Small Oscillations of Magnetizable Ideal Fluid
The small oscillations of a magnetizable ideal fluid in partially filled vessel are considered. Solvability of the initial-boundary problem is proved and the generic properties of the frequencies spectrum of normal free oscillations of foluid are determined. The principle of minimum of the potential...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1066512016-10-02T03:02:50Z Small Oscillations of Magnetizable Ideal Fluid Borisov, I.D. Yatsenko, T.Yu. The small oscillations of a magnetizable ideal fluid in partially filled vessel are considered. Solvability of the initial-boundary problem is proved and the generic properties of the frequencies spectrum of normal free oscillations of foluid are determined. The principle of minimum of the potential energy in the problem on the stability of the fluid equilibrium states is proved. 2010 Article Small Oscillations of Magnetizable Ideal Fluid / I.D. Borisov T.Yu. Yatsenko // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 4. — С. 383-395. — Бібліогр.: 16 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106651 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The small oscillations of a magnetizable ideal fluid in partially filled vessel are considered. Solvability of the initial-boundary problem is proved and the generic properties of the frequencies spectrum of normal free oscillations of foluid are determined. The principle of minimum of the potential energy in the problem on the stability of the fluid equilibrium states is proved. |
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Article |
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Borisov, I.D. Yatsenko, T.Yu. |
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Borisov, I.D. Yatsenko, T.Yu. Small Oscillations of Magnetizable Ideal Fluid Журнал математической физики, анализа, геометрии |
| author_facet |
Borisov, I.D. Yatsenko, T.Yu. |
| author_sort |
Borisov, I.D. |
| title |
Small Oscillations of Magnetizable Ideal Fluid |
| title_short |
Small Oscillations of Magnetizable Ideal Fluid |
| title_full |
Small Oscillations of Magnetizable Ideal Fluid |
| title_fullStr |
Small Oscillations of Magnetizable Ideal Fluid |
| title_full_unstemmed |
Small Oscillations of Magnetizable Ideal Fluid |
| title_sort |
small oscillations of magnetizable ideal fluid |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2010 |
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http://dspace.nbuv.gov.ua/handle/123456789/106651 |
| citation_txt |
Small Oscillations of Magnetizable Ideal Fluid / I.D. Borisov T.Yu. Yatsenko // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 4. — С. 383-395. — Бібліогр.: 16 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT borisovid smalloscillationsofmagnetizableidealfluid AT yatsenkotyu smalloscillationsofmagnetizableidealfluid |
| first_indexed |
2025-07-07T18:49:18Z |
| last_indexed |
2025-07-07T18:49:18Z |
| _version_ |
1837015141332287488 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2010, v. 6, No. 4, pp. 383–395
Small Oscillations of Magnetizable Ideal Fluid
I.D. Borisov
V.N. Karazin Kharkiv National University
4 Svoboda Sq., Kharkiv, 61077, Ukraine
E-mail:borisov@univer.kharkov.ua
T.Yu. Yatsenko
Institute for Radiophysics and Electronics of NAS of Ukraine
12 Proskura Str., Kharkiv, 61085, Ukraine
E-mail:t.dream@gmx.net
Received October 25, 2009
The small oscillations of a magnetizable ideal fluid in partially filled vessel
are considered. Solvability of the initial-boundary problem is proved and the
generic properties of the frequencies spectrum of normal free oscillations of
fluid are determined. The principle of minimum of the potential energy in
the problem on the stability of the fluid equilibrium states is proved.
Key words: magnetizable capillary fluid, equilibrium state, stability of
equilibrium, small oscillations, spectrum of eigenfrequencies.
Mathematics Subject Classification 2000: 76W05.
Introduction
The magnetization of fluid in sufficiently strong magnetic field produces a va-
riety of interesting physical effects. These effects are clearly observed in mag-
netic fluids (dispersions of ferro- or ferrimagnetic nanoparticles)and often called
ferrofluids. The unique combination of ferromagnetic properties and fluidity gene-
rates a variety of forms of the free surface of ferrofluid in an external magnetic
field.
Many authors considered stability and bifurcation of equilibrium forms of
the free surface and the motion of magnetic fluids near the equilibrium state.
The main results were obtained while studying the plane horizontal layers of
ferrofluid in a homogeneous magnetic field, the axisymmetric forms of free surface
in an azimuthal field and some other cases of consistent symmetry of the magnetic
field and equilibrium configurations of ferrofluids. A complete review of these
results can be found in [1–11] and bibliography therein.
c© I.D. Borisov and T.Yu. Yatsenko, 2010
I.D. Borisov and T.Yu. Yatsenko
In this paper we present a general statement of the problem of small fluc-
tuations in magnetizable ideal (nonviscous) capillary fluid based on the ferro-
hydrodynamics equations [1]. We also prove the solvability of the initial-boundary
problem and study the spectral problem of normal oscillations of fluid near equi-
librium state.
1. Problem Statement
Let us consider a closed vessel at rest in a uniform gravitational field filled
with homogeneous incompressible capillary ideal fluid and gas, which are also
placed in a magnetic field. Assume that the fluid and gas are nonconductive and
their ponderomotive interaction with the magnetic field is caused by magneti-
zation of mediums. We neglect any motion of gas as well as the fluid viscosity.
The oscillations of magnetizable viscous fluid were considered earlier [12].
We denote Ω1 and Ω2 to be the volumes filled with fluid and gas at equilibrium
state, Ω3 = R3\Ω to be the unbounded volume outside the vessel, Ω := Ω1∪Γ∪Ω2,
where Γ is the fluid equilibrium free surface. Let S be the closed surface of the
vessel Ω; S1 and S2 be the surfaces of contact of the fluid and gas with the vessel
wall.
For definiteness, we assume that the magnetic field is generated by currents
distributed in the domain Ω3, and the current density ~j(~x), ~x ∈ Ω3 remains
invariant during fluid oscillations. For simplicity, the medium in domain Ω3 is
considered to be homogeneously magnetizable. Relation between the induction
~B(k) and the field strength ~H(k) in each of the domains Ωk may be written in the
form:
~B(k) = µ0µk(H(k)) ~H(k) = µ0( ~H(k) + ~M (k)), k = 1, 3,
where µ0 is the magnetic constant, µk is the relative magnetic permeability of
the k-medium, ~M (k) is the magnetization of the k-medium. The functions µk :
H → µk(H), k = 1, 3, are considered to be given ones.
Let ~x = ~X(ξ1, ξ2) be the equation of equilibrium free surface of the fluid Γ,
ξ1 and ξ2 be the coordinate parameters of this surface. In the neighborhood
of Γ we define the curvilinear coordinates Oξ1ξ2ξ3 connected to the Cartesian
coordinates by relation: ~x = ~X(ξ1, ξ2) + ξ3~n(ξ1, ξ2), where ~n is the unit normal
to the surface Γ. We agree the normals to the surface interface to be directed to
the domain with a larger index.
We write the equation of oscillations of free surface of fluid Γ(t) in the form:
ξ3 = ζ(t, ξ1, ξ2). Let ~v(t, ~x) and ψ(t, ~x) be respectively the field of fluid velocities
and the potential of perturbation of intensity of magnetic field generated by the
fluid oscillations. The functions ζ(t, ξ1, ξ2), ~v(t, ~x), ψ(t, ~x) are thought to be small
quantities of first order.
384 Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 4
Small Oscillations of Magnetizable Ideal Fluid
In the linear approximation the functions ζ(t, ξ1, ξ2), ~v(t, ~x), ψ(t, ~x) should
satisfy the following system of equations, boundary and initial conditions:
∂~v
∂t
= −1
ρ
∇(p + µ0
~M · 5ψ) + ~f, div~v = 0 in Ω1; (1)
∂ζ
∂t
= vn (:= ~n · ~v) on Γ; (2)
p(1) + µ0
~M (1) · ∇ψ(1) = σ(−4Γ + a)ζ
+
{
Bn(~n· ∧∇ ψ)− ~Bτ · ∇Γψ
}
Γ
on Γ; (3)
∂ζ
∂ν
+ κζ = 0 on ∂Γ; (4)
vn = 0 on S1; (5)
div(µ(k)
∧
∇ (k)ψ(k)) = 0 in Ωk, k = 1, 3; (6)
{ψ}Γ = {Hn}Γ ζ,
{
µ0µ ~n· ∧∇ ψ
}
Γ
= −
{
divΓζ ~Bτ
}
Γ
on Γ; (7)
{ψ}S = 0,
{
µ~n· ∧∇ ψ
}
S
= 0 on S; (8)
ψ(t, ~x) → 0 at |~x| → ∞; (9)
~v(0, ~x) = ~v0(~x), ζ(0, ξ1, ξ2) = ζ0(ξ1, ξ2); (10)
κ :=
kΓ cosα + kS
sinα
,
∧
∇ (k)(·) :=
(
∇+
µ
(k)
H
~H(k)
µ(k)H(k)
( ~H(k) · ∇)
)
(·),
µ(k)(~x) := µk(H(k)(~x)), µ
(k)
H (~x) :=
dµk(H(k)(~x))
dH
a := −ρ
σ
~g · ~n− (k2
1 + k2
2)
+
1
σ
2∑
α,β=1
bαβ
(
∂ ~X
∂ξα
· ~H
)(
∂ ~X
∂ξβ
· ~B
)
− (k1 + k2)HnBn
Γ
.
Here ρ is the fluid density; ~g is the uniform acceleration of gravitational force;
~f is the volume density perturbations of the external field of mass forces; σ is the
coefficient of surface tension on Γ; Hn and ~Hτ are respectively the projection on
the normal and the tangent component of the magnetic field strength on Γ; k1,
k2 are the main curvatures of surface Γ; bαβ are the components of the second
fundamental form of surface Γ; kΓ and kS are the curvatures calculated on ∂Γ of
the sections of surfaces Γ and S by the plane perpendicular to ∂Γ; ∂/∂ν is the
derivative along the ~ν to ∂Γ in the tangent plane to Γ; α is the contact angle
(dihedral angle formed by fluid at the contour points ∂Γ); ∇Γ(·) is the gradient
of the scalar functions given on Γ; divΓ(·) is the surface divergence of the tangent
field of vectors on Γ; ∆Γ = divΓ∇Γ is the Laplace–Beltrami operator on Γ.
Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 4 385
I.D. Borisov and T.Yu. Yatsenko
The curvatures of the surfaces Γ and S are considered to be positive if the
corresponding normal sections are convex in the direction of Ω1. Curly brackets
in (3), (7) and everywhere below denote the jump of the respective enclosed
expression on the interface of the two mediums, {A}Γ := (A(2) − A(1))|Γ. Top
index numbers are used to denote the mediums to which the quantity relates.
The first equation in (1) is a linearized equation of the oscillations of magne-
tizable fluid [1, 2]. The kinematic condition (2) and the condition of preservation
of the contact angle (4) have the same form as in the case of ordinary capillary
fluid (see, e.g., [13]). The condition (3) is obtained by linearization of the dy-
namic conditions for the jump of normal stresses on the fluid free surface caused
by surface tension and polarization of magnetic forces field.
In the mathematical model used in [1], the magnetic field is determined by
the following equations:
∇× ~H(k)(~x, t) = ~j(~x), ∇ · ~B(k)(~x, t) = 0, in Ωk(t), k = 1, 3.
Linearization of these equations under the assumption of invariance of current
density ~j(~x) during the fluid oscillations leads to equations (6). The equalities (7)
and (8) in the linear approximation express the conditions of continuity of tan-
gential components of tension and normal component of the magnetic field on
the interface surfaces of the mediums Γ and S.
The function ζ(t, ξ1, ξ2) should satisfy the condition
∫
Γ
ζdΓ = 0 ∀t ≥ 0, (11)
which follows easily from the second equation (1) (expressing the condition of
incompressibility of the fluid) and conditions of nonpenetrability of the vessel
wall (5). Note also that the vector-function ~v0(~x) in the initial condition (10)
should satisfy the second equations of (1) and condition (5).
2. Operator-Differential Formulation of Evolutionary Problem
Suppose that the fluid completely wets the vessel wall such that ∂Ω1 = Γ
⋃
S
and ∂Ω2 = Γ (in this the case condition (4) is omitted). Assume that the surfaces
Γ and S are homeomorphic to the sphere. Such an equilibrium state of ferrofluid
can be easily seen during the experiments.
For brevity, the case when the surfaces Γ and S intersect is not considered
here. However, the obtained below results on the solvability of evolution prob-
lem (1)–(10) and the structure of spectrum of eigenfrequencies of fluid oscillations
can be extended on this case as well.
386 Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 4
Small Oscillations of Magnetizable Ideal Fluid
Further, the surfaces Γ, S and the vector-function ~H(k)(~x), k = 1, 3, are as-
sumed to be sufficiently smooth. We also assume that for equations (6) the
conditions of uniform ellipticity are satisfied
m0|~ξ|2 ≤ µ(k)(~x)|~ξ|2 +
µ
(k)
H (~x)
H(k)(~x)
(~ξ · ~H(k)(~x))2 ≤ m0|~ξ|2, (12)
∀~x ∈ Ωk, ∀~ξ ∈ R3, k = 1, 3,
where m0, m0 are some positive constants.
The vector-functions in (1) for ∀t ≥ 0 are the elements of the Hilbert space
~L2(Ω1). As known from [13], there is the orthogonal decomposition
~L2(Ω1) = ~J0(Ω1)⊕ ~Gh,S(Ω1)⊕ ~G0,Γ(Ω1),
where ~J0(Ω1) is the subspace of the solenoidal vector fields with zero normal
component on ∂Ω1, ~Gh,S(Ω1) is the subspace of the potential harmonic fields
with zero normal component on S, and ~G0,Γ(Ω1) is the subspace of the potential
harmonic fields whose potential vanishes on the surface Γ.
By (1), (5), the velocity fields ~v belong to the subspace ~J0,S(Ω1) := ~J0(Ω1)⊕
~Gh,S(Ω1) such that
~v = ~u +5ϕ, ~u ∈ ~J0(Ω1), 5ϕ ∈ ~Gh,S(Ω1). (13)
Let P0 be the operator of orthogonal projection on the medium ~J0(Ω1) in
~L2(Ω1). Applying P0 to both sides of equation (1) and to initial condition (10),
we get
d
dt
~u = P0
~f (t > 0), ~u(0) = P0~v0. (14)
Hence there can be found a solenoidal component of ~u of the fluid field velo-
cities
~u := P0~v = P0~v0 +
t∫
0
(P0
~f )(τ)dτ. (15)
Let I be the unit operator in ~L2(Ω1), and I −P0 be the orthogonal projector
on the subspace of the potential fields ~G(Ω1) := ~Gh,S(Ω1) ⊕ ~G0,Γ(Ω1). Extract
the potential component ∇χ := (I−P0)~f of the perturbations ~f of external force
field. Applying operator (I − P0) to (1), we get the Cauchy–Lagrange integral
for small potential movements of magnetizable fluid
ρ
∂ϕ
∂t
+ p + µ0
~M · 5ψ − ρχ = c(t), (16)
Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 4 387
I.D. Borisov and T.Yu. Yatsenko
where c(t) is an arbitrary function of time. By the second equation in (1), the
kinematic condition (2) and the nonpenetrability condition of the solid wall (5),
the velocity potential ϕ satisfies the following equation and the boundary condi-
tions:
4ϕ = 0 in Ω1,
∂ϕ
∂n
=
∂ζ
∂t
on Γ,
∂ϕ
∂n
= 0 on S,
∫
ϕdΓ = 0. (17)
Let L2(Γ) be the Hilbert space of the square-summable scalar functions de-
fined on Γ, and H0(Γ) := L2(Γ) ª {1} be the subspace of the functions or-
thogonal to constants in L2(Γ). Note that ζ ∈ H0(Γ) ∀t ≥ 0 (11). Denote
by L the elliptic differential operator of the second order defined by equality:
Lζ := (−4Γ + a((X))ζ. Define the operator B0
B0ζ := PHLζ = Lζ − (mes Γ)−1
∫
Γ
(Lζ)dΓ, (18)
D(B0) = H2
0(Γ) := H2(Γ) ∩H0(Γ),
where PH is the orthogonal projector on the subspace H0(Γ) in L2(Γ), H2(Γ) is
the Sobolev space of the scalar functions belonging to L2(Γ) together with the
generalized derivatives up to the second order inclusively. As known [13], B0 is
the selfadjoint bounded from below operator in H0(Γ).
Denote by γ
(k)
Γ the trace operator assigning to an arbitrary function ψ(k) ∈
Hs(Ωk) its value on the surface Γ
γ
(k)
Γ ψ(k) := ψ(k)
∣∣
Γ
∈ Hs−1/2(Γ), k = 1, 2,
where Hs(Ωk), Hs(Γ), s ≥ 0 are the Sobolev–Slobodetsky spaces [15].
The operator γ
(k)
Γ is bounded from Hs(Ωk) to Hs−1/2(Γ) [15, 16].
Define the operators ∂
(k)
Γ ,
∧
∂
(k)
Γ on a set of the smooth functions by the equa-
lities
∂
(k)
Γ ψ(k) := ~n · ∇ψ(k)
∣∣
Γ
,
∧
∂
(k)
Γ := ~n · ∧∇
(k)
ψ(k)
∣∣
Γ
, k = 1, 2.
The operators ∂
(k)
Γ ,
∧
∂
(k)
Γ are extended by continuity to the bounded ones from
Hs(Ωk) to Hs−3/2(Γ) [15, 16].
Define the Hilbert space Hs
0(Γ) := Hs(Γ) ∩ H0(Γ), s > 0, and the space
(Hs
0(Γ))∗ := H−s
0 (Γ) adjoint to Hs
0(Γ). Note that the equations (6) with the
conditions (7)–(9) define uniquely the perturbation of the magnetic field potential
ψ := (ψ(1), ψ(2), ψ(3)) for the given function ζ ∈ H3/2
0 (Γ) and
ψ(k) ∈ H2(Ωk), k = 1, 2, ψ(3) ∈ H2
loc(Ω3) ∩ D(Ω3),
388 Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 4
Small Oscillations of Magnetizable Ideal Fluid
where D(Ω3) is the closure of smooth functions by the Dirichlet norm ‖u‖D(Ω3) :=
‖∇u‖~L2(Ω3). The operator M assigning the function ζ to the function ψ denote
by ψ = Mζ.
Define the operator B1
B1ζ := PH
{
Bn(
∧
∂Γ Mζ)− ~Bτ · ∇Γ(γΓMζ)
}
Γ
=
{
Bn
∧
∂Γ ψ − ~Bτ · ∇Γψ
}
Γ
− (mes Γ)−1
∫
Γ
{
Bn
∧
∂Γ ψ − ~Bτ · ∇Γψ
}
Γ
dΓ. (19)
The operator B1 is the bounded operator from H3/2
0 (Γ) to H1/2
0 (Γ) [12]. B1 is the
unbounded symmetric operator inH0(Γ) with the domain D(B1) := H3/2
0 (Γ) [12].
Define the operator B in H0(Γ)
B := B0 + B1, D(B) = H2(Γ) ∩H0(Γ). (20)
Note that operator B can be Friedrichs extended to the selfadjoint semibounded
operator H0(Γ) with the domain H1
0(Γ), [12]. Below this extension will be again
denoted by B.
Let Hs
0(Ω1) be the subspace of the functions ϕ ∈ Hs(Ω1) such that γ
(1)
Γ ϕ ∈
Hs−1/2
0 (Γ). The unique solution ϕ ∈ H1
0(Ω1) to the Neumann problems (17) for
∀(dζ/dt|Γ) ∈ H−1/2
0 (Γ) exists [13]. Let T be the operator that assigns the value of
normal derivative of Γ to the solution of problem (17). Following [13], we define
the operator C := γ
(1)
Γ T ,
ϕ|Γ = γ
(1)
Γ T
(
∂ϕ
∂n
∣∣∣∣
Γ
)
= C dζ
dt
. (21)
The operator C is a bounded operator from H−1/2
0 (Γ) to H1/2
0 (Γ); the restriction
of C on the space H0(Γ) is the selfadjoint positive compact operator inH0(Γ) [13].
Writing the Cauchy–Lagrange integral (16) on the free surface Γ by using the
dynamic condition (3), equalities (18)–(20) and the initial conditions (10), we get
the Cauchy problem in the Hilbert space H0(Γ) which describes small potential
fluid motions near the equilibrium state
ρC d2ζ
dt2
+ σBζ = ρχ0(t) (t > 0); (22)
ζ(0) = ζ0,
dζ(0)
dt
= ζ ′0 := ((I − P0)~v0 · ~n)|Γ; (23)
χ0(t) := χ(t) + c(t), c(t) := − 1
mesΓ
∫
Γ
χdΓ.
Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 4 389
I.D. Borisov and T.Yu. Yatsenko
Thus, the general evolutionary problem on small motions of the magnetizable
capillary fluid (1)–(10) reduces to two independent problems (14) and (22), (23).
As known [3], the equilibrium state of the system corresponds to the stationary
value of the functional of potential energy Π such that δΠ(Γ; ζ) = 0 ∀ζ ∈ H1
0(Γ),
where δΠ is the first variation of the functional Π. It is easy to show that the
second variation of the potential energy δ2Π(Γ; ζ) coincides (up to the factor)
with the quadratic form of the operator B. Thus, if the deviations of the free
surface of fluid from its equilibrium state are small, there can be applied the
equality
Π =
1
2
δ2Π(Γ; ζ) =
σ
2
(Bζ, ζ)0 . (24)
Here and below by (·, ·)0 we denote the continuation of the scalar product in
L2(Γ) to the adjoint spaces H−s
0 (Γ) and Hs
0(Γ).
The kinetic energy K of the potential motions of fluid is represented as
a quadratic form associated with the operator C [13]
K =
ρ
2
∫
Ω1
|∇ϕ|2dΩ =
ρ
2
(
C dζ
dt
,
dζ
dt
)
0
. (25)
The operators B and C in equalities (24) and (25) are naturally called the
operators of potential and kinetic energy, respectively.
3. Eigenfrequency Oscillations of Magnetizable Fluid.
Solvability of the Cauchy Problem
Consider the normal eigenoscillations of the magnetizable fluid, described by
solutions of homogeneous equation (22), depending on time t according to the law
ζ = eiωtu(ξ1, ξ2),
where ω is the circular frequency of oscillations, u(ξ1, ξ2) is the mode oscillations
of the free surface of fluid. The equation (22) leads to a spectral problem
Bu = λCu, λ := ω2ρ/σ. (26)
The operator of potential energy B is the semibounded from below operator
with a discrete real spectrum; generically B has a countable set of positive eigen-
values of λk(B) and, possibly, a finite number of negative and zero eigenvalues
[13]. Let æ be the number of negative eigenvalues of the operator B (counting
their multiplicity), æ0 be the multiplicity of zero eigenvalue. The eigenvalues of
the operator B are numbered, as usual, in the ascending order
λ1(B) ≤ λ2(B) ≤ . . . ≤ λæ(B) < 0, λæ+1(B) = . . . = λæ+æ0(B) = 0,
0 < λæ+æ0+1(B) ≤ λæ+æ0+2(B) ≤ λæ+æ0+3(B) ≤ . . . . (27)
390 Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 4
Small Oscillations of Magnetizable Ideal Fluid
Using the results of [13], we formulate the generic properties of the eigenvalue
and eigenfunction of problem (26).
Theorem 1. Let the eigenvalues of the operator of potential energy B
satisfy (27). Then the problem (26) has a discrete spectrum {λk}∞k=1 consisting of
eigenvalues λk of finite multiplicity; all eigenvalues λk are real and λk := λ−k < 0
∀k = 1, æ, λæ+k := λ0
k = 0 ∀k = 1,æ0, λæ+æ0+k := λ+
k > 0 ∀k = 1,∞,
and λk → +∞ when k → ∞. The set of eigenfunctions {uk}∞k=1 := {u−k }æ
k=1 ∪
{u0
k}æ0
k=1∪{u+
k }∞k=1 (u±k , u0
k are the eigenfunctions corresponding to the eigenvalues
λ±k , λ0
k, respectively: uk := u−k ∀k = 1, æ, uæ+k := u0
k ∀k = 1,æ0, uæ+æ0+k := u+
k
∀k = 1,∞) is complete in H0(Γ) . It also forms the Riesz basis and can be chosen
to satisfy the relation
(Buj , uk)0 = δjkλk, (Cuj , uk)0 = δjk. (28)
It can be shown that for a magnetizable fluid the asymptotics of the spectrum,
given in [13] for a regular capillary fluid, remains valid
λk = λk(B, C) =
(
mesΓ
4π
)−3/2
k3/2(1 + o (1)), k →∞.
Note also that the spectrum of eigenfrequencies of fluid oscillations contains æ
pairs of purely imaginary frequencies ωk = ±i|λkσ/ρ|1/2 ∀k = 1,æ and a count-
able set of real frequencies ωk = ±(λkσ/ρ)1/2 ∀k > (æ + æ0).
The function ζ(t), which is continuous on t ∈ [0, T ] in the norm of H1
0(Γ)
with continuous first derivative on t ∈ [0, T ] in the norm of the space H−1/2
0 (Γ),
ζ(t) ∈ C([0, T ];H1
0(Γ)), ζ ′(t) ∈ C([0, T ];H−1/2
0 (Γ)) (′:= d/dt),
satisfies the integral identity
T∫
0
(
ρ(Cζ ′(t), η′(t))0 − σ(Bζ(t), η(t))0 + ρ(χ0(t), η(t))0
)
dt + ρ(Cζ ′0, η(0))0 = 0,
∀ η(t) ∈ L2(0, T ;H1
0(Γ)), η′(t) ∈ L2(0, T ;H−1/2
0 (Γ)), η(T ) = 0, (29)
and is called the generalized solution of the Cauchy problem (22) and
(23).
Theorem 2. Let the conditions ζ0 ∈ H1
0(Γ), ζ ′0 ∈ H−1/2
0 (Γ) and χ0(t) ∈
L2(0, T ;H0(Γ)) are satisfied. Then there exists a unique weak solution to the
Cauchy problem (22) and (23).
Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 4 391
I.D. Borisov and T.Yu. Yatsenko
P r o o f of the theorem for the case of a positive definite potential energy
operator B is given in [13]. For the general case, when negative and zero eigenval-
ues λj(B) of the operator B exist, the theorem can be easily proved analogously
to [16, Ch. 3].
The generalized solution to the problem (22) and (23) can be written in
the form:
ζ(t) = ζ−(t) + ζ0(t) + ζ+(t) =
æ∑
j=1
c−j (t)u−j +
æ0∑
j=1
c0
j (t)u
0
j +
∞∑
j=1
c+
j (t)u+
j . (30)
The coefficients c±j (t), c0
j (t) in (30) are defined as follows:
c−j (t) = α−j cosh (|ωj |t) + β−j sinh (|ωj |t)
+ |ωj |−1
t∫
0
sinh (|ωj |(t− τ))χ−j (τ)dτ ∀j = 1,æ,
c+
j (t) = α+
j cos(|ωj |t) + β+
j sin(|ωj |t)
+ |ωj |−1
t∫
0
sin (|ωj |(t− τ))χ+
j (τ)dτ ∀j = (æ + æ0 + 1),∞, (31)
c0
j (t) = α0
j + β0
j t +
t∫
0
dτ
τ∫
0
χ0
j (s)ds ∀j = (æ + 1), (æ + æ0),
where
α±j := (Cζ0, u
±
j )0, β±j := (Cζ ′0, u±j )0, χ±j (t) := (χ0(t), u±j )0,
α0
j := (Cζ0, u
0
j )0, β0
j := (Cζ ′0, u
0
j )0, χ0
j (t) := (χ0(t), u0
j )0.
The set of functions ~v(t, ~x), ζ(t, ξ1, ξ2), ψ(k)(t, ~x), k = 1, 3, where the function
ζ is the generalized solution to the Cauchy problem (22) and (23), the velocity
field ~v has the form: ~v = ~u +∇ϕ, the vortex component ~u is defined in (15), and
the potential component ∇ϕ = ∇(Tζ ′) and the potential ψ := (ψ(1), ψ(2), ψ(3)) of
perturbations of the magnetic field are determined by perturbations ζ of the free
surface of the fluid by equality: ψ = Mζ, is called the generalized solution to
the initial-boundary problem (1)–(10).
The previous discussion leads to the following theorem.
392 Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 4
Small Oscillations of Magnetizable Ideal Fluid
Theorem 3. Let the conditions ~f(t) ∈ C([0, T ]; ~L2(Ω1)), ζ0 ∈ H1
0(Γ) and
~v0 ∈ ~J0,S(Ω1) are satisfied. Then there exists a unique solution to the initial-
boundary value problem (1)–(10). The balance equation of (kinetic + potential)
energy
1
2
(ρ‖~v(t)‖2
~L2(Ω1)
+ σ(Bζ(t), ζ(t))0)
=
1
2
(ρ‖~v0‖2
~L2(Ω1)
+ σ(Bζ0, ζ0)0) + ρ
t∫
0
(~f(τ), ~v(τ))~L2(Ω1)d τ (32)
is satisfied for the generalized solution.
We consider the stability conditions of equilibrium states of magnetizable
capillary fluid. The equilibrium state is called stable if for any ε > 0
there exists δ > 0 such that for any initial perturbations ζ0 and ~v0, satis-
fying ‖ζ0‖H1
0(Γ) < δ and ‖~v0‖~L2(Ω1) < δ, the inequalities ‖ζ(t)‖H1
0(Γ) < ε and
‖~v(t)‖~L2(Ω1) < ε for ∀t > 0 hold true.
The stability (or instability) of the equilibrium states of magnetizable fluid
is determined by the sign of the smallest eigenvalue λ1(B) of the operator of
potential energy B.
The spectral criterion of stability. If there are no perturbations of exter-
nal field of mass forces (~f ≡ 0), then the equilibrium state of the magnetizable
capillary fluid is stable if the lowest eigenvalue λ1(B) of the operator of potential
energy B is positive, λ1(B) > 0, and is unstable if λ1(B) < 0.
Thus, if λ1(B) > 0, then by (32) and (15), it easy to see that under condition
~f ≡ 0 the equilibrium state is stable. If λ1(B) < 0, then by (30) and (31),
there are arbitrarily small initial perturbations (in the norm of H1
0(Γ)) of the
equilibrium state such that ‖ζ(t)‖H1
0(Γ) →∞ when t →∞.
In some cases, the spectral criterion of stability can be checked rather easily.
In [3], there are given the examples of using it for constructing the boundary of
stability in the space of physical parameters characterizing the equilibrium states
of fluid.
By (24), the functional of the potential energy has a minimum value in the
equilibrium state of magnetizable fluid if the operator of potential energy B is
positive definite and, consequently, λ1(B) > 0. From (24) it follows that if
λ1(B) < 0, then the second variation of potential energy may take negative
values. Thus, we have the following theorem.
Theorem 4. If there are no perturbations of external field of mass forces
(~f ≡ 0), then the equilibrium state of magnetizable capillary fluid is stable if it
corresponds to an isolated local minimum of potential energy. If the equilibrium
Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 4 393
I.D. Borisov and T.Yu. Yatsenko
state corresponds to the stationary value of the functional of potential energy but
it is not a local minimum, and the second variation of potential energy can take
negative values, then the equilibrium state is unstable.
This theorem is analogous to the well-known Lagrange theorem (and its
reverse) on the stability of equilibrium of the conservative systems with finite
number of degrees of freedom.
Conclusion
The generic formulation of the problem on the small motions of magnetizable
ideal fluid near the equilibrium state is given. The problem is reduced to the
evolution problem for a system of operator-differential equations in the Hilbert
space. The solvability of the evolutionary problem is proved.
The spectral problem on normal eigen-oscillations of magnetizable capillary
fluid is reduced to the studying of eigenvalues and vectors of the linear bunch of
operators of kinetic and potential energies. The main qualitative properties of the
spectrum of natural frequencies and modes of normal vibrations of magnetizable
capillary fluid are found. In particular, it is proved that the system of normal
modes of oscillations forms a basis in certain functional spaces. This allows one
to provide the solutions to evolution equations in the form of expansions in series
of the eigenfunctions of the problem.
The principle of minimum potential energy in the problem of stability of equi-
librium states of the magnetizable fluid is provided in the linear approximation.
The spectral criterion of stability of equilibrium states is formulated. Efficient
methods of calculating the stability of equilibrium states of magnetizable fluid
can be based on the the spectral criterion.
References
[1] R.E Rosensweig, Ferrohydrodynamics. Cambridge Univ. Press, Cambridge, 1985.
[2] E. Blums, A. Cebers, and M.M. Maiorov, Magnetic Fluids. Walter de Gruyter,
Berlin, 1997.
[3] I.D. Borisov, Stability of Equilibrium of a Magnetic Capillary Fluid. — Magneto-
hydrodynamics 19 (1983), No. 2, 45–54.
[4] H. Kikuraa, T. Sawadaa, T. Tanahashia, and L.S. Seob, Propagation of Surface
Waves of Magnetic Fluids in Travelling Magnetic Fields. — J. Magnetism and
Magnetic Materials 85 (1990), No. 1–3, 167–170.
[5] V.M. Korovin and A.A. Kubasov, Instability of the Flat Surface of a Magnetic Fluid
in a Cylindrical Cavity in the Presence of a Vertical Magnetic Field. — Techn. Phys.
46 (1998), No. 1, 23–30.
394 Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 4
Small Oscillations of Magnetizable Ideal Fluid
[6] V.M. Korovin, Capillary Instability of the Cylindrical Interface between Ferrofluids
in a Magnetic Field with Circular Field Lines. — Techn. Phys. 71 (2001), No. 12,
2–25.
[7] R. Friedrichs, Pattern and Wave Number Selection in Ferrofluid. — Phys. Rev. E
64 (2001), 021406, 1–10.
[8] V.V. Mekhonoshin and A. Lange, Faraday Instability on Viscous Ferrofluid in a
Horizontal Magnetic Feld: Oblique Rolls of Orbitary Orientation. — Phys. Rev. E
65 (2002), 1–7.
[9] V.M. Korovin, Capillary Disintegration of a Configuration Formed by Two Viscous
Ferrofluids Surrounding a Current-Carrying Conductor and Having a Cylindrical
Interface. — Techn. Phys. 49 (2004), No. 12, 8–15.
[10] Ch. Gollwitzer, G. Matthies, R. Richter, I. Rehberg, and L. Tobiska, The Surface
Topography of a Magnetic Fluid – a Quantitative Comparison between Experiment
and Numerical Simulation. — J. Fluid Mech. 571 (2007), 455–474.
[11] V.M. Korovin, Capillary Disintegration of a Ferrofluid Cylindrical Film Magnetized
to Saturation by an Axial Magnetic Field. — Techn. Phys. 53 (2008), No. 5, 22–28.
[12] I.D. Borisov and T.Yu. Yatsenko, On Small Motions of a Magnetizable Viscous
Liquid. — Mat. Fiz., Anal., Geom. 10 (2003), 135–146. (Russian)
[13] Nikolay D. Kopachevsky and Selim G. Krein, Operator Approach to Linear Prob-
lems of Hydrodynamics. Birkhauser Verlang, Basel, Boston, Berlin, 2001.
[14] O.A. Ladyzhenskaya and N.N. Ural’tseva, Linear and Quasi-Linear Elliptic Equa-
tions. 1st Ed. Acad. Press, New York, London, 1968.
[15] S.M. Nikol’skii, Approximation of Functions of Several Variables and Embedding
Theorems. 2nd Ed., Revised and Augmented. Nauka, Moscow, 1977. (Russian)
[16] J.-L. Lions and E. Magenes, Problèmes aux Limits Non Homogènes et Application.
Dunod, Paris, 1968.
Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 4 395
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