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Viscosity measurements of a ferrofluid: comparison with various hydrodynamic equations
Journal of Colloid and Interface Science 263 (2003) 661–664 www.elsevier.com/locate/jcis
Viscosity measurements of a ferrofluid: comparison with various hydrodynamic equations Rajesh Patel, R.V. Upadhyay, and R.V. Mehta ∗ Laboratory of Ferrofluids, Department of Physics, Bhavnagar University, Bhavnagar, 364 002, India Received 23 September 2002; accepted 19 March 2003
Abstract Effective viscosity of a magnetic fluid as a function of applied magnetic field oriented in the perpendicular direction of the capillary flow is determined. Close agreement with the Shliomis expression derived on the basis of effective field method is observed. 2003 Elsevier Inc. All rights reserved. Keywords: Magnetohydrodynamics; Rotational viscosity; Magnetic fluids
1. Introduction A ferrofluid is a heterodispersed system in which a large number of nanomagnets are colloidally suspended in a host liquid. Flow properties of even a normal colloid are sometimes unusual and distinctive, while in a ferrofluid additional complexity is introduced by subjecting it to an external magnetic field. Such a fluid exhibits pronounced non-Newtonian effects under the action of field. Theoretical as well as experimental studies of these magnetically induced rheological effects are essential for designing certain magnetofluidic devices. McTague [1] measured the viscosity of a magnetic fluid by measuring capillary flow under the action of a magnetic field applied in parallel and perpendicular directions of the flow. He showed that the viscosity increases with the field in both the configurations and increment in parallel configuration (η ) is nearly double that in perpendicular configuration (η⊥ ) [1]. Hall and Busenberg [2] developed a theoretical model but their predicted increments were significantly smaller than that observed by McTague [1]. Using the concept of the internal rotation Shliomis [3] gave the phenomenological equation of magnetization and derived expressions of viscosities for two configurations. Theoretical calculations of the viscosities agree well with those observed by McTague.
* Corresponding author.
E-mail address: [email protected] (R.V. Mehta). 0021-9797/03/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0021-9797(03)00325-4
Recently Felderhof [4] revised the conventional hydrodynamic equations for ferrofluids and proposed some modifications in the equation of ferrofluid magnetization. Later Shliomis [5] showed that this phenomenological magnetization equation looks similar to a corresponding Shliomis equation but leads to an erroneous value of viscosity at infinite field. Using the effective field method Shliomis [6] derived the magnetization equation microscopically and obtained more accurate expression for rotational viscosity. In this work we have experimentally determined the incremental viscosity in perpendicular configuration (η⊥ ) at different fields and compared with the expressions given by Shliomis [6] with that of Felderhof [4]. A brief description of the theory involved is given.
2. Theory A ferrofluid resembles an electrical dipolar fluid. Therefore to describe the motion of the dipolar magnetization one can use Debye formalism. When a ferrofluid flows the magnetic dipoles tend to follow the flow direction, because particles are assumed to be rigid magnetic dipoles and its reorientation is possible only with the rotation of the particles. This gives rise to dissipation due to friction. This dissipation further increases if the movement of the dipoles is restricted by applying an external magnetic field. Shliomis [3] derived the magnetization relaxation equation considering a local reference frame (Σ ), in which av-
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R. Patel et al. / Journal of Colloid and Interface Science 263 (2003) 661–664
erage angular velocity of the particle ωp = 0, given by d M 1 = − (M − M0 ), dt τ
leads to an incorrect limiting value given by (1)
where M0 = nµ[coth α − α −1 ], τ is the Brownian time for rotational particle diffusion = 3ηV /(kT ), and α = µH /(kT ). Here, n is the number density of the particles, η is the viscosity, µ is the magnetic moment of the particle, V is the volume of the particle, H is the applied magnetic field, k is Boltzmann’s constant, and T is absolute temperature. If the local frame of reference rotates with the angular velocity, ωp , with respect to the laboratory frame of reference, Eq. (1) can be written as [5] 1 dM 1 = × M − (M − M0 ) − M × (M × H), dt τ 6ηφ
(2)
where is the local angular velocity of the fluid and φ is the volume fraction of the dispersed phase. Considering rotational motion of the particle relative to the carrier liquid and Eq. (2), Shliomis [3] derived an equation for rotational viscosity for Poiseuille or planar Couette flow under the magnetic field H⊥ given by (3)
It is evident from Eq. (3) that in the absence of the field particles will roll freely along corresponding shear surface with ωp = , so that ηr (α) = 0. Similarly for α → ∞, one can consider a rolling of the particle to be slipping and one gets limiting values of ηr (α) as 3ηφ . (4) 2 Hall and Busenberg [2] also obtained the same limiting value given by Eq. (4). It should be noted here that the value of ηr (∞) does not depend on a form of magnetization the equation, but follows the equation of ferrofluid motion [5]. Recently Felderhof [4] modified Eq. (3) using irreversible thermodynamics and showed that the rate of change of magnetization can be written as ηrS1 (∞) =
dM 1 = × M − γH (Hl − H) − M × (M × H), dt 6ηφ
(8)
where relaxation times of the components of magnetization are given by d ln L(α) τ, d ln α
τ⊥ =
2L(α) τ. α − L(α)
(9)
Using these equations, one can write rotational viscosity for H⊥ as
1 (M × H) + , 6ηφ
3 α − tanh α . ηrS1 (α) = ηφ 2 α + tanh α
H[H(M − M0 )] H × (M × H) dM =×M− − , dt τ H 2 τ⊥ H 2
τ =
where ωp =
9ηφ . (7) 10 In the same paper, Shliomis showed that the “local field” Hl considered in the Felderhof [4] model can be replaced by an effective field as discussed in his modified model [5]. In this model, the equation of rate of change of magnetization was derived using the Fokker–Plank equation and the effective field concept. This is given by
ηrF (∞) =
αL2 (α) 3 ηrS2 (α) = ηφ . 2 α − L(α)
(10)
Equation (10) reduces the correct limiting value given in Eq. (4). Figure 1 shows the reduced rotational viscosity (ηr (α)/ ηr (∞)) as a function of α using Eqs. (3), (6), and (10). Equations (3), (6), and (10) are derived for H⊥, but for an arbitary orientation of these vectors the above equations reduce to ηri (α, θ ) = ηri (α) sin2 θ,
(11)
where θ is the angle between H and and ηri (α) are given by Eqs. (3), (6), and (10). In the present case we have determined the viscosity from the time required for the suspension to flow through a capillary placed in a homogeneous magnetic field. Thus we are dealing with Poiseuille flow. Hence isolines of the velocity curl ( = const) are concentric circles in the capillary cross-section plane. In the case H
(5)
where γH is a positive phenomenological constant and Hl is the local magnetization. Using the above equation, Felderhof [4] derived the equation for rotational viscosity for H⊥ as L2 (α) 9 ηrF (α) = ηφ . 2 2 + 3L2 (α)
(6)
Shliomis, in his recent study [5], showed that the limiting value of rotational viscosity (i.e., α → ∞) using Eq. (6)
Fig. 1. Reduced rotational viscosity as a function of α given by Eqs. (3) (dotted line S1 ), (6) (dashed line F ), and (10) (solid line S2 ).
R. Patel et al. / Journal of Colloid and Interface Science 263 (2003) 661–664
perpendicular to the fluid flow direction θ assumes all values from 0 to 2π , so Eq. (11) will be 1 ηri (α, θ ) = ηri (α)sin2 θ = ηri (α). (12) 2 Therefore Eqs. (3), (6), and (10) can be written for H perpendicular to flow direction as 3 α − tanh α , ηrS1 (α) = ηφ 4 α + tanh α
(13)
L2 (α) 9 ηrF (α) = ηφ , 4 2 + 3L2 (α)
(14)
αL2 (α) 3 . ηrS2 (α) = ηφ 4 α − L(α)
(15)
3. Experimental 3.1. Preparation of ferrofluid We followed the well-established chemical coprecipitation technique to synthesize fine ferromagnetic particles of iron ferrite. GPR grades FeCl3 ·6H2 O and FeSO4 ·7H2 O were used to obtain ions of Fe3+ and Fe2+ in aqueous solution. This aqueous solution is added to a 25% NH4 OH so that the pH of the final solution reaches 10. Oleic acid was then added to the precipitates and heated at 95 ◦ C for 5 min. The pH 10 was maintained during the heating process. In order to coagulate the oleic acid coated particles, a dilute acid solution was added. After decantation, the product was washed a number of times with hot distilled water so as to remove impurities. Finally, water was removed by washing the product with acetone. This acetone-wet slurry was dispersed in kerosene and homogenized. The resulting fluid was centrifuged at 16000 g for 20 min. The XRD pattern of the both bare and the coated ferrite particle showed single phase fcc spinel structure.
663
gap of a magnet. The capillary diameter was 0.1 cm and length was approximately 8 cm. Viscosity was measured at room temperature, by varying the magnetic field between 0 and 1000 Gauss.
4. Result and discussion Figure 2 shows the reduced magnetization of the above fluid versus an applied field at room temperature. To describe the magnetization of the magnetic fluid, several theoretical models are used, but the simplest one is Langevin’s model. According to this model the reduced magnetization i.e., M/Ms , is given by M µH , (16) = coth α − α −1 , α = Ms kT where Ms is the saturation magnetization of the fluid, which is given by Ms = φm Md , where φm is the magnetic volume fraction and Md is the domain magnetization of the particle. From the Langevin fit the parameters obtained are median diameter of the log-normal volume distribution Dm = 129 Å, the standard deviation of the logarithmic of the diameter σ = 0.3, Md = 320 emu/cc, and the saturation magnetization Ms = 258 Gauss. Figure 3 shows the reduced rotational viscosity as a function of α for a magnetic field perpendicular to fluid flow direction. Initially, the reduced viscosity increases with field and for high field it tends to saturate. In order to compare this result with the theory of Shliomis [5] and Felderhof [4] we have assumed that the particles dispersed in magnetic fluid are mono-disperse. Assuming this and using Eqs. (14) and (15), we have fitted the experimental data and same is shown in Fig. 3. The good fit was obtained using the parameters D = 150 Å and Md = 320 emu/cc. The fit to the data clearly shows that the Shliomis model agrees well with the results compared to Felderholf model. As magnetic fluid is a polydisperse system, the lognormal size distribution of α (this means the moment distribution and if one assumes that domain magnetization of
3.2. Magnetization measurement A search coil method was used to measure the magnetization of the fluid. A pick-up coil with an inner diameter of 1 cm was wound on a nonmagnetic former and was concentrically placed in an air-core solenoid. Output of the pickup coil was connected with a storage oscilloscope. The solenoid was energized by a constant current power supply and the field was monitored by an axial Hall probe. This assembly was calibrated with a fluid of known magnetization. 3.3. Viscosity measurement Assembly for the rotational viscosity measurement is similar to that used by McTague [1]. The capillary section was horizontal in order to permit investigations within the
Fig. 2. Reduced magnetization versus magnetic field for Fe3 O4 fluid. The line through the data point is fit to Eq. (16) using Dm = 129 Å, σ = 0.3, and Md = 320 emu/cc.
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5. Conclusion The present study shows that the new hydrodynamic equation given by Shliomis is in good agreement with the capillary viscosity measurements of ferrofluids. It is also possible to find the size and size distribution from the fielddependent viscosity measurement.
Acknowledgments
Fig. 3. Experimental verification of reduced viscosity variation with α using effective model of Shliomis (Eq. (15), solid line) and Felderhof (Eq. (14), dashed line) for a monodispersed system. The symbol is experimental points. Inset: Experimental data were fitted using the Shliomis model and log-normal moment distribution (see text for details).
The authors are thankful to the DST-M-15 project, Department of Physics, Bhavnagar University, for providing the ferrofluid for this study. The authors are also thankful to IUC-DAEF for providing financial support in terms of the Project CRS-M-91.
References the particle does not change with size, then one gets the size and size distribution) was used to fit the experimental data. The inset in Fig. 3 shows the fit to Eq. (13) considering the distribution. The best fit was obtained for the parameters Dm = 130 Å, σ = 0.28, and Md = 320 emu/cc. These values are very close to that obtained from magnetization measurements.
[1] [2] [3] [4] [5] [6]
J.P. McTague, J. Chem. Phys. 51 (1969) 133. W.F. Hall, S.N. Busenberg, J. Chem. Phys. 51 (1969) 137. M. Shliomis, Phys. JEPT 34 (1972) 1291. B.U. Felderhof, Phys. Rev. E 62 (2000) 3848. M. Shliomis, cond-mat/0106414, v1 20 Jun 2001. M.A. Martsenyuk, Yu.L. Raikher, M.I. Shliomis, Sov. Phys. JEPT 38 (1974) 413.
Viscosity measurements of a ferrofluid: comparison with various hydrodynamic equations Rajesh Patel, R.V. Upadhyay, and R.V. Mehta ∗ Laboratory of Ferrofluids, Department of Physics, Bhavnagar University, Bhavnagar, 364 002, India Received 23 September 2002; accepted 19 March 2003
Abstract Effective viscosity of a magnetic fluid as a function of applied magnetic field oriented in the perpendicular direction of the capillary flow is determined. Close agreement with the Shliomis expression derived on the basis of effective field method is observed. 2003 Elsevier Inc. All rights reserved. Keywords: Magnetohydrodynamics; Rotational viscosity; Magnetic fluids
1. Introduction A ferrofluid is a heterodispersed system in which a large number of nanomagnets are colloidally suspended in a host liquid. Flow properties of even a normal colloid are sometimes unusual and distinctive, while in a ferrofluid additional complexity is introduced by subjecting it to an external magnetic field. Such a fluid exhibits pronounced non-Newtonian effects under the action of field. Theoretical as well as experimental studies of these magnetically induced rheological effects are essential for designing certain magnetofluidic devices. McTague [1] measured the viscosity of a magnetic fluid by measuring capillary flow under the action of a magnetic field applied in parallel and perpendicular directions of the flow. He showed that the viscosity increases with the field in both the configurations and increment in parallel configuration (η ) is nearly double that in perpendicular configuration (η⊥ ) [1]. Hall and Busenberg [2] developed a theoretical model but their predicted increments were significantly smaller than that observed by McTague [1]. Using the concept of the internal rotation Shliomis [3] gave the phenomenological equation of magnetization and derived expressions of viscosities for two configurations. Theoretical calculations of the viscosities agree well with those observed by McTague.
* Corresponding author.
E-mail address: [email protected] (R.V. Mehta). 0021-9797/03/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0021-9797(03)00325-4
Recently Felderhof [4] revised the conventional hydrodynamic equations for ferrofluids and proposed some modifications in the equation of ferrofluid magnetization. Later Shliomis [5] showed that this phenomenological magnetization equation looks similar to a corresponding Shliomis equation but leads to an erroneous value of viscosity at infinite field. Using the effective field method Shliomis [6] derived the magnetization equation microscopically and obtained more accurate expression for rotational viscosity. In this work we have experimentally determined the incremental viscosity in perpendicular configuration (η⊥ ) at different fields and compared with the expressions given by Shliomis [6] with that of Felderhof [4]. A brief description of the theory involved is given.
2. Theory A ferrofluid resembles an electrical dipolar fluid. Therefore to describe the motion of the dipolar magnetization one can use Debye formalism. When a ferrofluid flows the magnetic dipoles tend to follow the flow direction, because particles are assumed to be rigid magnetic dipoles and its reorientation is possible only with the rotation of the particles. This gives rise to dissipation due to friction. This dissipation further increases if the movement of the dipoles is restricted by applying an external magnetic field. Shliomis [3] derived the magnetization relaxation equation considering a local reference frame (Σ ), in which av-
662
R. Patel et al. / Journal of Colloid and Interface Science 263 (2003) 661–664
erage angular velocity of the particle ωp = 0, given by d M 1 = − (M − M0 ), dt τ
leads to an incorrect limiting value given by (1)
where M0 = nµ[coth α − α −1 ], τ is the Brownian time for rotational particle diffusion = 3ηV /(kT ), and α = µH /(kT ). Here, n is the number density of the particles, η is the viscosity, µ is the magnetic moment of the particle, V is the volume of the particle, H is the applied magnetic field, k is Boltzmann’s constant, and T is absolute temperature. If the local frame of reference rotates with the angular velocity, ωp , with respect to the laboratory frame of reference, Eq. (1) can be written as [5] 1 dM 1 = × M − (M − M0 ) − M × (M × H), dt τ 6ηφ
(2)
where is the local angular velocity of the fluid and φ is the volume fraction of the dispersed phase. Considering rotational motion of the particle relative to the carrier liquid and Eq. (2), Shliomis [3] derived an equation for rotational viscosity for Poiseuille or planar Couette flow under the magnetic field H⊥ given by (3)
It is evident from Eq. (3) that in the absence of the field particles will roll freely along corresponding shear surface with ωp = , so that ηr (α) = 0. Similarly for α → ∞, one can consider a rolling of the particle to be slipping and one gets limiting values of ηr (α) as 3ηφ . (4) 2 Hall and Busenberg [2] also obtained the same limiting value given by Eq. (4). It should be noted here that the value of ηr (∞) does not depend on a form of magnetization the equation, but follows the equation of ferrofluid motion [5]. Recently Felderhof [4] modified Eq. (3) using irreversible thermodynamics and showed that the rate of change of magnetization can be written as ηrS1 (∞) =
dM 1 = × M − γH (Hl − H) − M × (M × H), dt 6ηφ
(8)
where relaxation times of the components of magnetization are given by d ln L(α) τ, d ln α
τ⊥ =
2L(α) τ. α − L(α)
(9)
Using these equations, one can write rotational viscosity for H⊥ as
1 (M × H) + , 6ηφ
3 α − tanh α . ηrS1 (α) = ηφ 2 α + tanh α
H[H(M − M0 )] H × (M × H) dM =×M− − , dt τ H 2 τ⊥ H 2
τ =
where ωp =
9ηφ . (7) 10 In the same paper, Shliomis showed that the “local field” Hl considered in the Felderhof [4] model can be replaced by an effective field as discussed in his modified model [5]. In this model, the equation of rate of change of magnetization was derived using the Fokker–Plank equation and the effective field concept. This is given by
ηrF (∞) =
αL2 (α) 3 ηrS2 (α) = ηφ . 2 α − L(α)
(10)
Equation (10) reduces the correct limiting value given in Eq. (4). Figure 1 shows the reduced rotational viscosity (ηr (α)/ ηr (∞)) as a function of α using Eqs. (3), (6), and (10). Equations (3), (6), and (10) are derived for H⊥, but for an arbitary orientation of these vectors the above equations reduce to ηri (α, θ ) = ηri (α) sin2 θ,
(11)
where θ is the angle between H and and ηri (α) are given by Eqs. (3), (6), and (10). In the present case we have determined the viscosity from the time required for the suspension to flow through a capillary placed in a homogeneous magnetic field. Thus we are dealing with Poiseuille flow. Hence isolines of the velocity curl ( = const) are concentric circles in the capillary cross-section plane. In the case H
(5)
where γH is a positive phenomenological constant and Hl is the local magnetization. Using the above equation, Felderhof [4] derived the equation for rotational viscosity for H⊥ as L2 (α) 9 ηrF (α) = ηφ . 2 2 + 3L2 (α)
(6)
Shliomis, in his recent study [5], showed that the limiting value of rotational viscosity (i.e., α → ∞) using Eq. (6)
Fig. 1. Reduced rotational viscosity as a function of α given by Eqs. (3) (dotted line S1 ), (6) (dashed line F ), and (10) (solid line S2 ).
R. Patel et al. / Journal of Colloid and Interface Science 263 (2003) 661–664
perpendicular to the fluid flow direction θ assumes all values from 0 to 2π , so Eq. (11) will be 1 ηri (α, θ ) = ηri (α)sin2 θ = ηri (α). (12) 2 Therefore Eqs. (3), (6), and (10) can be written for H perpendicular to flow direction as 3 α − tanh α , ηrS1 (α) = ηφ 4 α + tanh α
(13)
L2 (α) 9 ηrF (α) = ηφ , 4 2 + 3L2 (α)
(14)
αL2 (α) 3 . ηrS2 (α) = ηφ 4 α − L(α)
(15)
3. Experimental 3.1. Preparation of ferrofluid We followed the well-established chemical coprecipitation technique to synthesize fine ferromagnetic particles of iron ferrite. GPR grades FeCl3 ·6H2 O and FeSO4 ·7H2 O were used to obtain ions of Fe3+ and Fe2+ in aqueous solution. This aqueous solution is added to a 25% NH4 OH so that the pH of the final solution reaches 10. Oleic acid was then added to the precipitates and heated at 95 ◦ C for 5 min. The pH 10 was maintained during the heating process. In order to coagulate the oleic acid coated particles, a dilute acid solution was added. After decantation, the product was washed a number of times with hot distilled water so as to remove impurities. Finally, water was removed by washing the product with acetone. This acetone-wet slurry was dispersed in kerosene and homogenized. The resulting fluid was centrifuged at 16000 g for 20 min. The XRD pattern of the both bare and the coated ferrite particle showed single phase fcc spinel structure.
663
gap of a magnet. The capillary diameter was 0.1 cm and length was approximately 8 cm. Viscosity was measured at room temperature, by varying the magnetic field between 0 and 1000 Gauss.
4. Result and discussion Figure 2 shows the reduced magnetization of the above fluid versus an applied field at room temperature. To describe the magnetization of the magnetic fluid, several theoretical models are used, but the simplest one is Langevin’s model. According to this model the reduced magnetization i.e., M/Ms , is given by M µH , (16) = coth α − α −1 , α = Ms kT where Ms is the saturation magnetization of the fluid, which is given by Ms = φm Md , where φm is the magnetic volume fraction and Md is the domain magnetization of the particle. From the Langevin fit the parameters obtained are median diameter of the log-normal volume distribution Dm = 129 Å, the standard deviation of the logarithmic of the diameter σ = 0.3, Md = 320 emu/cc, and the saturation magnetization Ms = 258 Gauss. Figure 3 shows the reduced rotational viscosity as a function of α for a magnetic field perpendicular to fluid flow direction. Initially, the reduced viscosity increases with field and for high field it tends to saturate. In order to compare this result with the theory of Shliomis [5] and Felderhof [4] we have assumed that the particles dispersed in magnetic fluid are mono-disperse. Assuming this and using Eqs. (14) and (15), we have fitted the experimental data and same is shown in Fig. 3. The good fit was obtained using the parameters D = 150 Å and Md = 320 emu/cc. The fit to the data clearly shows that the Shliomis model agrees well with the results compared to Felderholf model. As magnetic fluid is a polydisperse system, the lognormal size distribution of α (this means the moment distribution and if one assumes that domain magnetization of
3.2. Magnetization measurement A search coil method was used to measure the magnetization of the fluid. A pick-up coil with an inner diameter of 1 cm was wound on a nonmagnetic former and was concentrically placed in an air-core solenoid. Output of the pickup coil was connected with a storage oscilloscope. The solenoid was energized by a constant current power supply and the field was monitored by an axial Hall probe. This assembly was calibrated with a fluid of known magnetization. 3.3. Viscosity measurement Assembly for the rotational viscosity measurement is similar to that used by McTague [1]. The capillary section was horizontal in order to permit investigations within the
Fig. 2. Reduced magnetization versus magnetic field for Fe3 O4 fluid. The line through the data point is fit to Eq. (16) using Dm = 129 Å, σ = 0.3, and Md = 320 emu/cc.
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5. Conclusion The present study shows that the new hydrodynamic equation given by Shliomis is in good agreement with the capillary viscosity measurements of ferrofluids. It is also possible to find the size and size distribution from the fielddependent viscosity measurement.
Acknowledgments
Fig. 3. Experimental verification of reduced viscosity variation with α using effective model of Shliomis (Eq. (15), solid line) and Felderhof (Eq. (14), dashed line) for a monodispersed system. The symbol is experimental points. Inset: Experimental data were fitted using the Shliomis model and log-normal moment distribution (see text for details).
The authors are thankful to the DST-M-15 project, Department of Physics, Bhavnagar University, for providing the ferrofluid for this study. The authors are also thankful to IUC-DAEF for providing financial support in terms of the Project CRS-M-91.
References the particle does not change with size, then one gets the size and size distribution) was used to fit the experimental data. The inset in Fig. 3 shows the fit to Eq. (13) considering the distribution. The best fit was obtained for the parameters Dm = 130 Å, σ = 0.28, and Md = 320 emu/cc. These values are very close to that obtained from magnetization measurements.
[1] [2] [3] [4] [5] [6]
J.P. McTague, J. Chem. Phys. 51 (1969) 133. W.F. Hall, S.N. Busenberg, J. Chem. Phys. 51 (1969) 137. M. Shliomis, Phys. JEPT 34 (1972) 1291. B.U. Felderhof, Phys. Rev. E 62 (2000) 3848. M. Shliomis, cond-mat/0106414, v1 20 Jun 2001. M.A. Martsenyuk, Yu.L. Raikher, M.I. Shliomis, Sov. Phys. JEPT 38 (1974) 413.