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Energy and hydrodynamic approaches to magnetocapillary instability of nonconducting jets
Journal of Colloid and Interface Science 281 (2005) 209–217 www.elsevier.com/locate/jcis
Energy and hydrodynamic approaches to magnetocapillary instability of nonconducting jets Y. Zimmels, L.G. Fel ∗ Department of Civil and Environmental Engineering, Technion, Haifa 32000, Israel Received 19 May 2004; accepted 6 August 2004 Available online 21 September 2004
Abstract The field energy and magnetocapillary instability of isothermal incompressible and inviscid nonconducting liquid jets in a uniform magnetic field are considered. The equivalence between static and dynamic approaches at the onset of instability and cutoff wavelength is shown and its implications are discussed. A new dispersion relation for magnetocapillary instability in such jets is derived. This relation differs from that given in the literature. The existence of a critical magnetic field that stabilizes jets with finite susceptibility is established. It is shown that the jet is stabilized by the field irrespective of its being para- or diamagnetic, but the extent of stabilization is different. 2004 Elsevier Inc. All rights reserved. Keywords: Plateau problem; Magnetocapillary instability; Nonconducting liquid jet
1. Introduction Joseph Antoine Ferdinand Plateau (1801–1883) was a Belgian physicist who is best remembered in mathematics for the Plateau problem. He wrote his seminal book [1] when he was already blind, as he was for the last 40 years of his life. The Plateau problem, in its brief formulation, is to find a surface of minimum area S given its boundary ∂Ω. The problem relates to the principle of minimum free energy at equilibrium. In this context it may be required to find a surface with isotropic tension σ which provides a minimum surface free energy σ S. Rayleigh [2] gave a theoretical explanation for the instability of liquid cylinders that are longer than their circumference. Further generalization is called for if the excess free energy W of the cylinder comprises different types of energy that reflect a more complex structure of the liquid (e.g., elasticity [3]) as well as its capacity to interact with external fields. The threshold of static instability, which is associated with the change of sign of W , when the disturbed configuration becomes more preferable, is defined * Corresponding author.
E-mail address: [email protected] (L.G. Fel). 0021-9797/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2004.08.024
by W (kR, p) = 0 → ks = ks (p),
(1)
where k is a wavenumber assigned to a small surface disturbance, R is a radius of cylinder, and the parameter p stands for the effect of an external interaction, which contributes to the jet evolution, e.g., fluid polarization in the presence of external fields. Note that the static cutoff wavenumber ks is related to the static cutoff wavelength Λs by Λs (p) = 2π/ks . Plateau instability being a static problem has also a dynamic aspect. Consider Rayleigh instability in liquid jets of radius R and its corresponding dispersion equation s = s(kR, p). The latter determines the exponential evolution in time Θ(r, t) ∼ est of all hydrodynamic functions Θ(r, t) of the jet with the growth rate s. Rayleigh’s theory of capillary instability in liquid jets states that the maximum of the dispersion function s(kR, p), which corresponds to the wavenumber kmax (p), gives rise to evolution of the largest capillary instability. The range of wavenumbers k which contribute to the evolution of the instability is given by s > 0. Thus, the threshold of instability follows as s(kR, p) = 0 → kd = kd (p).
(2)
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Note that the dynamic cutoff wavenumber kd is related to the dynamic cutoff wavelength Λd by Λd (p) = 2π/kd . The value Λd (0), which corresponds to free-jet evolution, is equal to 2πR according to Rayleigh [2]. If there exists a critical parameter p = pcr such that for all p pcr the expression s(kR, p) 0 holds, then for p pcr the jet is stable for all wavelengths. This means that the liquid jet preserves its initial shape, which is unaffected by small perturbations, irrespective of the jet velocity. Thus, this conclusion must hold also in the limiting case of a motionless fluid; i.e., Λd (p) ≡ Λs (p).
(3)
This identity reflects a deep equivalence between the static approach (the threshold of static instability concerned with an excess free energy) and the dynamic approach (the bifurcation of the first nontrivial steady state of the inviscid hydrodynamic system). Thus, we come to the Plateau problem for a liquid cylinder with a free surface in an external field. The capillary instability of magnetizable liquid jets in the presence of an axial magnetic field constitutes a classical example. Chandrasekhar [4] treated the hydrodynamics of nonpermeable conducting liquid jets and derived their dispersion relation in implicit form. The relation reads particularly simply for superconducting ssc ( ) and nonconducting snc ( ) liquids (see [4, pp. 545–549]), σ I1 ( ) 2 (1 − 2 ) ssc ( ) = 3 R ρ I0 ( ) 2 H − , Hs I0 ( )K1 ( ) σ I1 ( ) (1 − 2 ), (4) R 3 ρ I0 ( ) √ where = kR and Hs = σ/µ0 R is designated by Chandrasekhar as the characteristic field. σ, ρ stand for the isotropic surface tension and density of the liquid, respectively, and µ0 denotes the permeability of free space. Im (x) and Km (x) are the modified Bessel functions of order m of the 1st and 2nd kind, respectively. In the superconducting limit the relation (4) discloses the existence of a critical √ magnetic field Hcr = Hs / 2 beyond which the jet is stable. In the nonconducting limit, this dispersion relation coincides with that of Rayleigh [2]. In 1975, Taktarov [5] considered the magnetocapillary instability of nonconducting permeable jets and derived a dispersion relation in explicit form (see also Rosensweig [6]). This relation accounts for the coupling effect between the magnetization of the liquid and the external field. Being nondissipative, this linear problem facilitates its examination by either the energy or the hydrodynamic approach, and both must yield the same cutoff wavelength. In comparing our results, we have found a discrepancy with [5]. The aim of this paper is to give an accurate solution, first for the case of Plateau instability (using the energy 2 ( ) = snc
approach; see Sections 2, 3, and 4), and then regarding the dynamic case. In this context, the energy and hydrodynamic approaches must give identical results. In Section 5 we solve, using the hydrodynamic approach, the magnetocapillary instability problem of isothermal, incompressible, inviscid, and nonconducting jets, in the presence of a uniform magnetic field. The new dispersion relation accounts for the effect of magnetic fields and conforms with the solution of Plateau instability for magnetizable liquids.
2. Free energy of a liquid cylinder in the presence of a magnetic field The Plateau problem of static instability of a nonconducting liquid cylinder which is subjected to a uniform magnetic field appears, at first glance, to be a simple generalization of its counterpart in the absence of external fields. However, deeper consideration shows that the presence of the field complicates considerably the physical picture and computational procedure. A fundamental question arises concerning the correct definition of the excess free energy W which must be minimized via variation of the cylinder shape. Consider an isothermal liquid cylinder in a uniform magnetic field H0 that is applied in free space along the cylinder axis. The magnetic susceptibility χ of the cylinder is assumed to be isotropic, independent of magnetic field, and satisfying the thermodynamic condition χ > −1 [7]. When the liquid cylinder is undisturbed the total free energy F 0 of the system is given by χµ0 H02 · F 0 = Es0 − (5) dv, 2 0 Ωcyl
where the integral represents the volume πR 2 L, enclosed 0 , which is occupied by the undisturbed by the area ∂Ωcyl cylinder. The term Es0 = σ ∂Ω 0 ds = 2πσ RL stands for cyl
the surface free energy of the undisturbed cylinder, where R, L, and σ denote its radius, length, and surface tension, respectively. Deformation of the cylinder shape changes the field H(r) over all space R3 , i.e., in both the internal domain Ωcyl and its complement (the exterior domain) R3 \ Ωcyl . Following Plateau, we assume conservation of the cylinder volume dv = dv. (6) 0 Ωcyl
Ωcyl
The total free energy F of the disturbed cylinder takes the form µ0 in 2 · F = Es − (1 + χ) H (r) dv 2 µ0 · − 2
Ωcyl
R3 \Ωcyl
ex
µ0 · H (r) dv + 2
2
H02 dv, R3
(7)
Y. Zimmels, L.G. Fel / Journal of Colloid and Interface Science 281 (2005) 209–217
where in H(r) and ex H(r) denote the internal and external magnetic fields, and Es = σ ∂Ωcyl ds is surface energy. The second and third terms on the right-hand side of (7) reflect the contribution of the distorted permeable cylinder to the magnetic free energy, stored in the internal domain Ωcyl (second term) and the exterior domain R3 \ Ωcyl (third term), corrected by the energy of the uniform magnetic field in free space (forth term). The excess free energy W of the system is defined as W = F − F 0.
where according to the assumption 1 the following approximations apply (see Section 3): in 1 ex 1 in 1 ex 1 in 1 ex 1 in 1 ex 1
Hr , Hr , Hz , Hz = hr , hr , hz , hz × χH0 .
(10)
The dimensionless fields in,ex h1r,z (r, z) are dependent on the coordinates. Inserting (10) into (7), and performing the integration, we evaluate W given in (8) as µ0 U, (11) 2 where use was made of (5) and U is given by (see Appendix A) in 1 2 in 1 2
dv U = (1 + χ) Hz + Hr W = Es − Es0 −
Ωcyl
+
ex
Hz1
2
+
ex
Hr1
2
R3 \Ωcyl
+ 2H0 (1 + χ)
dv
in
Hz1 dv +
Ωcyl
ex
Hz1 dv
. (12)
R3 \Ωcyl
We specify the commonly used harmonic deformation of the cylinder as r(z) = R + ζ0 cos kz, where k = 2π/Λ, Λ being the disturbance wavelength. By virtue of translational invariance of the problem hr,z (r, z + Λ) = in,ex h1r,z (r, z)
in,ex 1
we set L = Λ and evaluate the free energy per unit wavelength. For this type of deformation the term Es − Es0 can be evaluated πζ 2 ds − 2πσ RL = σ 0 L( 2 − 1), Es − Es0 = σ 2R ∂Ωcyl
= kR.
(14)
Next we proceed to solve the boundary problem and get the distribution of the magnetic fields.
(8)
From the mathematical standpoint, the variational problem for minimization of W , supplemented with constraint (6) for all smooth surfaces ∂Ωcyl , is known as the isoperimetric problem. The cylinder instability can be studied, assuming small perturbation in shape. In this case the Plateau problem becomes solvable in closed form. Let the extent of deformation be characterized by a length ζ0 , such that ζ0 /R = 1. The fields in H(r) and ex H(r) which must satisfy Maxwell’s equations can be represented as small perturbations of H0 , in H(r) = H0 + in H1 (r) = H0 + in Hz1 , in Hr1 , ex (9) H(r) = H0 + ex H1 (r) = H0 + ex Hz1 , ex Hr1 ,
211
(13)
3. Boundary problem and its solution The magnetostatics of the disturbed liquid cylinder is governed by Maxwell equations for the internal in H(r) and external ex H(r) magnetic fields, rot in H = rot ex H = 0,
div in B = div ex B = 0,
(15)
where in B = µ0 (1 + χ)in H and ex B = µ0 ex H denote internal and external magnetic inductions, respectively. Equations (15) must be supplemented with boundary conditions (BC) at the interface r = R, in H, t = ex H, t,
in B, e = ex B, e,
e, t = 0,
(16)
where , denotes a scalar product and t and e stand for tangential and normal unit vectors to the surface, respectively. Since the surface deformation is small, linearization can be applied, tz = er = 1 − (∂ζ /∂z)2 1. (17) tr = −ez = ∂ζ /∂z, A standard way to solve the problem is to introduce the magnetic potentials Φin (r) and Φex (r), which are defined as in H1 (r) = −∇Φin , ex H1 (r) = −∇Φex , where |∇Φin |, |∇Φex | H0 . These potentials satisfy the first two equations in (15). The last two equations in (15) yield ∂ 2 Φex ∂ 2 Φin +
Φ = 0, + 2 Φex = 0, 2 in ∂z2 ∂z2 ∂2 1 ∂ ,
2 = 2 + (18) r ∂r ∂r where 2 is the two-dimensional Laplacian. Reformulation of the BC (16) for Φex (r), Φin (r) gives ∂Φex ∂Φin ∂ζ ∂Φex ∂Φin = , − (1 + χ) = χH . (19) ∂z ∂z ∂r ∂r ∂z Using Φin (r, z) = φin (r) sin kz and Φex (r, z) = φex (r) sin kz we find ( 2 − k 2 )φex = 0,
( 2 − k 2 )φin = 0
(20)
∂φin ∂φex − = χH kζ0. ∂r ∂r
(21)
with BC at r = R, φex = φin ,
(1 + χ)
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The solutions of (20), which satisfy BC (21) and are finite at r = 0 and r = ∞, are obtained as b(, χ) I0 (kr), c(, χ) φex (r) = ζ0 χH0 (22) K0 (kr), where K0 ( ) I0 ( ) b(, χ) = 2 , c(, χ) = 2 , T (, χ) T (, χ) T (, χ) = 1 + χ I1 ( )K0 ( ). (23)
φin (r) = ζ0 χH0
Recalling the definition (10) of the dimensionless fields hr,z (r, z), the following final expressions are obtained:
in,ex 1
Fig. 1. A plot of hcr versus kR for large positive susceptibilities: χ = 1, 10, 102 , 103 , 106 , from right to left, respectively.
hz = −b(, χ)I0 (kr) cos kz,
in 1
hr = −b(, χ)I1 (kr) sin kz,
in 1
hz = −c(, χ)K0 (kr) cos kz,
ex 1
hr = c(, χ)K1 (kr) sin kz.
ex 1
(24)
Bearing in mind that the function Q( ) = I1 ( )K0 ( ) is monotone growing at the positive half axis and is bounded, 0 Q( ) < 1/2, we conclude that the fields in,ex h1r,z (r, z) are free of singularities in the thermodynamically relevant region χ > −1.
4. Plateau instability In this section we calculate via Eq. (10) the excess free energy W in terms of the dimensionless fields ex h1z,r , in h1z,r . Inserting (24) into (12) we get (see Appendix A) W=
πLσ R 2 · f (, χ, H0 ), 2
µ0 RH02 I0 ( )K0 ( ) , σ T (, χ) (25) where the dimensionless excess free energy f (, χ, H0 ) can be defined by introducing the characteristic field Hch , I0 ( )K0 ( ) H0 2 f (, χ, H0 ) = 2 − 1 + χ 2 , T (, χ) Hch σ . Hch = (26) µ0 χR f (, χ, H0 ) = 2 − 1 + χ 2 2
Notice that Hch differs from Hs , introduced by Chandrasekhar in [4]. Formula (26) serves for both paramagnetic (χ > 0) and diamagnetic (χ < 0) liquids, since the latter case does not lead to an imaginary expression in (26) due to the term χ 2 in (25). Consider the field Hcr (, χ) satisfying f (, χ, Hcr ) = 0 and call it critical field. Beyond this field, H0 Hcr , the cylinder is stable. The expression for the critical field is
Hch 1 − 2 T (, χ) . Hcr (, χ) = (27) χ I0 ( )K0 ( )
Fig. 2. A plot of hcr versus kR for small positive and negative susceptibilities: χ = ±10−4 , ±10−3 , ±10−2 , ±10−1 , from right to left, respectively. The plots corresponding to diamagnetic and paramagnetic cases nearly coincide.
Figs. 1 and 2 show plots of hcr = Hcr /Hch for strong (χ 1) and weak (|χ| 1) magnetic liquids. The corresponding asymptotics for Hcr (, χ), in the case of weak χ 1 and strong χ 1 magnetic susceptibilities and in a long wave limit → 0, which are derived in Appendix B, are presented below: χ 1 Hcr (, χ) 1 − 2 ln , = √ Hch 2 −χ ln → 0, |χ| 1, (28) 9 2 1 Hcr (, χ) = √ 1 − , → 0, χ → ∞, (29) Hch 16 2
1 Hcr (, χ) = B1 B2 + 1 − 2 , → 1, (30) Hch χ √ where B1 = 1/ I0 (1)K0 (1) 1.3697, B2 = I1 (1)K0 (1) 0.2379. Fig. 1 shows that increase of χ results in curves that approach a parabolic shape; see (29). The asymptotic tendency of large χ curves is to the one shown for χ = 106 . At → 0 there is a fast increase in hcr that appears as a jump. Furthermore, the high χ range is associated with a relatively
Y. Zimmels, L.G. Fel / Journal of Colloid and Interface Science 281 (2005) 209–217
213
low hcr . This is realized upon comparing Figs. 1 and 2 where the range of hcr in Fig. 2 is considerably extended. At the end of this section we present physical arguments which justify our conclusion regarding stabilization of the permeable jets for both paramagnetic and diamagnetic liquids. The orientation of the dipole moments along the applied magnetic field in the paramagnetic liquid enforces the rigidity of the jet under small disturbances. In the case of diamagnetic liquid the increase of the jets’ stabilization is due to orientation of the dipole moments in the opposite direction which also enforces the rigidity of the jet. The similar increase of the rigidity, and its corresponding influence on the stabilization, was found recently [3] in the liquid crystalline jet due to elasticity of the media. Although the critical fields for paramagnetic Hcr (, χ) and diamagnetic Hcr (, −χ) liquids are different, the following universal relation holds:
potentials Φin (r, z, t), Φex (r, z, t),
Hcr2 (, χ) Hcr2 (, −χ) 1 − 2 I1 ( ) . − =2 2 2 I0 ( ) Hch Hch
It is necessary to apply boundary conditions (16), which are imposed on H and B, as well as those for the hydrodynamic variables. First, the velocity Vr must be compatible, at r = R, with the assumed form of the deformed boundary ∂ζ /∂t. Second, at the free surface of a liquid jet the jump in stress must be balanced by Laplace pressure [7],
(31)
5. Hydrodynamics of nonconducting jet in a magnetic field Consider an isothermal, incompressible, inviscid, and nonconducting jet in the presence of a magnetic field H0 applied along its z-axis. The deviation from initial values of the pressure is defined as P1in = P in − P0in , where P0in is the unperturbed pressure within the cylindrical jet. The deviations of the internal and external magnetic fields are defined as in 1 H = in H − H0 and ex H1 = ex H − H0 , respectively. The governing equations of magnetohydrodynamics which are given by div Vin = 0, ∂Vjin
(32)
∂t
α = in, ex,
(33)
(34)
can be simplified considerably by applying the Maxwell equations (34) for nonconducting media (γ in = γex = 0) to the Navier–Stokes equation (33). γ in and γex denote conductivities of the jet’s interior and exterior, respectively, Tj k stands for the magnetic stress tensor, and Vin is local fluid velocity. Finally the magnetohydrodynamic problem is decoupled into the hydrodynamic and magnetostatic parts: div Vin = 0, div α H = 0,
α = in, ex.
1 ∂Vin = − grad P in , and ∂t ρ rot α H = 0, α = in, ex.
(35)
A standard way to solve the boundary problem (35) is to introduce the Stokes stream function Ψ (r, z, t) and magnetic
(36)
This gives the following governing equations: 1 ∂P in 1 ∂ 2 Ψ 1 ∂P in 1 ∂ 2 Ψ + = 0, − = 0, ρ ∂z r ∂r∂t ρ ∂r r ∂z∂t ∂2
2 + 2 Φα = 0, α = in, ex. ∂z
(37)
These equations must be supplemented by four boundary conditions, which are derived in Section 5.1. 5.1. Boundary conditions
in [Trz ]in ex ez + [Trr ]ex er = 2σ Her , in [Tzz ]in ex ez + [Tzr ]ex er = 2σ Hez ,
[Tj k ]in ex
(38)
where = Tj k − Tj k and H is mean surface curvature, decomposed as in
ex
H = H0 + H1 , 1 1 ζ ∂ 2ζ ζ0 , H1 = − H0 = + ∝ = . 2R 2 R2 ∂z2 R
(39)
By virtue of (17) we get ∂ζ , ∂z ∂ζ in , or [Tzr ]in ex = [Tzz ]ex − 2σ H ∂z ∂ζ 2 in [Trr ]in − 2σ H = [T ] − 2σ H , zz ex ex ∂z ∂ζ in , [Tzr ]in (40) ex = [Tzz ]ex − 2σ H ∂z where
µ in Trrin = P in + ( Hz )2 − (in Hr )2 , 2
µ in in in Tzz = P + ( Hr )2 − (in Hz )2 , 2
µ0 ex in in Tzr = −µ Hr in Hz , ( Hz )2 − (ex Hr )2 , Trrex = 2
µ0 ex Tzzex = ( Hr )2 − (ex Hz )2 , Tzrex = −µ0 ex Hr ex Hz . 2 Recalling that ∂ζ /∂z ∝ and in [Trr ]in ex − 2σ H = [Trz ]ex
∂Tjink
=− , ∂xk in 2 H in in Tj k = P + µ δj k − µin Hj in Hk , 2 div α H = 0, rot α H = γα Eα + µα [V × α H] ,
ρ
1 ∂Ψ 1 ∂Ψ , Vz = , r ∂z r ∂r α α 1 H = H0 + α H1 , H = −∇Φα , Vr = −
in
Hz − H0 = in Hz1 ∝ ,
in
Hr = in Hr1 ∝ ,
ex
ex
Hz − H0 = ex Hz1 ∝ ,
Hr = ex Hr1 ∝ ,
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(in Hz )2 = H02 + 2H0in Hz1 + O( 2 ), ex
( Hz )
2
= H02
+ 2H0
ex
Hz1 2
+ O( ), 2
( Hr ) = ( Hr ) = O( ) in
2
ex
2
we get within the first-order approximation in µ in µ0 ex in in 2 2 [Trr ]ex − 2σ H = P − 2σ H + ( Hz ) − ( Hz ) 2 2 µ0 ex µ ( Hr )2 − (in Hr )2 + 2 2 µ0 χ 2 in H0 = P − 2σ H + 2 in 1 + H0 µ Hz − µ0 ex Hz1 + O( 2 ), (41) µ in µ0 ex in in 2 2 [Tzz ]ex − 2σ H = P − 2σ H + ( Hr ) − ( Hr ) 2 2 µ0 ex µ ( Hz )2 − (in Hz )2 + 2 2 χ µ 0 H02 = P in − 2σ H − 2 − H0 µin Hz1 − µ0 ex Hz1 + O( 2 ), (42) ex ex in in [Tzr ]in ex = µ0 Hr Hz − µ Hr Hz ex 1 = H0 µ0 Hr − µin Hr1 + O( 2 ).
(43)
Assuming P1in ∝ and combining the first equation in (40) with (41) we obtain µ0 χ 2 P in − 2σ H + H0 + H0 µin Hz1 − µ0 ex Hz1 = 0 2 P0in − 2σ H0 + 12 µ0 χH02 = 0, → P1in − 2σ H1 + H0 µin Hz1 − µ0 ex Hz1 = 0. This gives the unperturbed pressure within the cylindrical jet as µ0 H02 σ −χ . (44) R 2 Combining the second boundary condition in (40) with (42) gives ex µ0 Hr − µin Hr H0 µ0 χ 2 ∂ζ in H0 . = P0 − 2σ H0 − (45) 2 ∂z P0in =
Both Eqs. (44) and (45) lead to the conclusion that ∂ζ (46) , ∂z which coincides with the second static boundary conditions in (16). Thus we arrive at ζ ∂ 2ζ in + 2 + H0 µin Hz1 − µ0 ex Hz1 = 0, P1 + σ 2 R ∂z ∂ζ , Vr = (47) ∂t ∂ζ in 1 (1 + χ)in Hr − ex Hr = χH0 , (48) Hz = ex Hz1 , ∂z µ0 ex Hr − µin Hr = −µ0 χH0
or using the notations of (36) ∂Φin ∂Φex ζ ∂ 2ζ − µ = 0, + H µ + P1in + σ 0 0 ∂z ∂z R2 ∂z2 1 ∂Ψ ∂ζ (49) + = 0, r ∂z ∂t ∂Φin ∂ζ ∂Φex ∂Φin ∂Φex − (1 + χ) = χH , = . (50) ∂r ∂r ∂z ∂z ∂z Assuming that an axisymmetrical disturbance, characterized by a wavelength 2π/k, increases exponentially in time with the growth rate s gives
Φin , Φex , Ψ, ζ, P1in = iφin(r), iφex (r), iψ(r), ς (r), p(r) × est +ikz .
(51)
Consequently, the following boundary conditions hold: 1 2 p+σ − k ς + kH0 (µ0 φin − µφex ) = 0, R2 ψ sς = k , r ∂φin ∂φex − (1 + χ) = kχH0 ς, (52) φex = φin . ∂r ∂r The presence of the field-dependent term in the first equation of (52) indicates that the coupling of hydrodynamics and magnetostatics in the boundary conditions must change the Rayleigh dispersion relation (4) for a nonconducting jet. 5.2. Dispersion relation Inserting (51) into (37) results in the following amplitude equations: 1 ∂p ψ + sρ = 0, k ∂r r ( 2c − k 2 )φin = 0,
1 ∂ψ = 0, r ∂r ( 2c − k 2 )φex = 0.
kp + sρ
(53)
The latter have fundamental solutions that are finite at r = 0 and r = ∞, ψ(r) = A1 krI1 (kr), φin (r) = A2 I0 (kr),
p(r) = −A1 sρkI0 (kr), φex (r) = A3 K0 (kr),
(54)
where Ai are three indeterminate coefficients. Substituting (54) into (52), we get the dispersion relation σ I1 ( ) (1 − 2 ) R 3 ρ I0 ( ) µ0 H 2 I1 ( )K0 ( ) − χ 2 3 2 0 (55) T (, χ) R ρ (see Fig. 3). The last expression although of similar form clearly differs1 from results obtained in [5]. Moreover, (55) s 2 ( ) =
1 The corresponding dispersion relation of [5] can be reduced in the µ H2 I ( ) notation of our paper to s 2 ( ) = σ3 I 1( ) (1 − 2 ) − χ 2 3 02 0 × R ρ 0 R ρ I ( )K ( )
0 0 T (,χ ) . In this expression I0 ( ) appears instead of I1 ( ) in the numerator of the second right-hand-side term.
Y. Zimmels, L.G. Fel / Journal of Colloid and Interface Science 281 (2005) 209–217
215
Appendix A. Contribution of the magnetic field to the free energy Evaluate the contribution µ0 U/2 of the magnetic field inside Ωcyl and outside R3 \ Ωcyl of the disturbed liquid cylinder to the excess free energy W (see Eq. (11)), in 2 2 in 2 U =χ H (r) dv − H0 dv + H (r) dv Ωcyl
0 Ωcyl
+ Fig. 3. A plot of the dispersion relation s( ) for positive susceptibilities χ = 0, 1, 3, 10, from right to left.
H2 (r) dv −
R3 \Ωcyl
=χ
H02 dv R3
H0 +
in
2 Hz1
+
in
(56)
where f (, χ, H0 ) can be recognized as the dimensionless excess free energy found in (26). Expression (56) confirms the equivalence (3) between the static and dynamic approaches in the problem of cutoff wavelength since both nontrivial zeroes, d and s , of snc (d ) and f (s , χ, H0 ), respectively, coincide. Fig. 3 shows that the magnetic effect is to decrease the cutoff wavenumber and the magnitude of the growth rate s. This agrees with the increased stability due to the magnetic field. It is noteworthy that in the limit χ → ∞, H 2 σ I1 ( ) 2 2 2 snc ( ) 3 , (57) (1 − ) − Hch R ρ I0 ( ) which coincides with the dispersion relation (4) for a superconducting jet in the longwave limit → 0.
H0 + in Hz1
+
in
Hr1
2
dv
2 2
H0 + ex Hz1 + ex Hr1 dv −
+ R3 \Ωcyl
dv − Ωcyl
H02 dv
dv
0 Ωcyl
2 2
2H0 in Hz1 + in Hz1 + in Hr1 dv
+
R3
= χ H02
Ωcyl
2 2
2H0 in Hz1 + in Hz1 + in Hr1 dv
+ Ωcyl
+
2H0ex Hz1 +
ex
Hz1
2
+
ex
Hr1
2
dv
R3 \Ωcyl
6. Conclusion
2 2
2H0 in Hz1 + in Hz1 + in Hr1 dv
Ωcyl
+
Acknowledgment
2
Ωcyl
= (1 + χ)
• The field energy and magnetocapillary instability of isothermal incompressible and inviscid nonconducting liquid jets in a uniform magnetic field, are considered and a new dispersion relation (Eq. (55)) for magnetocapillary instability in such jets is derived. • The equivalence between the energy and hydrodynamic approaches at the onset of instability and cutoff wavelength is demonstrated by Eq. (56).
H02 dv
0 Ωcyl
+
2
Hr1 dv −
Ωcyl
can be recasted as σ I1 ( ) 2 f (, χ, H0 ), snc ( ) = − 3 R ρ I0 ( )
ex
Ωcyl
2H0ex Hz1 +
ex
Hz1
2
+
ex
Hr1
2
dv
R3 \Ωcyl
= (1 + χ)
in
Hz1
2
+
in
Hr1
2
dv
Ωcyl
+
ex
Hz1
2
+
ex
Hr1
2
R3 \Ωcyl
+ 2H0 (1 + χ)
dv
in
Hz1 dv
Ωcyl
+
ex
Hz1 dv
.
R3 \Ωcyl
Hence, we conclude that The comments of R.E. Rosensweig are hereby acknowledged.
U = (C1 2 χ 2 + 2C2 χ)H02 ,
(A.1)
216
Y. Zimmels, L.G. Fel / Journal of Colloid and Interface Science 281 (2005) 209–217
where
1+χ 1 C1 = πL πL
in
h1z
2
+
in
h1r
2
I0 ( )K0 ( ) 2 K1 ( ) , = 3 R2 T ( ) K0 ( ) 1 in 1 hz dv πL
dv
0 Ωcyl
1 + πL
ex
2 h1z
+
0 R3 \Ωcyl
1 1+χ C2 = πL πL
hz dv +
in 1
1 πL
ex
2
dv, h1r
Ωcyl
(A.2)
ex 1
hz dv.
=−
2b R L
Recall the expression (11) for W , µ0 H02 π W = ( 2 − 1)σ 2 RL − C1 2 χ 2 + 2C2 χ 2 2 πLσ R 2 2 ( − 1) = 2 C2 χµ0 H02 2 C1 χ + 2 . − (A.4) πL πL σR Making use of the integration u
2 I0 (v) + I12 (v) v dv = uI0 (u)I1 (u),
0 ∞
2 K0 (v) + K12 (v) v dv = uK0 (u)K1 (u),
u
u
∞ I0 (v)v dv = uI1 (u),
K0 (v)v dv = uK1 (u), u
0
and inserting (24) into (A.2) and (A.3) gives in 1 2 in 1 2
1 dv hz + hr πL 0 Ωcyl
kR b2 L R2 2 = 2π 2 I0 (kr) + I12 (kr) (kr) d(kr) πL 2 0
b2 R2 I0 ( )I1 ( ) 2 I0 ( )K0 ( ) 2 I1 ( ) , = 3 R2 T ( ) I0 ( ) ex 1 2 ex 1 2
1 dv hz + hr πL =
0 R3 \Ωcyl
c2 L R2 = 2π 2 πL 2
∞
K02 (kr) + K12 (kr) (kr) d(kr)
kR
=
c2 R 2 K0 ( )K1 ( ) 2
L
kr(z)
dz cos kz 0
(A.3)
R3 \Ωcyl
Ωcyl
R2 b 2π 2 =− πL
I0 (kr)(kr) d(kr) 0
L cos kz(R + ζ0 cos kz) 0
1 πL
× I1 (kR + kζ0 cos kz) dz,
ex 1
hz dv
R3 \Ωcyl
=−
R2 c 2π 2 πL
L
∞ dz cos kz
0
2c R =− L
K0 (kr)(kr) d(kr)
kr(z)
L cos kz(R + ζ0 cos kz) 0
× K1 (kR + kζ0 cos kz) dz. Expanding I1 (x + ε) and K1 (x + ε) in the vicinity of x = 0, up to the first order in ε, gives ε I1 (x + ε) = I1 (x) + I0 (x) + I2 (x) , 2 ε K1 (x + ε) = K1 (x) − K0 (x) + K2 (x) . 2 Finally we arrive at 1 in 1 hz dv πL Ωcyl
2b R =− L
L
dz cos kz RI1 ( ) + ζ0 cos kz 0
I0 ( ) + I2 ( ) × I1 ( ) + 2 L 2b R I0 ( ) + I2 ( ) ζ0 I1 ( ) + =− dz cos2 kz L 2 0
= − b(, χ)R I0 ( ), 1 ex 1 hz dv πL 2
R3 \Ωcyl
2c R =− L
L
dz cos kz RK1 ( ) + ζ0 cos kz
0
Y. Zimmels, L.G. Fel / Journal of Colloid and Interface Science 281 (2005) 209–217
K0 ( ) + K2 ( ) × K1 ( ) − 2 K0 ( ) + K2 ( ) 2c R ζ0 K1 ( ) − =− L 2 L × dz cos2 kz = c(, χ)R 2 K0 ( ).
2 , K0 ( ) ∼ β( ) 1 + 4
• → 0, χ → ∞: (A.5)
(A.6)
πLσ R 2 · f (, χ), 2 C2 χµ0 H02 C1 2 χ +2 . (A.7) f (, χ) = − 1 − πL πL σR Inserting (A.5) and (A.6) in the latter expression, we obtain µ0 RH02 I0 ( )K0 ( ) . (A.8) σ T (, χ)
Appendix B. Asymptotics of expressions Consider the expressions obtained in Appendix A and evaluate their asymptotics:
I0 ( ) ∼ 1 +
H0 2 2 f (, χ) ∼ − 1 + 2 1 + 8 Hch 9 Hch → Hcr (, χ) = √ 1 − 2 . 16 2 2
(B.3)
• → 1:
W=
• → 0, |χ| 1: 2 I1 ( ) ∼ 1+ , 2 8
γ , 2
f (, χ) = 2 − 1 + χ 2 β( ) H0 2 χ 2 (B.1) , × 1 − β( ) 2 Hch χ Hch 1 1 − 2 ln . (B.2) Hcr (, χ) = √ 2 −χ ln
0
f (, χ) = 2 − 1 + χ 2 2
β( ) = − ln
where γ = 0.577216 is Euler’s constant. The corresponding asymptotics for the dimensionless excess free energy f (, χ) and the critical field Hcr (, χ) read
Thus, we get
1 I0 ( )K0 ( ) 2 C1 = 3 R 2 πL T (, χ) I1 ( ) K1 ( ) + × (1 + χ) I0 ( ) K0 ( ) I0 ( )K0 ( ) = 2 R2 , T (, χ) I0 ( )K0 ( ) 1 C2 = − 2 R 2 (1 + χ) πL T (, χ) I0 ( )K0 ( ) − T (, χ) I ( )K0 ( ) 0 , = − χ 2 R 2 T (, χ) which consequently gives
217
2 , 4
χI0 (1)K0 (1) H0 2 f (, χ) ∼ − 1 + , 1 + χI1 (1)K0 (1) Hch
1 Hcr (, χ) = B1 Hch B2 + (B.4) 1 − 2, χ √ where B1 = 1/ I0 (1)K0 (1) 1.3697, B2 = I1 (1) × K0 (1) 0.2379. 2
References [1] J. Plateau, Statique Expérimentale et Théorique des Liquides Soumis aux Seules Forces Moléculaires, Gauthier–Villars, Paris, 1873. [2] L. Rayleigh, Proc. London Math. Soc. 10 (1879) 4. [3] L.G. Fel, Y. Zimmels, Sov. Phys. JETP 98 (5) (2004) 960. [4] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Instability, Oxford Univ. Press, Oxford, 1961. [5] N.G. Taktarov, Magnetohydrodynamics 11 (2) (1975) 156. [6] R.E. Rosensweig, Ferrohydrodynamics, Cambridge Univ. Press, Cambridge, 1985. [7] L. Landau, E. Lifshitz, Electrodynamics of Continuous Media, Pergamon, Oxford, 1984.
Energy and hydrodynamic approaches to magnetocapillary instability of nonconducting jets Y. Zimmels, L.G. Fel ∗ Department of Civil and Environmental Engineering, Technion, Haifa 32000, Israel Received 19 May 2004; accepted 6 August 2004 Available online 21 September 2004
Abstract The field energy and magnetocapillary instability of isothermal incompressible and inviscid nonconducting liquid jets in a uniform magnetic field are considered. The equivalence between static and dynamic approaches at the onset of instability and cutoff wavelength is shown and its implications are discussed. A new dispersion relation for magnetocapillary instability in such jets is derived. This relation differs from that given in the literature. The existence of a critical magnetic field that stabilizes jets with finite susceptibility is established. It is shown that the jet is stabilized by the field irrespective of its being para- or diamagnetic, but the extent of stabilization is different. 2004 Elsevier Inc. All rights reserved. Keywords: Plateau problem; Magnetocapillary instability; Nonconducting liquid jet
1. Introduction Joseph Antoine Ferdinand Plateau (1801–1883) was a Belgian physicist who is best remembered in mathematics for the Plateau problem. He wrote his seminal book [1] when he was already blind, as he was for the last 40 years of his life. The Plateau problem, in its brief formulation, is to find a surface of minimum area S given its boundary ∂Ω. The problem relates to the principle of minimum free energy at equilibrium. In this context it may be required to find a surface with isotropic tension σ which provides a minimum surface free energy σ S. Rayleigh [2] gave a theoretical explanation for the instability of liquid cylinders that are longer than their circumference. Further generalization is called for if the excess free energy W of the cylinder comprises different types of energy that reflect a more complex structure of the liquid (e.g., elasticity [3]) as well as its capacity to interact with external fields. The threshold of static instability, which is associated with the change of sign of W , when the disturbed configuration becomes more preferable, is defined * Corresponding author.
E-mail address: [email protected] (L.G. Fel). 0021-9797/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2004.08.024
by W (kR, p) = 0 → ks = ks (p),
(1)
where k is a wavenumber assigned to a small surface disturbance, R is a radius of cylinder, and the parameter p stands for the effect of an external interaction, which contributes to the jet evolution, e.g., fluid polarization in the presence of external fields. Note that the static cutoff wavenumber ks is related to the static cutoff wavelength Λs by Λs (p) = 2π/ks . Plateau instability being a static problem has also a dynamic aspect. Consider Rayleigh instability in liquid jets of radius R and its corresponding dispersion equation s = s(kR, p). The latter determines the exponential evolution in time Θ(r, t) ∼ est of all hydrodynamic functions Θ(r, t) of the jet with the growth rate s. Rayleigh’s theory of capillary instability in liquid jets states that the maximum of the dispersion function s(kR, p), which corresponds to the wavenumber kmax (p), gives rise to evolution of the largest capillary instability. The range of wavenumbers k which contribute to the evolution of the instability is given by s > 0. Thus, the threshold of instability follows as s(kR, p) = 0 → kd = kd (p).
(2)
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Y. Zimmels, L.G. Fel / Journal of Colloid and Interface Science 281 (2005) 209–217
Note that the dynamic cutoff wavenumber kd is related to the dynamic cutoff wavelength Λd by Λd (p) = 2π/kd . The value Λd (0), which corresponds to free-jet evolution, is equal to 2πR according to Rayleigh [2]. If there exists a critical parameter p = pcr such that for all p pcr the expression s(kR, p) 0 holds, then for p pcr the jet is stable for all wavelengths. This means that the liquid jet preserves its initial shape, which is unaffected by small perturbations, irrespective of the jet velocity. Thus, this conclusion must hold also in the limiting case of a motionless fluid; i.e., Λd (p) ≡ Λs (p).
(3)
This identity reflects a deep equivalence between the static approach (the threshold of static instability concerned with an excess free energy) and the dynamic approach (the bifurcation of the first nontrivial steady state of the inviscid hydrodynamic system). Thus, we come to the Plateau problem for a liquid cylinder with a free surface in an external field. The capillary instability of magnetizable liquid jets in the presence of an axial magnetic field constitutes a classical example. Chandrasekhar [4] treated the hydrodynamics of nonpermeable conducting liquid jets and derived their dispersion relation in implicit form. The relation reads particularly simply for superconducting ssc ( ) and nonconducting snc ( ) liquids (see [4, pp. 545–549]), σ I1 ( ) 2 (1 − 2 ) ssc ( ) = 3 R ρ I0 ( ) 2 H − , Hs I0 ( )K1 ( ) σ I1 ( ) (1 − 2 ), (4) R 3 ρ I0 ( ) √ where = kR and Hs = σ/µ0 R is designated by Chandrasekhar as the characteristic field. σ, ρ stand for the isotropic surface tension and density of the liquid, respectively, and µ0 denotes the permeability of free space. Im (x) and Km (x) are the modified Bessel functions of order m of the 1st and 2nd kind, respectively. In the superconducting limit the relation (4) discloses the existence of a critical √ magnetic field Hcr = Hs / 2 beyond which the jet is stable. In the nonconducting limit, this dispersion relation coincides with that of Rayleigh [2]. In 1975, Taktarov [5] considered the magnetocapillary instability of nonconducting permeable jets and derived a dispersion relation in explicit form (see also Rosensweig [6]). This relation accounts for the coupling effect between the magnetization of the liquid and the external field. Being nondissipative, this linear problem facilitates its examination by either the energy or the hydrodynamic approach, and both must yield the same cutoff wavelength. In comparing our results, we have found a discrepancy with [5]. The aim of this paper is to give an accurate solution, first for the case of Plateau instability (using the energy 2 ( ) = snc
approach; see Sections 2, 3, and 4), and then regarding the dynamic case. In this context, the energy and hydrodynamic approaches must give identical results. In Section 5 we solve, using the hydrodynamic approach, the magnetocapillary instability problem of isothermal, incompressible, inviscid, and nonconducting jets, in the presence of a uniform magnetic field. The new dispersion relation accounts for the effect of magnetic fields and conforms with the solution of Plateau instability for magnetizable liquids.
2. Free energy of a liquid cylinder in the presence of a magnetic field The Plateau problem of static instability of a nonconducting liquid cylinder which is subjected to a uniform magnetic field appears, at first glance, to be a simple generalization of its counterpart in the absence of external fields. However, deeper consideration shows that the presence of the field complicates considerably the physical picture and computational procedure. A fundamental question arises concerning the correct definition of the excess free energy W which must be minimized via variation of the cylinder shape. Consider an isothermal liquid cylinder in a uniform magnetic field H0 that is applied in free space along the cylinder axis. The magnetic susceptibility χ of the cylinder is assumed to be isotropic, independent of magnetic field, and satisfying the thermodynamic condition χ > −1 [7]. When the liquid cylinder is undisturbed the total free energy F 0 of the system is given by χµ0 H02 · F 0 = Es0 − (5) dv, 2 0 Ωcyl
where the integral represents the volume πR 2 L, enclosed 0 , which is occupied by the undisturbed by the area ∂Ωcyl cylinder. The term Es0 = σ ∂Ω 0 ds = 2πσ RL stands for cyl
the surface free energy of the undisturbed cylinder, where R, L, and σ denote its radius, length, and surface tension, respectively. Deformation of the cylinder shape changes the field H(r) over all space R3 , i.e., in both the internal domain Ωcyl and its complement (the exterior domain) R3 \ Ωcyl . Following Plateau, we assume conservation of the cylinder volume dv = dv. (6) 0 Ωcyl
Ωcyl
The total free energy F of the disturbed cylinder takes the form µ0 in 2 · F = Es − (1 + χ) H (r) dv 2 µ0 · − 2
Ωcyl
R3 \Ωcyl
ex
µ0 · H (r) dv + 2
2
H02 dv, R3
(7)
Y. Zimmels, L.G. Fel / Journal of Colloid and Interface Science 281 (2005) 209–217
where in H(r) and ex H(r) denote the internal and external magnetic fields, and Es = σ ∂Ωcyl ds is surface energy. The second and third terms on the right-hand side of (7) reflect the contribution of the distorted permeable cylinder to the magnetic free energy, stored in the internal domain Ωcyl (second term) and the exterior domain R3 \ Ωcyl (third term), corrected by the energy of the uniform magnetic field in free space (forth term). The excess free energy W of the system is defined as W = F − F 0.
where according to the assumption 1 the following approximations apply (see Section 3): in 1 ex 1 in 1 ex 1 in 1 ex 1 in 1 ex 1
Hr , Hr , Hz , Hz = hr , hr , hz , hz × χH0 .
(10)
The dimensionless fields in,ex h1r,z (r, z) are dependent on the coordinates. Inserting (10) into (7), and performing the integration, we evaluate W given in (8) as µ0 U, (11) 2 where use was made of (5) and U is given by (see Appendix A) in 1 2 in 1 2
dv U = (1 + χ) Hz + Hr W = Es − Es0 −
Ωcyl
+
ex
Hz1
2
+
ex
Hr1
2
R3 \Ωcyl
+ 2H0 (1 + χ)
dv
in
Hz1 dv +
Ωcyl
ex
Hz1 dv
. (12)
R3 \Ωcyl
We specify the commonly used harmonic deformation of the cylinder as r(z) = R + ζ0 cos kz, where k = 2π/Λ, Λ being the disturbance wavelength. By virtue of translational invariance of the problem hr,z (r, z + Λ) = in,ex h1r,z (r, z)
in,ex 1
we set L = Λ and evaluate the free energy per unit wavelength. For this type of deformation the term Es − Es0 can be evaluated πζ 2 ds − 2πσ RL = σ 0 L( 2 − 1), Es − Es0 = σ 2R ∂Ωcyl
= kR.
(14)
Next we proceed to solve the boundary problem and get the distribution of the magnetic fields.
(8)
From the mathematical standpoint, the variational problem for minimization of W , supplemented with constraint (6) for all smooth surfaces ∂Ωcyl , is known as the isoperimetric problem. The cylinder instability can be studied, assuming small perturbation in shape. In this case the Plateau problem becomes solvable in closed form. Let the extent of deformation be characterized by a length ζ0 , such that ζ0 /R = 1. The fields in H(r) and ex H(r) which must satisfy Maxwell’s equations can be represented as small perturbations of H0 , in H(r) = H0 + in H1 (r) = H0 + in Hz1 , in Hr1 , ex (9) H(r) = H0 + ex H1 (r) = H0 + ex Hz1 , ex Hr1 ,
211
(13)
3. Boundary problem and its solution The magnetostatics of the disturbed liquid cylinder is governed by Maxwell equations for the internal in H(r) and external ex H(r) magnetic fields, rot in H = rot ex H = 0,
div in B = div ex B = 0,
(15)
where in B = µ0 (1 + χ)in H and ex B = µ0 ex H denote internal and external magnetic inductions, respectively. Equations (15) must be supplemented with boundary conditions (BC) at the interface r = R, in H, t = ex H, t,
in B, e = ex B, e,
e, t = 0,
(16)
where , denotes a scalar product and t and e stand for tangential and normal unit vectors to the surface, respectively. Since the surface deformation is small, linearization can be applied, tz = er = 1 − (∂ζ /∂z)2 1. (17) tr = −ez = ∂ζ /∂z, A standard way to solve the problem is to introduce the magnetic potentials Φin (r) and Φex (r), which are defined as in H1 (r) = −∇Φin , ex H1 (r) = −∇Φex , where |∇Φin |, |∇Φex | H0 . These potentials satisfy the first two equations in (15). The last two equations in (15) yield ∂ 2 Φex ∂ 2 Φin +
Φ = 0, + 2 Φex = 0, 2 in ∂z2 ∂z2 ∂2 1 ∂ ,
2 = 2 + (18) r ∂r ∂r where 2 is the two-dimensional Laplacian. Reformulation of the BC (16) for Φex (r), Φin (r) gives ∂Φex ∂Φin ∂ζ ∂Φex ∂Φin = , − (1 + χ) = χH . (19) ∂z ∂z ∂r ∂r ∂z Using Φin (r, z) = φin (r) sin kz and Φex (r, z) = φex (r) sin kz we find ( 2 − k 2 )φex = 0,
( 2 − k 2 )φin = 0
(20)
∂φin ∂φex − = χH kζ0. ∂r ∂r
(21)
with BC at r = R, φex = φin ,
(1 + χ)
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Y. Zimmels, L.G. Fel / Journal of Colloid and Interface Science 281 (2005) 209–217
The solutions of (20), which satisfy BC (21) and are finite at r = 0 and r = ∞, are obtained as b(, χ) I0 (kr), c(, χ) φex (r) = ζ0 χH0 (22) K0 (kr), where K0 ( ) I0 ( ) b(, χ) = 2 , c(, χ) = 2 , T (, χ) T (, χ) T (, χ) = 1 + χ I1 ( )K0 ( ). (23)
φin (r) = ζ0 χH0
Recalling the definition (10) of the dimensionless fields hr,z (r, z), the following final expressions are obtained:
in,ex 1
Fig. 1. A plot of hcr versus kR for large positive susceptibilities: χ = 1, 10, 102 , 103 , 106 , from right to left, respectively.
hz = −b(, χ)I0 (kr) cos kz,
in 1
hr = −b(, χ)I1 (kr) sin kz,
in 1
hz = −c(, χ)K0 (kr) cos kz,
ex 1
hr = c(, χ)K1 (kr) sin kz.
ex 1
(24)
Bearing in mind that the function Q( ) = I1 ( )K0 ( ) is monotone growing at the positive half axis and is bounded, 0 Q( ) < 1/2, we conclude that the fields in,ex h1r,z (r, z) are free of singularities in the thermodynamically relevant region χ > −1.
4. Plateau instability In this section we calculate via Eq. (10) the excess free energy W in terms of the dimensionless fields ex h1z,r , in h1z,r . Inserting (24) into (12) we get (see Appendix A) W=
πLσ R 2 · f (, χ, H0 ), 2
µ0 RH02 I0 ( )K0 ( ) , σ T (, χ) (25) where the dimensionless excess free energy f (, χ, H0 ) can be defined by introducing the characteristic field Hch , I0 ( )K0 ( ) H0 2 f (, χ, H0 ) = 2 − 1 + χ 2 , T (, χ) Hch σ . Hch = (26) µ0 χR f (, χ, H0 ) = 2 − 1 + χ 2 2
Notice that Hch differs from Hs , introduced by Chandrasekhar in [4]. Formula (26) serves for both paramagnetic (χ > 0) and diamagnetic (χ < 0) liquids, since the latter case does not lead to an imaginary expression in (26) due to the term χ 2 in (25). Consider the field Hcr (, χ) satisfying f (, χ, Hcr ) = 0 and call it critical field. Beyond this field, H0 Hcr , the cylinder is stable. The expression for the critical field is
Hch 1 − 2 T (, χ) . Hcr (, χ) = (27) χ I0 ( )K0 ( )
Fig. 2. A plot of hcr versus kR for small positive and negative susceptibilities: χ = ±10−4 , ±10−3 , ±10−2 , ±10−1 , from right to left, respectively. The plots corresponding to diamagnetic and paramagnetic cases nearly coincide.
Figs. 1 and 2 show plots of hcr = Hcr /Hch for strong (χ 1) and weak (|χ| 1) magnetic liquids. The corresponding asymptotics for Hcr (, χ), in the case of weak χ 1 and strong χ 1 magnetic susceptibilities and in a long wave limit → 0, which are derived in Appendix B, are presented below: χ 1 Hcr (, χ) 1 − 2 ln , = √ Hch 2 −χ ln → 0, |χ| 1, (28) 9 2 1 Hcr (, χ) = √ 1 − , → 0, χ → ∞, (29) Hch 16 2
1 Hcr (, χ) = B1 B2 + 1 − 2 , → 1, (30) Hch χ √ where B1 = 1/ I0 (1)K0 (1) 1.3697, B2 = I1 (1)K0 (1) 0.2379. Fig. 1 shows that increase of χ results in curves that approach a parabolic shape; see (29). The asymptotic tendency of large χ curves is to the one shown for χ = 106 . At → 0 there is a fast increase in hcr that appears as a jump. Furthermore, the high χ range is associated with a relatively
Y. Zimmels, L.G. Fel / Journal of Colloid and Interface Science 281 (2005) 209–217
213
low hcr . This is realized upon comparing Figs. 1 and 2 where the range of hcr in Fig. 2 is considerably extended. At the end of this section we present physical arguments which justify our conclusion regarding stabilization of the permeable jets for both paramagnetic and diamagnetic liquids. The orientation of the dipole moments along the applied magnetic field in the paramagnetic liquid enforces the rigidity of the jet under small disturbances. In the case of diamagnetic liquid the increase of the jets’ stabilization is due to orientation of the dipole moments in the opposite direction which also enforces the rigidity of the jet. The similar increase of the rigidity, and its corresponding influence on the stabilization, was found recently [3] in the liquid crystalline jet due to elasticity of the media. Although the critical fields for paramagnetic Hcr (, χ) and diamagnetic Hcr (, −χ) liquids are different, the following universal relation holds:
potentials Φin (r, z, t), Φex (r, z, t),
Hcr2 (, χ) Hcr2 (, −χ) 1 − 2 I1 ( ) . − =2 2 2 I0 ( ) Hch Hch
It is necessary to apply boundary conditions (16), which are imposed on H and B, as well as those for the hydrodynamic variables. First, the velocity Vr must be compatible, at r = R, with the assumed form of the deformed boundary ∂ζ /∂t. Second, at the free surface of a liquid jet the jump in stress must be balanced by Laplace pressure [7],
(31)
5. Hydrodynamics of nonconducting jet in a magnetic field Consider an isothermal, incompressible, inviscid, and nonconducting jet in the presence of a magnetic field H0 applied along its z-axis. The deviation from initial values of the pressure is defined as P1in = P in − P0in , where P0in is the unperturbed pressure within the cylindrical jet. The deviations of the internal and external magnetic fields are defined as in 1 H = in H − H0 and ex H1 = ex H − H0 , respectively. The governing equations of magnetohydrodynamics which are given by div Vin = 0, ∂Vjin
(32)
∂t
α = in, ex,
(33)
(34)
can be simplified considerably by applying the Maxwell equations (34) for nonconducting media (γ in = γex = 0) to the Navier–Stokes equation (33). γ in and γex denote conductivities of the jet’s interior and exterior, respectively, Tj k stands for the magnetic stress tensor, and Vin is local fluid velocity. Finally the magnetohydrodynamic problem is decoupled into the hydrodynamic and magnetostatic parts: div Vin = 0, div α H = 0,
α = in, ex.
1 ∂Vin = − grad P in , and ∂t ρ rot α H = 0, α = in, ex.
(35)
A standard way to solve the boundary problem (35) is to introduce the Stokes stream function Ψ (r, z, t) and magnetic
(36)
This gives the following governing equations: 1 ∂P in 1 ∂ 2 Ψ 1 ∂P in 1 ∂ 2 Ψ + = 0, − = 0, ρ ∂z r ∂r∂t ρ ∂r r ∂z∂t ∂2
2 + 2 Φα = 0, α = in, ex. ∂z
(37)
These equations must be supplemented by four boundary conditions, which are derived in Section 5.1. 5.1. Boundary conditions
in [Trz ]in ex ez + [Trr ]ex er = 2σ Her , in [Tzz ]in ex ez + [Tzr ]ex er = 2σ Hez ,
[Tj k ]in ex
(38)
where = Tj k − Tj k and H is mean surface curvature, decomposed as in
ex
H = H0 + H1 , 1 1 ζ ∂ 2ζ ζ0 , H1 = − H0 = + ∝ = . 2R 2 R2 ∂z2 R
(39)
By virtue of (17) we get ∂ζ , ∂z ∂ζ in , or [Tzr ]in ex = [Tzz ]ex − 2σ H ∂z ∂ζ 2 in [Trr ]in − 2σ H = [T ] − 2σ H , zz ex ex ∂z ∂ζ in , [Tzr ]in (40) ex = [Tzz ]ex − 2σ H ∂z where
µ in Trrin = P in + ( Hz )2 − (in Hr )2 , 2
µ in in in Tzz = P + ( Hr )2 − (in Hz )2 , 2
µ0 ex in in Tzr = −µ Hr in Hz , ( Hz )2 − (ex Hr )2 , Trrex = 2
µ0 ex Tzzex = ( Hr )2 − (ex Hz )2 , Tzrex = −µ0 ex Hr ex Hz . 2 Recalling that ∂ζ /∂z ∝ and in [Trr ]in ex − 2σ H = [Trz ]ex
∂Tjink
=− , ∂xk in 2 H in in Tj k = P + µ δj k − µin Hj in Hk , 2 div α H = 0, rot α H = γα Eα + µα [V × α H] ,
ρ
1 ∂Ψ 1 ∂Ψ , Vz = , r ∂z r ∂r α α 1 H = H0 + α H1 , H = −∇Φα , Vr = −
in
Hz − H0 = in Hz1 ∝ ,
in
Hr = in Hr1 ∝ ,
ex
ex
Hz − H0 = ex Hz1 ∝ ,
Hr = ex Hr1 ∝ ,
214
Y. Zimmels, L.G. Fel / Journal of Colloid and Interface Science 281 (2005) 209–217
(in Hz )2 = H02 + 2H0in Hz1 + O( 2 ), ex
( Hz )
2
= H02
+ 2H0
ex
Hz1 2
+ O( ), 2
( Hr ) = ( Hr ) = O( ) in
2
ex
2
we get within the first-order approximation in µ in µ0 ex in in 2 2 [Trr ]ex − 2σ H = P − 2σ H + ( Hz ) − ( Hz ) 2 2 µ0 ex µ ( Hr )2 − (in Hr )2 + 2 2 µ0 χ 2 in H0 = P − 2σ H + 2 in 1 + H0 µ Hz − µ0 ex Hz1 + O( 2 ), (41) µ in µ0 ex in in 2 2 [Tzz ]ex − 2σ H = P − 2σ H + ( Hr ) − ( Hr ) 2 2 µ0 ex µ ( Hz )2 − (in Hz )2 + 2 2 χ µ 0 H02 = P in − 2σ H − 2 − H0 µin Hz1 − µ0 ex Hz1 + O( 2 ), (42) ex ex in in [Tzr ]in ex = µ0 Hr Hz − µ Hr Hz ex 1 = H0 µ0 Hr − µin Hr1 + O( 2 ).
(43)
Assuming P1in ∝ and combining the first equation in (40) with (41) we obtain µ0 χ 2 P in − 2σ H + H0 + H0 µin Hz1 − µ0 ex Hz1 = 0 2 P0in − 2σ H0 + 12 µ0 χH02 = 0, → P1in − 2σ H1 + H0 µin Hz1 − µ0 ex Hz1 = 0. This gives the unperturbed pressure within the cylindrical jet as µ0 H02 σ −χ . (44) R 2 Combining the second boundary condition in (40) with (42) gives ex µ0 Hr − µin Hr H0 µ0 χ 2 ∂ζ in H0 . = P0 − 2σ H0 − (45) 2 ∂z P0in =
Both Eqs. (44) and (45) lead to the conclusion that ∂ζ (46) , ∂z which coincides with the second static boundary conditions in (16). Thus we arrive at ζ ∂ 2ζ in + 2 + H0 µin Hz1 − µ0 ex Hz1 = 0, P1 + σ 2 R ∂z ∂ζ , Vr = (47) ∂t ∂ζ in 1 (1 + χ)in Hr − ex Hr = χH0 , (48) Hz = ex Hz1 , ∂z µ0 ex Hr − µin Hr = −µ0 χH0
or using the notations of (36) ∂Φin ∂Φex ζ ∂ 2ζ − µ = 0, + H µ + P1in + σ 0 0 ∂z ∂z R2 ∂z2 1 ∂Ψ ∂ζ (49) + = 0, r ∂z ∂t ∂Φin ∂ζ ∂Φex ∂Φin ∂Φex − (1 + χ) = χH , = . (50) ∂r ∂r ∂z ∂z ∂z Assuming that an axisymmetrical disturbance, characterized by a wavelength 2π/k, increases exponentially in time with the growth rate s gives
Φin , Φex , Ψ, ζ, P1in = iφin(r), iφex (r), iψ(r), ς (r), p(r) × est +ikz .
(51)
Consequently, the following boundary conditions hold: 1 2 p+σ − k ς + kH0 (µ0 φin − µφex ) = 0, R2 ψ sς = k , r ∂φin ∂φex − (1 + χ) = kχH0 ς, (52) φex = φin . ∂r ∂r The presence of the field-dependent term in the first equation of (52) indicates that the coupling of hydrodynamics and magnetostatics in the boundary conditions must change the Rayleigh dispersion relation (4) for a nonconducting jet. 5.2. Dispersion relation Inserting (51) into (37) results in the following amplitude equations: 1 ∂p ψ + sρ = 0, k ∂r r ( 2c − k 2 )φin = 0,
1 ∂ψ = 0, r ∂r ( 2c − k 2 )φex = 0.
kp + sρ
(53)
The latter have fundamental solutions that are finite at r = 0 and r = ∞, ψ(r) = A1 krI1 (kr), φin (r) = A2 I0 (kr),
p(r) = −A1 sρkI0 (kr), φex (r) = A3 K0 (kr),
(54)
where Ai are three indeterminate coefficients. Substituting (54) into (52), we get the dispersion relation σ I1 ( ) (1 − 2 ) R 3 ρ I0 ( ) µ0 H 2 I1 ( )K0 ( ) − χ 2 3 2 0 (55) T (, χ) R ρ (see Fig. 3). The last expression although of similar form clearly differs1 from results obtained in [5]. Moreover, (55) s 2 ( ) =
1 The corresponding dispersion relation of [5] can be reduced in the µ H2 I ( ) notation of our paper to s 2 ( ) = σ3 I 1( ) (1 − 2 ) − χ 2 3 02 0 × R ρ 0 R ρ I ( )K ( )
0 0 T (,χ ) . In this expression I0 ( ) appears instead of I1 ( ) in the numerator of the second right-hand-side term.
Y. Zimmels, L.G. Fel / Journal of Colloid and Interface Science 281 (2005) 209–217
215
Appendix A. Contribution of the magnetic field to the free energy Evaluate the contribution µ0 U/2 of the magnetic field inside Ωcyl and outside R3 \ Ωcyl of the disturbed liquid cylinder to the excess free energy W (see Eq. (11)), in 2 2 in 2 U =χ H (r) dv − H0 dv + H (r) dv Ωcyl
0 Ωcyl
+ Fig. 3. A plot of the dispersion relation s( ) for positive susceptibilities χ = 0, 1, 3, 10, from right to left.
H2 (r) dv −
R3 \Ωcyl
=χ
H02 dv R3
H0 +
in
2 Hz1
+
in
(56)
where f (, χ, H0 ) can be recognized as the dimensionless excess free energy found in (26). Expression (56) confirms the equivalence (3) between the static and dynamic approaches in the problem of cutoff wavelength since both nontrivial zeroes, d and s , of snc (d ) and f (s , χ, H0 ), respectively, coincide. Fig. 3 shows that the magnetic effect is to decrease the cutoff wavenumber and the magnitude of the growth rate s. This agrees with the increased stability due to the magnetic field. It is noteworthy that in the limit χ → ∞, H 2 σ I1 ( ) 2 2 2 snc ( ) 3 , (57) (1 − ) − Hch R ρ I0 ( ) which coincides with the dispersion relation (4) for a superconducting jet in the longwave limit → 0.
H0 + in Hz1
+
in
Hr1
2
dv
2 2
H0 + ex Hz1 + ex Hr1 dv −
+ R3 \Ωcyl
dv − Ωcyl
H02 dv
dv
0 Ωcyl
2 2
2H0 in Hz1 + in Hz1 + in Hr1 dv
+
R3
= χ H02
Ωcyl
2 2
2H0 in Hz1 + in Hz1 + in Hr1 dv
+ Ωcyl
+
2H0ex Hz1 +
ex
Hz1
2
+
ex
Hr1
2
dv
R3 \Ωcyl
6. Conclusion
2 2
2H0 in Hz1 + in Hz1 + in Hr1 dv
Ωcyl
+
Acknowledgment
2
Ωcyl
= (1 + χ)
• The field energy and magnetocapillary instability of isothermal incompressible and inviscid nonconducting liquid jets in a uniform magnetic field, are considered and a new dispersion relation (Eq. (55)) for magnetocapillary instability in such jets is derived. • The equivalence between the energy and hydrodynamic approaches at the onset of instability and cutoff wavelength is demonstrated by Eq. (56).
H02 dv
0 Ωcyl
+
2
Hr1 dv −
Ωcyl
can be recasted as σ I1 ( ) 2 f (, χ, H0 ), snc ( ) = − 3 R ρ I0 ( )
ex
Ωcyl
2H0ex Hz1 +
ex
Hz1
2
+
ex
Hr1
2
dv
R3 \Ωcyl
= (1 + χ)
in
Hz1
2
+
in
Hr1
2
dv
Ωcyl
+
ex
Hz1
2
+
ex
Hr1
2
R3 \Ωcyl
+ 2H0 (1 + χ)
dv
in
Hz1 dv
Ωcyl
+
ex
Hz1 dv
.
R3 \Ωcyl
Hence, we conclude that The comments of R.E. Rosensweig are hereby acknowledged.
U = (C1 2 χ 2 + 2C2 χ)H02 ,
(A.1)
216
Y. Zimmels, L.G. Fel / Journal of Colloid and Interface Science 281 (2005) 209–217
where
1+χ 1 C1 = πL πL
in
h1z
2
+
in
h1r
2
I0 ( )K0 ( ) 2 K1 ( ) , = 3 R2 T ( ) K0 ( ) 1 in 1 hz dv πL
dv
0 Ωcyl
1 + πL
ex
2 h1z
+
0 R3 \Ωcyl
1 1+χ C2 = πL πL
hz dv +
in 1
1 πL
ex
2
dv, h1r
Ωcyl
(A.2)
ex 1
hz dv.
=−
2b R L
Recall the expression (11) for W , µ0 H02 π W = ( 2 − 1)σ 2 RL − C1 2 χ 2 + 2C2 χ 2 2 πLσ R 2 2 ( − 1) = 2 C2 χµ0 H02 2 C1 χ + 2 . − (A.4) πL πL σR Making use of the integration u
2 I0 (v) + I12 (v) v dv = uI0 (u)I1 (u),
0 ∞
2 K0 (v) + K12 (v) v dv = uK0 (u)K1 (u),
u
u
∞ I0 (v)v dv = uI1 (u),
K0 (v)v dv = uK1 (u), u
0
and inserting (24) into (A.2) and (A.3) gives in 1 2 in 1 2
1 dv hz + hr πL 0 Ωcyl
kR b2 L R2 2 = 2π 2 I0 (kr) + I12 (kr) (kr) d(kr) πL 2 0
b2 R2 I0 ( )I1 ( ) 2 I0 ( )K0 ( ) 2 I1 ( ) , = 3 R2 T ( ) I0 ( ) ex 1 2 ex 1 2
1 dv hz + hr πL =
0 R3 \Ωcyl
c2 L R2 = 2π 2 πL 2
∞
K02 (kr) + K12 (kr) (kr) d(kr)
kR
=
c2 R 2 K0 ( )K1 ( ) 2
L
kr(z)
dz cos kz 0
(A.3)
R3 \Ωcyl
Ωcyl
R2 b 2π 2 =− πL
I0 (kr)(kr) d(kr) 0
L cos kz(R + ζ0 cos kz) 0
1 πL
× I1 (kR + kζ0 cos kz) dz,
ex 1
hz dv
R3 \Ωcyl
=−
R2 c 2π 2 πL
L
∞ dz cos kz
0
2c R =− L
K0 (kr)(kr) d(kr)
kr(z)
L cos kz(R + ζ0 cos kz) 0
× K1 (kR + kζ0 cos kz) dz. Expanding I1 (x + ε) and K1 (x + ε) in the vicinity of x = 0, up to the first order in ε, gives ε I1 (x + ε) = I1 (x) + I0 (x) + I2 (x) , 2 ε K1 (x + ε) = K1 (x) − K0 (x) + K2 (x) . 2 Finally we arrive at 1 in 1 hz dv πL Ωcyl
2b R =− L
L
dz cos kz RI1 ( ) + ζ0 cos kz 0
I0 ( ) + I2 ( ) × I1 ( ) + 2 L 2b R I0 ( ) + I2 ( ) ζ0 I1 ( ) + =− dz cos2 kz L 2 0
= − b(, χ)R I0 ( ), 1 ex 1 hz dv πL 2
R3 \Ωcyl
2c R =− L
L
dz cos kz RK1 ( ) + ζ0 cos kz
0
Y. Zimmels, L.G. Fel / Journal of Colloid and Interface Science 281 (2005) 209–217
K0 ( ) + K2 ( ) × K1 ( ) − 2 K0 ( ) + K2 ( ) 2c R ζ0 K1 ( ) − =− L 2 L × dz cos2 kz = c(, χ)R 2 K0 ( ).
2 , K0 ( ) ∼ β( ) 1 + 4
• → 0, χ → ∞: (A.5)
(A.6)
πLσ R 2 · f (, χ), 2 C2 χµ0 H02 C1 2 χ +2 . (A.7) f (, χ) = − 1 − πL πL σR Inserting (A.5) and (A.6) in the latter expression, we obtain µ0 RH02 I0 ( )K0 ( ) . (A.8) σ T (, χ)
Appendix B. Asymptotics of expressions Consider the expressions obtained in Appendix A and evaluate their asymptotics:
I0 ( ) ∼ 1 +
H0 2 2 f (, χ) ∼ − 1 + 2 1 + 8 Hch 9 Hch → Hcr (, χ) = √ 1 − 2 . 16 2 2
(B.3)
• → 1:
W=
• → 0, |χ| 1: 2 I1 ( ) ∼ 1+ , 2 8
γ , 2
f (, χ) = 2 − 1 + χ 2 β( ) H0 2 χ 2 (B.1) , × 1 − β( ) 2 Hch χ Hch 1 1 − 2 ln . (B.2) Hcr (, χ) = √ 2 −χ ln
0
f (, χ) = 2 − 1 + χ 2 2
β( ) = − ln
where γ = 0.577216 is Euler’s constant. The corresponding asymptotics for the dimensionless excess free energy f (, χ) and the critical field Hcr (, χ) read
Thus, we get
1 I0 ( )K0 ( ) 2 C1 = 3 R 2 πL T (, χ) I1 ( ) K1 ( ) + × (1 + χ) I0 ( ) K0 ( ) I0 ( )K0 ( ) = 2 R2 , T (, χ) I0 ( )K0 ( ) 1 C2 = − 2 R 2 (1 + χ) πL T (, χ) I0 ( )K0 ( ) − T (, χ) I ( )K0 ( ) 0 , = − χ 2 R 2 T (, χ) which consequently gives
217
2 , 4
χI0 (1)K0 (1) H0 2 f (, χ) ∼ − 1 + , 1 + χI1 (1)K0 (1) Hch
1 Hcr (, χ) = B1 Hch B2 + (B.4) 1 − 2, χ √ where B1 = 1/ I0 (1)K0 (1) 1.3697, B2 = I1 (1) × K0 (1) 0.2379. 2
References [1] J. Plateau, Statique Expérimentale et Théorique des Liquides Soumis aux Seules Forces Moléculaires, Gauthier–Villars, Paris, 1873. [2] L. Rayleigh, Proc. London Math. Soc. 10 (1879) 4. [3] L.G. Fel, Y. Zimmels, Sov. Phys. JETP 98 (5) (2004) 960. [4] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Instability, Oxford Univ. Press, Oxford, 1961. [5] N.G. Taktarov, Magnetohydrodynamics 11 (2) (1975) 156. [6] R.E. Rosensweig, Ferrohydrodynamics, Cambridge Univ. Press, Cambridge, 1985. [7] L. Landau, E. Lifshitz, Electrodynamics of Continuous Media, Pergamon, Oxford, 1984.