Mason Number Research Articles

Mason numbers for magnetorheology

The electric field strength and shear rate dependence of the apparent shear viscosity of electrorheological (ER) suspensions can often be represented by a function of only the Mason number. A Mason number defined for magnetorheological (MR) suspensions by direct substitution of magnetostatic variables for electrostatic variables does not produce a similar collapse of shear viscosity data for MR suspensions. We show that a Mason number defined in terms of the suspension magnetization can be employed to produce a collapse of experimental data at various magnetic field strengths and shear rates. As for ER suspensions, this Mason number can be calculated from experimentally measured quantities.

Dynamics of rotating paramagnetic particle chains simulated by particle dynamics, Stokesian dynamics and lattice Boltzmann methods

Paramagnetic particles, when subjected to external unidirectional rotating magnetic fields, form chains which rotate along with the magnetic field. In this paper three simulation methods, particle dynamics (PD), Stokesian dynamics (SD) and lattice Boltzmann (LB) methods, are used to study the dynamics of these rotating chains. SD simulations with two different levels of approximations—additivity of forces (AF) and additivity of velocities (AV)—for hydrodynamic interactions have been carried out. The effect of hydrodynamic interactions between paramagnetic particles under the effect of a rotating magnetic field is analyzed by comparing the LB and SD simulations, both of which include hydrodynamic interactions, with PD simulations in which hydrodynamic interactions are neglected. It was determined that for macroscopically observable properties like average chain length as a function of Mason number, reasonable agreement is found between all the three methods. For microscopic properties like the force distribution on each particle along the chain, inclusion of hydrodynamic interaction becomes important to understand the underlying physics of chain formation.

Rheology and structure of a suspension of particles subjected to Quincke rotation

We report some experimental results that show that, if the particles of a suspension rotate faster than the surrounding liquid, the apparent viscosity is lessened. The increase of the particles’ spin rate is induced by Quincke rotation, which is the spontaneous rotation of an insulating particle immersed in a slightly conducting liquid when submitted to a high enough DC electric field. We study the flow of such a suspension with internal spin rate in various geometries, and we observe, depending on the values of the shear rate and of the electric field, that the suspension can form a layered structure or present a decomposition into two phases. To predict the appearance of the layered structure, we propose to introduce a modified Mason number that takes into account the rotation of the particles.

On the theory of rheological properties of magnetic suspensions

The paper deals with a theoretical study of influence of magnetic field on effective viscosity of suspension of non-Brownian magnetizable particles. It is supposed that experimentally observed magnetorheological effects are provided by chain-like aggregates, consisting of the particles. Unlike previous works on this subject, we take into account that the chains cannot be identical and estimate their size distribution. The following power law ( η - η 0 ) / η 0 ∼ Mn - Δ , detected in many experiments, is obtained theoretically ( η and η 0 are the suspension effective viscosity and the carrier liquid viscosity, respectively, Mn is the so-called Mason number, proportional to the shear rate and inversely proportional to the square of magnetic field). The calculated magnitude of the exponent Δ increases with the applied magnetic field from approximately 0.66 to 0.8–0.9 and slowly increases with the volume concentration ϕ of the particles. These results are in agreement with known experiments.

Transient behaviour of magnetic micro-bead chains rotating in a fluid by external fields

Magnetic micro-beads can facilitate many functions in lab-on-a-chip systems, such as bio-chemical labeling, selective transport, magnetic sensing and mixing. In order to investigate potential applications of magnetic micro-beads for mixing in micro fluidic systems, we developed a pin-jointed mechanism model that allows analysing the behaviour of rotating superparamagnetic bead chains. Our numerical model revealed the response of the chains on a rotating magnetic field over time. We could demonstrate that the governing parameters are the Mason number and number of beads in the chain. The results are in agreement with the simplified analytical model, assuming a straight chain, but also allow prediction of the transient chain shape. The modelled chains develop an anti-symmetric S-shape that is stable, if the Mason number for a given chain length does not surpass a critical value. Above that value, rupture occurs in the vicinity of the chain centre. However, variations in bead susceptibility can shift the location of rupture. Moreover, we performed experiments with superparamagnetic micro-beads in a small fluid volume exposed to a uniform rotating magnetic field. Our simulation could successfully predict the observed transient chain form and the time for chain rupture. The developed model can be used to design optimised bead based mixers in micro fluidic systems.

Thermal transport in sheared electro- and magnetorheological fluids

Thermal energy transport in sheared electrorheological and magnetorheological (ER and MR) fluids is analyzed. Although energy production by viscous dissipation can be significant, energy transport on the particle length scale can be analyzed by ignoring viscous dissipation. For typical situations, energy transport normal to the flow direction is dominated by conduction. Particle-level simulations were employed to determine the suspension structure as a function of Mason number and volume fraction. A self-consistent mean-field dipole model is used to estimate the effective thermal conductivities for these simulated structures. The field-induced chain-like aggregates that form at small Mason number result in a larger effective thermal conductivity at small Mason number than at large Mason number. Effects of higher-order multipoles are estimated by analyzing effective thermal conductivities of model structures. For highly conducting particles, the effective thermal conductivity of a sheared ER or MR suspension is predicted to roughly double as the Mason number is decreased from the large to the small Mason number limits.

Dynamics of rotating paramagnetic particles simulated by lattice Boltzmann and particle dynamics methods

Novel biochemical sensors consisting of rotating chains of microscale paramagnetic particles have been proposed that would enable convenient, sensitive analyte detection. Predicting the dynamics of these particles is required to optimise their design. The results of lattice Boltzmann (LB) and particle dynamics (PD) simulations are reported, where the LB approach provides a verified solution of the complete Navier-Stokes equations, including the hydrodynamic interactions among the particles. On the other hand, the simpler PD approach neglects hydrodynamic interactions, and does not compute the fluid motion. It is shown that macroscopic properties, like the number of aggregated particles, depend only on the drag force and not on the total hydrodynamic force, making PD simulations yield reasonably accurate predictions. Relatively good agreement between the LB and PD simulations, and qualitative agreement with experimental data, are found for the number of aggregated particles as a function of the Mason number. The drag force on a rotating cylinder is significantly different from that on particle chains calculated from both simulations, demonstrating the different dynamics between the two cases. For microscopic quantities like the detailed force distributions on each particle, the complete Navier-Stokes solution, here represented by the LB simulation, is required.

Doublet dynamics of magnetizable particles under frequency modulated rotating fields

We study experimentally the dynamics of an isolated pair of paramagnetic micro-particles when applying rotating and frequency modulated magnetic fields. In the case of purely rotating field, the inter-particle distance performs radial oscillations above a given threshold. The oscillation threshold depends on the viscosity and the amplitude of the magnetic field. When the data are represented in terms of the relative distance to the critical Mason number all curves collapse nicely. The master curve obtained shows the same features that are obtained in numerical simulations. For frequency modulated rotating fields, complex and irregular dynamics is obtained. This dynamics is again in qualitative agreement with numerical simulations.

FRICTION FACTOR OF MAGNETO-RHEOLOGICAL FLUID FLOW IN GROOVED CHANNELS

The focus of this work is to study surface effects on the friction factor a magneto-rheological (MR) fluid flowing through a grooved channel under various magnetic fields and volumetric flow rates. Based on the experimental data, a relation is developed for the friction factor of MR fluid in channel flow in terms of Mason number evaluated at the surface, and the depth of the grooves. Using this relation, the pressure loss of a MR fluid flowing through a channel with grooved walls can be determined without implementing a constitutive model for MR fluids or utilizing the concept of shear yield stress. Several grooves with different configurations in channel walls have been considered. From the experimental results it has been demonstrated that under an applied magnetic field, the grooved surface would increase the friction factor of MR fluid flow significantly when comparing to the surface without grooving. The depth of grooves plays an important role in this increment.

Transient response of magnetorheological fluids: Shear flow between concentric cylinders

An experimental investigation of the rheological response of magnetorheological suspensions subjected to step changes in applied magnetic field strength at fixed shear rate is reported. For small applied field strengths, the shear stress increases rapidly to a steady value. Above a critical field strength, the rapid initial increase in shear stress is followed by a slow, transient increase in stress. The critical Mason number corresponding to the critical magnetic field strength at the onset of this transient depends on the particle volume fraction as well as the shear rate. This is in contrast to a previous analysis where the critical Mason number was predicted to depend on only the particle volume fraction. The discrepancy is attributed to colloidal forces that are significant in our experimental system, but were not included in the analysis. Further comparison with the previous analysis requires either including the effects of colloidal forces, or performing experiments with systems in which colloidal forces are not important.

Shear flow behavior of confined magnetorheological fluids at low magnetic field strengths

The non-linear properties of iron based magneto-rheological (MR) fluids are investigated at low magnetic field strengths (0–1.7 kA/m) and different gap thickness (0–500 μm) in a plate-plate configuration. Single-width chain models qualitatively predict the low-shear flow behavior when plotting the field-specific viscosity, ηF, as a function of the Mason number, Mn: a slope close to −1 is observed in log-log representations. Wall depletion effects are observed when the suspensions are sheared under the presence of low external magnetic fields applied and/or large gap distances. These results are correlated to frictional yield stress measurements and chain length distribution calculations in the presence of the external magnetic field. Finally, an equivalent slip layer thickness is calculated using the method of Yoshimura and Prud’homme.

Effects of electric fields and volume fraction on the rheology of hematite/silicone oil suspensions

Electrorheological (ER) fluids composed of iron(III) oxide (hematite) particles suspended in silicone oil are studied in this work. The rheological response has been characterized as a function of field strength, shear rate and volume fraction. The dielectric properties of the suspensions were first studied in order to get information about the conductivity of the solid. Rheologi- cal tests under a.c. electric fields elucidated the influence of the elec- tric field strength and volume frac- tion on the field-dependent yield stress. It was found that this quan- tity scales as a square power law in both cases. The viscosities of elec- trified suspensions were found to increase by several orders of magni- tude as compared to the unelectrified suspension at low shear rates, al- though at high shear rates hydrody- namic effects become dominant and no effects of the electric field on the viscosity are observed. The ER behaviour of the suspensions was analysed by considering the funda- mental forces (of hydrodynamic and electrostatic origin) acting on the particles and it is found that, at a given volume fraction, all the dependencies of relative viscosity on shear rate and field strength can be described by a single function of the Mason number, Mn. Finally, two different chain models were used to explain the shear-thinning behaviour observed: rheological measurements showed a power-law dependence of relative viscosity decrease on the Mason number, gFMn D , with D� )0.95.

ER 유체의 채널유동에 대한 직접수치해석

Steady flow of an ER (electro-rheological) fluid in a two-dimensional electrode channel is studied by using FEM. Hydrodynamic interactions between the particles and the fluid are calculated by solving the Navier-Stokes equation combined with the equation of motion for each particle, where the multi-body electrostatic interaction is described by using point-dipole model. Motion of the particles in the ER fluid is elucidated in conjunction with the mechanisms of the flow resistance and the increase of viscosity. The ER effects have been studied by varying the Mason number and volume fraction of particles. These parameters have an influence on the formation of the chains resulting in the changes of the fluid velocity and the effective viscosity of ER fluids.

One-dimensional Analysis Of A Continuum Model For Structure In Electrorheological Fluids

A continuum model is used to describe the evolution of nonuniform particle concentration distributions in electrorheological and magnetorheological fluids. Particles are shown to migrate relative to the bulk mixture motion as the result of the particle-phase contribution to the suspension stress. The particle stress results from both electrostatic and hydrodynamic contributions; the ratio of hydrodynamic to electrostatic stress is characterized by the Mason number, Mn. The model predicts structures which are in agreement with those observed experimentally for both unsheared and sheared conditions. The emphasis here is on the sheared condition, where particle-rich stripes with normal along the vorticity direction of a simple shear flow are observed when a field is applied in the direction of the velocity gradient. The linear stability and steady-state structures are analyzed for the one spatial dimensional model with variation in the vorticity direction. The linear stability analysis predicts a critical Mason number below which the uniformly distributed suspension demixes to a striped structure. The particle concentration in the steady state stripe structures is found to depend on Mn, with concentration decreasing with increasing Mn.

Chain model of a magnetorheological suspension in a rotating field

We develop an athermal chain model for magnetorheological suspensions in rotating magnetic fields. This model is based on a balance of hydrodynamic and magnetostatic forces and focuses on the mechanical stability of chains. Using a linear approximation of the chain shape, we compute the orientation and size of a critical chain in a rotating magnetic field as a function of the Mason number Mn, which is the ratio of dipolar to hydrodynamic forces between two particles in contact. The critical chain length is found to scale with the inverse square root of Mn, and its orientation relative to the instantaneous field is independent of Mn. The actual nonlinear shape of a chain in a rotating field is then computed self-consistently. Finally, the effect of local fields on the dipolar interaction force is considered, leading to predictions for the chain shape and orientation that depend rather strongly on the magnetic permeability of the particles. A principal finding is the possibility of brittle or ductile chain fracture, depending on the permeability of the particles. Single-chain simulations confirm this prediction, as do experimental measurements.

The effect of dielectric properties on the electrorheology of suspensions of silica particles coated with polyaniline

The flow behaviour of silicone-oil suspensions of five types of silica particles coated with a polyaniline base in a DC electric field has been linked to their dielectric properties. The relaxation frequencies corresponding to the position of the dielectric-loss maxima in the frequency spectra identify the interfacial polarization of suspension particles as a controlling factor for a strong electrorheological effect. The yield stresses of suspensions under the influence of electric field and critical shear rates, at which the chains of polarized particles were broken by shear forces, is correlated with the difference between the limit values of dielectric constants above and below the relaxation frequency. The analysis of particle dipole coefficient β showed that particle polarizability is the main factor affecting rigidity of the electrorheological structure. In contrast with this, particle shape and size, controlling the field-off suspension viscosity, become unimportant after the electric field has been applied. The plots of the relative viscosity of studied suspensions vs. Mason number characterizing the relation between shear and polarization forces have been discussed. While the results obtained at different shear rates and field strengths were reduced to a single dependence, for various particle suspensions these dependences differed.

Dynamics of simple magnetorheological suspensions under rotating magnetic fields with modulated Mason number

We study theoretically the dynamics of a system of two magnetizable particles suspended in a non-magnetic fluid subject to a rotating magnetic field when a modulation on the Mason number (ratio of viscous to magnetic forces) is applied. We find, using a periodic modulation, that a resonant-like phenomenon between the periodic modulation of the Mason number and the intrinsic radial oscillation of the system without modulation occurs. For a random perturbation of the Mason number, we obtain an optimum noise strength at which the average interparticle distance reaches the lowest value. When a weak periodic modulation and a noise source are included in the Mason number, stochastic resonance (SR) is found for different frequencies and amplitudes of the modulation. An interpretation of this SR phenomenon is made by means of a threshold crossing mechanism.

CHAIN ROTATIONAL DYNAMICS IN MR SUSPENSIONS

The dynamics of induced dipolar chains in magnetorhelogical suspensions subject to rotating magnetic fields has been experimentally studied combining scattering dichroism and video microscopy experiments. When a rotating field is imposed the chainlike aggregates rotate synchronously with the magnetic field. We found that the average size of the aggregates decreases with Mason number (ratio of viscous to magnetic forces) following a power law with exponent -0.5 being the hydrodynamic friction forces the cause of the chains break up. However the total number of aggregated particles shows two different behaviors depending on Mason number. For low Mason numbers, the total number of aggregated particles remains almost constant and above a critical Mason number, the rotation of the field prevents the particle aggregation process from taking place so the number of aggregated particles decreases with Mason number following a power law behavior with exponent -1. Athermal molecular dynamics simulations are also reported, showing good agreement with the experiments.

EFFECT OF MAGNETIC HYSTERESIS OF THE SOLID PHASE ON THE RHEOLOGICAL PROPERTIES OF MR FLUIDS

An experimental investigation is described concerning the effect of the existence of a remanent magnetization of the dispersed particles on the rheological properties of magnetorheological fluids (MRF). Two MRF's were used: (1) solid phase: cobalt ferrite particles + silica gel (1.5% w/w); liquid phase: silicone oil (viscosity 20 mPa·s); and (2) solid phase: carbonyl iron + silica gel; liquid phase; silicone oil. The cobalt ferrite particles were synthetized as monodisperse colloidal spheres with an average diameter of 850 nm. The dependence of the dimensionless shear stress (τ⋆/ϕ) vs. Mason number (Mn) fails to scale when a "magnetorheological hysteresis procedure" is followed, specially for the higher volume fractions used (≈ 7.5%). The yield stress (τy) is first estimated from successive rheograms obtained decreasing the external field (H0) values for different ϕ. A more precise determination can be done by applying a stress ramp in the oscillatory regime. The critical stress amplitude (τc) needed to exceed the viscoelastic linear region (VLR) is obtained. It is found that both τy and τc strongly depend on the magnetic history of the sample. As expected, the previous results were not obtained in a classical MRF of carbonyl iron particles since they do not present magnetic hysteresis. We conclude that cobalt ferrite suspensions are an other kind of MRF which works at low fields (0 – 17.8 kA/m) with the opposite effect: decrease of the yield stress with the field. This property can be improved using particles with stronger remanent magnetization.

Polarizable Particle Aggregation Under Rotating Magnetic Fields Using Scattering Dichroism

We used scattering dichroism to study the dynamics of dipolar chains induced in magnetorheological suspensions under rotating magnetic fields. Both the dichroism (proportional to the total number of aggregated particles) and the phase lag show different behavior below and above a cross-over frequency. The cross-over frequency depends linearly on both the square of the magnetization and the inverse of the viscosity. The Mason number (ratio of viscous to magnetic forces) governs the dynamics. Therefore, there is a cross-over Mason number below which the dichroism remains almost constant and above which the rotation of the field prevents the particle aggregation process from taking place. Our experimental results have been compared with particle dynamics simulations showing good agreement.