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Acoustic Velocity and Attenuation in Magnetorhelogical Fluids Based on an Effective Density Fluid Model

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MATEC Web of Conferences 45, 001 01 (2016) DOI: 10.1051/matecconf/201645001 01 C Owned by the authors, published by EDP Sciences, 2016 Acoustic Velocity and Attenuation in Magnetorhelogical fluids based on an effective density fluid model Min Shen1,2 , Qibai Huang 1 1 Huazhong University of Science and Technology in Wuhan,China 2 Wuhan Textile University in Wuhan,China Abstract. Magnetrohelogical fluids (MRFs) represent a class of smart materials whose rheological properties change in response to the magnetic field, which resulting in the drastic change of the acoustic impedance. This paper presents an acoustic propagation model that approximates a fluid-saturated porous medium as a fluid with a bulk modulus and effective density (EDFM) to study the acoustic propagation in the MRF materials under magnetic field. The effective density fluid model derived from the Biot’s theory. Some minor changes to the theory had to be applied, modeling both fluid-like and solid-like state of the MRF material. The attenuation and velocity variation of the MRF are numerical calculated. The calculated results show that for the MRF material the attenuation and velocity predicted with this effective density fluid model are close agreement with the previous predictions by Biot’s theory. We demonstrate that for the MRF material acoustic prediction the effective density fluid model is an accurate alternative to full Biot’s theory and is much simpler to implement. 1 Introduction Magnetorheological fluids (MRFs) are materials whose properties change when an external electro-magnetic field is applied. They are considered as “smart materials” because their physical characteristics can be adapted to different conditions. Hence, MRFs material can be used (a)No magnetic field in design of active sound barriers to control noise transmission loss and for modification of noise absorption characteristics of sound absorbing materials [1]. The MRFs consist of micron size magnetically permeable particles suspended in a carrying liquid such (b) Applied Magnetic field H as the mineral or silicon oil. From the point of view of the inner structure, the particles are scattered randomly in the Figure.1 The MR effect:(a) the particles in the absence of liquid carrier in the absence of magnetic fields shown in magnetic fields;(b) particles magnetized and form columns Figure.1(a). When a magnetic field is applied to the fluid, when applied field; the particles magnetized and chain-like structures that align in the direction of the applied field as depicted in Since the discovery of the magnetorheology, only a Figure.1(b). In the presence of shear strain, the particles few works have been published in acoustic properties of aligned as columnar microstructure, in turn dramatically the MRF. Malinovsky et al.[2-3] designed actively change its acoustic impedance control magnetic field. electro-magnetically controlled acoustic metamaterial. However, if the magnetic field is removed, the They construct negative mass employs mass-spring suspension turns to a Newtonian fluid and the transition oscillator build into another mass. A lattice of such between these two phases is highly reversible, which “mass-in-mass” elements could be highly attenuated for provides a unique feature of electric- or magnetic-field acoustic/vibration waves. An one dimensional chain controllability of acoustic impedance of MRF. This model discussed based on discrete elements with feature has inspired the design of new active sound effective mass density and elastic modulus. In their study, barriers to control noise transmission loss or as acoustic the theoretical model of acoustic wave propagation in metamaterials with negative effective dynamic bulk [ MRF is based on one dimensional wave equation in the modulus 2-3]. elastic medium. This is an Open Access article distributed under the terms of the Creative Commons Attribution License 4.0, which permits distribution, and reproduction in any medium, provided the original work is properly cited. Article available at http://www.matec-conferences.org or http://dx.doi.org/10.1051/matecconf/20164501001 MATEC Web of Conferences Howarth et al[4] conducted a series of acoustic where is the toal mass density , f is the pore fluid experiments at the National High Magnetic Field Laboratory in Tallahassee, Florida. The acoustic sound mass density and s is the particle mass density. speed of MR fluids was measured as functions of applied The material coefficients are magnetic field strength, normal and orthogonal field HKKDKK 2 (6) orientations, and acoustic frequency. Their presentation [(rb ) / ( b )] b (4 / 3) discussed measurement methodology and preliminary C( KKKDKrbr ) /( b )ˈ (7) results about the MRF material. 2 In previous work, Shen and Huang [5] present the M/() KDKrbˈ (8) acoustic model of MRF materials based on the Biot-Stoll DK KK model[6-8]. The Biot’s model is an useful extension of rrr[1 ( / 1)] (9) the equivalent fluids description, which takes into We note that K is the bulk modulus of individual account the stiffness of the frame and demonstrates that r two compressional waves and one shear wave can particles and K f is th bulk moulus of the base fluid.The propagate in porous materials. For the case where a F strong coupling exists between the fluid and the frame, parameter representsthe deviation from Poiseuille flow the two compressional wave exhibit very different as frequency increases. Biot derived an expression for properties, and are identified as the slow wave and the F that in the present notation is fast wave. The Biot-Stoll model is a comprehensive T()/4 F() (10) model. It is necessary to derive an approximate equation 12 iT ()/ for simplify practical application acoustic wave propagation in MRF suspensions. with This paper presents an effective density fluid model ()() iJ i that studies the acoustic propagation in the MRF material T( ) 1 (11) Ji under magnetic field to simple the Biot’s model. The 0 () effective density fluid model (EDFM) approximates a where J and J are cylindrical Bessel functions and fluid-saturated porous medium as a fluid with a bulk 1 0 modulus and effective density derived from Biot’s theory. f a (12) 2 Acoustic Model The parameter a is the pore size parameter. Johnson et al[9]. Examined the problem of a pressure gradient 2.1 The Biot’s model applied across a porous sample and found that for a porous media consisting of nonintersecting canted tubes the When an acousitc wave propagates in a porous, elastic,solid relationship for a medium saturated by a visous fluid. The Biot’s model considers that two compressionl waves and one shear wave 8 a (13) can propagate in a porous medium simultaneously[6-8]. Let u is displacement of the frame, U is displacement of the pore fluid relative to the frame. Using Eqs.(3) and (3),a fourth-order polynomial for the complex wave number k can be found. The soltion gives u .ss . (1) two different wave numbers corresponding to the Biot fast ()uU ..ff and slow waves. Both these waves show significant (2) dispersion. where is the porosity. Biot’s equations for the scalar potensials are then given as (using a plane wave solution 2.2 The effetive density fluid model exp[ikx ( . t )]) 22 2 2 Taking the Kb ==0 limit of Biot theory is () kHsf kC () s f (3) motivated by the fact that for magnetroheological fluids the frame and shear moduli are much lower than the other kC22 kM 2 2 ()sf () fsff / moduli[10-12]. (4) K iF Using Eqs.(6-8), it is easy to shown that, when b and f / are set to zero. in Eqs.(3) and (4), kc / is the acoustic wave number, HCM 1 1 f is the frequency, c is the speed of sound, is the () (14) KKrf structure constant or tortusotity, is the pore fluid The result states that the compressibility is a linear viscosity, is the permeability. function of the concentration of the particles in a suspension fs(1 ) (5) as assumed by Urick in 01001-p.2 ICMM 2016 2 22 2 ˄˅-()kHsf kH sff (15) 2 f ˄˅-()kH2 kH22 sf fsf (16) iF 2 + f Using Eqs.(15) and (16)ˈa second-order equation for the wave number k can be found. This solution and the relation from Eq.(3-4) that kc=/ can be used to find that 2 () k 2 eff (17) H where iF (1) (1) Figure.2 The real (solid curve) and imaginary (dot-dashed) sf f parts of the effective density given in Eq.(12) for the MRFs in ()Table 1. eff() f iF (1 ) ( 22 ) sf Table.1 The EDFM model Parameters of MRF-132DG (18) is a complex effective density. Parameters Symbol(Unit) MRF 3 Solid density ˄NJr / m ˅ 3160 3 Numerical Results and Discussion 3 Fluid density ˄NJf / m ˅ 1000 To verify the correctness of this model, the longitudinal Grain bulk 2 wave velocity and attenuation coefficient of the MRF are K Nm/ 11 modulus ˄˅r 1.26 10 calculated. Using the equations in the preceding section, the Volumetric longitudinal wave velocity and attenuation coefficient are 0.4 Specitic loss kr calculated for MRF-132DG material. Table 1 lists the Fluid bulk 2 parameters of the model for the MRF investigated which is K˄˅Nm/ 8 modulus f 9.8 10 typical of MRF-132DG manufactured by Lord Corporation. Grain d -6 All parameters of MRF in braces are either directly supplied m (m) 1 10 10 by the manufacturer or have been calculated[5]. The diameter Fluid 2 calculations for the linear equations are numerically (Ns/) m 0.07510 6 performed. For the MRF description, the fluid bulk viscosity modulus can be assumed purely elastic, because in our case Frame shear Nm2 6 G/˄˅b 0 810 the base fluid is low-viscosity silicone oil.
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