SOUND SPEED IN A NON-UNIFORMLY

MAGNETOHYDRODYNAMICS Vol. 47 (2011), No. 1, pp. 3–13

SOUND SPEED IN A NON-UNIFORMLY MAGNETIZED MAGNETIC FLUID S.G. Yemelyanov, V.M. Polunin, A.M. Storozhenko, E.B. Postnikov, P.A. Ryapolov South West State University, Kursk, Russia

Experimental results of the study on sound speed distribution in a non-uniformly magnetized magnetic fluid are considered. The sound speed in the magnetic liquid within the measurement error does not depend on the magnetic field gradient, which it is exposed to. This fact does not completely preclude the possibility of the sound speed dependence on the magnetic field gradient predicted by the well-known theory. However, the estimates and the known literature data do not support the statement on the exclusive role of this parameter.

Introduction. The frequency-dependent expressions for the velocity distribution and sound absorption in the magnetic fluid (MF), determining the force action on ferromagnetic particles (FP) from the non-uniform magnetic field, have been found in [1]. With the uniform magnetic field, theory predicts the zero effect of this dependence. Unfortunately, within the framework of the proposed model, absolute values of the increments of the acoustic parameters are not given. In [2], when discussing the physical nature of the sound speed field dependence, magnetic fluids are assumed as dispersion media, in which, in addition to the mechanisms typical for continuous media (induced magnetic field inhomogeneity in the acoustic wave [3]), mechanisms determined by the structure peculiarities of magnetic colloids (“slippage” of particles relative to the liquid matrix [4, 5]) exist; the latter result in a much more significant change of the sound speed in the magnetic field. It is shown that the magnetic field non-uniformity can affect the sound speed in magnetic fluids only in very strong non-uniform magnetic fields, with the level of modern measurement error being ∼ 1 m/s. Assuming the problem to be still undecided, we attempted to experimentally determine this dependence. With this purpose in mind, the dependence of the acoustic wave distribution in the MF on the non-uniformity of a magnet magnetic field from a permanent magnet was investigated. 1. Measurement method and the experimental setup. The results of the sound speed spread study in a non-uniformy magnetized magnetic fluid based on the acousto-magnetic effect (AME) are considered in this paper. The idea of the method is the emission of electromagnetic waves by the column of a magnetized MF, with sound waves spreading in it [2]. The electro-motive force (e.m.f.) induced in the loop is proportional to the amplitude of the fluid magnetization fluctuations caused mainly by the concentration fluctuations of nano-particles in the disperse phase. The experimental setup for sound speed measurements in the MF, filling the tube, in a transverse uniform magnetic field induced by a permanent magnet is schematically presented in Fig. 1. Sound vibrations of 20–70 kHz frequency are introduced into the fluid through the plane-parallel bottom of the tube. A magnetic fluid 5 fills a glass tube 10 located between the poles of a permanent magnet 13. The signal from a sound vibration generator 1 passes parallel to a 3

S.G. Yemelyanov, V.M. Polunin, A.M. Storozhenko, E.B. Postnikov, P.A. Ryapolov 12

8 11

6 9

7 10

N

S

5 4

13 1 2

Fig. 1.

3

Schematic presentation of the AME-based experimental setup.

frequency meter 2, voltmeter 3 and a piezoelectric plate 4. The sound signal forms a standing wave passing through the column of the magnetic fluid 5 and reflects from its free surface. A semicircle inductor coil 6 located close to the outer tube surface is tightly connected to the kinematic unit of a cathetometer 12. The voltage from the inductor 6 enters a broadband amplifier 7, from the outlet of which it passes to an oscilloscope 8 and to an external analog-to-digit converter NI USB6251 BNC 9 connected to a laptop 11. The code developed in NI LabView allows filtration of the obtained signals as well as their range decomposition providing the noise level control, definition of the frequency and amplitude of the AME and data storage in the MS Excel format. The copper winding frame of rounded shape is used as an induction probe. It is moved along the tube by the cathetometer with the 0.01 mm accuracy. Additionally, along with the central zone of the interpole gap, the movement zone covers the adjacent area of the uniform field. The permanent magnet 13 is a part H, kA/m 100

50

–100 –50

0

Z, mm

50

100 150

200

Fig. 2. Comparison between the calculated and experimental values of the magnetic field projection. 4

Sound speed in a non-uniformly magnetized magnetic fluid Z, mm 100

50

0

Fig. 3.

Lines of the magnetic field intensity.

θ2 0

ϕ θ

S Bn dL

H0 N

Fig. 4.

Problem schematic presentation.

of the setup. Its magnetic field in the interpole gap and in the neighbourhood has been investigated in detail. The normal to the vertical axis 0Z component of the magnetic field is measured; the axis is located symmetrically to the poles between them. The experimental data and the line, along which the graph is plotted against the overall computational area, are shown in Fig. 2. In addition, the magnetic field was theoretically analyzed using the program for numerical solutions of differential equations in partial derivatives FlexPDE v. 5.0.22 (comparison with the experimental data is shown in Fig. 2). The resulting distribution of the force lines of the non-uniform magnetic field is illustrated in Fig. 3. Let us derive a relation, describing the AME amplitude dependence on a corner ϕ formed between the magnetic field and the normal to the frame (Fig. 4), assuming that the rounded frame is close to the tube surface and the tube axis is perpendicular to the intensity vector. 5

S.G. Yemelyanov, V.M. Polunin, A.M. Storozhenko, E.B. Postnikov, P.A. Ryapolov With static deformation (the tube is rigid) of he liquid column, the magnetic field increment within the model [6] is: δB = μ0 (δM − N δM ) = μ0 (1 − N ) δM,

(1)

where δM is the equilibrium value of the magnetization increment. The demagnetization factor is N = 0.5 in this case. Due to the constant normal component of magnetic induction, at the magnetic boundaries we have δBn(i) = Bn(e) , (i)

(2)

(e)

where δBn and δBn are the normal components of the magnetic induction increment inside the tube and on its surface. δBn(e) = δB · cos θ = μ0 (1 − N ) δM · cos θ,

(3)

where θ is the angle between the magnetic field and the beam bounding the frame. The increment of the magnetic flux through a strip of width δL is δΦ = NB hδL · δBn(e) = NB hδL · μ0 (1 − N ) δM · cos θ,

(4)

where L is the frame length, h is its height (h < < λ); NB is the number of turns. On the other hand, dθ = 2dL/d, where d is the tube diameter. Then, d δΦ = μ0 (1 − N ) NB h · δM · cos θ · dθ, 2

(5)

The magnetic flux penetrating the loop frame is ⎡θ ⎤ 1 θ2 d Φ = Φ1 + Φ2 = μ0 (1 − N ) NB hδM ⎣ cos θ · dθ + cos θ · dθ⎦ . 2 0

(6)

0

In this case, ϕ = (θ2 − θ1 )/2, hence ⎡

(L/d)−ϕ 

d ⎢ Φ = μ0 (1 − N ) NB hδM ⎣ 2

(L/d)+ϕ 

cos θ · dθ + 0

⎤ ⎥ cos θ · dθ⎦

0

(7)

= μ0 (1 − N ) NB dh · δM sin(L/d) cos ϕ. The amplitude of the e.m.f. induced in the loop is e=−

d(ΔΦ) − μ0 (1 − N ) NB dh · δM sin(L/d) cos ϕ. dt

(8)

The used magnetic field gradient ΔH/ΔZ is the main parameter of this problem. It was calculated on the experimental data obtained with the step ΔZ = 5 mm. The dependence of the magnetic field gradient on the coordinate along the 0Z-axis is illustrated in Fig. 5. The magnetic fluid fills the glass tube; the lower part is located between the poles of the permanent magnet. The MF column height is 355 mm. The parameters of the used glass tube are the following: glass NS-3; the Young modulus E = 7.26 · 1010 Pa, the Poisson ratio is 0.21, the density is ρt = 2400 kg/m3 , the longitudinal wave velocity cp = 5500 m/s; the inner and outer radii R1 = 8 mm, R2 = 6.9 mm, the wall thickness h = 1.1 mm. 6

Sound speed in a non-uniformly magnetized magnetic fluid dH/dZ, kA/m2 1000 500

Z, mm –100

–50

0

50

100

–500

–1000

Fig. 5.

Dependence of the magnetic field gradient projection on the coordinate.

1 2

3 4 5

8

6

9 10 11

Fig. 6.

7

Schematic of the acoustic cell.

Ultrasound into the MF column is introduced by the acoustic cell schematically shown in Fig. 6. The acoustic cell used in the experimental setup provides a variable e.m.f. to the piezoelectric plate and its mechanical protection. In addition, the acoustic cell design makes it possible to fix the lower tube end as well as to seal the filling cavity, dismantle, clean and reassembly it. A magnetic fluid 1 fills the glass tube 2 (see Fig. 6). The platen ring 4, lid 5, body 6 are made of a nonmagnetic material (aluminum), fastening screws 3 are made of brass. A sound wave generator 7 induces an alternating voltage of a given frequency, which passes through the spring 10 to the piezoelectric plate 9. The filling cavity is isolated with the rubber ring 8 partly recessed into the annular groove. The lower end of the spring impinges on the bottom of the teflon cup 11. The piezoelectric plate is tightened to the bottom of the cover from below. With this design, the elastic waves are introduced into the MF through the thin bottom plane parallel to the cover. With the restriction Rc > 0.61λ (being satisfied, this contributes to the excitation of zero (piston) oscillation modes in the fluid–cylindrical shell system (here Rc is the tube radius, λ is the sound wavelength). 7

S.G. Yemelyanov, V.M. Polunin, A.M. Storozhenko, E.B. Postnikov, P.A. Ryapolov The gap between the bottom and the piezoelectric plate is filled with a thin layer of contact lubricant for better sound wave passing into the fluid. The sound speed is calculated as follows. At some distance from the bottom of the pipe, not less than 3λ/2, the maximum amplitude of sound waves is recorded and the coordinate Zd is measured several times. Upon countdown of the coordinate of the lower maximum Zd , the cathetometer carriage moves up to some upper maximum amplitude Zu of the e.m.f. The number of half-waves n between Zd and Zu is registered at the same time. When using the oscilloscope in the regime of external synchronization, the phase change is observed on the oscilloscope screen, when the coil passes through the standing wave nodes. The coordinate of the upper peak is also defined several times. The sound wavelength is calculated from the measured coordinates Zu and Zd by the formula: λ2 (Zu − Zd ) /n (9) and its error is

Δλ = λ ·

ΔZu ΔZd + Zu − Zd Zu − Zd

=λ·

2ΔZ Zu − Zd

(10)

where ΔZ is the standard quadratic deviation value of Zu and Zd from the sample average. Then the sound speed in the magnetic fluid, which fills the pipe, is calculated by the formula c = λν (11) where ν is the frequency of sound vibrations introduced into the system. The velocity measurement error was determined from the formula for error calculation in indirect measurements: c =

Δλ Δν Δc = + . c λ ν

(12)

The error in sound speed definition in the MF does not exceed 0.5%. 2. Object of the study. The objects of the current study are two magnetic fluid samples MF1 and MF2, which are colloidal solution of magnetite in a hydrocarbon medium – kerosene stabilized by oleic acid. MF2 sample was produced by diluting the sample with kerosene to an appropriate solid phase concentration. The main MF physical parameters are presented in Table 1 (ρ is the density, ϕ is the solid phase concentration, Ms is the saturation magnetization, χ is the initial magnetic susceptibility). Magnetization curves of MF1 and MF2 obtained by the ballistic method are shown in Fig. 7. The length of the ampoule with the magnetic fluid is much more than its diameter that allows us to neglect the demagnetizing field in this case.

Table 1.

8

Sample

MF1

MF2

ρ, kg/m3 ϕ, % Ms , kA/m χ c, m/s

1360 13.0 57 4.2 930

1028 5.4 25 2.3 1030

Sound speed in a non-uniformly magnetized magnetic fluid 60

M , kA/m

50 40 30 20 10

0

50

Fig. 7.

100

150

200

250

H, kA/m

300

350

400

M (H) dependencies:  – MF1,  - MF2. β , βH

+1

Z, mm 0

200

−1

Fig. 8.

Comparison of β (Z) and βH (Z) curves.

The vertical dotted line in Fig. 7 shows the part of the magnetic field with the largest gradient of intensity. 3. Measurements and analysis. The dependence of the e.m.f. oscillations relative amplitude β on the coordinate Z in the circuit is shown in Fig. 8 (the point with the coordinate Z = 0 coincides with the gap center between the poles of the permanent magnet). The relative strength of the magnetic field βH (Z) is shown in the same figure. One can see the qualitative likeness of the shown dependences. This fact is also in line with the developed theoretical model. At the same time, there are some differences: first, the numerical value of β (Z) is much higher than βH (Z) at large distances from the point Z = 0; second, with the distance, small alternating “steps” are superimposed on monotonic changes of adjacent peaks of the β (Z) dependence. The first difference is explained by the nonlinearity of the dependence β (H), and the second one is explained by the presence of the travelling wave tapped through the holder of the tube and the long-wave oscillations modes. The distance between adjacent peaks is the length of the standing wave. 9

S.G. Yemelyanov, V.M. Polunin, A.M. Storozhenko, E.B. Postnikov, P.A. Ryapolov

, mV

4

3

2 1

0

Fig. 9.

20

40

60

H, kA/m

80

100

The AME amplitude dependence on the magnetic field.

With the existing maxima coordinates and field intensity data at these points, the AME amplitude dependences on the magnetic field have been plotted. One of them is presented in Fig. 9. Dots mark the experimental data and the solid line is the approximation of this dependence by the Microsoft Excel. The nonlinearity of the dependence β (H) is very pronounced there. The acousto-magnetic effect study and the sound speed measurements in the MF using a permanent magnet have some advantages. One may get the field dependence of the induced electromotive force amplitude avoiding the gradual field increase by the electromagnet. It is possible to combine the study of the AME in the non-uniform magnetic field with the measurements of the sound speed in the MF. The magnetic material concentration redistribution may occur in the nonuniform magnetic field, which may be registered due to the varying amplitude of magnetization perturbation, which, in turn, can be used to characterize the MF stability. The results of sound speed measuring in samples MF1 and MF2 at the 41 kHz frequency and 32◦ C are presented in Table 2. The experimental dependence of the sound velocity on the magnetic field gradient shown in Fig. 10 was plotted basing on the measurements results. The measurements were made in the direction of the decreasing magnetic field for

Table 2. Parameters Number of half-waves (the distance between the peaks) Wavelength Sound speed Absolute error of measurement of coordinates of upper and lower peaks Absolute error in determining the wavelength Absolute error in determining the sound speed Relative error in determining the sound speed

10

MF1

MF2

23 22.58 mm 930 m/s

21 24.59 mm 1030 m/s

0.3 mm

0.4 mm

0.06 mm

0.08 mm

2 m/s

3 m/s

0.3%

0.3%

Sound speed in a non-uniformly magnetized magnetic fluid c, m/s

1200

800

600

400

200

–1

–0.5

0

0.5

ΔH/ΔZ · 10−3 , kA/m2 Dependence of the sound velocity on the magnetic field gradient.  - MF1 sample,  - MF2 sample.

Fig. 10.

the samples MF1 and MF2. Values c obtained by formula (11) are represented by points, and the wavelength is calculated as twice the distance between two adjacent antinodes, i.e. by formula (9) for n = 1. The mean velocity values calculated using formulas (9) and (11), where, respectively, n = 23 and n = 21, are shown straight; the upper line was obtained for MF2, the lower one for MF1. We use the Korteweg formula [7] to calculate the sound speed c0 in an unbounded ferrofluid. The fluid compressibility β is expressed from the relation  1 , c0 = ρf β and after simple transformations we derive Eh c0 = c . E  h − 2Rρf c2

(13)

By substituting the parameter values into this formula, we find the following values of the sound speed c0 in the unbounded medium: for MF1 – 1093 m/s, for MF2 – 1194 m/s, which are close to the earlier obtained results for this MF type [2]. Thus, due to the obtained data, the sound speed within the measurement error does not depend on the magnetic field gradient. This fact does not completely preclude the existence of the sound velocity dependence on the magnetic field gradient predicted by theory [1], but does not permit to accept the exclusive role of this parameter. However, this result is consistent with the estimates of the possible influence of the magnetic field non-uniformity on the distribution velocity of sound vibrations in the MF [2]. The largest contribution to the sound velocity increment in nano-dispersed magnetic fluids due to the magnetic field non-uniformity may be obtained in a high-gradient magnetic field as a results of a pressure drop Δp in the fluid. The static equilibrium condition in this case takes the form [8]: ∇p = μ0 M ∇H + ρg.

(14) 11

S.G. Yemelyanov, V.M. Polunin, A.M. Storozhenko, E.B. Postnikov, P.A. Ryapolov By neglecting the hydrostatic pressure and taking into account the fact that, with reference to our data (Fig. 7), the sample magnetizations at the points of the maximum magnetic field gradient are MMF1 = 47 kA/m and MMF2 = 21 kA/m, for the two MF samples we find ∇pMF1 = 0.66 · 105 Pa and ∇pMF1 = 0.29 · 105 Pa. According to [9], the pressure coefficient of the sound speed in this MF type depending on the concentration of the dispersed phase is within (0.34 − 0.38) · 10−5 m/s·Pa, i.e. the average of 0.36 m/s·Pa. Hence, under the experimental conditions, in the ∇H direction the velocity of the average sample length is Δc incremented, respectively, ∼ 0.2 m/s and ∼ 0.1 m/s that is within the measurement error. Dominant contribution to the sound speed increment (up to ∼ 10 m/s) at magnetic fluid magnetization is made by the relative motion of disperse system phases (“slippage” of particles) considered by the dynamic theory [2, 4, 5]. The presented values should be considered as an upper (?) estimate based on the model theory. 4. Conclusions. The acousto-magnetic effect studies and the measurements of the sound speed in magnetic fluids exposed to a non-uniform magnetic field allow the following conclusions. (i) The sound speed within the measurement error does not depend on the magnetic field gradient. This fact does not completely preclude the possibility of the sound speed dependence on the magnetic field gradient predicted by the well-known theory. However, the estimates and the known literature data do not support the statement on the exclusive role of this parameter. (ii) Under the non-uniform magnetic field, it is possible to obtain the AME field dependence avoiding the gradual field intensity increase by using an electromagnet. (iii) It is possible to combine the investigation of the acousto-magnetic effect in the non-uniform magnetic field with the sound velocity measurements in magnetic fluids. (iv) Redistribution of the magnetic material concentration occurs in the nonuniform magnetic field, which can be found from the variation of the amplitude of magnetization perturbation, which, in turn, may be used to characterize the stability of the MF. Acknowledgements. The study presented in this article was supported by the Federal Target Program “Research and Scientific-Pedagogical Personnel of Innovation in Russia” (grant NK-410P, contract no. 2311P). REFERENCES [1] S. Odinaev K. Komilov. Frequency dependences of the velocity and absorption coefficient of sound waves in a magnetic fluid. Acoustic Journal , vol. 54 (2008), no. 6, pp. 796–802 (in Russian). [2] V.M. Polunin. Acoustic Effects in Magnetic Fluids (Moscow, FIZMATLIT, 2008), 208 p. (in Russ.). [3] B.I. Pirozhkov, M.I. Shliomis. The relaxational sound absorption in the ferrosuspension. Porc. the 9th National Acoustics Conf., Section G (Nauka, Moscow, 1977), pp. 123–126 (in Russian). [4] V.V. Gogosov, S.I. Martynov, S.N. Tsurikov, G.A. Shaposhnikova. Ultrasound propagation in a magnetic fluid. I. Inclusion of particle aggrega12

Sound speed in a non-uniformly magnetized magnetic fluid tion: derivation and analysis of the dispersion equation. Magnetohydrodynamics, vol. 23 (1987), no. 2, pp. 131–139. [5] V.V. Gogosov, S.I. Martynov, S.N. Tsurikov, G.A. Shaposhnikova. Propagation of ultrasound in a magnetic liquid. II. Analysis of experiments. Determination of the sizes of aggregates. Magnetohydrodynamics, vol. 23 (1987), no. 3, pp. 241–248. [6] V.M. Polunin, N.S. Kobelev, A.M. Storozhenko, I.A. Shabanova, P.A. Ryapolov. On the estimation of physical parameters of magnetic nanoparticles in magnetic fluid. Magnetohydrodynamics, vol. 46 (2010), no. 1, pp. 31–40. [7] E. Skuchik. Fundamentals of Acoustics (Foreign Literature, Moscow, 1959), vol. 2, 1959, 397 p. (in Russ.). [8] E. Blums, A. Cebers, M. Maiorov. Magnetic Fluids (Walter De Gruyter, 1996), 416 p. [9] S.P. Dmitriev, V.V. Sokolov. Speed of sound in magnetic fluids under high pressures. Mater. 3th School-Seminar on Magnetic Fluids, Ples, 1983) (Moscow State University, 1983), pp. 86–87 (in Russian). Received 15.12.2010

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