Lange, Adrian

Reimann, Bert

Richter, Reinhard

English

EPub Bayreuth

MAGNETOHYDRODYNAMICS Vol. 37 (2001), No. 3, pp. 261{267WAVE NUMBER OF MAXIMAL GROWTHIN VISCOUS FERROFLUIDSA. Lange1, B. Reimann2, R. Richter21Universitat Magdeburg, Institut fur Theoretische Physik, Postfach 4120,D-39106 Magdeburg, Germany2Universitat Bayreuth, Physikalisches Institut, Experimentalphysik V,D-95440 Bayreuth, GermanyWithin the frame of linear stability theory an analytical method is presented for thenormal eld instability in magnetic uids. It allows to calculate the maximal growthrate and the corresponding wave number for arbitrary values of the layer thickness andviscosity. Applying this method to magnetic uids of nite depth, the results are quanti-tatively compared to the wave number of the transient pattern observed experimentallyafter a jumplike increase of the eld. The wave number grows linearly with increasinginduction where the theoretical and the experimental data agree well.Introduction. The most striking phenomenon of pattern formation in mag-netic uids is the Rosensweig or normal eld instability [1]{[4]. Above a thresholdBcof the induction, the initially at surface exhibits a stationary hexagonal pat-tern of peaks. Typically, patterns are characterized by a wave vector q whoseabsolute value gives the wave number q = jqj. There are few but contradictory ex-perimental observations. In experiments where the eld is increased continuously,there are reports about constant [1, 5] as well as about varying wave numbers [6].Notably, all these observations are of entirely qualitative character.All qualitative observations refer to the nal arrangement of peaks. The nalstable pattern, resulting from nonlinear interactions, does not generally correspondto the most unstable linear pattern. Such a pattern should grow with the maxi-mal growth rate and should display the corresponding wave number. Since bothquantities are calculated by the linear theory, the most unstable linear pattern hasto be detected and measured experimentally for a meaningful comparison betweentheory and experiment.1. Theory. A horizontally unbounded layer of an incompressible, noncon-ducting, and viscous magnetic uid of thickness h and constant density  is con-sidered. The uid is bounded from below by the bottom of a container made ofa magnetically impermeable material and has a free surface with air above. Theelectrically insulating uid justies the stationary form of the Maxwell equations,which reduce to the Laplace equation for the magnetic potentials in each of thethree dierent regions. It is assumed that the magnetization of the magnetic uiddepends linearly on the applied magnetic eld, M = (r  1)H, where ris therelative permeability of the uid.In a linear stability analysis, all small disturbances from the basic state areanalysed into normal modes, i.e., they are proportional to exp[ i(! t   q r)]. IfIm(!) > 0, initially small undulations will grow exponentially and the originallyhorizontal surface is unstable. Therefore it has been established to denote ! asgrowth rate, which is in fact true only for its imaginary part in the chosen normalmode ansatz. Following the standard procedure, the linear stability analysis leads261to the dispersion relation [7]{[9] (all formulas in the references are equivalent toeach other)2~q coth(~qh)  q coth(qh)~q4q4+ (q2+ ~q2)2coth(~qh) q4q2~q2+ (q2+ ~q2)2tanh(qh) 4q2~q(q2+ ~q2)cosh(qh) sinh(~qh)+tanh(qh)gq +q3 0rM2(qh) q2= 0 ; (1)where 0is the permeability of free space,  the kinematic viscosity, ~q =pq2  i!=,and(qh) =eqh(1 + r) + e qh(1  r)eqh(1 + r)2  e qh(1  r)2: (2)The condition of marginal stability, ! = 0, denes the threshold where ! changesits sign and therefore the normal eld or Rosensweig instability appears. With! = 0, Eq. (1) reduces to g+ q2 (r  1)2B20rq(qh) = 0 : (3)In the limit of an innitely thick layer (h ! 1), the critical induction and thecritical wave number are [1]B2c;1=20r(r+ 1)p  g(r  1)2qc;1=r g: (4)These critical values apply to both viscous and inviscid magnetic uids due to thestatic character of the instability.1.1. Innite layer. The starting point of the analysis is the determination ofthe parameters for which the dispersion relation (1) for an innitely thick layer[10]1 i!2q22+142q4gq + q3 (r  1)2(r+ 1)0rB2q2=s1 i!q2(5)has solutions of purely imaginary growth rates. Dimensionless quantities are in-troduced for all lengths, the induction, the time, and the viscosityl = qc;1l ;B =BBc;1; (6)t =g3=41=41=4t =ttc;  =g1=43=43=4 ; (7)where tcis the so-called capillary time. The real part of Eq. (5) reduces for ! = i!2tof(q; j!j; ;B) := j!j2q22+q + q3  2B2q24q4  24s1j!jq22= 0 ; (8)where the  sign corresponds to !2? 0. The parameters  andB determine thesolution of this implicit equation for the variables q and j!j. For supercritical in-ductions, the solutions of Eq. (8) have a maximum in the growth rate !m= i !2;m2621.0 1.1 1.2 1.3 1.4 1.5induction B (Bc,    )0.51.01.52.02.53.0maximal wave number qm (q c)0.428ν80ν0.037Fig. 1. Maximal wave number qmas a function of the supercritical inductionB fordierent viscosities. qmis a monotonously increasing function ofB with the exceptionqm= 1 [10] in the case of innitely large viscosities (lower dot-dashed line). In thelimit of an inviscid uid (upper dot-dashed line) the dependence of qmonB is given byqm= (1=3)(2B2+p4B4  3 ) [7].at qm. The wave number with the maximal growth rate is dened by @!2=@q =@j!j=@q = 0. Since j!j is given implicitly by f(q; j!j; ;B) = 0, the maximalgrowth rate results from @qf+= 0. This condition gives a second implicit functionwhose specic form is given in [11]. The cross section of the solutions of f+= 0and @qf+= 0 gives j!mj and qm, which is shown for dierent viscosities in Fig. 1.For all nite viscosities, the wave number qmis not constant, i.e., for nite viscosi-ties qmdepends on the external control parameterB. With increasing viscosity qmvaries less with increasing induction. For small viscosities qmdepends linearly onB ifB is not too large. The analysis reveals that only in the case of innitely largeviscosities a constant wave vector of maximal growth qm= 1 can be expected. Theexperimental observation in [1, 5] cannot be explained by the result of an asymp-totic analysis [10] in which the case of an innitely viscous uid was considered.Experimental uids [6, 12] show a viscosity of  = 0:037 ( ' 6:4 10 6m2s 1),i.e., realistic magnetic uids tend rather to the limit  ! 0.1.2. Finite layer. Since the experiments are performed with a vessel of nitedepth, the analysis presented in the preceding section has to be applied to magneticuids of nite thickness. The implicit equationf+(q; !; ;B; h) :=2[cosh(2~q1h)  1]~q1sinh(2~q1h)  q coth(qh) [cosh(2~q1h)  1]~q1sinh(2~q1h)(5q4+ 2q2~q21+ ~q41)cosh(2~q1h)  1  q tanh(qh) 5q2~q21+ q4+ ~q41 4q2~q1(q2+ ~q21)cosh(qh) sinh(~q1h)+ tanh(qh)gq +q3 (r  1)2B20r(qh) q2= 0 ;(9)2630.00 0.05 0.10 0.15 0.20scaled induction B0.00.20.40.60.8scaled maximal wave number qm0.00 0.040.000.13><(a)>0.00 0.05 0.10 0.15 0.20scaled induction B0.00.51.0scaled maximal growth rate ω2,m><(b)Fig. 2. Scaled maximal wave number q^m(a) and scaled maximal growth rate !^2;m(b)as a function of the scaled supercritical induction^B. The data are calculated for h = 100mm (Æ), 50 mm (), 10 mm (+), 4 mm (), 2 mm (). For h  4 mm, the data aretted by q^m= 3:32^B   0:10p^B for q^m[solid line (a)] and by !^2;m= 1:16p^B + 2:95^Bfor !^2;m[solid line (b)]. Small deviations from the generic behaviour can be seen forh = 2 mm (inset). Material parameters of EMG 901: r= 4:0,  = 1:53  103kgm 3, = 6:54  10 6m2s 1, and  = 2:27  10 2kgs 2.for the variables q and ! contains now the additional parameter h. The inuenceof the layer thickness h on the solution were studied in detail in [11]. This studydisplayed that the wave number of maximal growth qmis less sensitive to changesin the layer thickness than the maximal growth rate !2;m.To analyse the behaviour of !2;mand qmon B and h, again the cross sectionof the solutions of f+= 0 and @qf+= 0 has to be determined. Through theimplicit character of the functions, a general analytical expression cannot be givenfor the dependence of qmand !2;mon , B and h. Alternatively, a two-parametert is tested, which describes the generic behaviour of qmand !2;mon B and h overa wide range of layer thicknesses. An excellent agreement is achieved for h  4mm264byq^m= 3:32^B   0:10p^B for 0:005 ^B  0:4 ; (10)!^2;m= 1:16p^B + 2:95^B for 0:005 ^B  0:4 (11)(see Fig. 2), where^B = (B   Bc;h)=Bc;hand q^m= (qm  qc;h)=qc;hdenote thescaled distances from the critical values. The dimensionless growth rate is givenby !^2;m= !2;mtc. For small^B, the behaviour of q^mis only weakly nonlinearwhereas the behaviour of !^2;mis determined by the square-root term. A carefulinspection of the data reveals that for h = 2 mm (lled circles), small deviationsfrom the proposed ts appear: q^mgrows linearly over the entire^B region [see insetin Fig. 2 (a)]. Thus h = 2 mm indicates the lower limit of the validity of (4, 5)with respect to the layer thickness.2. Experiment and Comparison with Theory. Let us start with asketch of the experimental setup. A more detailed description can be found in[11]. We place a cylindrical Teonrvessel with a diameter of d=12 cm and adepth of 2 mm, completely lled with magnetic uid (EMG 909), in the centerof a pair of Helmholtz coils. A CCD-camera is positioned above the vessel in thecenter of a ring of LEDs. By this construction only an inclined surface of properangle will reect light into the camera. In the theoretical analysis the supercriti-cal magnetic eld is assumed to be instantly present, thus in the experiment themagnetic eld has to be increased jump-like from a subcritical value B0to the de-sired value B. For all measurements B0was xed to 133 10 4T. The induction isrecorded by a Siemens Hall-probe (KSY 13) with short relaxation time, positionedimmediately under the vessel.Fig. 3. Plot of the wave number q versus the magnetic induction B. The opensquares give the experimental values extracted from the circular deformations, examplesof which are given in the insets a), c). The solid line displays the theoretical results forthe material parameters of EMG 909:  = 1:53  103kg m 3;  = 6:54 10 6m2 s 1, and = 2:27  10 2kg  s 2, using r' 1:85 as a t{parameter. The open circles denote thewave number of the nal hexagonal patterns (see inset b), c) e.g.) calculated via a t ofthe central hexagonal structure.265Fig. 3 a) shows circular surface deformations taken 180 ms after a jump likeincrease of the induction. After this transient concentric arrangement a hexagonalpattern of Rosensweig peaks evolves (see Fig. 3 b)). The evolution of the patterncan be understood as a transition from a one dimensional unstable solution [13]to a stable hexagonal one (see Fig. 2 in [3]).Next we focus on the experimental results displayed in Fig. 3, where the wavenumber q is plotted versus the magnetic induction B. Each open square denotesthe wave number extracted from a picture taken during a jump{like increase of themagnetic eld to B > Bc. The estimated maximal errors for q of 4:2% and forB of 0:9% are not plotted for the purpose of clarity. Using ras a t{parametergives the solid line with r' 1:85. The tted value for rdiers by 2:8% fromthe value given by Ferrouidics, a deviation which is well within the tolerance ofproduction specied by Ferrouidics. Obviously there is a rather good agreementbetween the experimental results and the theoretical graph. In contrast to thislinear dependence we nd a constant behaviour for the wave number of the nalhexagonal pattern, which is marked by the open circles. This measured constantvalue conrms the qualitative observations for the nal pattern in [1, 5]. Theexperimental data in Fig. 3 show convincingly the dierence between the linearand nonlinear stages of the pattern forming process.To conclude, we have presented an analytical method which allows to cal-culate the wave number of maximal growth for any combination of experimentalparameters. It has been applied to a liquid layer of 2mm thickness. We havedemonstrated that the transient pattern is the most suitable one to be comparedto the linear theory. The linear increase in the appearing wave number, bothin experiment and in theory, is our main outcome. The induction independentbehaviour of the nal wave number is in agreement with previous observations.However, it is correlated with a nonlinear state which should not be comparedwith a linear theory.Acknowledgments. The authors prot from stimulating discussions withJ. Berg, A. Engel, R. Friedrichs, H. W. Muller, I. Rehberg and A. Rothert. Thiswork was supported by the Deutsche Forschungsgemeinschaft under Grant EN278/2 and LA 1182/2-1.REFERENCES1. M. D. Cowley and R. E. Rosensweig. The interfacial stability of a ferromagneticuid. J. Fluid Mech., vol. 30 (1967), no. 4, pp. 671-688.2. R. E. Rosensweig. Ferrohydrodynamics (Cambridge University Press, Cambridge,1985).3. A. Gailitis. Formation of the hexagonal pattern of the surface of a ferromagneticuid in an applied magnetic eld. J. Fluid Mech., vol. 82 (1977), no. 3, pp. 401{413.4. V. G. Bashtovoi, M. S. Krakov, and A. G. Recks. Instability of a at layerof magnetic liquid for supercritical magnetic elds. Magnetohydrodynamics, vol. 21(1985), no. 1, pp. 14{19.5. J.-C. Bacri and D. Salin. First-order transition in the instability of a magneticuid interface. J. Phys. (France), vol. 45 (1984), no. 11, pp. L559{L564.6. T. Mahr. Oberacheninstabilitaten bei Ferrouiden. Ph.D. thesis, UniversitatMagdeburg, 1998.7. B. Abou, G. Neron de Surgy, and J. E. Wesfreid. Dispersion relation in aferrouid layer of any thickness and viscosity in a normal magnetic eld; asymptoticregimes. J. Phys. II, vol. 7 (1997), no. 8, pp. 1159{1171.2668. J. Weilepp and H. R. Brand. Competition between the Benard-Marangoni andthe Rosensweig instability in magnetic uids. J. Phys. II, vol. 6 (1996), no. 3, pp. 419{441.9. H. W. Muller. Parametrically driven surface waves on viscous ferrouids. Phys.Rev. E, vol. 58 (1998), no. 5, pp. 6199{6205.10. D. Salin. Wave vector selection in the instability of an interface in a magnetic orelectric eld. Europhys. Lett., vol. 21 (1993), no. 6, pp. 667{670.11. A. Lange, B. Reimann, and R. Richter. Wave number of maximal growth inviscous magnetic uids of arbitrary depth. Phys. Rev. E, vol. 61 (2000), no. 5,pp. 5528{5539.12. T. Mahr and I. Rehberg. Nonlinear dynamics of a single ferrouid-peak in anoscillating magnetic eld. Physica D, vol. 111 (1998), pp. 335{346.13. V. M. Zaitsev, and M. I. Shliomis. Doklady Akad. Nauk S.S.S.R., vol. 188 (1969),pp. 1261{1264; english translation: V.M. Zaitzev and M.I. Shliomis. Nature ofthe instability of the interface between two liquids in a constant eld. Sov. Phys.-Doklady, vol. 14 (1970), no. 10, pp. 1001-1002.Received 22.05.2001267

Wave number of maximal growth in viscous ferrofluids

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Wave number of maximal growth in viscous ferrofluids

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