We study the influence of surface tension on the shape of the conical
miniscus built up by a magnetic fluid surrounding a current-carrying wire.
Minimization of the total energy of the system leads to a singular second order
boundary value problem for the function $\zeta(r)$ describing the axially
symmetric shape of the free surface. An appropriate transformation regularizes
the problem and allows a straightforward numerical solution. We also study the
effects a superimposed second liquid, a nonlinear magnetization law of the
magnetic fluid, and the influence of the diameter of the wire on the free
surface profile

Engel, Andreas

John, Thomas

Rannacher, Dirk

English

arXiv

We study the influence of surface tension on the shape of the conical miniscus built up by a magnetic fluid surrounding a current-carrying wire. Minimization of the total energy of the system leads to a singular second order boundary value problem for the function zeta(r) describing the axially symmetric shape of the free surface. An appropriate transformation regularizes the problem and allows a straightforward numerical solution. We also study the effects a superimposed second liquid, a nonlinear magnetization law of the magnetic fluid, and the influence of the diameter of the wire on the free surface profile

Engel, Adreas

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Mathematics - Faculty Scholarship Mathematics 3-6-2006 Influence of Surface Tension on the Conical Miniscus of a Magnetic Fluid in the Field of a Current-Carrying Wire Thomas John Carl von Ossietzky Universtitat Dirk Rannacher Carl von Ossietzky Universitat Adreas Engel Carl von Ossietzky Universitat Follow this and additional works at: https://surface.syr.edu/mat  Part of the Mathematics Commons Recommended Citation John, Thomas; Rannacher, Dirk; and Engel, Adreas, "Influence of Surface Tension on the Conical Miniscus of a Magnetic Fluid in the Field of a Current-Carrying Wire" (2006). Mathematics - Faculty Scholarship. 90. https://surface.syr.edu/mat/90 This Article is brought to you for free and open access by the Mathematics at SURFACE. It has been accepted for inclusion in Mathematics - Faculty Scholarship by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. arXiv:cond-mat/0603136v1  [cond-mat.mtrl-sci]  6 Mar 2006Influence of surface tension on the conical miniscus of a magnetic fluid in the field of acurrent-carrying wireThomas John, Dirk Rannacher, and Andreas EngelInstitut für Physik, Carl von Ossietzky Universtität, 26111 Oldenburg, GermanyWe study the influence of surface tension on the shape of the conical miniscus built up by amagnetic fluid surrounding a current-carrying wire. Minimization of the total energy of the systemleads to a singular second order boundary value problem for the function ζ(r) describing the axiallysymmetric shape of the free surface. An appropriate transformation regularizes the problem andallows a straightforward numerical solution. We also study the effects a superimposed second liquid,a nonlinear magnetization law of the magnetic fluid, and the influence of the diameter of the wireon the free surface profile.PACS numbers: 75.50.Mm, 68.03.Cd, 02.60.Lj,I. INTRODUCTIONThe shape of the free surface of a ferrofluid in a staticmagnetic field is one of the prominent examples for a non-trivial interplay between magnetic and hydrodynamic de-grees of freedom in ferrohydrodynamics: On the one handthe magnetic stresses contribute to the force balance de-termining the surface profile whereas on the other handthe local magnetic field depends on this profile due to themagnetic boundary conditions at the surface of the mag-netically permeable material [1]. A particularily popularsetup is the conical meniscus of a ferrofluid surrounding avertical current-conducting wire [1], see Fig. 1. Neglect-ing surface tension the conical shape can be determinedanalytically from the balance between gravitational andmagnetic force [1].In the present note we investigate theoretically the in-fluence of surface tension on the shape of the conicalmeniscus. Although surface tension is known to mod-ify free surface profiles in ferrohydrodynamics and its in-fluence has been studied, e.g., for small gaps [2, 3] andcapillaries [4, 5] to our knowledge no systematic inves-tigation has been done so far for the conical meniscusproblem. Clearly, surface tension is present in all exper-iments in ferrohydrodynamics and a full understandingof its impact on a basic experiment in the field is hencedesirable.As expected no analytical expression for the free sur-face profile can be derived when surface tension is in-cluded. The numerical determination can be reduced to asingular boundary value problem for an ordinary differen-tial equation which is, however, not completely straight-forward to solve. Still, using an appropriate transforma-tion, accurate solutions can be obtained. The numeri-cal effort is much smaller than in alternative proceduressuch as finite difference, Galerkin, or collocation methodswhich is for example used from commercial software likeANSYS or COMSOL Multiphysics [6, 7]The paper is organized as follows. In section II we col-lect the basic equations describing the system. Section IIIcontains the analysis of a somewhat idealized situation.In section IV we discuss the modification of the resultsif the idealizations of section III are removed. Finallysection V provides some conclusions.II. BASIC EQUATIONSThe setup to be investigated is sketched in Fig. 1. Aninfinitely long, straight wire of radius R oriented alongthe z-axis of a cylindrical coordinate system (r, φ, z) car-ries an electric current I. The wire is surrounded by twosuperimposed liquids. The lower one with density ̺1 isa ferrofluid with a given magnetization law ~M = ~M( ~H).The upper one has density ̺2 < ̺1 and is a non-magneticfluid. The interface between the two fluids is character-ized by an interface tension σ1,2. For I = 0 there is nomagnetic field and the flat interface between the two flu-ids is taken as the z = 0–plane of the coordinate system.For I 6= 0 a magnetic field~H(r) =I2πr~eφ (1)builds up which induces a force density in the lowerfluid. This gives rise to an axis-symmetric conical inter-face parametrized by a function ζ(r) the determination ofwhich is the central aim. Note that the magnetic field (1)is everywhere tangential to the surface and is thereforeindependent of the detailed shape of the surface whichmakes the analysis of this case particularily transparent.One way to determine ζ(r) is by minimizing the to-tal energy Etot of the system. It is convenient thento use the energy of the flat interface configuration asreference state and hence to minimize the energy differ-ence between states with a non-trivial ζ(r) and ζ ≡ 0.Equivalently one may start with the ferrohydrodynamicBernoulli equation [1, 8].The total energy difference Etot is the sum of threeparts, the gravitational, the surface, and the magneticenergy,Etot = Eg + Es + Em. (2)2FIG. 1: Sketch of the setup and definition of the main vari-ables. The free surface ζ(r) of the ferrofluid is axis-symmetric.The different parts are given, respectively, byEg =∫Vd3r (̺1 − ̺2)gz = π(̺1 − ̺2)g∫∞Rdr r ζ(r)2 (3)Es =∫∂Vd2r σ1,2 = 2πσ1,2∫∞Rdr r(√1 + ζ′(r)2 − 1)(4)and [1, 9]Em = −∫Vd3r µ0∫ H(~r)0d ~H · ~M( ~H)= −2πµ0∫∞Rdr r ζ(r)∫ H(r)0dHM(H) (5)Here V denotes that part of the volume occupied by theferrofluid for which z > 0, ∂V is the area of the interfacebetween the two fluids and µ0 is the permeability of freespace. Note that in the last equality of (5) we have as-sumed that the magnetization ~M and the magnetic field~H are parallel as is the case in ferrofluids. Note also thatH(r) is given by (1).The Euler-Lagrange equationδEtotδζ(r)= 0 (6)corresponding to the minimization of Etot[ζ(r)] is a non-linear ordinary differential equation for the desired inter-face profile ζ(r). For a unique solution this equation hasto be complemented by appropriate boundary conditions.These arelimr→∞ζ(r) = 0 (7)andζ′(R) =σ1,3 − σ2,3√σ21,2 − (σ1,3 − σ2,3)2, (8)where the prime denotes differentiation with respect to r.The second boundary condition results from the Youngequation [10]σ2,3 = σ1,3 + σ1,2 cos θ, (9)for the contact angle θ of the fluids at the wire and thefact that ζ′(R) = − tan(π/2 − θ) (cf. Fig. 1).III. SIMPLIFIED MODELWe first analyse a somewhat simplified version of theboundary value problem derived in the last section. Thesimplifications are the following. We assume that thenon-magnetic fluid is absent, ̺2 = 0, that the magneti-zation law of the ferrofluid is linear, ~M(H) = χ ~H whereχ denotes the magnetic susceptibility, and that the di-ameter of the wire is negligible, R = 0.Since the magnetic field H(r) diverges for r → 0 sodoes the magnetic force density. In order to get a stableinterface profile the magnetic force has to be counterbal-anced by gravitation and surface tension which is possibleonly iflimr→0ζ(r) = ∞. (10)This equation replaces the boundary condition (8) in thepresent case. Consequently σ1,3 and σ2,3 are irrelevantand only σ1,2 remains which will be denoted simply by σin the present section.Using the simplifying assumptions we find for the mag-netic energy (5) the expressionEm = −µ0 χ I24π∫∞0drζ(r)r, (11)which allows to explicitly perform the variation in eq.(6).As a result we get the following differential equation forζ(r)g̺1ζ −µ0χI28π21r2− σζ′ + ζ′3 + rζ′′(1 + ζ′2)3/21r= 0. (12)It is convenient to introduce dimensionless quantities bymeasuring both r and ζ in units ofa = 3√µ0χI28π2g̺1(13)and σ in units of ̺1ga2. Eq. (12) then acquires the formζ −1r2− σζ′ + ζ′3 + rζ′′(1 + ζ′2)3/21r= 0. (14)This equation is easily solved for σ = 0 yielding the well-known hyperbola ζ(r) = 1/r2 [1]. A perturbative solu-tion for σ 6= 0 by expanding ζ(r) in powers of σ wasfound to yield satisfactory results only for values of σ3much smaller than those relevant in experiments. Wetherefore turned to a numerical solution.Eq. (14) together with the boundary conditions (7) and(10) represents a singular boundary value problem of sec-ond kind (see, e.g., [11]). The main problem that makesa straightforward numerical solution inexpedient are theinfinite intervals for r and ζ. We therefore transformboth quantities according to (cf. also Fig. 1)ζ(ρ, ϕ) =sin ϕρ, r(ρ, ϕ) =cosϕρ(15)and describe the interface profile by ρ = ρ(ϕ). Theboundary conditions (7) and (10) then transform intoρ(ϕ = 0) = 0 and ρ(ϕ =π2)= 0 (16)respectively which are much more convenient for the sub-sequent numerical solution. Substituting the derivativesin (14) according toζ′ =dζdr=ρ′ sin ϕ − ρ cosϕρ sin ϕ + ρ′ cosϕ, (17)ζ′′ =d2ζdr2= −ρ3(ρ + ρ′′)(ρ sin ϕ + ρ′ cosϕ)3, (18)we find as differential equation for ρ(ϕ)(sin ϕ − ρ3 sec2 ϕ) (ρ2 + ρ′2)3/2+ σρ2(ρρ′2 − ρ′3 tanϕ − ρ2ρ′ tan ϕ + ρ2 (2ρ + ρ′′))= 0. (19)Rewriting this equation in the form ρ′′(ϕ) =f(ϕ, ρ(ϕ), ρ′(ϕ)) the boundary value problem (19), (16)can now easily be solved numerically using, e. g., thenonlinear finite-difference method described in [12]. Asinitial guess for the solution which is needed in thisprocedure the transformation of the analytical solutionζ0(r) = 1/r2 for σ = 0 given by ρ0(ϕ) = cos(ϕ)3√tan(ϕ)may be used.Fig.2 shows the resulting interface profile for param-eter values corresponding to the ferrofluid EMG909 ofFerrotec [13]. For comparison the shape for σ = 0 is alsoshown. It is clearly seen that the free surface profile ismarkedly modified by the influence of the surface ten-sion. As expected the inclusion of surface tension makesthe profile narrower since an additional force pointingradially inward builds up.IV. MORE REALISTIC SITUATIONSThe above solution of the idealized problem forms aconvenient starting point for the discussion of the in-fluence of those feature in the original setup that wereneglected in the previous section.A simple modification is the case in which the ferrofluidis superimposed by a non-magnetic liquid. Then the den-sity ̺1 has to substituted by the density difference ̺1−̺2and the surface tension σ is to be replaced by the interfacetension σ1,2. The gravitational contribution to the to-tal energy gets reduced and consequently larger displace-ments from the flat interface are to be expected. Thesituation can be analyzed quantitatively without furthereffort by mapping it to the case analyzed in the previoussection. In fact the interface profile is again determined0 1 2 301230 ¼/4 ¼/20.00.5(a)Elongation ³(r)/mmDistance r/mm(b)Angle 'Inverse distance ½(')FIG. 2: (a) Free surface profile ζ(r) as determined numericallyfor R = 0 and I = 30 A for the ferrofluid EMG909 withparameters ̺1 = 1120 kgm−3, σ = 0.0259 Nm−1, and χ = 0.8(full line). Also shown is the result for σ = 0 and otherwiseidentical parameters (dashed line). (b) Same profiles as in (a)in terms of the transformed function ρ(ϕ).by eq. (14) with only the dimensionless units and thevalue of σ being modified. As an explicit example weshow in Fig. 3(a) how the interface profile of Fig. 2 getsmodified for ̺2 = 103 kgm−3 and σ1,2 = σ.Deviations form the linear magnetization law whichbecome relevant in particular for small values of r wherethe magnetic field gets strong can also be dealt with.An improved approximation is provided by the Langevin4relation [1],M(H) = Ms(coth(α) −1α), α =mHkBT, (20)with the saturation magnetization of the fluid MS, themagnetic moment m of a single ferromagnetic parti-cle, the Boltzmann-constant kB and the temperatureT . The zero-field susceptibility is then given by χ =mMS/(3kBT ). The density of the magnetic energy canagain be determined analytically∫ H(~r)0d ~H · ~M( ~H) =MSkBTmlnsinhαα. (21)This expression gives rise to a somewhat more compli-cated equation for ζ(r), however, the overall procedureof section III remains valid. As shown in Fig. 3(a) themodification of the result for linear magnetization law isusually small in accordance with the fact that the frac-tion of the ferrofluid volume for which the field is largeis rather small.Finally, to elucidate the influence of a non-zero radiusof the wire we consider the problem for R > 0 but with̺2 = 0 and a linear magnetization law. The main qual-itative difference is that instead of (10) we have to use(8) as boundary condition for small r. This implies thatζ(r) is not singular at the lower boundary. Hence al-ready the simple transformation r = 1/u is sufficient toget a numerically tractable boundary value problem forthe function ζ(u). As initial guess a linear dependenceζ0(u) = λu may be used.Note that λ fixes the value of ζ(r) at r = R whereasthe boundary condition (8) involves the derivative ζ′(r)at r = R. Therefore λ or equivalently ζ(R) has to bemodified until the required value for ζ′(R) is obtained.This procedure leads to a monotonic relation betweenthe prescribed contact angle θ and ζ(R), cf. Fig. 3(c).With the help this value for ζ(R) the complete surfaceprofile ζ(r) can then be determined. Fig. 3(b) showsa collection of surface profiles with different ζ(R) andcorresoponding contact angles θ. In the limiting caseR → 0, the surface profile is independent of the contactangle θ as demonstrated in Fig. 3(d) for a sequence ofprofiles with different values of R and a fixed contactangle θ = 45°.V. CONCLUSIONSIn the present investigation we have determined thefree surface profile of a magnetic fluid surrounding a ver-tical current carrying wire with special emphasis on theeffects of interface and surface tension respectively. Wehave found that for experimentally relevant parametervalues there is a strong influence of the surface tensiongiving rise to a more slender profile as compared to thewell-studied case without surface tension [1]. A more gen-eral magnetization law including saturation as well as theprescription of the contact angle at the wire changes theprofile only in the vicinity of the wire. A superimposednon-magnetic liquid changes the overall scale of the pro-file due to the reduction of the gravitational energy.[1] R. E. Rosensweig, Ferrohydrodynamics, Dover Publica-tions, New York, 1997.[2] C. Flament, S. Lacis, J.-C. Bacri, A. Cebers, S. Neveu,R. Perzynski, Measurements of ferrofluid surface tensionin confined geometry, Phys. Rev. E 53 (96) 4801.[3] V. Polevikov, L. Tobiska, Instability of magnetic fluid ina narrow gap between plates, Journal of Magnetism andMagnetic Materials 289 (2005) 379.[4] V. Bashtovoi, G. Bossis, P. Kuzhir, A. Reks, Magneticfield effect on capillary rise of magnetic fluids, Journal ofMagnetism and Magnetic Materials 289 (2005) 376.[5] V. Bashtovoi, P. Kuzhir, A. Reks, Capillary ascensionof magnetic fluids, Journal of Magnetism and MagneticMaterials 252 (2002) 265.[6] ANSYS, http://www.ansys.com.[7] COMSOL Multiphysics, http://www.femlab.de.[8] B. Berkovski, Fluid mechanical phenomena, in: V. Bash-tovoi (Ed.), Magnetic Fluids and Applications Hand-book, Begell House, New York, 1996, Ch. 3, p. 395.[9] L. D. Landau, E. M. Lifshitz, Electrodynamics of Con-tinuous Media, Vol. 8, Pergamon Press, Oxford, 1980.[10] L. D. Landau, E. M. Lifshitz, Statistical Physics, Vol. 5,Pergamon Press, Oxford, 1980.[11] U. M. Asher, R. M. M. Mattheij, R. D. Russel, Numer-ical Solution of Boundary Value Problems for OrdinaryDifferential Equations, Siam, Philadelphia, 1995.[12] J. D. Faires, R. L. Burden, Numerical Methods, PWS -Publishing Company, Boston, 1993.[13] FerroTec, http://www.ferrofluidics.de.50 1 2 30123450 1 2 30120120 1 2 30123(a)Elongation ³(r)/mmDistance r/mmµ(b)Elongation ³(r)/mmDistance r/mmR(c)0            ¼/2            ¼Contact angle µHeight ³(R)/mm(d)Elongation ³(r)/mmDistance r/mmFIG. 3: Modifications of the idealized results of section 3. If not explicitly stated otherwise the parameters are the same as inFig. 2. The dashed line shows always the same interface profile as the full line in Fig. 2. (a) The dotted line shows the interfaceprofile if a non-magnetic fluid with density ̺2 = 103 kgm−3 is superimposed to the ferrofluid. Also in (a), the solid line showsthe interface profile for a ferrofluid with magnetization law of Langevin type with Ms = 16 kAm−1 and the same zero-fieldsusceptibility χ as in Fig. 2. (b) Dependence of the interface profile on the contact angle θ for non-zero radius R = 0.5 mm.(c) Relation between the contact angle θ and the height of the interface at the wire ζ(R). Marked points correspond to theprofiles shown in (b). (d) Sequence of surface profiles with fixed contact angle θ = 45° for decreasing diameter of the wire,R = 2; 1.5; 1; 0.5; 0.2 mm. For R→ 0 the profile of Fig. 2 is almost reproduced.

Influence of surface tension on the conical miniscus of a magnetic fluid in the field of a current-carrying wire

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