of drop, the distance between drops and frequency of drop generation in Co-flowing microchannels are investigated in detail. The results reveal that a magnetic field can be used to control drop size, distance between drops, and drop formation frequency effectively. The outcome of this study further point out, a hybrid lattice-Boltzmann, with Shan–Chen model-finite volume method is an effective way to simulate drop formation in microchannels under the action of magnetic field.

The influence of a magnetic field on the mechanical behavior of a fluid interface

This work focuses on a theoretical investigation of the shape and equilibrium height of a magnetic liquid–liquid interface formed between two vertical flat plates in response to vertical magnetic fields. The formulation is based on an extension of the so called Young–Laplace equation for an incompressible magnetic fluid forming a two-dimensional free interface. A first order dependence of the fluid susceptibility with respect to the magnetic field is considered. The formulation results in a hydrodynamic-magnetic coupled problem governed by a nonlinear second order differential equation that describes the liquid–liquid meniscus shape. According to this formulation, five relevant physical parameters are revealed in this fluid static problem. The standard gravitational Bond number, the contact angle and three new parameters related to magnetic effects in the present study: the magnetic Bond number, the magnetic susceptibility and its derivative with respect to the field. The nonlinear governing equation is integrated numerically using a fourth order Runge-Kutta method with a Newton–Raphson scheme, in order to accelerate the convergence of the solution. The influence of the relevant parameters on the rise and shape of the liquid–liquid interface is examined. The interface shape response in the presence of a magnetic field varying with characteristic wavenumbers is also explored. The numerical results are compared with asymptotic predictions also derived here for small values of the magnetic Bond number and constant susceptibility. A very good agreement is observed. In addition, all the parameters are varied in order to understand how the scales influence the meniscus shape. Finally, we discuss how to control the shape of the meniscus by applying a magnetic field.

Microfluidic magnetic self-assembly at liquid-liquid interfaces.

We present a microfluidic method that controllably self-assembles microparticles into clusters at an aqueous two-phase liquid-liquid interface. The liquid-liquid interface is formed between converging flows of aqueous dextran and polyethylene glycol, in a microfluidic cross-slot device. We control the size of the self-assembled particle clusters as they pass through the liquid-liquid interface, by systematically varying the applied magnetic field gradient, and the interfacial tension of the liquid-liquid interface. We find that upon penetration through the interface, the number of particles within a cluster increases with increasing interfacial tension, and decreasing magnetic field gradient. We also develop a scaling model of the number of particles within a cluster, and observe an inverse scaling of the number of particles within a cluster with the dimensionless magnetic Bond number. Upon cluster penetration across the liquid-liquid interface, we find magnetic Bond number regimes where the fluid coating drains away from the surface of the cluster, and where the clusters are encapsulated inside a thin film coating layer. This self-assembly technique may find application in controlling the size of microscale self-assemblies, and coating such assemblies; for example, in clustering and coating of cells for immunoisolated cell transplants.

CFD simulation of equilibrium shape and coalescence of ferrofluid droplets subjected to uniform magnetic field

Abstract In this study a computational framework has been presented to simulate the equilibrium shape and coalescence behavior of suspended ferrofluid droplets subjected to uniform magnetic field. A finite volume based CFD Open source solver library (OpenFOAM®) is used for the simulations. Coupled flow field equations and magnetostatic equations are numerically solved and a coupled scheme of Volume of Fluid (VOF) and Level-Set (LS) methods called S-CLSVOF is implemented for interface capturing. The simulation results are compared with the experimental data reported in the literature. The effects of different parameters, such as magnetic field intensity, magnetic susceptibility, surface tension and initial size of droplet on the deformation of magnetic droplet are investigated. Aspect ratio of droplet at the equilibrium state is calculated and a cubic polynomial is proposed to formulate its dependency to the magnetic Bond number and magnetic susceptibility. Finally the coalescence of two falling ferrofluid droplets under the effect of uniform magnetic field is simulated. The results reveal that the application of magnetic field has a significant effect on the coalescence of droplets and it can be used as a promising method for emulsion breaking.

Solitary waves on a ferrofluid jet

AbstractThe propagation of axisymmetric solitary waves on the surface of an otherwise cylindrical ferrofluid jet subjected to a magnetic field is investigated. An azimuthal magnetic field is generated by an electric current flowing along a stationary metal rod which is mounted along the axis of the moving jet. A numerical method is used to compute fully nonlinear travelling solitary waves, and the predictions of elevation waves and depression waves made by Rannacher and Engel (New J. Phys., vol. 8, 2006, pp. 108–123) using a weakly nonlinear theory are confirmed in the appropriate ranges of the magnetic Bond number. New nonlinear branches of solitary wave solutions are identified. As the Bond number is varied, the solitary wave profiles may approach a limiting configuration with a trapped toroidal-shaped bubble, or they may approach a static wave (i.e. one with zero phase speed). For a sufficiently large axial rod, the limiting profile may exhibit a cusp.

Numerical Simulation of a Falling Ferrofluid Droplet in a Uniform Magnetic Field by the VOSET Method

This article presents a two-dimensional numerical simulation of the dynamics of a ferrofluid droplet falling in a non-magnetic fluid under a uniform magnetic field. A new interface tracking method called VOSET (a coupled volume-of-fluid and level set method) is used for capturing the evolution of interface. The Continuum Surface Force (CSF) model is used to model the surface tension force. A linearly magnetizable ferrofluid is assumed, and the magnetic force caused by the jump of magnetic permeability across the interface is incorporated into the Navier-Stokes equation as a body force. The effects of the magnetic Bond number, susceptibility, Weber number, Reynolds number, and magnetic field direction on the motion and deformation of the droplet are investigated. The results show that the increase in magnetic Bond number or susceptibility leads to the larger deformation of the droplet. For a high Weber number, even the formation of a teardrop-like droplet and an oblate elliptical-cap droplet occurs in the vertical and horizontal magnetic fields, respectively.

EQUILIBRIUM MODELS OF CORONAL LOOPS THAT INVOLVE CURVATURE AND BUOYANCY

We construct magnetostatic models of coronal loops in which the thermodynamics of the loop is fully consistent with the shape and geometry of the loop. This is achieved by treating the loop as a thin, compact, magnetic fibril that is a small departure from a force-free state. The density along the loop is related to the loop's curvature by requiring that the Lorentz force arising from this deviation is balanced by buoyancy. This equilibrium, coupled with hydrostatic balance and the ideal gas law, then connects the temperature of the loop with the curvature of the loop without resorting to a detailed treatment of heating and cooling. We present two example solutions: one with a spatially invariant magnetic Bond number (the dimensionless ratio of buoyancy to Lorentz forces) and the other with a constant radius of curvature of the loop's axis. We find that the density and temperature profiles are quite sensitive to curvature variations along the loop, even for loops with similar aspect ratios.

Deformation of Ferrofluid Marbles in the Presence of a Permanent Magnet

This paper investigates the deformation of ferrofluid marbles in the presence of a permanent magnet. Ferrofluid marbles are formed using a water-based ferrofluid and 1 μm hydrophobic polytetrafluoride particles. A marble placed on a Teflon coated glass plate deforms under gravity. In the presence of a permanent magnet, the marble is further deformed with a larger contact area. The geometric parameters are normalized by the radius of an undistorted spherical marble. The paper first discusses a scaling relationship between the dimensionless radius of the contact area as well as the dimensionless height and the magnetic Bond number. The dimensionless contact radius is proportional to the fourth root of the magnetic bond number. The dimensionless height scales with the inverse square root of the magnetic Bond number. In the case of a moving marble dragged by a permanent magnet, the deformation is evaluated as the difference between advancing and receding curvatures of the top view. The dimensionless height and the contact diameter of the marble do not significantly depend on the speed or the capillary number. The scaling analysis and experimental data show that the deformation is proportional to the capillary number.

Numerical study of the formation process of ferrofluid droplets

This paper numerically investigates the influence of a uniform magnetic field on the droplet formation process at a microfluidic flow focusing configuration. The mathematical model was formulated by considering the balance of forces such as interfacial tension, magnetic force, and viscous stress across the liquid/liquid interface. A linearly magnetizable fluid was assumed. The magnetic force acts as a body force where the magnetic permeability jumps across the interface. The governing equations were solved with finite volume method on a Cartesian fixed staggered grid. The evolution of the interface was captured by the particle level set method. The code was validated with the equilibrium steady state of a ferrofluid droplet exposed to a uniform magnetic field. The evolution of the droplet formation in a flow focusing configuration was discussed. The paper mainly analyzes the effects of magnetic Bond number and the susceptibility on the velocity field and the droplet size. The droplet size increased with increasing magnetic strength and susceptibility.

Edge pinch instability of oblate liquid metal drops in a transverse AC magnetic field

This paper considers the stability of liquid metal drops subject to a high-frequency AC magnetic field. An energy variation principle is derived in terms of the surface integral of the scalar magnetic potential. This principle is applied to a thin perfectly conducting liquid disk, which is used to model the drops constrained in a horizontal gap between two parallel insulating plates. Firstly, the stability of a circular disk is analysed with respect to small-amplitude harmonic edge perturbations. Analytical solution shows that the edge deformations with the azimuthal wavenumbers m = 2, 3, 4, . . . start to develop as the magnetic Bond number exceeds the critical threshold Bmc = 3π(m + 1)/2. The most unstable is m = 2 mode, which corresponds to an elliptical deformation. Secondly, strongly deformed equilibrium shapes are modelled numerically by minimising the associated energy in combination with the solution of a surface integral equation for the scalar magnetic potential on an unstructured triangular mesh. The edge instability is found to result in the equilibrium shapes of either two- or threefold rotational symmetry depending on the magnetic field strength and the initial perturbation. The shapes of higher rotational symmetries are unstable and fall back to one of these two basic states. The developed method is both efficient and accurate enough for modelling of strongly deformed drop shapes.

Generation of liquid metal structures of high aspect ratio by application of an ac magnetic field

We study how the shape of parts obtained through the LASER cladding process can be controlled by application of an ac magnetic field by means of two simple physical models: a numerical and an experimental one. More specifically, we show that straight metallic joints of high aspect ratio can be obtained by using inductors of triangular cross-section that concentrate electromagnetic forces at the bottom of the joint. The effect is first demonstrated on a numerical model for an infinitely long joint such as: we illustrate how the joint shape can be controlled by varying the inclination of the inductor and for a magnetic Bond number Bom=60 (which measures the ratio of electromagnetic to capillary forces), we obtain a joint of aspect ratio up to 7.2. We further find that inductor angles in the range 15°–25° lead to joint side faces that are close to vertical. These findings are then verified experimentally by placing a liquid metal drop in a purpose built inductor of triangular cross-section. We find a good agreement between the theoretical prediction of our two-dimensional model and the real three-dimensional drop. For the highest magnetic Bond number our generator could deliver, Bom=20.19, we achieved a drop aspect ratio of 2.73.

Magnetically assisted gas–solid fluidization in a tapered vessel: Part II: Dimensionless bed expansion scaling

The article presents an effort to create dimensionless scaling correlations of the overall bed porosity in the case of magnetically assisted fluidization in a tapered vessel with external transverse magnetic field. This is a stand of portion of new branch in the magnetically assisted fluidization recently created concerning employment of tapered vessels. Dimensional analysis based on “pressure transform” of the initial set of variables and involving the magnetic granular Bond number has been applied to develop scaling relationships of dimensionless groups representing ratios of pressures created by the fluid flow, gravity and the magnetic field over an elementary volume of the fluidized bed. Special attention has been paid on the existing data correlations developed for non-magnetic beds and the links to the new ones especially developed for tapered magnetic counterparts. A special dimensionless variable X p = ( A r Δ D b L ) 1 / 3 R g M Q combining Archimedes and Rosensweig numbers has been conceived for porosity correlation. Data correlations have been performed by power-law, exponential decay and asymptotic functions with analysis of their adequacies and accuracies of approximation.

Magnetically assisted gas–solid fluidization in a tapered vessel: Part I. Magnetization-LAST mode

This article presents further experimental results of the Magnetization-LAST mode in magnetically assisted gas-fluidized tapered beds, including external transverse magnetic field control of solid phase movement, central channel formation, spout depth and the pressure drop across the bed. Phase diagrams similar to those recently reported for the Magnetization-FIRST mode were also developed. Dimensional analysis based on “pressure transform” of the initial set of variables and involving the magnetic granular Bond number pertinent to particle aggregate formation was applied to develop the scaling relationships.

Field-induced motion of ferrofluid droplets through immiscible viscous media

The motion of a hydrophobic ferrofluid droplet placed in a viscous medium and driven by an externally applied magnetic field is investigated numerically in an axisymmetric geometry. Initially, the drop is spherical and placed at a distance away from the magnet. The governing equations are the Maxwell equations for a non-conducting flow, momentum equation and incompressibility. A numerical algorithm is derived to model the interface between a magnetized fluid and a non-magnetic fluid via a volume-of-fluid framework. A continuum-surface-force formulation is used to model the interfacial tension force as a body force, and the placement of the liquids is tracked by a volume fraction function. Three cases are studied. First, where inertia is dominant, the magnetic Laplace number is varied while the Laplace number is fixed. Secondly, where inertial effects are negligible, the Laplace number is varied while the magnetic Laplace number is fixed. In the third case, the magnetic Bond number and inertial effects are both small, and the magnetic force is of the order of the viscous drag force. The time taken by the droplet to travel through the medium and the deformations in the drop are investigated and compared with a previous experimental study and accompanying simpler model. The transit times are found to compare more favourably than with the simpler model.

Magnetically assisted gas–solid fluidization in a tapered vessel: First report with observations and dimensional analysis

AbstractThe article presents first experimental results on gas–solid fluidization in a tapered bed in presence of an external transverse magnetic field that creates a novel branch in magnetically assisted fluidization. Phase diagrams similar to those used to describe cylindrical beds have been created to distinguish the bed regimes occurring under the action of two principle macroscopic variables such as field intensity and gas flow rate. A detailed analysis and parallelism to the bed behaviour exhibited by non‐magnetic spouted beds of cohesive particles have been performed. Principle process variables such as bed depth, field intensity, particle size, cone angle have been detected. A dimensional analysis utilizing a “pressure transform” of the initial set of variables has been applied to develop scaling relationships. Examples of scaling experimental data pertinent to the minimum spouting point and involving the magnetic granular Bond number have been developed.