Analyzed is Two-dimensional thermomagnetic convection followed by a steady- state natural convection of magnetic fluid as shown in Figure 1, where a conductive wire of radius R₀ with a direct current I₀ is placed in the center of an annular space filled with a magnetic fluid of magnetic permeability μ₀. It is assumed that magnetic fluid is magnetized by a magnetic field H due to an external magnetic field from a wire current and an induction magnetic field of the magnetic fluid, thosemagnetic fluid is assumed to be at rest up to time t = 0 with a uniform temperature T_l, and at t > 0 a half part of the outside circumference is assumed to be fixed at a high temperature (T_h), and the rest half part of the outside circumference at a low temperature(T_l).

The following assumption applies:

1) The radius of wire is much smaller than that of the inside circle ( R 0 ≪ R 1 ).

2) The effect of magnetic field is limited within the distance 50R₂.

3) The magnetic permeability of magnetic fluid is equal to that of vacuum.

4) The magnetic fluid is incompressible fluid.

The equation of continuity is

∇ ⋅ ν = 0 , (1)

The momentum equation neglecting gravity force is

ρ D ν D t = − ∇ p + η ∇ 2 v + ∇ ⋅ ( B H ) + I t s ∇ × ( Ω − ω ) , (2)

where t is time, v: velocity, ρ: mass density of magnetic fluid, p: pressure, η: viscosity of magnetic fluid in absence of a magnetic field, I: average inertia moments of particles per unit volume, t_s: the relaxation time of internal spin rotation, H: magnetic field, B: magnetic induction, Ω: internal spin rate, ω: effective rate of rotation of a fluid element ≡ ( 1 / 2 ) ∇ × ν .

The internal angular momentum equation is

I D Ω D t = γ I ∇ 2 Ω − I t s ( Ω − ω ) + μ 0 M × H , (3)

where γ is a dissipation coefficient of inertia spin moment, M: magnetization of the magnetic fluid. The magnetization relaxation equation, Shliomis (1974) [9] , is

D M D t = Ω × M − 1 t b ( M − M 0 ) , (4)

where t_b is a relaxation time of the particle rotation by Brownian rotation motion, and M₀ is the equilibrium magnetization of the magnetic fluid, and magnitude is given by

| M 0 | ( | H | , T ) = ϕ M s ( T ) L ( π μ 0 M s ( T ) d 3 | H | 6 k T ) , (5a)

where L ( x ) = coth x − 1 / x is a Langevin function, T: temperature, k: Boltzmann constant, d: diameter of the particle, ϕ : volume fraction of ferromagnetic particles, M_s: magnetic moment of a particle, if T ≤ 0.8T_c (T_c: temperature of Curie point),

M s ( T ) ≈ M s ( 0 ) ( 1 − k T n M B H M ) , (5b)

where M_B: Bohrmagneton, n: number of spin perunitatom of ferromagntic metal, H_M: molecular field.

The energy equation can be derived by Cowley and Rosensweig [10] as

ρ C m D T D t − μ 0 ρ T Κ ( D | H | D t ) = k 1 ∇ 2 T + Φ , (6)

where C_m is the heat capacity at constant volume with a magnetic field, K: pyromagnetic coefficient of magnetic fluid, k₁: the thermal conductivity, Φ: a viscous dissipation term usually negligible. The Maxwell’s equation for a non-conducting fluid with no displacement current becomes

∇ × H = 0, ∇ ⋅ B = 0 , (7)

B = μ 0 ( H + M ) . (8)

Cartesian coordinate (x, y) is fixed in the center of an annular pipe, the y-axis being vertically upward, Relationship between cylindrical and Cartesian coordinates is

x + i y = R e i θ , (9)

where, a spectral finite difference method is applied for solving Equation (1) - Equation (8), and primary variables are expressed in a Fourier series to the circumferential θ-direction.

Introduced are a magnetic induction stream functions, ψ i m :

B i x = ∂ ψ i m ∂ y , B i y = − ∂ ψ i m ∂ x , (10)

ψ i m ≡ ψ i m ¯ − μ 0 I 0 2 π ln R , (11)

where i = 1, 2, 3 correspond to the following three regions:

・ R₀ ≤ R ≤ R₁:

∇ 2 ψ 1 m ¯ = 0 , (12a)

・ R₁ ≤ R ≤ R₂:

1 μ 0 ∇ 2 ψ 2 m ¯ + ( M θ R + ∂ M θ ∂ R − 1 R ∂ M R ∂ θ ) = 0 , (12b)

・ R₂ ≤ R ≤ 50R₂:

∇ 2 ψ 3 m ¯ = 0. (12c)

where M_R and M_θ are R- and θ-component of magnetization of the magnetic fluid, M.

The boundary conditions used are as follows:

・ At R = R₁, no-slip, no-spin, and thermal insulation:

[ v ] R = R 1 = 0, [ Ω ] R = R 1 = 0, ( ∂ T ∂ R ) R = R 1 = 0 , (13)

・ At R=R₂, no-slip, no-spin, and Dirichlet temperature condition:

[ v ] R = R 2 = 0 , [ Ω ] R = R 2 = 0 , (14)

T ( θ , R 2 ) = T h   for   θ ∈ ( π 2 , 3 2 π ) , (15a)

T ( θ , R 2 ) = T l for   θ ∈ [ 0 , π 2 ] ∪ [ 3 2 π , 2 π ] . (15b)

The normal component of the magnetic induction and the tangential component of the magnetic field are continuous across the interface between dissimilar regions,

at   R = R 1 : [ B 1 ] n = [ B 2 ] n , [ H 1 ] t = [ H 2 ] t , (16a)

at   R = R 2 : [ B 2 ] n = [ B 3 ] n , [ H 2 ] t = [ H 3 ] t , (16b)

where the subscript n and t denote normal and tangential component. Then the boundary condition of Equations 12(a)-(c) are given by

φ 1 m ¯ = 0 , ∂ φ 1 m ¯ ∂ R = 0     ( R = 0 ) , (17a)

∂ φ 1 m ¯ ∂ R = ∂ φ 2 m ¯ ∂ R + μ 0 M θ     ( R = R 1 ) , (17b)

∂ φ 3 m ¯ ∂ R = ∂ φ 2 m ¯ ∂ R + μ 0 M θ     ( R = R 2 ) , (17c)

φ 3 m ¯ = 0     ( R = 5 0 R 2 ) . (17d)

Since the relaxation time t_s (~10⁻¹¹ s) of internal spin rotation and the dissipation coefficient γ ( ≈ t b / 2 ~ 10 − 7   s − 1 ) of inertia spin moment is very small, Equation (3) can be reduced as

I t s ( Ω − ω ) = μ 0 M × H , (18)

and substituting Equation (18) into Equation (2) leads to an approximate momentum equation:

ρ D v D t = − ∇ p + η ∇ 2 v + μ 0 ( M ⋅ ∇ ) H , (19)

and then the curl of the magnetic body force is given by

∇ × μ 0 ( M ⋅ ∇ ) H = μ 0 ∂ χ ∂ T [ ∇ T × ( H ⋅ ∇ ) H ] , (20)

where χ (= χ(T, |H|)) is the magnetic susceptibility of magnetic fluid, and the magnetization of the magnetic fluid can be written as M = χH. As for the Equation (20), the producing condition of thermomagnetic convection is derived in the gravity-free space as ∇ T × ∇ | H | ≠ 0 .

In the case of I₀ =10 A, T_h − T_l = 20 K and R₁/R₂ = 0.2, the isotherms, streamlines and dimensionless magnetization field strengths with the position of electric current wire varies are displayed as shown in Figures 2-4, where all quantities are made dimensionless base on a length L₀ (=R₂), a temperature T₀ (=T_h − T_l),

and reference velocity V₀ ( = H 0 μ 0 / ρ ), H₀ ( = I 0 2 π L 0 ) is representative mag-

netic field. It shows that the convection pattern depends on electric current distribution, depending on which multiple or single circulation is produced. Thus, it is found that thermomagnetic convection is controlled by changing relative direction of temperature gradient to that of a magnetic field.