The shape of the free surface of a magnetic fluid above a spherical ferromagnetic body immersed in it in a uniform magnetic field is investigated experimentally. The effect of the direction and magnitude of the magnetic field on the deformation characteristics of the free surface of the magnetic fluid with various magnetic properties and geometrical parameters is established

Bashtovoi, V. G.

Motsar, A. A.

Naletova, V. A.

Reks, A. G.

Repository of Belarusian National Technical University (BNTU)

1437ISSN 1063-7842, Technical Physics, 2015, Vol. 60, No. 10, pp. 1437–1442. © Pleiades Publishing, Ltd., 2015.Original Russian Text © V.G. Bashtovoi, A.A. Motsar, V.A. Naletova, A.G. Reks, 2015, published in Zhurnal Tekhnicheskoi Fiziki, 2015, Vol. 85, No. 10, pp. 23–28.The Shape of the Magnetic Fluid Surface above a Magnetizable Sphere in a Uniform Magnetic FieldV. G. Bashtovoia, A. A. Motsara, V. A. Naletovab, and A. G. Reksaa Belarus National Technical University, pr. Nezavisimosti 65, Minsk, 220013 Belarusb Research Institute of Mechanics, Moscow State University, Michurinskii pr. 1, Moscow, 119192 Russiae-mail: alexfx20@yandex.ruReceived June 26, 2014Abstract—The shape of the free surface of a magnetic f luid above a spherical ferromagnetic body immersedin it in a uniform magnetic field is investigated experimentally. The effect of the direction and magnitude ofthe magnetic field on the deformation characteristics of the free surface of the magnetic f luid with variousmagnetic properties and geometrical parameters is established.DOI: 10.1134/S1063784215100060INTRODUCTIONThe interest in studying the shape of the magneticfluid surface is primarily due to its possible practicalapplication in some device, associated with the abilityof a magnetic f luid to change the shape of its free sur-face under the action of a magnetic field. In particular,this property makes it possible to control heat andmass transfer during hardening of solids in a magneticfluid [1] and to use magnetic f luids in various switchesand valves.The introduction of a magnetizable body into auniform magnetic field leads to its distortion and theemergence of local nonuniformities. If the body is in amagnetic f luid, the field nonuniformities caused bydistortions may change the shape of its free surface. Itwas found in [2] that a bounded volume of a magneticfluid is distributed irregularly on a spherical steel bodyin a uniform magnetic field.The problem of the shape of a magnetic f luid withbodies of various shapes immersed in it was consideredin a number of publications. Theoretical and experi-mental investigations of deformation of the magneticfluid surface with a magnetizable cylinder immersedinto it under the action of an external uniform mag-netic field were carried out in [3, 4]. The deformationof the magnetic f luid surface by a magnetizable spherein a uniform magnetic field was considered in [5–7];the conformity between the theoretical and experi-mental results was established in [7].Here, we report on the results of investigationsrevealing some peculiarities in the formation of thefree surface of magnetic f luids of various magnetiza-tions around a spherical ferromagnetic body, whichcontinue the studies started in [6].1. EXPERIMENTAL SETUPIn the experiment, the cuvette with a magneticfluid and with a steel sphere immersed in it was placedinto the uniform magnetic field produced by Helm-holtz coils. The cuvette had the shape of a paral-lelepiped with a size of 81 × 82 mm. The diameter ofthe steel sphere was 6 mm. The amplitude of heightvariation Δh of the free surface above the body wasmeasured by a micrometer with a thin magnetic needleand was photographed. For the origin of coordinates,we took the point on the surface of the f luid in zeromagnetic field, when its entire surface was f lat. Theobservation of the shape of the magnetic f luid surfacewas carried out for various values of uniform magneticfield strength H, the variation of which can be assumedto be quasi-static. The initial thickness h of the layer ofthe magnetic f luid above the body was determined formagnetic field strength H = 0 and was varied by fillingthe cuvette with different volumes of the magneticfluid.We considered several magnetic field directions. Inthe first case, the horizontal direction was parallel tothe f luid surface. In the second case, the magneticfield vector formed an angle of 30° with the horizontal.In the third case, the direction of the field was verticaland perpendicular to the initially f lat equilibrium sur-face of the magnetic f luid.In our experiments, we used magnetic f luids basedon MMTr-10 and MMTr-44 transformer oil and MK-27and MK-51 kerosene (the properties of these f luids aregiven in the table).GASES AND LIQUIDS1438TECHNICAL PHYSICS  Vol. 60  No. 10  2015BASHTOVOI et al.Properties of f luidsProperties of f luids ММТr-44 ММТr-10Fluid base Transformer oilSaturation magnetization Ms, kA/m43.8 9.6Initial magnetic susceptibility χ0 2.7 0.1Density ρf , kg/m3 1400 1115Surface tension σ, mN/m 28 30Wetting angle of the cuvette θw, deg46 31Wetting angle of the body θc, deg 46 31Fig. 1. Shape of the free surface of a magnetic f luid with partial submergence of the sphere into it; H, kA/m: (a) 0, (b) 9.7, and(c) 23.9.(a)Magnetic fluidMagnetic fluidMagnetic fluidThe bodyThe bodyThe bodyHH(b)(c)2. HORIZONTAL MAGNETIC FIELDIn an external uniform magnetic field, the shape ofthe f luid surface was determined by the existence ofregions of local nonuniformity of the field around thesphere. In the case of horizontal direction of the field,its distortion occurs so that regions with the maximalvalue of the field are formed at the “end” faces of thesphere relative to the field, while regions with the min-imal value of the field are formed at the “lateral” faces.If the magnetizable sphere is partly immersed intoa magnetic f luid, the field distortions induced by itlead to the f low of the f luid to the region with the max-imal field strength (end faces of the sphere). The levelof liquid in this case falls in the regions of lateral facesof the sphere, where the field is weaker. This is illus-trated by a set of photographs in Fig. 1.The direction of the horizontal field is indicated byan arrow in the figure. In zero field, the f luid is distrib-uted axisymmetrically around the body (Fig. 1a), andits surface is deformed insignificantly due to wetting ofthe ball surface. In a magnetic field, the surface of thefluid is deformed so that it f lows to the ends regions ofthe sphere and forms a curvilinear surface of an intri-cate shape (Fig. 1b). On the lateral surface of thesphere, a central annular segment not covered with thefluid appears, and in this region dips are formed on thefluid surface. With increasing field strength, the defor-mation amplitude increases, as well as the size of thedips at the lateral surfaces (Fig. 1c).If the depth of the layer of the f luid is equal to thediameter of the sphere, and the thickness of the layerof the magnetic f luid above the body is h = 0, even insmall fields of about 1 kA/m, a dip appears above thesphere with the shape asymmetric relative to its centralaxis (Fig. 2a). In the direction of the field, the f luidflows onto the end regions of the sphere, leaving thecenter of its top open. In the transverse direction, inthe lateral regions of the sphere, zones with a lowerlevel of the f luid are formed. Upon a further increasein the field, the deformation of the surface becomesstronger, and the dip acquires a more complex shape(Fig. 2b). As in the case of the sphere coated with thefluid by half, a zone free of f luid is formed near thecentral transverse segment.If the sphere is submerged so that a layer of themagnetic f luid of thickness h > 0 is formed above it, anasymmetric dip appears in a horizontal magnetic fieldlike in the previous case. An increase in the fieldstrength increases the amplitude of deformation Δh ofthe free surface of the f luid above the body.The type of deformation of the surface and the rateof variation of its deformation amplitude Δh upon achange in the field strength depends on the magneticTECHNICAL PHYSICS  Vol. 60  No. 10  2015THE SHAPE OF THE MAGNETIC FLUID SURFACE 1439properties of the f luid and on initial thickness h of thelayer of the f luid above the sphere.For small layer thicknesses, the deformation of thesurface is smooth: the amplitude of the dip increasesmonotonically with the field until it becomes equal tothe thickness of the f luid layer, and the top of thesphere is open in this case. With increasing thicknessof the f luid layer above the sphere, the rate of variationof the deformation amplitude decreases with increas-ing field (Fig. 3, curves for MMTr-44 fluid, layerthickness is 2.1 and 3 mm). This is due to the fact thatfor a large thickness, the surface of the f luid is locatedin the region at the larger distance from the sphere,where the magnetic field distortions are weaker.It is found that there exists a certain thickness of thefluid layer above the sphere, above which an abruptrupture of the f luid layer down to the sphere surfacemay occur. For such thicknesses of the layer, thereexists the critical value of magnetic field strength Hcrτ1at which the amplitude of the dip on the f luid surfacebecomes jumpwise equal to the thickness of the layer(see Fig. 3, the curve for MMTr-4 f luid, thickness ofthe layer is 3 mm). A vertical channel of an intricateshape changing with increasing field is formed abovethe sphere.For the opposite direction of field variation, thechannel above the sphere disappears also abruptly infields Hcrτ2 smaller than Hcrτ1. The solid arrow in Fig. 3for MMTr-44 fluid shows the variation of deformationamplitude Δh upon an increase in the field, while thedotted arrow shows the variation upon a decrease inthe field. In the range of fields smaller than Hcrτ2, thedependences of the deformation amplitude in the for-ward and reverse directions of the field variation coin-cide (i.e., there is now hysteresis of the shape). Theshape hysteresis is not observed for small thicknessesof the f luid layer above the sphere either.Thus, it can be seen that in the range of fields fromHcrτ2 to Hcrτ1, the hysteresis in the shape of the free sur-face is observed, and the shape depends on the direc-tion of field variation.With increasing thickness h, critical field Hcrτ1increases, as well as the difference between criticalfields Hcrτ1 and Hcrτ2 [7].Figure 3 shows for comparison the characteristicsof deformation for f luids with different magnetiza-tions, MMTe-44 (Ms = 43.8 kA/m) and MMTr-10(Ms = 9.6 kA/m). The deformation of the surfaceoccurs under the action of bulk magnetic forceμ0M grad H, which is proportional to magnetization Mof the f luid. For this reason, for the same magneticfield strength, the amplitude of deformation of thesurface is smaller for the f luid with a lower magnetiza-tion.For MMTr-10 fluid with a small magnetization, incontrast to MMTr-44 fluid with stronger magneticproperties, no jumpwise rupture of the layer of thefluid is observed in the range of magnetic fields underinvestigation up to 30 kA/m, and there is no hysteresisof the shape of the surface. The amplitude of deforma-tion of the free surface of the f luid varies monotoni-Fig. 2. Shape of the free surface of a magnetic f luid with complete submergence of the sphere into it; H, kA/m: (a) 10.2 and(b) 23.9.(a) (b)Magnetic fluid Magnetic fluidThe bodyThe bodyHHFig. 3. Dependence of the amplitude of deformation of themagnetic f luid surface on the external magnetic fieldstrength. 50 10 15 20 25 H, kA/m0∆h, mm−2.5−2.0−3.0−0.5−1.0−1.52.12.13MMTr-10, h (mm)MMTr-44, h (mm)2.93.441440TECHNICAL PHYSICS  Vol. 60  No. 10  2015BASHTOVOI et al.cally with the field. Upon an increase in the thicknessof the f luid layer, the rate of variation of the deforma-tion amplitude decreases.To visualize the variation of the shape of the freesurface in the horizontal field, we performed a set ofexperiments with a magnetizable sphere placed into anarrow slit with a size of 6 × 54 mm. The sphere is atequal distances from the faces of the slit; for this rea-son, in contrast to the case investigated in [6], the f luidwas not only above the sphere, but also under it. Figure 4shows photographs for f luids based on MK-27 andMK-51 kerosene with the saturation magnetizationsof 27 and 51 kA/m, respectively. For the situationsdepicted in the figure, the magnetic field is directedalong the slit.Figures 4a–4d illustrate the shape of the free sur-face of MK-27 fluid, while Figs. 4e and 4f show it forMK-51 fluid with stronger magnetic properties. Fig-ures 4a–4c illustrate the case with fields H < Hcrτ1 priorto rupture of the f luid surface above the sphere. Pho-tographs in Figs. 4d–4f show the rupture of the surfaceof the magnetic f luid in fields exceeding the criticalvalue (H > Hcrτ1).It was found that in fields exceeding Hcrτ1, after theformation of an air channel above the sphere, furtherdeformation of the surface occurs with the formationof a vertical annular air cavity around the sphere. Thiscavity can clearly be seen in Fig. 4f.3. TILTED MAGNETIC FIELDIn a tilted magnetic field, we considered the partialand complete submergence of a magnetizable sphereinto a magnetic f luid. In our experiment, the tilt angleof the magnetic field vector to the horizontal was 30°.The shapes of the f luid surface in a tilted magneticfield are shown in Fig. 5. Figures 5a and 5b show theFig. 4. Shape of the free surface of a magnetic f luid in a horizontal magnetic field for MK-27 magnetic f luid; H, kA/m: (a) 0,(b) 5.7, (c) 8.41, and (d) 19.9; for MK-51 magnetic f luid: (e) 16; (f) 21.8.(a)Field HField HThe bodyThe bodyThe bodyAir cavityFree surfaceof magnetic fluidFree surfaceof magnetic fluidFree surfaceof magnetic fluidField HField HField H(c)(e)(b)(d)(f)Fig. 5. Shape of the free surface of a magnetic f luid in a tilted magnetic field H, kA/m: (a) 16.7, (b) 23, (c) 13.3, and (d) 21.7.The bodyThe body The bodyThe body HHHHTECHNICAL PHYSICS  Vol. 60  No. 10  2015THE SHAPE OF THE MAGNETIC FLUID SURFACE 1441case with a partial submergence of the sphere into thefluid. As in the case with the horizontal field, the f luidflows to the sphere in the end regions of its surfacewhere the magnetic field is stronger. On the one sideof the sphere, the level of the f luid is higher and resem-bles the wave crest.With increasing field strength H, this region of thesurface is transformed into a tilted conical peak (Fig. 5b).Due to the action of the force of gravity, the angle ofinclination of the peak to the horizontal is slightlysmaller than the tilt angle of field H. In the region oflateral surfaces of the sphere, dips with an intricateshape are formed on fluid surface.When the body is completely covered with thefluid, the surface above the sphere is deformed uponthe application of the field, and at a certain value offield strength H, the top of the sphere becomes open,and the f luid f lows to the sphere along the field in thesame way as in the previous case (Figs. 5c, 5d). Withincreasing field, the deformation of the surfaceincreases in the direction of the field. In the range offields under investigation up to 30 kA/m, no conicalpeak was formed along the field.4. VERTICAL MAGNETIC FIELDIn a vertical uniform magnetic field, the variationof the shape of the f luid depends on the depth of sub-mergence of the sphere into the magnetic f luid.If the sphere is half-submerged into the magneticfluid, the application of the external uniform magneticfield leads to the formation of an annular depressioncone is formed around the sphere with a depthincreasing with the field; the height of wetting of thesphere by the f luid changes thereby.When the layer of the f luid covers the sphere com-pletely, the vertical field leads to the emergence of aprotrusion of the f luid above the sphere, the deforma-tion amplitude Δh and the shape of the protrusiondepending on field strength H. For example, it wasestablished in [6] that there exists Hcr1 at which theshape of the protrusion is abruptly transformed from asmooth spherical hill to a conical vertical peak. In thiscase, the rate of variation of the deformation ampli-tude sharply increases with the field. The critical fieldHcr1 increases with thickness h of the layer of the f luid.The characteristics of deformation of the f luid surfacein the vertical field are described in [6, 7].At a certain critical value of field Hcr2, an instabilitymanifested in the jumpwise formation of a set of peaksover the entire surface develops in the region of thefluid surface at large distances from the sphere. It wasestablished earlier [8] that critical field Hcr2 is deter-mined by the depth of the f luid layer. It decreasesmonotonically with increasing depth of the layer. Inthe range of fields H > Hcr2, the rate of variation of thedeformation amplitude above the sphere decreasesupon an increase in the field.With increasing depth of the layer of the f luid,thickness h of the layer above the sphere increases. Thesurface of the liquid is now at a larger distance from thesphere; accordingly, field nonuniformities appearingin this case become smaller. As a result, the influenceof the presence of the sphere in the f luid on the shapeof its surface becomes weaker, and critical fields Hcr1and Hcr2 become closer and tend to the critical field fora thick layer of the f luid.Figure 6 shows the experimental dependences ofthe critical fields for the surface of MMTr-44 mag-netic f luid above the sphere and at a large distancefrom it on thickness h of the layer of the f luid.CONCLUSIONSThe results of the investigation show that zones oflocal inhomogeneity are formed around a sphericalmagnetizable body in a uniform magnetic field as aresult of field redistribution. When this body is placedonto a plane layer of a magnetic f luid in a uniformmagnetic field, the f lat surface of the f luid isdeformed. The characteristics of deformation of thesurface are determined by the magnitude and orienta-tion of the magnetic field.ACKNOWLEDGMENTSThis study was supported by the Foundation forBasic Research of the Belarus Republic and the Rus-sian Foundation for Basic Research (project no. 14-01-90003).Fig. 6. Dependence of critical fields for MMTr-44 on thethickness of the layer of the f luid above the body. 3 4 5 6 7MMTr-44Hcr1Hcr28H, mm7.7H, kA/m7.67.57.47.37.27.17.06.91442TECHNICAL PHYSICS  Vol. 60  No. 10  2015BASHTOVOI et al.REFERENCES1. V. V. Gogosov, V. V. Kiryushin, and A. Ya. Simon-ovskii, Magnetohydrodynamics 30, 163 (1994).2. V. V. Gogosov, V. A. Naletova, and G. A. Shaposh-nikova, Itogi Nauki Tekh., Ser.: Mekh. Zhidk. Gaza 16,76 (1981).3. K. Zimmermann, V. A. Naletova, I. Zeidis, V. A. Turkov,D. A. Pelevina, V. Bohm, and J. Popp, Magnetohydro-dynamics 44, 175 (2008).4. K. Zimmermann, I. Zeidis, V. A. Naletova, V. A. Turkov,J. Popp, A. V. Rozin, and D. A. Pelevina, J. Magn.Magn. Mater. 324, 1253 (2012).5. V. A. Naletova, D. A. Pelevina, and V. A. Turkov, FluidDyn. 44, 797 (2009).6. V. Bashtovoi, A. Motsar, V. Naletova, A. Reks, andD. Pelevina, Magnetohydrodynamics 49, 592 (2013).7. D. Pelevina, V. Naletova, V. Bashtovoi, A. Motsar,andA. Reks, Magnetohydrodynamics 50, 83 (2014).8. V. G. Bashtovoi, B. M. Berkovskii, and A. N. Vislovich,Introduction to Thermomechanics of Magnetic Liquids(IVTAN SSSR, Moscow, 1985).Translated by N. Wadhwa

The Shape of the Magnetic Fluid Surface above a Magnetizable Sphere in a Uniform Magnetic Field

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The Shape of the Magnetic Fluid Surface above a Magnetizable Sphere in a Uniform Magnetic Field

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