We report on the response of a thin layer of ferrofluid to a spatially
modulated magnetic field. This field is generated by means of a constant
current in a special arrangement of aluminum wires. The full surface profile of
the liquid layer is recorded by means of the absorption of X-rays. The outcome
is analyzed particularly with regard to the magnetic self focusing effect under
a deformable fluid layer

Friedrich, Thomas

Lange, Adrian

Rehberg, Ingo

Richter, Reinhard

English

arXiv

Th. Friedrich

A. Lange

I. Rehberg

R. Richter

Crossref

A note on the magnetic spatial forcing of a ferrofluid layer

Name not available

MAGNETOHYDRODYNAMICS Vol. 47 (2011), No. 2, pp. 167–173A NOTE ON THE MAGNETIC SPATIAL FORCINGOF A FERROFLUID LAYERTh.Friedrich1, A. Lange2, I. Rehberg1, R. Richter11 Experimentalphysik V, Universita¨t Bayreuth, D-95440, Germany2 Lehrstuhl Magnetofluiddynamik, TU Dresden, D-01062, GermanyWe report on the response of a thin layer of ferroﬂuid to a spatially modulated magneticﬁeld. This ﬁeld is generated by means of a constant current in a special arrangement ofaluminum wires. The full surface proﬁle of the liquid layer is recorded by means of theabsorption of X-rays. The outcome is analyzed particularly with regard to the magneticself focusing eﬀect under a deformable ﬂuid layer.Introduction. A beautiful example of spatial forcing in a pattern formingsystem was studied experimentally in electroconvection [1]. More recently, inclinedlayer convection was measured under the inﬂuence of lamellar surface corrugations[2]. In both cases, stripes are the ﬁrst convection pattern beyond a threshold. TheRosensweig instability in a layer of ferroﬂuid can provide a primary instability tohexagons if a homogeneous magnetic ﬁeld normal to the ﬂat surface is applied[3, 4]. In case of a tilted magnetic ﬁeld, a primary instability to stripes can beobserved [5]. This system allows to study the response of both conﬁgurations to astripe like modulation of the magnetic induction.As a start, here we present and characterize the inﬂuence of the modulationon the layer of ferroﬂuid in the subcritical regime. One method of generating aspatially modulated magnetic ﬁeld was described in [6, 7] and uses rods of metalliciron to modulate a homogeneous magnetic ﬁeld. It lacks the possibility to controlthe amplitude of modulation independently from the ﬁeld oﬀset, a nuisance whichis overcome in the experiments presented in this paper.1. Setup. To produce a spatially modulated magnetic ﬁeld, we developed anarray of nonmagnetic conducting wires shown in Fig. 1a. The array is machinedfrom a plate of aluminum by cutting slits from opposite directions using electricaldischarge machining. Aluminum has been chosen because of its low mass absorp-tion coeﬃcient for X-rays. The rectangular cross-section of the wires is crucialto guarantee a spatially homogeneous absorption. By design, the current in thealigned conductors is reversing its direction from one to the next conductor. Theresulting induction μ0Hz is depicted in Fig. 1b. It was measured using a Hallprobe for a certain region of the array as indicated by the arrow in Fig. 1a. Toachieve a spatial forcing with a wavelength of λc = 9.6mm, which correspondsto the critical wavelength [4] of the used ferroﬂuid, the center-to-center distancebetween adjacent conductors has to be λc/2. The components of the generatedmagnetic ﬁeld H can be well approximated byHx = −ΔH cos(kx)e−kz and Hz = H0 +ΔH sin(kx)e−kz , (1)where ΔH represents the amplitude of the ﬁeld modulation, k = 2π/λ is thewave number and H0 states an oﬀset. As Fig. 1b shows, a ﬁt of Hz in Eq. (1)to the experimental data reveals a good agreement. In addition to the spatially167Th. Friedrich, A. Lange, I. Rehberg, R.Richter(a)(b)μ0Hz[mT]0.20.10–0.1–0.2x [mm]0 20 40 60 80Fig. 1. (a) Photograph of the ﬁeld modulation array made of aluminum. Length:14.6 cm, width: 13 cm, thickness: 2mm. (b) Blue: magnetic induction Bz = μ0Hzgenerated by the modulation array at a constant current of 10A, measured via a Hallprobe along the black arrow in (a). The distance between the array and the Hall probewas 2mm. Red: Hz from Eq. (1) ﬁtted to the experimental data.modulated ﬁeld from the array described above, we superimpose a homogeneousmagnetic ﬁeld H0 created by a pair of Helmholtz coils. Thus, H0 and ΔH can becontrolled by two independent currents.The described modulation array is mounted below the bottom of a containermachined from PerspexTM. It contains a block-shaped cavity with a length of120mm, a width of 100mm and a height of 25mm. This cavity is ﬁlled with 30mlof ferroﬂuid EMG909 from Ferrotec Corporation (density ρ = 994.5kg/m3, sur-face tension 23.37mN/m). The ferroﬂuid EMG909 consists of magnetite particlesdispersed in kerosene. Its nonlinear magnetization curve is plotted in Fig. 2. Wemeasured M(H) using a commercial vibrating sample magnetometer (LakeShoreVSM 7404). To represent M(H) for modeling, we used the approximation byVislovich et al. for a nonlinear magnetization [8]. It describes the magnetizationasMvis(H) = MsatHH + Msat/χi. (2)The ﬁt yields the two parameters Msat = 15.2kA/m and χi = 0.99.168A note on the magnetic spatial forcing of a ferrofluid layer0 5 10 15 200246810M[kA/m]H [kA/m]Fig. 2. Magnetization of the ferroﬂuid EMG909 as a function of the internal magneticﬁeld. The points denote experimental data, and the curve is a ﬁt of Eq. (2).2. Experimental results. The application of the modulated magnetic ﬁeldleads to a periodic deformation of the initially ﬂat liquid’s surface. This deforma-tion is driven by the Kelvin force since the magnetized ferroﬂuid is subjected toa periodically varying ﬁeld gradient. To characterize the deformation, we recordX-ray transmission images of the ﬂuid layer to measure its thickness for the wholesurface at the same time. This method has been described in detail in [9, 10]. Thedeformation amplitude of the ferroﬂuid’s free surface depends on ΔH and H0.To resolve this dependance, the ﬁeld oﬀset is kept constant at μ0H0 = 20.8mT,while the modulation amplitude ΔH is successively increased from zero. As theamplitude of the ﬁeld modulation increases, liquid ridges build up. Fig. 3a depictsthe deformed surface for a modulation of μ0ΔH = 0.27mT.As the black symbols in Fig. 3b indicate, the surface already shows someperiodic deformation in the absence of a spatial modulation of the applied magneticﬁeld. In order to measure the amplitude of the deformation, the data are ﬁttedby a phase shifted sine with two higher harmonics (wave numbers k, 2k and 3k)and a constant oﬀset. As a measure for the amplitude of deformation, only theamplitude of the fundamental mode k is taken into account. The resulting datasetis depicted in Fig. 4a.3. The self-focusing eﬀect. Under the deformed surface of a polarizableﬂuid, a self focusing of the magnetic ﬁeld occurs. To characterize this eﬀect with aconvenient parameter, we ﬁrst calculate the ﬁeld Hflat under the artiﬁcial assump-tion of a ﬂat surface. The parameter is then deﬁned as the dimensionless ratiobetween the realistic ﬁeld and Hflat.To derive a relation between the amplitude of the surface deformation and themodulation of the realistic ﬁeld, we use the static form of the ferrohydrodynamicBernoulli equation. The free surface must then be in an equilibrium of pressuresand the thickness of the ferroﬂuid layer is given by ζ(x). Thus, the Bernoulliequation readsC = ρgζ(x)− μ0∫ H(ζ)0MdH ′ + σK − μ02(Mn)2 . (3)Here C is a constant, g represents the gravitational acceleration and σ is the surface169Th. Friedrich, A. Lange, I. Rehberg, R.Richter(a)x[mm]y [mm](b)0.00mT0.07mT0.13mT0.27mTζ[mm]2.82.62.42.2x [mm]0 20 40 60Fig. 3. (a) Measured height proﬁle of the ﬂuid layer under spatial forcing. The ﬁeldoﬀset is μ0H0 = 20.8mT, while the amplitude of modulation is μ0ΔH = 0.27mT. Thelayer thickness of the liquid is color coded as shown by the indicator bar. The drawnisolines are equally spaced each 0.3mm. (b) Average ﬂuid height of the innermost tencolumns (3.6mm) for four diﬀerent values of the modulation amplitudes as nominatedby the inset. The two vertical black lines in (a) mark the displayed region.tension. The normal vector to the surface and the associated curvature readn =1√1 + (∂xζ)2( −∂xζ1)and K = div (n) . (4)Equation (3) has to be fulﬁlled for all points along the surface. In approxi-mation, the surface shape shall be given byζ(x) = h +Δh sin(kx) , (5)where h denotes the thickness of the undisturbed ﬂuid layer, and Δh is its am-plitude of modulation. Now Δh can be numerically determined as a function ofΔH . The results are depicted in Fig. 4a. The dashed red line gives the result fora linear magnetization law of the form M = χH , the dotted blue line shows thesame for the nonlinear Vislovich approximation given by Eq. (2). Clearly this doesnot describe the observed amplitudes as the deformation of the magnetic liquid’ssurface causes a focusing of the magnetic ﬁeld towards the maxima of the ﬂuidheight and accordingly a decrease of the ﬁeld strength at the minimafconc =Hreal(ζmax)Hflat(ζmax), freduc = f−1conc . (6)170A note on the magnetic spatial forcing of a ferrofluid layer(a)Δh[mm]ΔB [mT](b)f conc,f reducΔB [mT]fconcfreducFig. 4. (a) Solid squares: measured amplitude of the surface deformation forΔB = μ0ΔH (measured in the absence of the ferroﬂuid) and μ0H0 = 20.8mT. Thered dashed (blue dotted) line shows the calculated amplitudes without focusing for a lin-ear (nonlinear) magnetization law. The overlapping red solid and blue dot-dashed linesunderneath the data points are calculated by using the values of f shown in (b) for alinear magnetization and nonlinear magnetization, respectively. (b) Values of f neededto ﬁt the measured amplitudes in (a) for linear (red solid lines) and nonlinear (blue dash-dotted lines) magnetization. The red dashed and blue dotted lines at fconc = freduc = 1correspond to the case, where the focusing by a deformed surface was not taken intoaccount.Here, Hreal denotes the ﬁeld strength obtained from the pressure equilibrium inorder to get to the experimentally measured deformation amplitude. To make aproper model, this focusing has to be taken into account. Without solving themagnetostatic problem together with the hydrodynamics, we can estimate theinﬂuence of the deformed surface on the magnetic ﬁeld. Therefore, we calculatethe dimensionless parameters fconc and freduc from the experimental data via thepressure equilibrium. The outcome is shown in Fig. 4b.The focusing of the magnetic ﬁeld has been investigated earlier for fully grownRosensweig cusps in [11]. There, a factor of 1.5 in the alteration of the ﬁeld hasbeen found. As our deformations are about one order of magnitude smaller, achange in the magnetic ﬁeld of about 3% seems realistic.171Th. Friedrich, A. Lange, I. Rehberg, R.RichterExperimentally, we observe a deformation of the surface even without anyﬁeld modulation. This can be understood as a spatial oscillatory decay inducedby the meniscus at the walls of the ﬂuid container in the advent of the Rosensweiginstability. Such an eﬀect has been measured previously [9, 10]. The wave numberkc is favored by the nonmonotonous dispersion relation for surface waves on aferroﬂuid in a magnetic ﬁeld. Note that the width of the container is close to tentimes the critical wavelength (λc = 9.6mm) of the used ferroﬂuid.4. Conclusion and outlook. The described experiment could successfullyproduce a well controlled spatially modulated magnetic ﬁeld and allowed to studythe response of a layer of ferroﬂuid to this ﬁeld. Using radioscopy, accurate dataabout the deformed surface could be gathered. The measured amplitude of theresulting deformation serves to estimate the amount of ﬁeld focusing. This givesa conception of the impact of this eﬀect, as in this case the focusing ampliﬁesthe ﬁeld modulation by almost one order of magnitude compared to the case ofa ﬂat ﬂuid layer. The question how this ampliﬁcation depends on H0 is to beinvestigated. Since the presented calculation uses experimental data to determinethe inﬂuence of the deformed surface on the magnetic ﬁeld further studies arenecessary in order to calculate the equilibrium shape of the ﬂuid ab initio. Themodulation device described in this paper is ideally suited to force Rosensweigpatterns which will be exploited in forthcoming investigations.Acknowledgements. We are grateful to DFG FOR 608 for ﬁnancial sup-port, Ch.Kruelle for inspiring discussions and K.Oetter for making the ﬁeld mod-ulation device possible.REFERENCES[1] M.Lowe, J.Gollub, and T. Lubensky. Commensurate and incommensu-rate structures in a nonequilibrium system. Phys. Rev. Lett., vol. 51 (1983),no. 9, pp. 786–789.[2] G.Seiden, et al. Pattern forming system in the presence of diﬀerentsymmetry-breaking mechanisms. Phys. Rev. Lett., vol. 101 (2008), no. 21,p. 214503.[3] M.D.Cowley and R.E.Rosensweig. The interfacial stability of a ferro-magnetic ﬂuid. J. Fluid Mech., vol. 30 (1967), p. 671.[4] R.E.Rosensweig. Ferrohydrodynamics (Cambridge University Press, Cam-bridge, New York, Melbourne, 1985).[5] B.Reimann, et al. Hexagons become the secondary pattern if symmetry isbroken. Phys. Rev. E , vol. 71 (2005), p. 055202(R).[6] A.Beetz, C.Gollwitzer, R.Richter, and I.Rehberg. Response ofa ferroﬂuid to traveling-stripe forcing. J. Phys.: Condens. Matter , vol. 20(2008), no. 20, p. 204109.[7] T.Friedrich, I.Rehberg, and R.Richter. Magnetic traveling-stripeforcing: Enhanced transport in the advent of the Rosensweig instability. Phys.Rev. E , vol. 82 (2010), no. 3, p. 036304.[8] A.N.Vislovich. Phenomenological equation of static magnetization of mag-netic ﬂuids. vol. 26 (1990), no. 2, pp. 178–183.172A note on the magnetic spatial forcing of a ferrofluid layer[9] R.Richter and J. Bla¨sing. Measuring surface deformations in magneticﬂuid by radioscopy. Rev. Sci. Instrum., vol. 72 (2001), pp. 1729–1733.[10] C.Gollwitzer, et al. The surface topography of a magnetic ﬂuid: a quan-titative comparison between experiment and numerical simulation. J. FluidMech., vol. 571 (2007), pp. 455–474.[11] G.Matthies. Finite element methods for free boundary value problems withcapillary surfaces (Ph.D. thesis, 2002).Received 09.06.2011173

EPub Bayreuth

https://epub.uni-bayreuth.de/3953/1/mahyd-47.2-167.pdf

A note on the magnetic spatial forcing of a ferrofluid layer

Abstract

Similar works

Full text

Available Versions

Crossref

Name not available

EPub Bayreuth