This article describes the results of a theoretical study of magnetic fluid two-dimensional Couett flow in magnetic fluid seal model in view of mass transfer processes. It has been shown that very inhomogeneous magnetic field in seal gap lead to magnetic particle concentration rearrangement due to magnetophoresis and Brownian diffusion. In turn, it lead to inhomogeneous magnetic fluid viscosity and change in local and integral shearing force at channel walls. Integral shearing force has been shown to depend on magnetic field and magnetic fluid parameters. Closely-packed fluid density distribution conditions have been defined. Proposed theory covers real magnetic fluid seal performance features adequately.
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Bashtovoi, V.G.

Kuzhir, R.

Pogirnitskaya, S.G.

Polunin, V.M.

Ryapolov, P.A.

Shabanova, I.A.

Storozhenko, A.M.

Electronic Sumy State University Institutional Repository

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ Vol. 5 No 4, 04011(3pp) (2013) Том 5 № 4, 04011(3cc) (2013)   The article was reported at the International Conference «Physics and Technology of Nanomaterials and Structures», Kursk, 21-22 November, 2013 2077-6772/2013/5(4)04011(3) 04011-1  2013 Sumy State University Influence of Mass Transfer Processes on Couette Flow of Magnetic Fluid  V.G. Bashtovoi1, S.G. Pogirnitskaya1, R. Kuzhir2, V.M. Polunin3, P.A. Ryapolov3,  I.A. Shabanova3, A.M. Storozhenko3  1 Belarusian National Technical University, 65, Nezavisimosti Ave., 220013 Minsk, Belarus 2 Univeristy of Nice-Sophia Antipolis, Parc Valrose 06108 Nice Cedex 2, France 3 Southwest State University, 94, 50 Let Oktyabrya Str., 305040 Kursk, Russia  (Received 27 October 2013; published online 10 December 2013)  This article describes the results of a theoretical study of magnetic fluid two-dimensional Couett flow in magnetic fluid seal model in view of mass transfer processes. It has been shown that very inhomogene-ous magnetic field in seal gap lead to magnetic particle concentration rearrangement due to magnetopho-resis and Brownian diffusion. In turn, it lead to inhomogeneous magnetic fluid viscosity and change in lo-cal and integral shearing force at channel walls. Integral shearing force has been shown to depend on magnetic field and magnetic fluid parameters. Closely-packed fluid density distribution conditions have been defined. Proposed theory covers real magnetic fluid seal performance features adequately.  Keywords: Couett flow, Magnetic fluid, Magnetic particles, Brownian diffusion, Magnetophoresis, concentration, Viscosity, Shearing force.   PACS numbers: 47.11. – j, 47.15. – x   Magnetic fluids are basically colloidal liquids of  solid ferromagnetic particles suspended in a carrier fluid by Brownian motion, i.e. magnetic fluid can be considered as homogenous fluid with magnetic moment (magnetization) M per unit volume. Macroscopic mag-netic fluid behavior in an external gradient magnetic field is determined by the force acting on each ferro-magnetic particle. Also this force cause Brownian mag-netic particles to move relative to the carrier fluid (magnetophoresis phenomenon) resulting in particle concentration rearrangement. Particle volume fraction φ increases along with increasing field strength. In opposition to this, magnetic particles move down the concentration gradient. Both processes result in equi-librium distribution which depends on magnetic field configuration and magnetic fluid properties. As such, the fluid magnetization increases along with the parti-cle concentration. Usually, maximum magnetization regions feature maximum field gradient. It has a signif-icant effect on ponderomotive magnetic force distribu-tion, flotation conditions and all the hydrostatics [1, 2]. These processes are most pronounced in the mag-netic fluid seal [3-5] where magnetic field gradient is extremely high and magnetic fluid film thickness is extremely low. Particle concentration rearrangement inter alia results in increased peak differential pres-sure that the seal holds. This article also notice that the magnetic fluid vis-cosity is significantly determined by the concentration. This fact can be successfully analyzed within the framework developed by articles [1, 2]. Therefore mag-netophoresis and Brownian diffusion can result in local viscosity inhomogeneities as well as changes in shear elasticity on solid surfaces including temporal changes in seal frictional moment. Hereinafter, equations from the articles [1, 2] are cit-ed in order to deal the problem of interaction between mass transfer and magnetic fluid viscosity and shear elasticity on Couett flow in magnetic fluid seal model.   Fig. 1 – Problem geometry  In Fig. 1, Newtonian magnetic fluid of volume (MF) (0  x  a, 0  y  h) fills the gap between two flat solid surfaces y  0 and y  h and remains in its place due to inhomogeneous magnetic field H(x) which is created by sharp constant magnet pole (MP). Such magnetic fluid position can be caused by the differential pressure Δp between side surfaces and its value matches the peak differential pressure that the magnetic fluid seal holds. Due to symmetry, obtained results are valid if the magnetic fluid position is symmetric with respect to pole axis. Side surfaces are assumed to be flat. Lower surface (rotating shaft) (y  0) moves relative to the upper one (y  h) at a rate of v0 along z axis. In such a case, the Couett flow with velocity profile of vz  v0(y/h – 1) starts. Viscous friction stress τ on solid surfaces equals to 0( / ) /zv y v h       and depends on dynamic viscosity coefficient η which in turn de-pends on the particle concentration φ(x):   [φ(x)]. Therefore the shearing force at fluid film border 00 0( / ) ( )a aF w dx wv h x dx          depends on the particle concentration distribution. (w is fluid film length along the surface move direction z).  x y z 0 a h MF M  P  V.G. BASHTOVOI, S.G. POGIRNITSKAYA, R. KUZHIR, ET AL.  J. NANO- ELECTRON. PHYS. 5, 04011 (2013)   04011-2 As a matter of convenience, the following non-dimensional values shall be used: coordinate x'  x/a, magnetic field strength H'  H/H0, relative concentration С  φ / С0, non-dimensional viscosity '  /0, viscous stress 0 0/h v    , viscous friction '0 0/F F h wa v    using typical length a, field strength H0, viscosity 0, and average absolute concentration С0. For classic diffusion [1, 2] processes, static magnetic particle distribution is determined by zero mass flow resulting in the following equation for relative particle concentration (hereinafter, primes will be omitted for non-dimensional values):   ( )C HUC Hx x    (1)  and constant average particle concentration   10( ) 1C x dx   (2)  Furthermore,  10 ( )F x dx    . The problem is described by non-dimensional pa-rameter 0 0 /mU m H kT  (0 – vacuum permeability, mm – particle magnetic moment, k – Boltzmann con-stant, T – absolute temperature) as well as Langevin function ( ) cth( ) 1 / )H UH UH   . Magnetic fluid diffu-sion becomes significant if magnetic parameter U  1. If the particles are of typical magnetic moment of mm  2.5·10 – 19 J/T and the temperature is T  300 K then U  1 takes place for magnetic field strength of about 10 kA/m. In strong high-gradient magnetic fields in magnetic fluid seal gaps this parameter can even achieve two-digit valuesв. Exact solution of the equations (1), (2) is as follows:   110sh( ) sh( )UH UHC dxH H      (3)  Magnetic force Fm for the entire fluid volume is de-termined by magnetization M which in turn depends on magnetic particle concentration C and saturation magnetic moment Ms: M  MsC(x)Λ(H).   0 00 0m sa adH dHF wh M dx wh M C dxdx dx     .   This equation for the non-dimensional values is as follows:   00 01/ ( ) ( )m sdHF wh M H H x C x dxdx      (4)  With regard to (1) the equation can also be as fol-lows:   0 0/ (1 / ) (0) (1)m sF wh M H U C C      (5)  For the homogenous fluid (С  1), this force will be as follows   011 (0)( 1) ln(1)mdH CF Ndz U C         (6)  The interaction between the concentration rear-rangement and magnetic force value can be character-ized by relative magnetic force coefficient:   ( ) (0) (1)( 1) ln (0) ln (1)mmmF С C CkF C C C   (7)  Similarly, the interaction between the concentra-tion rearrangement and the friction on channel walls shall be characterized by relative friction coefficient: ( ) / ( 1).k F С F C     At first, it can easily be shown that linear relations between viscosity coefficient and particle concentration (like the Einstein equation) (1 )С   result in the following equation   1 10 0(1 ) 1 1 ( 1)F С dx Сdx F C            and 1Fk  .  Thus, on the assumption of the linear relations be-tween viscosity and particle concentration the friction on the moving surface doesn’t depend on the concentra-tion distribution. If the relations between viscosity and particle con-centration are not linear kτ doesn’t equal to 1. Moreo-ver, positive nonlinearity results in kτ  1 (increasing friction) and negative nonlinearity results in kτ  1 (de-creasing friction). Further approximation of the relations between vis-cosity and particle concentration is performed using Krieger-Dougherty equation [6]:   max2,5max(1 )   , (8)  where φmax is peak allowable concentration which cor-responds to some type of close packing. For max 0,64   it will be as follows    1,6(1 1,56 )    .  (9)  Magnetic field model approximation H  H(x) de-scribes the fact that field strength quickly decreases along with increasing distance to pole. We can rely on commonly used solution of magnetostatic equations for hyperbolical pole [3]: 21 / 1H nx   or in broader terms:    ( ) 1 / (1 )q mH x nx   (10)  Calculation results for (3)-(10) are shown below in Fig. 2 and 3 and in Table 1. Referring to the Fig. 2, it will be seen that the par-ticle concentration in maximum magnetic field strength regions (x  0) can be several times of that in distant regions due to magnetophoresis and Brownian diffusion. The most significant increase in the particle concentration takes place in about third part of the fluid volume immediately under the magnet pole.  Maximum concentration increases along with increas-ing magnetic parameter U up to the value correspon-ding the close packing as shown in Fig. 2b. Definitely, closely-packed region (φ  φmax) cannot be assumed to  INFLUENCE OF MASS TRANSFER PROCESSES ON COUETTE FLOW… J. NANO- ELECTRON. PHYS. 5, 04011 (2013)   04011-3 be fluid. Closely-packed region volume increases along with the increasing U. The more average particle con-centration, the less U is necessary to close packing. Referring to the Table 1, it will be seen that the aver-age concentration of C0  0.15 requires U  5 to achieve the close packing while C0  0.2 requires U  4.  0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.190.1950.20.2050.210.2150.220.2250.230.235n=9 m=1 U=1 C0=0.2x  0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.7n=9 m=1 U=4 C0=0.2x  Fig. 2 – Typical concentration distribution φ profiles for the magnetic fluid on the assumption of the average concentration of C0  0.2 and two U values: a) U  1, b) U  4. (The magnetic field configuration is n  9, m  1, q  2)  Referring to the Table 1, it will be seen that excess par-ticle concentration on solid surfaces results in increasing friction and kτ  1. For the magnetic fluid seal, it means that processes in question result in increasing friction tor-ques. The proposed shear elasticity calculation model is limited by the infinite viscosity of closely-packed regions according to the Krieger-Dougherty equation. Therefore, there are dashes for such regions in the Table 1. In Fig. 3, the results for equation (7) are shown demonstrating increase in magnetic force coefficient due to magnetic particle concentration rearrangement in the seal gap. The above steady-state problem implies steady-state particle concentration distribution. Definitely, Table 1 – Maximum (C(0)) and minimum (C(1)) concentration and friction coefficient kτ values for the different average par-ticle concentration C0 (The magnetic field configuration is n  9, m  1, q  2)    U 1 2 3 4 5 6 C00.1 C(0) – 0.15 0.22 0.31 0.42 0.55 C(1) – 0.08 0.07 0.05 0.04 0.02 kτ – 1.00 1.02 1.06 1.19 1.67 C00.15 C(0) 0.17 0.22 0.33 0.47 0.64 – C(1) 0.14 0.07 0.1 0.07 0.05 – kτ 1.00 1.02 1.06 1.28 – – C00.2 C(0) 0.25 0.30 0.45 0.64 – – C(1) 0.19 0.17 0.14 0.10 – – kτ 1.00 1.02 1.17 – – –  1 2 3 4 51,01,52,02,53,03,54,0 m=1 m=2 m=3q=1, n=5Magnetic force coefficient kmMagnetic parameter U   Fig. 3 – The relations between magnetic force coefficient km  Fm/Fm(C  1) and magnetic parameter U in different magnetic field configurations  real magnetic fluid seal performance requires certain amount of time achieve the abovementioned steady-state values and the coefficients kτ and km monotonical-ly increase during this time.  CONCLUSIONS  The above results indicate that the proposed theory adequately describes particle concentration rear-rangement in the nanodispersed magnetic fluid due to magnetophoresis and Brownian diffusion and covers real magnetic fluid seal performance features namely increase in friction torque and peak differential pres-sure that the seal holds.  ACKNOWLEDGEMENT  The study was supported by The Ministry of educa-tion and science of Russian Federation, project № 14.В37.21.0906 "The research on interaction of physical fields in acoustomagnetic effect on nanodispersed mag-netic fluid".  REFERENCES  1. V.G. Bashtovoi, V.K. Polevikov, A.E. Suprun, A.V. Stroots, S.A. Beresnev, Magnetohydrodynamics 44, 2, 121 (2008). 2. A.Yu. Zubarev, L.Yu. Iskakova, Magnetohydrodynamics 43 No 1, 3 (2007). 3. S. Taketomi, Jpn. J. Appl. Phys. 19, 1929 (1980). 4. I. Anton, L. Vekash, I. Potents, D. Bika, Magnetohydrodynamics 21 No 3, 213 (1985). 5. V.G. Bashtovoi, V.K. Polevikov, A.M. Algadal, Proceedings of the National Academy of Sciences of Belarus 3, 42 (2006). 6. R.G. Larson, The Structure and Rheology of Complex Fluids (Oxford University Press: New York: 1999). a b 

Influence of Mass Transfer Processes on Couette Flow of Magnetic Fluid

This article describes the results of a theoretical study of magnetic fluid two-dimensional Couett flow in magnetic fluid seal model in view of mass transfer processes. It has been shown that very inhomogene-ous magnetic field in seal gap lead to magnetic particle concentration rearrangement due to magnetopho-resis and Brownian diffusion. In turn, it lead to inhomogeneous magnetic fluid viscosity and change in lo-cal and integral shearing force at channel walls. Integral shearing force has been shown to depend on magnetic field and magnetic fluid parameters. Closely-packed fluid density distribution conditions have been defined. Proposed theory covers real magnetic fluid seal performance features adequately

Influence of Mass Transfer Processes on Couette Flow of Magnetic Fluid

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