A RHEOLOGICAL MODEL FOR MAGNETO-RHEOLOGICAL FLUIDS

Transcription

1 A RHEOLOGICAL MODEL FOR MAGNETO-RHEOLOGICAL FLUIDS Daniela Susan-Resiga 1, Ladislau Vékás 2, Romeo Susan-Resiga 3 Introduction Magneto-rheological fluids (MRF) are suspensions of particles which can be magnetized. MRF exhibit fast, strong and reversible changes in their rheological properties when a magnetic field is applied. A commercially available damping system that takes full advantage of the controllable technology of MRF and delivers valveless damping control appeared on passenger vehicles in Further commercial systems are e.g. a MRF brake, steer-by-wire vehicle control, controllable friction damper that decreases the noise and vibrations in washing machines, above-the-knee prosthesis, and seismic mitigation MRF damping systems protecting buildings and bridges from earthquakes and windstorms. Goncalves et al. [6] present a recent review of the state of the art in magnetorheological technology. Since their invention in 1948, MR fluid development has made significant advancements. Throughout this time, MR fluids have been faced with many challenges and today MR fluid formulations appear to have overcome many of these challenges. Goncalves et al [6] focus thei review on strength (i.e. the achievable yield stress), stability and durability of the fluid, since when using a MR fluid in a particular application these three areas fall under the greatest scrutiny. The viscoelastic properties of a magneto-rheological fluid can be variably controlled using a magnetic field. Wollny et al. [13] have developed a new experimental method allowing an accurate determination of a magneto-rheological fluid s viscoelastic properties as a function of the preset magnetic field strength. Carletto and Bossis [2] attempt to correlate the rheological characteristics of a magnetic colloidal suspension subjected to a unidirectional magnetic field with the structures formed within the fluid. As far as the MRF yield stress is concerned, Bossis et al. conclude that the aggregates do not break but begin to slip on the wall, then they begin to rotate and no longer touch the plates. At some critical strain the hydrodynamic shear force will break the aggregates. This particular behaviour is found in the experimental results we present in this paper, and it is captured in the mathematical model we propose 1 PhD, Lecturer, West University of Timişoara, Romania 2 PhD, Senior Researcher, Romanian Academy Timişoara Branch, Romania 3 PhD, Professor, Politehnica University of Timişoara, Romania

2 142 3 rd GERMAN-ROMANIAN WORKSHOP on TURBOMACHINERY HYDRODYNAMICS, May 10-12, 2007 to describe the MRF rheology in the presence of the magnetic field. Experimental investigations on both rheological properties and dispersion stability of MRF have been performed by Chin et al. [4] as well, in an effort to develop reliable methods of measurement and analysis able to provide design parameters for practical devices. The micromechanical properties of a MR system are advocated by Furst and Gast [5] to be of fundamental importance to the suspension rheology, especially with regards to the yield stress behaviour. The same viewpoint is shared by Melle and Martin [8] who develop a chain model based on a balance of hydrodynamic and magnetostatic forces and focuses on the mechanical stability of chains. Most applications of MRF are focused on various damper configurations. Zhu [14] develop and studies a simple disk-type MRF damper, Fig. 1, and incorporates this technical solution in a flexible rotor system. He proves the effectiveness of the disk-type MRF damper for attenuating the vibration of the rotor systems with a simple open-loop on-off control based on the feedback of rotational speed. Fig. 1. Cross-section of the disk-type MR fluid damper developed by Zhu [14]. Sassi et al. [10] performed theoretical and experimental studies for the design, development and testing of a MRF damper to be used for semi-active control of automotive suspensions, Fig. 2. Various control algorithms and schemes for MRF suspension systems can be found in [12] and [11]. Our main goal is to develop magnetorheological devices for semi-active flow control, in particular for mitigating the effects of severe flow instabilities that occurs in Francis hydraulic turbines operating at part load. As a result, our efforts are focused on two main issues: i) investigate the flow instabilities and identify the most effective ways to damp the pressure fluctuations and associated vibrations, ii) develop special magnetorheological dampers suitable for large

3 143 3 rd GERMAN-ROMANIAN WORKSHOP on TURBOMACHINERY HYDRODYNAMICS, May 10-12, 2007 scale hydraulic turbine applications. As we have shown, the key issue is to determine experimentally and model mathematically the MRF rheological properties for variable magnetic field intensity, within a large range of shear strain values. The present paper presents our latest experimental results together with a novel mathematical model that accurately captures the MRF rheological behaviour. Fig. 2. Automotive MRF damper developed by Sassi et al [10]. Magnetorheological investigations We have investigated the rheological behaviour of the MRF-132DG suspension of micrometer-size particles, using the MCR-300 rheometer with a plate-plate magneto-rheological cell. A 0.5mm gap has been chosen for all investigations, corresponding to a 0.157ml probe volume. Both oscillatory (amplitude sweep and frequency sweep) and rotational (flow curve, creep test, relaxation test, temperature test) test have been performed. All tests have been performed for variable magnetic field intensity, corresponding to the electric current intensity in the cell coil of 0,0.25,0.5,0.75,1.0,1.5,2.0,and 3.0 A.

4 144 3 rd GERMAN-ROMANIAN WORKSHOP on TURBOMACHINERY HYDRODYNAMICS, May 10-12, 2007 Fig. 3 shows the primary experimental results from rotational tests, for the above electric current intensities. In order to capture the behaviour of the MRF at lower shear rates, oscillatory tests have been performed, as shown in Fig. 4. So far we have found that only the results from amplitude sweep oscillatory tests merge with the viscosity curves from rotational tests. These combined data are used in the next section to develop a mathematical model describing the MRF 6 3 rheological behaviour within a very large range of shear rate values γ = US Pa s 10 6 Pa 132DG, FlowC, 0.5mm, 0A 1 132DG, FlowC, 0.5mm, PP 20/MR d=0.5 mm η Viscosity τ Shear Stress PP 20/MR d=0.5 mm η Viscosity τ Shear Stress DG, FlowC, 0.5mm, 0.75A 1 PP 20/MR d=0.5 mm 132DG, FlowC, 0.5mm, 1 PP 20/MR d=0.5 mm η 10 3 τ η τ Viscosity Shear Stress η τ Viscosity Shear Stress DG, FlowC, 0.5mm, 0.25A 1 PP 20/MR d=0.5 mm 132DG, FlowC, 0.5mm, 2 PP 20/MR d=0.5 mm η Viscosity η Viscosity τ Shear Stress 132DG, FlowC, 0.5mm, 0.5A 1 τ Shear Stress 132DG, FlowC, 0.5mm, PP 20/MR d=0.5 mm η Viscosity PP 20/MR d=0.5 mm η Viscosity /s 10 3 Shear Rate γ. τ Shear Stress τ Shear Stress Fig. 3. Experimental data for flow curves and viscosity curves from rotational 3 3 experiments, with shear rate within the range γ = Rheological behaviour of the MRF-132DG Magneto-rheological fluids (MRF) have a complex behaviour in the presence of a magnetic field. Chadhuria et al. [3] investigate the rheological flow curves (shear stress vs. Shear rate) of a nanoparticle cobalt-based magnetorhelogical fluid, and found that either Bingham-plastic or Herschel-Bulkley models can be used to describe the experimental data.

5 145 3 rd GERMAN-ROMANIAN WORKSHOP on TURBOMACHINERY HYDRODYNAMICS, May 10-12, 2007 Fig. 4. Experimental data for viscosity curves from oscillatory experiments (amplitude sweep and frequency sweep) and rotational experiments. The experimental investigations we have performed on a MRF-132DG probe reveals a quasi-newtonian behaviour in the start-up phase, with very low

6 146 3 rd GERMAN-ROMANIAN WORKSHOP on TURBOMACHINERY HYDRODYNAMICS, May 10-12, 2007 shear rate, switching in a shear-thinning behaviour as the shear rate increases. As a result, the mathematical model to be used for fitting the experimental data should capture this transition as ( ) = ( ) W ( ) + ( ) W ( ) τ γ τ γ γ τ γ γ (1) where the Newtonian behaviour is given by N N 1 HB 2 ( ) = 0 τ γ η γ, (2) and the shear thinning behaviour is described by the Hershel-Bulkley model, n τ γ = τ + c γ. (3) HB ( ) 1 0 The Hershel-Bulkley model assumes a yield stress τ 0, and for shearthinning fluids n < 1. The consistency constant c is a fit parameter. The weighting functions W1 ( γ ) and W2 ( γ ) in Eq.(1) should be chosen such as W ( 1 γ ) W2( γ ) for very low shear rate, and W ( 1 γ ) W2( γ ) at large shear rate, with W ( 1 γ) + W2( γ) = 1. There are many continuous, with continuous derivatives, functions that meet the above general requirements. Note that the function and its derivatives continuity insures a smooth transition between the two different behaviours (2) and (3), as observed in experiments. Let us assume that γ is a shear rate value in the neighborhood of the transition between the Newtonian and Hershel-Bulkley models. This characteristic value will be used in our model to make the shear rate dimensionless, thus rewriting the Hershel-Bulkley model as ( ) 1 n γ = τhb γ τ τ. (4) γ Note that the consistency constant in Eq.(3) has been replaced by the parameter τ 1 with dimension of shear stress. The function that blends the two models will depend on the dimensionless x γ γ 1 exp x, variable /, and it can be chosen as tanh ( x ), or erf ( x ), or ( ) or any other function with the qualitative behaviour as in Fig. 5. For example, the 1 exp( x) function has been used by Papanastasiou [9] to modify the viscoplastic fluid models in order to avoid the discontinuity in the flow curve due to the incorporation of the yield criterion [7]. As a result, a single equation can be used for the entire flow curve, before and after yield, as follows τ y [ 1 exp( a γ) ] τ ( γ) = η+ γ. (5) γ

7 147 3 rd GERMAN-ROMANIAN WORKSHOP on TURBOMACHINERY HYDRODYNAMICS, May 10-12, 2007 blending function erf(x) tanh(x) 1 exp( x) x Fig. 5. Blending functions. All the above functions vanish as x 0 and reach the asymptotic value of 1 for x. However, one can control how fast the functions approach the asymptotic value by stretching the abscissa with a parameter multiplying x. The role of this parameter is played in out model by the characteristic shear rate γ. For the present analysis we have chosen γ γ W 1 = 1 tanh γ and W2 = tanh γ, (6) but similar results are obtained with the other choices mentioned above. As a result, the mathematical model used for representing the mahneto-rheological behaviour of the MRF-132DG samples reads as 1 n γ γ γ τ = η 0γ 1 tanh + τ0 + τ1 tanh γ γ γ. (7) Fig. 6 shows that the results obtained by replacing the tanh ( x ) in Eq.(7) either by erf ( x ) or 1 exp( x) are practically identical, but obviously there are some changes in the fit parameter values. The shear stress vanishes as γ 0, while the viscosity η τ / γ has the limit value for vanishing shear rate τ 0 η limη = η0 + γ 0 γ. (8) Equation (8) clearly reveals that the start-up viscosity η incorporates both the contribution from the Newtonian behaviour, η 0, and the contribution from the yield stress of the Hershel-Bulkley model, τ 0.

8 148 3 rd GERMAN-ROMANIAN WORKSHOP on TURBOMACHINERY HYDRODYNAMICS, May 10-12, 2007 Fig. 6. Flow curve fitted with various blending functions. The shear stress reaches the characteristic value of τ τ γ = η γ τ + τ (9) ( ) 0 ( 0 1) at the characteristic shear rate value, corresponding to the shift in the rheological behaviour from quasi-newtonian to shear-thinning. Although our model (7) has five parameters to be found by fitting the experimental data, four of them are relevant from practical point of view: the threshold shear rate γ and the corresponding shear stress τ, the start-up viscosity η, and the shear-thinning exponent n. The latest was found to have practically the same value, n = ( ± 3% ) for a wide range of electric current intensity in the magneto-rheological cell coil I = A, and it has been subsequently fixed to a value of n = Fig. 7 through Fig. 13 show the flow curves (shear stress vs. shear rate) and viscosity curves (dynamic viscosity vs. shear rate) for variable intensity of the electric current in the magnetorheological cell coil. On the flow curve plots both τ and γ are marked with dashed lines, since these values correspond to the quasi-yield point. On the viscosity curve plots we have marked both η and γ are shown explicitly, in order to emphasize the start-up viscosity level

9 149 3 rd GERMAN-ROMANIAN WORKSHOP on TURBOMACHINERY HYDRODYNAMICS, May 10-12, 2007 Fig. 7. Flow curve τ ( γ ) and viscosity curve ( ) η γ for I = 0.25 A.

10 150 3 rd GERMAN-ROMANIAN WORKSHOP on TURBOMACHINERY HYDRODYNAMICS, May 10-12, 2007 Fig. 8. Flow curve τ ( γ ) and viscosity curve ( ) η γ for I = 0.5 A.

11 151 3 rd GERMAN-ROMANIAN WORKSHOP on TURBOMACHINERY HYDRODYNAMICS, May 10-12, 2007 Fig. 9. Flow curve τ ( γ ) and viscosity curve ( ) η γ for I = 0.75 A.

12 152 3 rd GERMAN-ROMANIAN WORKSHOP on TURBOMACHINERY HYDRODYNAMICS, May 10-12, 2007 Fig. 10. Flow curve τ ( γ ) and viscosity curve ( ) η γ for I = 1.0 A.

13 153 3 rd GERMAN-ROMANIAN WORKSHOP on TURBOMACHINERY HYDRODYNAMICS, May 10-12, 2007 Fig. 11. Flow curve τ ( γ ) and viscosity curve ( ) η γ for I = 1.5 A.

14 154 3 rd GERMAN-ROMANIAN WORKSHOP on TURBOMACHINERY HYDRODYNAMICS, May 10-12, 2007 Fig. 12. Flow curve τ ( γ ) and viscosity curve ( ) η γ for I = 2.0 A.

15 155 3 rd GERMAN-ROMANIAN WORKSHOP on TURBOMACHINERY HYDRODYNAMICS, May 10-12, 2007 Fig. 13. Flow curve τ ( γ ) and viscosity curve ( ) η γ for I = 3.0 A.

16 156 3 rd GERMAN-ROMANIAN WORKSHOP on TURBOMACHINERY HYDRODYNAMICS, May 10-12, 2007 and the shear-rate where the rheological behaviour switches from quasi-newtonian (in fact weakly shear-thickening) to shear-thinning. A synoptic perspective of the viscosity variation with the intensity of the magnetic field (proportional with the intensity of the electric current in the coil) is shown in Fig. 14. One can easily see a large jump in viscosity values even when a rather weak magnetic field is applied, and a rapid saturation as the intensity is increased. Fig. 14. Viscosity curves for variable electric current intensity. The main parameters, namely the yield stress τ and start-up viscosity η are plotted in Fig. 15 and Fig. 16, respectively, versus the electric current intensity. Conclusions Extensive experimental investigations have been performed in order to describe the rheological behaviour of a magneto-rheological fluid within magnetic field of variable strenght. Both oscillatory and rotational tests are employed in order to explore a large range of shear rate values. A new rheological model is developed, by combining a Newtonian behaviour for very low shear rate with a Hershel-Bulkley model for large shear rate. Our five parameter model, Eq.(7), accurately captures the flow curve and automatically identifies the shear rate γ where the abrupt qualitative change in the rheological behaviour occurs. The corresponding shear stress τ can be considered to correspond to the so-called yield stress. However, for γ < γ the MRF displays a quasi constant (in fact a weak shear-thickening behaviour) large viscosity η,

17 157 3 rd GERMAN-ROMANIAN WORKSHOP on TURBOMACHINERY HYDRODYNAMICS, May 10-12, 2007 Fig. 15. Yield stress τ versus the electric current intensity. Fig. 16. Start-up viscosity η versus the electric current intensity. which confirms Bossis et al. [1] conclusions that the magnetic particle aggregates do not break but begin to slip on the wall, then they begin to rotate and no longer touch the plates, and at some critical strain the hydrodynamic shear force will break the aggregates.

18 158 3 rd GERMAN-ROMANIAN WORKSHOP on TURBOMACHINERY HYDRODYNAMICS, May 10-12, 2007 Acknowledgements The present work has been supported by the Romanian Government Ministry of Education and Research, National Authority for Scientific Research through project CEEX-M1-C contract No. 64/ ismart-flow and by the Swiss National Science Foundation through the SCOPES Joint Research Project IB References [1] Bossis, G., Khuzir, P., Lacis, S., and Volkova, O., 2003, Yield behaviour of magnetorheological suspensions, Journal of Magnetism and Magnetic Materials, , pp [2] Carletto, P., and Bossis, G., 2003, Field-induced structures and rheology of a magnetorheological suspension confined between two walls, J. Phys.: Condens. Matter, 15, pp [3] Chaudhuria, A., Wereleya, N.M., Kothab, S., Radhakrishnanb, R., and Sudarshanb N.M., 2005, Viscometric characterization of cobalt nanoparticle-based magnetorhelogical fluids using genetic algorithms, Journal of Magnetism and Magnetic Materials, Vol.293, Issue 1, pp [4] Chin, B.D., Park, J.H., Kwon, M.H., and Park, O.O., 2001, Rheological properties and dispersion stability of magnetorheological (MR) suspensions, Rheologica Acta, 40, pp [5] Furst, E.M., and Gast, A.P., 2000, Micromechanics of magnetorhelogical suspensions, Physical Review E, Vol.61, No.6, pp [6] Goncalves, F.D., Koo, J.-H., and Ahmadian, M., 2006, A Review of the State of the Art in Magnetorheological Fluid Technologies Part I: MR fluid and MR fluid models, The Shock and Vibration Digest, Vol.38, No.3, [7] Macosko, C.W., 1994, RHEOLOGY. Principles, Measurements, and Applications, Wiley-VCH, New York. [8] Melle, S., and Martin, J.E., 2003, Chain model of a magnetorheological suspension in a rotating field, Journal of Chemical Physics, Vol.118, No.21, pp [9] Papanastasiou, T.C., 1987, J. Rheology, 31, p.385. [10] Sassi, S., Cherif, K., Mezghani, L., Thomas, M., and Kotrane, A., 2005, An innovative magnetorheological damper for automotive suspension: from design to experimental characterization, Smart Materials and Structures, 14, pp [11] Shen, Y., Golnaraghi, M.F., and Heppler, G.R., 2006, Semi-active Vibration Control Schemes for Suspension Systems Using Magnetorheological Dampers, Journal of Vibration and Control, 12(1), pp [12] Song, X., Ahmadian M., Southward, S., and Miller, L.R., 2005, An Adaptive Semiactive Control Algorithm for Magnetorheological Suspension Systems, Journal of Vibration and Acoustics, Vol. 127, pp [13] Wollny, K., Läuger, J. And Huck, S., 2002, Magneto Sweep A New Method for Characterizing the Viscoelastic Properties of Magneto-Rheological Fluids, Applied Rheology, Vol.12, No.1, pp [14] Zhu, C., 2005, A disk-type magneto-rheological fluid damper for rotor system vibration control, Journal of Sound and Vibration, 283, pp