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Microfluidic Applications of Magnetic and Dielectric Fluid Suspensions Markus Zahn Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science Research Laboratory of Electronics Laboratory for Electromagnetic and Electronic Systems High Voltage Research Laboratory Hsin-Fu Huang National Taiwan University Department of Mechanical Engineering Research Workshop of the Israel Science Foundation on Electrokinetic Phenomena in Nano-colloids and Nano-Fluidics Technion-Israel Institute of Technology Haifa, Israel 19-23 December 2010 1 OUTLINE 1. 2. 3. 4. 5. Introduction to Ferrohydrodynamics Applications to Magnetic Resonance Imaging (MRI) Ferrohydrodynamic Instabilities Dielectric Analog: Von Quincke Electrorotation Use of Micro-particle Electrorotation in Enhancing the Macroscopic Suspension Flow Rate 6. Continuum Analysis of Poiseuille Flow with Internal Micro-particle Electrorotation 2 1. INTRODUCTION TO FERRODYNAMICS FERROHYDRODYNAMICS 1 2 (Linear Magnetization) J B H (Force Density) F 2 J H ( M ) H (Magnetizable Medium) o o (Magnetic Diffusion Time) 2 Rm 2 (Skin Depth) 2 /v v (Magnetic Reynolds Number) ELECTROHYDRODYNAMICS 1 2 E E f 2 F f E ( P ) E / / Re /v (Linear Dielectric) (Force Density) (Polarizable Dielectric) (Dielectric Relaxation Time) v (Electric Reynolds Number) 3 Ferrofluids Colloidal dispersion of permanently magnetized particles undergoing rotational and translational Brownian motion. Established applications rotary and exclusion seals, stepper motor dampers, heat transfer fluids Emerging applications MEMS/NEMS – motors, generators, actuators, pumps Bio/Medical – cell sorters, biosensors, magnetocytolysis, targeted drug delivery vectors, in vivo imaging, hyperthermia, immunoassays Separations – magneto-responsive colloidal extractants permanently magnetized core adsorbed dispersant N Rp Md S solvent molecule Rp ~ 5nm 4 Ferrofluid Magnetic Properties Water-based Ferrofluid 0Ms = 203 Gauss f = 0.036 ; c0 = 0.65, =1.22 g/cc, 7 cp dmin5.5 nm, dmax11.9 nm B=2-10 s, N=5 ns-20 ms Isopar-M Ferrofluid 0Ms = 444 Gauss f = 0.079 ; c0 = 2.18, =1.18 g/cc, 11 cp dmin7.7 nm, dmax13.8 nm B=7-20 s, N=100 ns-200 s 5 Langevin Equation [ M 1 coth ] Ms Measured magnetization (dots) for four ferrofluids containing magnetite particles (Md = 4.46x105 Ampere/meter or equivalently μoMd = 0.56 Tesla) plotted with the theoretical Langevin curve (solid line). The data consist of Ferrotec Corporation ferrofluids: NF 1634 Isopar M at 25.4o C, 50.2o C, and 100.4o C all with fitted particle size of 11 nm; MSG W11 water-based at 26.3o C and 50.2o C with fitted particle size of 8 nm; NBF 1677 fluorocarbon-based at 50.2o C with fitted particle size of 13 nm; and EFH1 (positive α only) at 27o C with fitted particle size of 11 nm. All data falls on or near the universal Langevin curve indicating superparamagnetic behavior. 6 Ferrofluid Magnetization Force A magnetizable liquid is drawn upward around a current-carrying wire. 7 2. Application of Magnetic Media to Magnetic Resonance Imaging (MRI) The fundamental elements of MRI are (a) dipole alignment, (b) RF excitation at Larmor frequency f B0 / 2 , (c) slice selection and (d) image reconstruction. T2 decay is characterized by the exponential decay envelope of the magnetization signal in the {xy} plane after excitation. T1 recovery is characterized by the exponential growth of the magnetization along the z axis as the vector returns to the equilibrium magnetization. Larmor Frequency f B0 / 2 = 42.576 MHz/Tesla for Protons ( = gyromagnetic ratio, 2 = 42.576 MHz/Tesla for 1H nucleus.) 8 Ferrofluids as Contrast Agents in MRI MRI Image of the Brain (A) Before and (B) After Gadalinium contrast enhancement BEFORE AFTER Two categories of contrast agents in MRI: • Ferrofluids called superparamagnetic iron oxide (SPIO) contrast agents (currently smaller market share) • Gadolinium-based contrast agents T1 weighted MRI of the brain (a) before and (b) after Gadolinium contrast enhancement, showing a metastatic deposit involving the right frontal bone with a large extracranial soft tissue component and meningeal invasion. This figure comes from a case report of suspected dormant micrometatasis in a cancer patient. N.S. El Saghir et al., BioMed Central Cancer, 5, 2005 9 Effect of Ferrofluid on Image Contrast T1 and T2 results for various water based ferrofluid solid volume fractions, f. At very high concentrations (f>5 x 10-6), the SNR was not sufficient to allow for the accurate determination of T1 relaxation times. A : f 4.7 107 F : f 1.4 10 5 f =Percentage solid volume Experiments at the Martinos Center for Biomedical Imaging (MGH) Pádraig Cantillon-Murphy, “On the Dynamics of Magnetic Fluids in Magnetic Resonance Imaging”, Massachusetts Institute of Technology PhD thesis, May 2008. 10 T2 Time Constant as Function of Ferrofluid Solid Volume Fraction f T 2 0.00062 0.889 R 5.6nm Comparison of the theoretical prediction for T2 at various superparamagnetic particle radii with the experimental results over a range of MSG W11 water-based ferrofluid concentrations of the original 2.75% solution by volume. Particle size fit: R 5.6 nm Pádraig Cantillon-Murphy, “On the Dynamics of Magnetic Fluids in Magnetic Resonance Imaging”, Massachusetts Institute of Technology PhD thesis, May 2008. 11 Conservation of Linear and Angular Momentum 0 g Ω0>M1/τ H p ( ) v 2 Increasing 2 H y M = Nm m – Viscous dominated regime – Particle spin-velocity is no longer given by vorticity – Magnetic torque density in Con of Ang Mom: x N = num. particles per m3 ζ = vortex viscosity η = kinematic viscosity 0 0M H 2 v 2 12 Effect of Spin-velocity on Dynamics Increasing Ω > Ω1/τ Increasing frequency > 1/τ H y m x M. Shliomis, JETP, 34, 1972 M and H are related by a complex susceptibility – – – – Ω τ ω v (electrical frequency) (ferrofluid time constant) (spin-velocity) (flow velocity) M 1 v )M M M M eq t 13 Comsol Problem Definition Concentric Cylinders of Water and Ferrofluid y 8 cm 0.04 0.02 Brot µ (r > R2) →∞ Ferrofluid 0 Water -0.02 z -0.04 -0.06 -0.04 -0.02 B0 x Water y x 0 0.02 0.04 0.06 Dimensions in meters Ferrofluid 14 Exporting Comsol Results to Matlab Instantaneous transverse B field y M and H are approx. collinear, H(t=0) is x-directed Negligible spinvelocity x Ωτ = 0.001 z 15 Measured magnetization using a vibrating sample magnetometer (VSM) for (a) Feridex and (b) Magnevist MRI commercial contrast agents. The units of magnetic moment are (a) Am2/kg Fe for Feridex and (b) Am2/kg Gd for Magnevist. 16 The measured phase map for the center coronal slice of a tube of Feridex agent surrounded by water. The phase map only shows the net field component along the x-axis, i.e., Bx. The B0 field is left to right (x-directed) and has a value of 3 T. 17 4 4 2 2 0 0 -2 -2 -4 -4 -4 -2 0 2 4 4 4 2 2 0 0 -2 -2 -4 -4 -4 -2 0 (a) 2 4 -4 -2 0 2 4 -4 -2 0 2 4 (b) Comparing (a) theoretical MRI results and (b) measured B0 maps from the scanner at 3 T for Feridex and Magnevist contrast agents. The black line indicates the line of maximum field variation. 18 3. Ferrohydrodynamic Instabilities In DC Magnetic Fields 19 Labyrinthine Instability in Magnetic Fluids Magnetic fluid in a thin layer with uniform magnetic field applied tangential to thin dimension. Stages in magnetic fluid labyrinthine patterns in a vertical cell, 75 mm on a side with 1 mm gap, with magnetic field ramped from zero to 535 Gauss. 20 Rotating Magnetic Fields Uniform Non-uniform v (r ) bg v r y RO Surface Current Distribution K z Re Ke j ( f t ) x bg y z r z Ferrofluid RO z x z (r) Surface Current Distribution Ferrofluid , , ' , ,' a. One pole pair stator K z Re Ke j ( f t 2 ) b. Two pole pair stator Observed magnetic field distribution in the 3 phase AC stator 21 Ferrofluid Drops in Rotating Magnetic Fields A Gallery of Instabilities Ferrohydrodynamic Drops 22 Ferrofluid Spiral / Phase Transformations 23 4. Dielectric Analog: Von Quincke’s Rotor (Electrorotation) Von Quincke’s Rotor (a) Von Quincke’s rotor consists of a highly insulating cylinder that is free to rotate and that is placed in slightly conducting oil between parallel plate electrodes. As DC high voltage is raised, at a critical voltage the cylinder spontaneously rotates in either direction; (b) The motion occurs because the insulating rotor charges like a capacitor with positive surface charge near the positive electrode and negative surface charge near the negative electrode. Any slight rotation of the cylinder in either direction results in an electrical torque in the same direction as the initial displacement. 24 More on Quincke’s Rotor (Electrorotation) Definition of Quincke Rotation: Spontaneous rotation of insulating particles (or cylinders) suspended in a slightly conducting liquid subjected to a DC electric field with the field strength exceeding some critical value (Jones, 1984, 1995) 0 when 2 1 2 1 where i i i is the charge relaxation time in each region Two Competing Forces (Torques): The electrical torque and the fluid viscous torque exerted on the particle 25 Torques Exerted on a Micro-particle The fluid viscous torque The electric torque Tv 80 R3 Te p E Re 1 61 R 3 E02 1 1 2 MW 2 1 21 2 1 2 2 1 1 2 MW For a small perturbation of rotation to grow, the equation of angular motion for the particle is re-written as (Jones, 1995): 61R3 E02 1 1 2 MW d 3 I 80 R 2 2 dt 1 21 2 1 2 2 1 1 MW The bracket term should have a value larger than zero for the small perturbation to grow (Jones, 1995), thus 2 1 26 Competition of the Viscous and Electric Torques 2 80 1 Ecrit 1 2 2 1 31 2 2 1 MW E0 1 Ecrit Steady Maxwell-Wagner Relaxation Time MW 21 2 2 1 2 Te @0.5Ecrit E0 Ecrit Tvis Te @ Ecrit Te @ 2 Ecrit 27 5. Flow Rate Enhancement using Electrorotation Experiments have shown that for a given pressure gradient, the Poiseuille flow rate can be increased (Lemaire et al., 2006) by introducing micro-particle electrorotation into the fluid flow via the application of an external direct current (DC) electric field. E0 E E 0 From Hsin-Fu Huang PhD Thesis research, supervised by M. Zahn 28 6. Continuum Analysis for Couette & Poiseuille Flows with Internal Micro-particle Electrorotation The Couette flow geometry U0 eff i T iy Stress balance s eff h MW z The Poiseuille flow geometry 2D volume flow rate Q u y z dz h 0 The Continuum Governing Equations EQS & Electro-neutrality (Haus & Melcher, 1989) Polarization Relaxation Equilibrium Polarization Continuity E 0 D 0 DP P 1 v P P P Peq Dt t MW Peq Peqy n, E0 , i , i iy Peqz n, E0 , i , i iz n MW 21 2 2 1 2 Particle # density v 0 Incompressible flow: treating as a single phase continuum Linear Momentum (Dahler & Scriven, 1961, 1963; Condiff & Dahler, 1964; Rosensweig, 1997) Angular Momentum (Dahler & Scriven, 1961, 1963; Condiff & Dahler, 1964; Rosensweig, 1997) Dv p Pt E 2 v e 2 v Dt 2 No-slip boundary conditions 1.5f0 e ' h 0 1 2.5f Zaitsev & Shliomis, (1969); I Rosensweig, (1997) D Pt E 2 v 2 ' ' 2 Dt Field conditions: free-to-spin, symmetry, stable rotation Lobry & Lemaire, 1999; He, 2006; Lemaire et al., 2008 Spin field BCs: 2 v 0 1 Kaloni, 1992; Lukaszewicz, 1999; Rinaldi, 2002; Rinaldi & Zahn, 2002 Polarization Relaxation & Equilibrium Polarization Shliomis’ First Magnetization Relaxation Equation (1972, 2002) DM M 1 v M M M M eq Dt t M mH coth fMd kT M eq kT mH Cebers’ Dipole Relaxation Equation (1980) dP 1 P P Peq dt MW 1 Peqz 41R3n 2 2 1 E0 2 1 2 21 2 DP P 1 v P P P Peq Dt t MW Peqy 0 DP P 1 v P P P Peq Dt t MW Debye form (Shliomis, 1972, 2002) + Dahler & Scriven (1963) Pz Py Peqz MW x Peqy 2 1 MW x2 Peqy MW x Peqz 2 1 MW x2 We only relax the part of polarization directly related to the surface charges on the microparticles, i.e., the retarding part of the fluid polarization instead of relaxing the whole total ER fluid polarization. Huang, (2010); Huang, Zahn, & Lemaire, (2010a,b) Polarization Relaxation & Equilibrium Polarization Electric potential and field solutions to a spherical particle subjected to a uniform DC electric field rotating at an angular velocity of . z R r r , , f f r 0 2 Laplace’s equation with spherical harmonics (Jackson, 1999) BCs (Cebers, 1980; Melcher, 1981; Pannacci, 2006) r , E E0 iz E0 cos ir sin i R , , f R , , f n J f f v K f R Kf fV y 2 , 2 x f 1 Er R , , f 2 Er R , , f t f n J f 1Er R , , f 2 Er R , , f f r 1 , 1 E † E0 iz The proposed “rotating coffee cup model” for the retarding polarization relaxation equations with its accompanying (quasi-static) equilibrium retarding polarization (Huang, 2010; Huang, Zahn, & Lemaire, 2010a, b): DP P 1 v P P P Peq Dt t MW 2 1 2 1 21 2 z 3 2 1 2 Peq 41 R n E0 2 1 MW 2x Peqy 41 R3n 2 1 2 1 2 1 2 21 2 E 0 2 1 MW 2x MW x Retaining macroscopic fluid spin Including microscopic particle rotation 1 MW Ec 1 2 2 1 2 E0 1, Ec 0, 80 1 31 2 2 1 E0 Ec x E0 Ec E0 Modeling Results for the Couette Flow Geometry Schematic diagram for the Couette geometry Stress balance U0 s eff eff iz T iy h MW Modeling Results for the Couette Flow Geometry High shear rates Low shear rates We use the continuum governing equations with the rotating coffee cup polarization relaxation model in the zero spin viscosity limit '0 We use the continuum governing equations with or without the rotating coffee cup polarization relaxation model in the finite spin viscosity small spin velocity limit 2 '0 MW x2 1 Figure from Fig. 7b of Lemaire et al., J. Rheol., pp.769, (2008) 1 Comparison of Couette Effective Viscosity Results Zero spin viscosity Couette effective viscosity results compared with experimental/theoretical results found from the literature (Lemaire et al., J. Rheo. pp.769, 2008) HT: Huang theory (solid line) LT: Lemaire theory (dashed line) LE: Lemaire experiment (dotted line) 1 1.5 108 S m f 0.05 Ec 0.83 kV mm '0 Lemaire theory/experimental results are from Fig. 7a of Lemaire et al. (2008) Zero spin viscosity results do not involve ad hoc fitting! eff e 2 h x U0 Huang, (2010); Huang, Zahn, & Lemaire, (2010a) Comparison of Couette Effective Viscosity Results Zero spin viscosity Couette effective viscosity results compared with experimental/theoretical results found from the literature (Lemaire et al., J. Rheo. pp.769, 2008) HT: Huang theory (solid line) LT: Lemaire theory (dashed line) LE: Lemaire experiment (dotted line) 1 1.5 108 S m Ec 0.83 kV mm f 0.1 '0 eff e 2 h x U0 Lemaire theory/experimental results are from Fig. 7b of Lemaire et al. (2008) Zero spin viscosity results do not involve ad hoc fitting! Huang, (2010); Huang, Zahn, & Lemaire, (2010a) Modeling Results for the Couette Flow Geometry High shear rates Low shear rates We use the continuum governing equations with the rotating coffee cup polarization relaxation model in the zero spin viscosity limit '0 We use the continuum governing equations with or without the rotating coffee cup polarization relaxation model in the finite spin viscosity small spin velocity limit 2 '0 MW x2 1 Figure from Fig. 7b of Lemaire et al., J. Rheol., pp.769, (2008) 2 Comparison of Couette Effective Viscosity Results Finite spin viscosity small spin velocity Couette effective viscosity results compared with experimental/theoretical results found from the literature (Lemaire et al., J. Rheo. pp.769, 2008) HT: Huang theory (solid line) LT: Lemaire theory (dashed line) LE: Lemaire experiment (dotted line) 1 1.5 108 S m 0 In equilibrium polarization Ec 0.83 kV mm Lemaire theory/experimental results are from Figs. 7a and 7b of Lemaire et al. (2008) Finite spin viscosity results do not involve ad hoc fitting! 1 ' h2 h20 1 2.5f e du* dz* * Huang, (2010) z 0 * * 2 z* 0 * Right sign, voltage dependence, level out or plateau Comparison of Couette Effective Viscosity Results Finite spin viscosity small spin velocity Couette effective viscosity results compared with experimental/theoretical results found from the literature (Lemaire et al., J. Rheo. pp.769, 2008) *p 0.1 HT: Huang theory (solid line) Spin viscosity order of magnitude range consistent with Elborai (2006) & He (2006) LT: Lemaire theory (dashed line) LE: Lemaire experiment (dotted line) 1 1.5 108 S m 0 In equilibrium polarization Ec 0.83 kV mm Lemaire theory/experimental results are from Figs. 7a and 7b of Lemaire et al. (2008) Physically legitimate ad hoc fitting range: d 20 1 2.5f ' h20 1 2.5f 103 *p ' h20 1 2.5f 1 e du* dz* * z * 0 * 2 z* 0 * Huang, Zahn, Lemaire, (2010b) 1 Right sign, voltage dependence *p 0.2 Comparison of Couette Effective Viscosity Results Finite spin viscosity small spin velocity Couette effective viscosity results compared with experimental/theoretical results found from the literature (Lemaire et al., J. Rheo. pp.769, 2008) *p 0.34 HT: Huang theory (solid line) LT: Lemaire theory (dashed line) LE: Lemaire experiment (dotted line) 1 1.5 108 S m 0 In equilibrium polarization Ec 0.83 kV mm Lemaire theory/experimental results are from Figs. 7a and 7b of Lemaire et al. (2008) Physically legitimate ad hoc fitting range: d 20 1 2.5f ' h20 1 2.5f 103 *p ' h20 1 2.5f 1 e du* dz* * z * 0 * 2 z* 0 Spin viscosity order of magnitude range consistent with Elborai (2006) & He (2006) * Huang, Zahn, Lemaire, (2010b) 1 Right sign, voltage dependence, plateau *p 0.6 Modeling Results for the Poiseuille Flow Geometry Schematic diagram for the Poiseuille geometry 2D volume flow rate Q u y z dz h 0 Modeling Results for the Poiseuille Flow Geometry High pressure gradients 1 Experimental Ec used in modeling We use the continuum governing equations with the rotating coffee cup polarization relaxation model in the zero spin viscosity limit '0 Low pressure gradients We use the continuum governing equations with the rotating coffee cup polarization relaxation model in 2 the finite spin viscosity small spin velocity limit ' 0 MW x2 1 Figure from Fig. 6 of Lemaire et al., J. Electrostat., pp.586 (2006) Comparison of Poiseuille Flow Rate Results '0 Zero spin viscosity Poiseuille flow rate results compared with experimental/theoretical results found from the Lemaire theory/experimental results are from Figs. 5 and 6 of Lemaire et al., J. Electrostat., literature (Lemaire et al., 2006) HT: Huang theory (solid line) LT: Lemaire theory (dashed line) LE: Lemaire experiment (dotted line) f 0.05 pp.586, (2006) 1 4 108 S m Ec 1.3 kV mm f 0.1 Note: Exp. value for Ec was used in LT modeling (38% from theoretical) Zero spin viscosity results do not involve ad hoc fitting! Huang, (2010); Huang, Zahn, & Lemaire, (2010a) Comparison of Poiseuille Velocity Profile Results Zero spin viscosity Poiseuille flow velocity profiles compared with experimental results found from the literature (Peters et al., 2010) Lemaire experimental results are from Fig. 9 of Peters et al., J. Rheol., pp.311, (2010) 1 5.4 108 S m Ec 1.8 kV mm '0 f 0.05 Cusp structure for zero spin viscosities p Pa 5974.6 L m The zero spin viscosity solutions of our present continuum mechanical field equations over predicts the value of the spin velocity profile and has a cusp in the mid-channel position, which is not consistent with experimental measurements done by Peters et al. (2010). However, the order of magnitude is correct. Huang, (2010) Zero electric field solution of Poiseuille parabolic profile Modeling Results for the Poiseuille Flow Geometry High pressure gradients Experimental Ec used in modeling We use the continuum governing equations with the rotating coffee cup polarization relaxation model in the zero spin viscosity limit '0 Low pressure gradients 2 We use the continuum governing equations with the rotating coffee cup polarization relaxation model in 2 the finite spin viscosity small spin velocity limit ' 0 MW x2 1 Figure from Fig. 6 of Lemaire et al., J. Electrostat., pp.586 (2006) Comparison of Poiseuille Flow Rate Results Finite spin viscosity small spin velocity Poiseuille flow rate results compared with experimental/theoretical results found from the literature (Lemaire et al., 2006) Lemaire theory/experimental results are from Figs. 5 and 6 of Lemaire et al., J. Electrostat., pp. 586, (2006) HT: Huang theory (solid line) LT: Lemaire theory (dashed line) LE: Lemaire experiment (dotted line) 1 4 108 S m Ec 1.3 kV mm f 0.05 f 0.1 Finite spin viscosity results do not involve ad hoc fitting! 1 ' h2 h20 1 2.5f Huang, (2010) Comparison of Poiseuille Velocity Profile Results Finite spin viscosity small spin velocity Poiseuille flow velocity profiles compared with experimental results found from the literature (Peters et al., 2010) Lemaire experimental results are from Fig. 9 of 1 5.4 10 S m 1 ' 0.012h 2.96 10 8 2 10 N s Best fit, within spin viscosity values calculated by He (2006) and Elborai (2006) for ferrofluids Ec 1.8483 106 V m 1.8 kV mm Ec 1.8 kV mm round and substitute to analytic solution Peters et al., J. Rheol., pp.311, (2010) p Pa 5974.6 L m f 0.05 Note: At this pressure gradient, MAX spin velocity is not necessarily small. We are kind of pushing the limit of small spin velocities Zero electric field solution of Poiseuille parabolic profile Huang Analytic Solutions V.S. Lemaire Numeric Solutions Agreement achieved for the all voltages considered! (Rounding of critical electric field strength is only within 3%) Note: if we use particle diameter for spin viscosity, ' d 2 6.7 1013 N s Three orders of magnitude less than best fit value. Likely supports ER fluid parcel physical picture Ultrasound velocity profile measurements from Prof. Lemaire’s group likely support our finite spin viscosity theory combined with our new rotating coffee cup polarization model. Huang, 2010 Some Applications of Phenomenon and Theory Potential applications of the negative electrorheology phenomenon 1. Micro or nano fluidic pumping 2. Electrically responsive smart materials 3. Electrically variable damping devices 4. Clutch or torque transmission devices Applications of the theory presented herein 1. Negative electrorheological fluid flow induced by internal microparticle electrorotation 2. Offers complementary insights towards the ferrofluid spin-up problem in magnetorheology 3. Electrically or magnetically controlled suspensions in bio systems This work is partially supported by the United States-Israel Binational Science Foundation (BSF) through grant No. 2004081. Hsin-Fu Huang is financially supported by the National Science Council (Taipei, Taiwan) through the Taiwan Merit Scholarships (contract No. TMS-094-2-A-029). References Boissy C., P. Atten, and J.N. Foulc 1995, Journal of Electrostatics, 35, pp.13-20. Brenner H. 1970, Annual Review of Fluid Mechanics, 2, pp.137-176. Cebers A. 1980, Mechanics of Liquid and Gas (in Russian), 2, pp.86-93. Cebers A., E. Lemaire, and L. Lobry 2000, Magnetohydrodynamics, 36, pp.347-364. Cebers A., E. Lemaire, and L. Lobry 2002, International Journal of Modern Physics B, 16, pp.2063-2069. 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