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Dynamic analysis on magnetic fluid interface validated by physical laws
Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Dynamic analysis on magnetic fluid interface validated by physical laws Yo Mizuta Division of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
art ic l e i nf o
a b s t r a c t
Article history: Received 30 June 2016 Received in revised form 12 August 2016 Accepted 5 September 2016
Numerical analyses of magnetic fluid especially for fast phenomena such as the transition among interface profiles require rigorous as well as efficient method under arbitrary interface profiles and applied magnetic field distributions. Preceded by the magnetic analysis for this purpose, the present research has attempted to investigate interface dynamic phenomena. As an example of these phenomena, this paper shows the wavenumber spectrum of the interface profile and the sum of interface stresses changing in time, since the change of the balance among the interface stresses causing the transition can be observed conveniently. As time advances, wavenumber components increase due to the nonlinear interaction of waves. It is further argued that such analyses should be validated by the law of conservation of energy, the relation between the interface energy density and the interface stress, and the magnetic laws. & 2016 Elsevier B.V. All rights reserved.
Keywords: Magnetic fluid Magnetic field Free surface Transition Nonlinear interaction Numerical analysis
1. Introduction On the interface of magnetic fluid, unique phenomena such as the transition among interface profiles are observed. The interface which was initially flat abruptly changes to the profile with a regular pattern, when the intensity of an applied magnetic field is increased as close as to the critical value. Just after the magnetic fluid was invented, linear analyses [1,2] or weakly nonlinear analyses [3–5] were used for this phenomenon, where quantities on the interface are approximated by finite power series of the interface elevation which is assumed to be small. During the transition process, various nonlinear interactions of waves are considered to work, but the above analyses investigated just the onset of the process. It is an interesting problem to follow the process of pattern formation as time advances. However, analyses for such problem require rigorous as well as efficient methods under arbitrary interface profiles and applied magnetic field distributions. Preceded by the magnetic analysis for general use developed for this purpose (Section 4) [6,7], the present research has attempted the analysis on interface dynamic phenomena, such as the bifurcation of the interface stability [7,8] and the wavenumber spectra of interface quantities changing in time (Section 3 of this paper). E-mail address: [email protected]
Numerical analyses should be accompanied by the confirmation that obtained fluid and magnetic quantities satisfy physical laws sufficiently. This is especially important for fast phenomena such as the transition process. As shown in Section 2, the interface elevation ζ is obtained by integrating in time the equation for interface motion, which includes the sum of interface stresses S. Then, the validation problem is divided into two: First is the time integration which can be verified by the law of conservation of energy; second is the correctness of S itself which is verified by the relation with the interface energy density (interface energy per unit area in Flat Space), as discussed in Section 5. In addition, with respect to magnetic quantities, the relation between the magnetic interface energy density and the Maxwell stress should be verified together with Ampére's law and Gauss's law (Section 6).
2. Equation for interface motion When we analyze free surface phenomena of incompressible, irrotational and inviscid magnetic fluid with all nonlinear effects but with no limitations on the interface profile, we use the following equation for interface motion which is derived by using the kinematic and dynamic conditions on the interface for the tangential component of the fluid equation of motion [7]:
http://dx.doi.org/10.1016/j.jmmm.2016.09.030 0304-8853/& 2016 Elsevier B.V. All rights reserved.
Please cite this article as: Y. Mizuta, Journal of Magnetism and Magnetic Materials (2016), http://dx.doi.org/10.1016/j. jmmm.2016.09.030i
Y. Mizuta / Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
ρ
∂φ + S = 0, ∂t
S ≡ D + G + C + T + p0 ,
(1)
T = − [1 μj ]{μ1μ2 (hX2 + hY2) + bZ2} 2,
(2)
where ρ, φ, D, G, C, T, and p0 are the fluid density, velocity potential, dynamic pressure, gravity potential, surface tension, magnetic stress difference and atmospheric pressure, respectively. In (1), φ is obtained from the vertical component of the fluid ve-
locity vz and the interface elevation ζ as φ =
ζ
∫−∞ dzvz . Furthermore,
interface profiles and applied magnetic field distributions, as shown in Section 4 [6,7]. In addition, we omit D and p0 in S hereafter supposing that the interface moves slowly enough, and the atmospheric pressure is homogeneous. The interface elevation ζ(R ) and the sum of interface stresses S (R ) as functions of the interface coordinate parameter R = (X , Y ) are expressed as the superposition of periodic functions with the wavenumber k:
ζ (R ) =
∑k ζk(R ),
S(R ) =
∑k Sk(R ).
(3)
Then, (1) is rewritten as
T represents the action from the magnetic field to the fluid, where μj denotes the permeability of the fluid ( j = 1) or the vacuum
0=
( j = 2), [⋯] the difference of the value across the interface ( 2 − 1). The tangential magnetic field hX , Y and the normal magnetic flux bZ can be obtained rigorously as well as efficiently under arbitrary
where ∇2 = ( ∂/∂X , ∂/∂Y ) is the partial derivative in the tangential direction, and k = |k|.
⎧ ∂ ⎛ ∂ζk ⎞ 1 2 ⎫ ⎜ρ ⎟ − ∇ S ⎬, ⎩ ∂t ⎝ ∂t ⎠ k 2 k⎭
∑k ⎨
(4)
Fig. 1. Interface profile of hexagonal lattice in (a) real space and (b) wavenumber space. (c) Temporal change in basic wavenumber component of interface elevation ζ˜k (red) and force caused by sum of interface stresses −(k ρ)S˜k (blue). (d), (e) Wavenumber spectrum of interface elevation |ζ˜k|2 and sum of interface stresses |S˜k|2 at time 20.0 × 10−2 s (specific permeability of magnetic fluid μ1 μ0 = 1.2, and initial amplitude of basic wavenumber component of interface elevationis ζ0 = 0.2 mm ). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Please cite this article as: Y. Mizuta, Journal of Magnetism and Magnetic Materials (2016), http://dx.doi.org/10.1016/j. jmmm.2016.09.030i
Y. Mizuta / Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎∎∎–∎∎∎
We multiply ∂ζ /∂t on both sides of (4), and average over the interface F. Furthermore, we use the equation for the periodic function ∇22Sk + k 2Sk = 0 and the orthogonality ⟨ζk1(R )ζk2(R )⟩ = 0 ( k1 ≠ k2). Then we obtain
∂ 0= ∂t
∑k
2 ρ ⎛⎜ ∂ζk ⎞⎟ 2k ⎝ ∂t ⎠
+
∂ζ S , ∂t
(5)
1 F
where ⟨⋯⟩ ≡ ∬ dX dY ⋯ is the average over F. Eq. (5) implies the F balance between the kinematic energy (first term on the righthand side) and the work done by S (second term) per unit time and unit area.
3. Temporal change of wavenumber spectra The transition of the interface profile is caused when the intensity of the magnetic field is increased, and the balance among the interface stresses is changed. This is conveniently observed by the wavenumber spectra of interface quantities. Fig. 1 shows an example of the temporal change of the wavenumber spectra of the interface elevation and the sum of interface stresses obtained by using the equation for interface motion (1). Initially, we set a profile of hexagonal lattice on the interface of magnetic fluid with the infinite depth as (a), and apply a homogeneous magnetic field vertically. In the wavenumber domain, this initial profile is composed of three basic interface elevation modes (open circles) located at the distance of the critical wavenumber kCL from the origin (arc) as shown in (b), where k0 is adjusted to kCL 10. Light and shade squares indicate the magnitude of ω2 for H0 = HCL in the linear dispersion relation, where ω, H0 , HCL are the angular frequency, the intensity of the applied magnetic field and the linear critical magnetic field intensity, respectively. Fig. 1(c) shows the amplitude of the interface elevation ζ˜k and the force due to the sum of interface stresses −(k ρ)S˜k at the basic modes for H0/HCL = 0.6, 0. 8. The spectra of the interface elevation |ζ˜k|2 and the sum of interface stresses |S˜k|2 at the time 20.0 × 10−2 s are shown in (d) and (e), respectively, where the colored point at each wavenumber indicates their value according to the color box on the right, and the component with the absolute value larger than 1 × 10−6 m for |ζ˜k| or 1 × 10−1 N/m for |S˜k| is encircled. During the period, the interface oscillates almost periodically. However, close to the critical magnetic field intensity, the oscillation becomes slower and irregular, since the sum of interface stresses becomes weaker. Furthermore, the wavenumber components other than the basic modes appear: “prominent modes” and “continuous modes” which bury the space among prominent modes. Their increase is due to the nonlinear interaction of waves since the surface tension and the magnetic stress difference are nonlinear with respect to the interface elevation. These aspects are more remarkable for H0 closer to HCL .
4. Magnetic analysis for general use [6,7] Each component of the interface magnetic fields is divided into the basic field hX0 , Y , bZ0 and the induced field hX1 , Y , bZ1 . Basic fields are given directly by a given applied magnetic field h0 as hX0 , Y = tX , Y ·h0 and bZ0 = tZ ·h0 P where tX , Y are the tangential unit vectors, tZ is the normal unit vector, and P ≡ 1 μ2 + 1 μ1 2. Induced fields are to be determined to satisfy, together with the basic fields, both the harmonic property and the interface conditions, as discussed in [7]. Instead of solving a set of three-dimensional interface magnetic field equations, induced fields are obtained equivalently but simply as
(
)
⎧ 1 ^ ^ ⎪ bZ = HZ (1 − HZ )−1bZ0 , ⎨ ^ ^ −1 0 1 ⎪ ⎩ hX, Y = HX, Y (1 − HZ ) bZ .
3
(6)
^ ( The three-dimensional Hilbert transform operators H I = X, Y , Z ) I are defined as follows:
M ^ ^ ^ ^ ^ HX, Y F (R ) ≡ − MtX, Y ·GF (R ), HZF (R ) ≡ − tZ ·GF (R ), GF (R ) P ≡2
(
∬F dS′(∇′ψ )F (R′),
(7)
)
where M ≡ 1 μ2 − 1 μ1 2, and F (R′) is an arbitrary function of the interface coordinate parameter R′ = (X ′, Y ′). The position vectors for the observation point and the source point are denoted by r = r (R ) and r′ = r (R′). The functions and the derivatives for the ^ source point are shown by “′”. The operator G is composed of the integral over the source point on the interface F and the basic solution of the three-dimensional Poisson equation Δ′ψ = δ (r′ − r ) ^ as ψ ( r′ − r ) = − 1 4π |r′ − r |. The operation of G is evaluated analytically instead of by the numerical integration if its operand is expanded into a series of periodic functions [7].
5. Interface energy density Law of conservation of energy (5) can be used for verifying whether the equation for interface motion is integrated in time correctly or not for a given S. Then, another problem is the correctness of S itself, which should be confirmed by other methods. For this purpose, we prepare the interface energy density (interface energy per unit area in Flat Space) U (ζ ) to be obtained at any interface elevation ζ, and confirm the following relation for an infinitesimal shift δζ of ζ:
δU ≡ U (ζ + δζ ) − U (ζ ) = S(ζ )δζ .
(8)
When we denote the gravity acceleration, capillary coefficient and the principal curvatures of the interface profile as g, γ, κ1,2, the gravity potential G and the surface tension C in S are expressed as
G(ζ ) = ρgζ ,
C (ζ ) = − γ (κ1 + κ2).
(9)
Then, the interface energy densities generating above G and C in the same way as (8) are
UG(ζ ) =
1 ρgζ 2, 2
UC(ζ ) = γ 1 + (∇2ζ )2 .
(10)
In the following, we discuss the interface energy density UT (ζ ) which generates T (ζ ) of (2) as
δUT ≡ UT(ζ + δζ ) − UT(ζ ) = T (ζ )δζ .
(11)
The interface energy density UT (ζ ) is the sum of the magnetic interface energy density in the fluid domain UT1 and that in the vacuum domain UT2. We suppose here that they are changed as much as δUT1 = T1δζ1 or δUT2 = T2δζ2 by the shift of the interface δζ1 or δζ2. Then,
δUT = Tδζ = δUT1 + δUT2 = T1δζ1 + T2δζ2.
(12)
Since each shift outward is defined positive, δζ1 = − δζ2 = δζ . As shown later, δUT1 is the work done by the normal component of the Maxwell stress T1 on the fluid domain:
δUT1 = T1δζ1 =
⎫ ⎧ ⎞⎪ ⎛ 1 ⎪ bZ2 ⎜⎜ hX2 + hY2⎟⎟⎬δζ1. ⎨ − μ 1 ⎪ 2⎪ ⎠⎭ ⎝ ⎩ μ1
(13)
By using similar T2 for the vacuum domain, T which agrees with (2) is obtained as T = T1 − T2.
Please cite this article as: Y. Mizuta, Journal of Magnetism and Magnetic Materials (2016), http://dx.doi.org/10.1016/j. jmmm.2016.09.030i
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4
6. Relation between magnetic interface energy density and Maxwell stress Eq. (13) is a general relation which was derived, for example, by the Joule loss and the Lorentz force [9,10], or by a virtual electric current generating the magnetic field and the volume of influence in contact with a flat interface [1]. In order to be used for validating the numerical interface analysis of magnetic fluid with the low electric conductivity in a more general setup as present, we derive the Maxwell stress again from the magnetic energy contained in a rectangular column in the Flat Space as shown in Fig. 2, which extends perpendicularly to the interface and infinitely into the fluid domain with the sectional area SZ . In the field with the magnetic flux vector B = (BX , BY , BZ ) and the permeability μ1, the magnetic energy per unit volume is B2 2μ1 ( B2 = |B|2 = BX2 + BY2 + BZ2). Then the magnetic interface energy density is
UT1 =
1 2μ1SZ
⎛ dSZ ⎜⎜ ⎝ Z
∬S
⎞
ζ
∫−∞ dZB2⎟⎟⎠.
(14)
When the height of the interface changes from Z = ζ to ζ + δζ , the change in the components of B are
⎧ ⎛ ∂B ∂B ⎞ ∂B ⎪ δBX = − ⎜ X + Z ⎟δζ = − 2 X δζ , ⎝ ∂Z ∂X ⎠ ∂Z ⎪ ⎪ ⎪ ⎛ ∂BY ∂BZ ⎞ ∂BY ⎨ δBY = − ⎜ δζ , + ⎟δζ = − 2 ⎝ ∂Z ∂Y ⎠ ∂Z ⎪ ⎪ ⎞ ⎛ ⎪ δB = − ⎜ ∂BX + ∂BY + ∂BZ ⎟δζ = 0, Z ⎪ ⎝ ∂X ∂Y ∂Z ⎠ ⎩
∂BZ ∂B = Y, ∂Y ∂Z
(15)
∂BX ∂B ∂B + Y + Z = 0. ∂X ∂Y ∂Z
(16)
Then, the change of the vertical integral in (14) is: ζ
δ
ζ
⎞
⎛
∫−∞ dZB2 = B02δζ + 2 ∫−∞ dZ ⎜⎝ BXδBX + BY δBY + BZ δBZ ⎟⎠ = B02δζ − 2
ζ
⎛
⎞
∫−∞ dZ ∂∂Z ⎜⎝ BX2 + BY2⎟⎠δζ
⎞ ⎛ = B02δζ − 2⎜ BX20 + BY20⎟δζ , ⎠ ⎝
In the present procedure of analysis, the interface magnetic field hX , Y , bZ are obtained by the magnetic analysis for general use in Section 4 for a given interface profile. These interface magnetic field s are used to calculate the magnetic stress difference T of (2) in the sum of interface stresses S. The sum of interface stresses is used to integrate the equation for interface motion (4) in time, and obtain a new interface elevationζ. These processes are shown schematically in the diagram of Fig. 3. Obtained quantities should be validated in two steps: First is the time integration of equation for interface motion (4) which we can verify by the law of conservation of energy (5), supposing S given is correct; second is the correctness of S itself which is verified by the relation between the interface energy density and the interface stress (8), as discussed in Section 5. This relation is especially important for magnetic fluids where the relation between the magnetic interface energy density and the Maxwell stress (11) should be verified for magnetic quantities, together with Ampére's law and Gauss's law (Section 6). Actually, magnetic fields obtained by the magnetic analysis for general use are confined to those on the interface for efficiency, and the information for the magnetic field within the fluid or vacuum domain is necessarily compensated by the harmonic property, when the magnetic interface energy density UT is calculated. Some numerical condition to refine the results is expected to be derived.
8. Conclusion
together with Ampére's law and Gauss's law
∂BZ ∂B = X, ∂X ∂Z
7. Validation of numerical analysis
(17)
The transition process of pattern formation on the interface of magnetic fluid following the advance of time is an interesting problem. However, numerical analyses of such a fast phenomenon require rigorous as well as efficient methods under arbitrary interface profiles and applied magnetic field distributions. Preceded by the magnetic analysis for general use developed for this purpose, this paper showed the wavenumber spectrum of the interface profile and the sum of interface stresses changing in time close to the critical intensity of the applied magnetic field, where increasing wavenumber components due to the nonlinear interaction of waves were observed. Numerical analyses should be validated in two steps: First is the time integration which can be verified by the law of conservation of energy; second is the correctness of S itself which is verified by the relation between the interface energy density and the interface stress together with the magnetic laws.
where the subscript 0 denotes the quantity on the interface, and the first line of (17) considers the change of both the column volume and B2 itself. If B = μ1H is assumed between the magnetic field vector H , (13) is derived.
Fig. 2. Rectangular column for deriving relation between interface energy density and interface stress.
Fig. 3. Validation of quantities obtained by numerical analysis.
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References [1] M.D. Cowley, R..E. Rosensweig, The interfacial stability of a ferromagnetic fluid, J. Fluid Mech. 30 (4) (1967) 671–688. [2] R.E. Zelazo, J.R. Melcher, Dynamics and stability of ferrofluids:: surface interactions, J. Fluid Mech. 39 (1) (1969) 1–24. [3] A. Gailitis, Formation of the hexagonal pattern on the surface of a ferromagnetic fluid in an applied magnetic field, J. Fluid Mech. 82 (3) (1977) 401–413. [4] E. Twombly, J.W. Thomas, Mathematical theory of non-linear waves on the surface of a magnetic fluid, IEEE Trans. Magn. MAG 16 (2) (1980) 214–220. [5] E.E. Twombly, J.W. Thomas, Bifurcating instability of the free surface of a
5
ferrofluid, SIAM J. Math. Anal. 14 (4) (1983) 736–766. [6] Y. Mizuta, Interface magnetic field analysis for free surface phenomena of magnetic fluid, Magnetohydrodynamics 44 (2) (2008) 155–165. [7] Y. Mizuta, Stability analysis on the free surface phenomena of a magnetic fluid for general use, J. Magn. Magn. Mater. 323 (10) (2011) 1354–1359. [8] Y. Mizuta, Interface stability analysis of magnetic fluid by using a method for general use and nonlinear response, Magnetohydrodynamics 49 (2–4) (2013) 191–195. [9] J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975. [10] W.K. Panofsky, M. Phillips, Classical Electricity and Magnetism, AddisonWesley, Reading, MA, 1962.
Please cite this article as: Y. Mizuta, Journal of Magnetism and Magnetic Materials (2016), http://dx.doi.org/10.1016/j. jmmm.2016.09.030i
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Dynamic analysis on magnetic fluid interface validated by physical laws Yo Mizuta Division of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
art ic l e i nf o
a b s t r a c t
Article history: Received 30 June 2016 Received in revised form 12 August 2016 Accepted 5 September 2016
Numerical analyses of magnetic fluid especially for fast phenomena such as the transition among interface profiles require rigorous as well as efficient method under arbitrary interface profiles and applied magnetic field distributions. Preceded by the magnetic analysis for this purpose, the present research has attempted to investigate interface dynamic phenomena. As an example of these phenomena, this paper shows the wavenumber spectrum of the interface profile and the sum of interface stresses changing in time, since the change of the balance among the interface stresses causing the transition can be observed conveniently. As time advances, wavenumber components increase due to the nonlinear interaction of waves. It is further argued that such analyses should be validated by the law of conservation of energy, the relation between the interface energy density and the interface stress, and the magnetic laws. & 2016 Elsevier B.V. All rights reserved.
Keywords: Magnetic fluid Magnetic field Free surface Transition Nonlinear interaction Numerical analysis
1. Introduction On the interface of magnetic fluid, unique phenomena such as the transition among interface profiles are observed. The interface which was initially flat abruptly changes to the profile with a regular pattern, when the intensity of an applied magnetic field is increased as close as to the critical value. Just after the magnetic fluid was invented, linear analyses [1,2] or weakly nonlinear analyses [3–5] were used for this phenomenon, where quantities on the interface are approximated by finite power series of the interface elevation which is assumed to be small. During the transition process, various nonlinear interactions of waves are considered to work, but the above analyses investigated just the onset of the process. It is an interesting problem to follow the process of pattern formation as time advances. However, analyses for such problem require rigorous as well as efficient methods under arbitrary interface profiles and applied magnetic field distributions. Preceded by the magnetic analysis for general use developed for this purpose (Section 4) [6,7], the present research has attempted the analysis on interface dynamic phenomena, such as the bifurcation of the interface stability [7,8] and the wavenumber spectra of interface quantities changing in time (Section 3 of this paper). E-mail address: [email protected]
Numerical analyses should be accompanied by the confirmation that obtained fluid and magnetic quantities satisfy physical laws sufficiently. This is especially important for fast phenomena such as the transition process. As shown in Section 2, the interface elevation ζ is obtained by integrating in time the equation for interface motion, which includes the sum of interface stresses S. Then, the validation problem is divided into two: First is the time integration which can be verified by the law of conservation of energy; second is the correctness of S itself which is verified by the relation with the interface energy density (interface energy per unit area in Flat Space), as discussed in Section 5. In addition, with respect to magnetic quantities, the relation between the magnetic interface energy density and the Maxwell stress should be verified together with Ampére's law and Gauss's law (Section 6).
2. Equation for interface motion When we analyze free surface phenomena of incompressible, irrotational and inviscid magnetic fluid with all nonlinear effects but with no limitations on the interface profile, we use the following equation for interface motion which is derived by using the kinematic and dynamic conditions on the interface for the tangential component of the fluid equation of motion [7]:
http://dx.doi.org/10.1016/j.jmmm.2016.09.030 0304-8853/& 2016 Elsevier B.V. All rights reserved.
Please cite this article as: Y. Mizuta, Journal of Magnetism and Magnetic Materials (2016), http://dx.doi.org/10.1016/j. jmmm.2016.09.030i
Y. Mizuta / Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
ρ
∂φ + S = 0, ∂t
S ≡ D + G + C + T + p0 ,
(1)
T = − [1 μj ]{μ1μ2 (hX2 + hY2) + bZ2} 2,
(2)
where ρ, φ, D, G, C, T, and p0 are the fluid density, velocity potential, dynamic pressure, gravity potential, surface tension, magnetic stress difference and atmospheric pressure, respectively. In (1), φ is obtained from the vertical component of the fluid ve-
locity vz and the interface elevation ζ as φ =
ζ
∫−∞ dzvz . Furthermore,
interface profiles and applied magnetic field distributions, as shown in Section 4 [6,7]. In addition, we omit D and p0 in S hereafter supposing that the interface moves slowly enough, and the atmospheric pressure is homogeneous. The interface elevation ζ(R ) and the sum of interface stresses S (R ) as functions of the interface coordinate parameter R = (X , Y ) are expressed as the superposition of periodic functions with the wavenumber k:
ζ (R ) =
∑k ζk(R ),
S(R ) =
∑k Sk(R ).
(3)
Then, (1) is rewritten as
T represents the action from the magnetic field to the fluid, where μj denotes the permeability of the fluid ( j = 1) or the vacuum
0=
( j = 2), [⋯] the difference of the value across the interface ( 2 − 1). The tangential magnetic field hX , Y and the normal magnetic flux bZ can be obtained rigorously as well as efficiently under arbitrary
where ∇2 = ( ∂/∂X , ∂/∂Y ) is the partial derivative in the tangential direction, and k = |k|.
⎧ ∂ ⎛ ∂ζk ⎞ 1 2 ⎫ ⎜ρ ⎟ − ∇ S ⎬, ⎩ ∂t ⎝ ∂t ⎠ k 2 k⎭
∑k ⎨
(4)
Fig. 1. Interface profile of hexagonal lattice in (a) real space and (b) wavenumber space. (c) Temporal change in basic wavenumber component of interface elevation ζ˜k (red) and force caused by sum of interface stresses −(k ρ)S˜k (blue). (d), (e) Wavenumber spectrum of interface elevation |ζ˜k|2 and sum of interface stresses |S˜k|2 at time 20.0 × 10−2 s (specific permeability of magnetic fluid μ1 μ0 = 1.2, and initial amplitude of basic wavenumber component of interface elevationis ζ0 = 0.2 mm ). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Please cite this article as: Y. Mizuta, Journal of Magnetism and Magnetic Materials (2016), http://dx.doi.org/10.1016/j. jmmm.2016.09.030i
Y. Mizuta / Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎∎∎–∎∎∎
We multiply ∂ζ /∂t on both sides of (4), and average over the interface F. Furthermore, we use the equation for the periodic function ∇22Sk + k 2Sk = 0 and the orthogonality ⟨ζk1(R )ζk2(R )⟩ = 0 ( k1 ≠ k2). Then we obtain
∂ 0= ∂t
∑k
2 ρ ⎛⎜ ∂ζk ⎞⎟ 2k ⎝ ∂t ⎠
+
∂ζ S , ∂t
(5)
1 F
where ⟨⋯⟩ ≡ ∬ dX dY ⋯ is the average over F. Eq. (5) implies the F balance between the kinematic energy (first term on the righthand side) and the work done by S (second term) per unit time and unit area.
3. Temporal change of wavenumber spectra The transition of the interface profile is caused when the intensity of the magnetic field is increased, and the balance among the interface stresses is changed. This is conveniently observed by the wavenumber spectra of interface quantities. Fig. 1 shows an example of the temporal change of the wavenumber spectra of the interface elevation and the sum of interface stresses obtained by using the equation for interface motion (1). Initially, we set a profile of hexagonal lattice on the interface of magnetic fluid with the infinite depth as (a), and apply a homogeneous magnetic field vertically. In the wavenumber domain, this initial profile is composed of three basic interface elevation modes (open circles) located at the distance of the critical wavenumber kCL from the origin (arc) as shown in (b), where k0 is adjusted to kCL 10. Light and shade squares indicate the magnitude of ω2 for H0 = HCL in the linear dispersion relation, where ω, H0 , HCL are the angular frequency, the intensity of the applied magnetic field and the linear critical magnetic field intensity, respectively. Fig. 1(c) shows the amplitude of the interface elevation ζ˜k and the force due to the sum of interface stresses −(k ρ)S˜k at the basic modes for H0/HCL = 0.6, 0. 8. The spectra of the interface elevation |ζ˜k|2 and the sum of interface stresses |S˜k|2 at the time 20.0 × 10−2 s are shown in (d) and (e), respectively, where the colored point at each wavenumber indicates their value according to the color box on the right, and the component with the absolute value larger than 1 × 10−6 m for |ζ˜k| or 1 × 10−1 N/m for |S˜k| is encircled. During the period, the interface oscillates almost periodically. However, close to the critical magnetic field intensity, the oscillation becomes slower and irregular, since the sum of interface stresses becomes weaker. Furthermore, the wavenumber components other than the basic modes appear: “prominent modes” and “continuous modes” which bury the space among prominent modes. Their increase is due to the nonlinear interaction of waves since the surface tension and the magnetic stress difference are nonlinear with respect to the interface elevation. These aspects are more remarkable for H0 closer to HCL .
4. Magnetic analysis for general use [6,7] Each component of the interface magnetic fields is divided into the basic field hX0 , Y , bZ0 and the induced field hX1 , Y , bZ1 . Basic fields are given directly by a given applied magnetic field h0 as hX0 , Y = tX , Y ·h0 and bZ0 = tZ ·h0 P where tX , Y are the tangential unit vectors, tZ is the normal unit vector, and P ≡ 1 μ2 + 1 μ1 2. Induced fields are to be determined to satisfy, together with the basic fields, both the harmonic property and the interface conditions, as discussed in [7]. Instead of solving a set of three-dimensional interface magnetic field equations, induced fields are obtained equivalently but simply as
(
)
⎧ 1 ^ ^ ⎪ bZ = HZ (1 − HZ )−1bZ0 , ⎨ ^ ^ −1 0 1 ⎪ ⎩ hX, Y = HX, Y (1 − HZ ) bZ .
3
(6)
^ ( The three-dimensional Hilbert transform operators H I = X, Y , Z ) I are defined as follows:
M ^ ^ ^ ^ ^ HX, Y F (R ) ≡ − MtX, Y ·GF (R ), HZF (R ) ≡ − tZ ·GF (R ), GF (R ) P ≡2
(
∬F dS′(∇′ψ )F (R′),
(7)
)
where M ≡ 1 μ2 − 1 μ1 2, and F (R′) is an arbitrary function of the interface coordinate parameter R′ = (X ′, Y ′). The position vectors for the observation point and the source point are denoted by r = r (R ) and r′ = r (R′). The functions and the derivatives for the ^ source point are shown by “′”. The operator G is composed of the integral over the source point on the interface F and the basic solution of the three-dimensional Poisson equation Δ′ψ = δ (r′ − r ) ^ as ψ ( r′ − r ) = − 1 4π |r′ − r |. The operation of G is evaluated analytically instead of by the numerical integration if its operand is expanded into a series of periodic functions [7].
5. Interface energy density Law of conservation of energy (5) can be used for verifying whether the equation for interface motion is integrated in time correctly or not for a given S. Then, another problem is the correctness of S itself, which should be confirmed by other methods. For this purpose, we prepare the interface energy density (interface energy per unit area in Flat Space) U (ζ ) to be obtained at any interface elevation ζ, and confirm the following relation for an infinitesimal shift δζ of ζ:
δU ≡ U (ζ + δζ ) − U (ζ ) = S(ζ )δζ .
(8)
When we denote the gravity acceleration, capillary coefficient and the principal curvatures of the interface profile as g, γ, κ1,2, the gravity potential G and the surface tension C in S are expressed as
G(ζ ) = ρgζ ,
C (ζ ) = − γ (κ1 + κ2).
(9)
Then, the interface energy densities generating above G and C in the same way as (8) are
UG(ζ ) =
1 ρgζ 2, 2
UC(ζ ) = γ 1 + (∇2ζ )2 .
(10)
In the following, we discuss the interface energy density UT (ζ ) which generates T (ζ ) of (2) as
δUT ≡ UT(ζ + δζ ) − UT(ζ ) = T (ζ )δζ .
(11)
The interface energy density UT (ζ ) is the sum of the magnetic interface energy density in the fluid domain UT1 and that in the vacuum domain UT2. We suppose here that they are changed as much as δUT1 = T1δζ1 or δUT2 = T2δζ2 by the shift of the interface δζ1 or δζ2. Then,
δUT = Tδζ = δUT1 + δUT2 = T1δζ1 + T2δζ2.
(12)
Since each shift outward is defined positive, δζ1 = − δζ2 = δζ . As shown later, δUT1 is the work done by the normal component of the Maxwell stress T1 on the fluid domain:
δUT1 = T1δζ1 =
⎫ ⎧ ⎞⎪ ⎛ 1 ⎪ bZ2 ⎜⎜ hX2 + hY2⎟⎟⎬δζ1. ⎨ − μ 1 ⎪ 2⎪ ⎠⎭ ⎝ ⎩ μ1
(13)
By using similar T2 for the vacuum domain, T which agrees with (2) is obtained as T = T1 − T2.
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6. Relation between magnetic interface energy density and Maxwell stress Eq. (13) is a general relation which was derived, for example, by the Joule loss and the Lorentz force [9,10], or by a virtual electric current generating the magnetic field and the volume of influence in contact with a flat interface [1]. In order to be used for validating the numerical interface analysis of magnetic fluid with the low electric conductivity in a more general setup as present, we derive the Maxwell stress again from the magnetic energy contained in a rectangular column in the Flat Space as shown in Fig. 2, which extends perpendicularly to the interface and infinitely into the fluid domain with the sectional area SZ . In the field with the magnetic flux vector B = (BX , BY , BZ ) and the permeability μ1, the magnetic energy per unit volume is B2 2μ1 ( B2 = |B|2 = BX2 + BY2 + BZ2). Then the magnetic interface energy density is
UT1 =
1 2μ1SZ
⎛ dSZ ⎜⎜ ⎝ Z
∬S
⎞
ζ
∫−∞ dZB2⎟⎟⎠.
(14)
When the height of the interface changes from Z = ζ to ζ + δζ , the change in the components of B are
⎧ ⎛ ∂B ∂B ⎞ ∂B ⎪ δBX = − ⎜ X + Z ⎟δζ = − 2 X δζ , ⎝ ∂Z ∂X ⎠ ∂Z ⎪ ⎪ ⎪ ⎛ ∂BY ∂BZ ⎞ ∂BY ⎨ δBY = − ⎜ δζ , + ⎟δζ = − 2 ⎝ ∂Z ∂Y ⎠ ∂Z ⎪ ⎪ ⎞ ⎛ ⎪ δB = − ⎜ ∂BX + ∂BY + ∂BZ ⎟δζ = 0, Z ⎪ ⎝ ∂X ∂Y ∂Z ⎠ ⎩
∂BZ ∂B = Y, ∂Y ∂Z
(15)
∂BX ∂B ∂B + Y + Z = 0. ∂X ∂Y ∂Z
(16)
Then, the change of the vertical integral in (14) is: ζ
δ
ζ
⎞
⎛
∫−∞ dZB2 = B02δζ + 2 ∫−∞ dZ ⎜⎝ BXδBX + BY δBY + BZ δBZ ⎟⎠ = B02δζ − 2
ζ
⎛
⎞
∫−∞ dZ ∂∂Z ⎜⎝ BX2 + BY2⎟⎠δζ
⎞ ⎛ = B02δζ − 2⎜ BX20 + BY20⎟δζ , ⎠ ⎝
In the present procedure of analysis, the interface magnetic field hX , Y , bZ are obtained by the magnetic analysis for general use in Section 4 for a given interface profile. These interface magnetic field s are used to calculate the magnetic stress difference T of (2) in the sum of interface stresses S. The sum of interface stresses is used to integrate the equation for interface motion (4) in time, and obtain a new interface elevationζ. These processes are shown schematically in the diagram of Fig. 3. Obtained quantities should be validated in two steps: First is the time integration of equation for interface motion (4) which we can verify by the law of conservation of energy (5), supposing S given is correct; second is the correctness of S itself which is verified by the relation between the interface energy density and the interface stress (8), as discussed in Section 5. This relation is especially important for magnetic fluids where the relation between the magnetic interface energy density and the Maxwell stress (11) should be verified for magnetic quantities, together with Ampére's law and Gauss's law (Section 6). Actually, magnetic fields obtained by the magnetic analysis for general use are confined to those on the interface for efficiency, and the information for the magnetic field within the fluid or vacuum domain is necessarily compensated by the harmonic property, when the magnetic interface energy density UT is calculated. Some numerical condition to refine the results is expected to be derived.
8. Conclusion
together with Ampére's law and Gauss's law
∂BZ ∂B = X, ∂X ∂Z
7. Validation of numerical analysis
(17)
The transition process of pattern formation on the interface of magnetic fluid following the advance of time is an interesting problem. However, numerical analyses of such a fast phenomenon require rigorous as well as efficient methods under arbitrary interface profiles and applied magnetic field distributions. Preceded by the magnetic analysis for general use developed for this purpose, this paper showed the wavenumber spectrum of the interface profile and the sum of interface stresses changing in time close to the critical intensity of the applied magnetic field, where increasing wavenumber components due to the nonlinear interaction of waves were observed. Numerical analyses should be validated in two steps: First is the time integration which can be verified by the law of conservation of energy; second is the correctness of S itself which is verified by the relation between the interface energy density and the interface stress together with the magnetic laws.
where the subscript 0 denotes the quantity on the interface, and the first line of (17) considers the change of both the column volume and B2 itself. If B = μ1H is assumed between the magnetic field vector H , (13) is derived.
Fig. 2. Rectangular column for deriving relation between interface energy density and interface stress.
Fig. 3. Validation of quantities obtained by numerical analysis.
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Please cite this article as: Y. Mizuta, Journal of Magnetism and Magnetic Materials (2016), http://dx.doi.org/10.1016/j. jmmm.2016.09.030i