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Thermomagnetic Convection of Magnetic Fluid in an Annular Space under a Non-Uniform Magnetic and Thermal Field

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1. Introduction

A familiar example of thermal convection is buoyancy-induced convection, where a driving body force resides in a force of gravity, another example is established in a heated magnetic fluid subject to a magnetic body force, which depends on a thermodynamic constitutive equation. The enhancement of convective heat trans- fer in magnetic fluids could be applied in cooling current-carrying conductors, and transmissionlines, especially in gravity-free space. Characteristics of thermomagnetic convection are investigated by many researchers, e.g. Finlayson (1970) [1] , Polevikov & Fertman (1977) [2] , Busse & Riahi (1982) [3] , Stiles & Kagan (1990) [4] , Zebib (1996) [5] , Fruh (2005) [6] , Yamaguchi (2009) [7] and Siddiqa (2013) [8] .

In this paper, two-dimensional thermomagnetic convection of magnetic fluid possessing internal spin and relaxation of magnetization with high thermal sensitivity is numerically analyzed under a non-uniform magnetic and thermal field, using a spectral finite difference schemes, and the thermomagnetic convection pattern of magnetic fluid within an annulus is numerical investigated with spatial distributions of magnetic field variations.

2. Analysis

2.1. Physical Model

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 R_{0} with a direct current I_{0} is placed in the center of an annular space filled with a magnetic fluid of magnetic permeability μ_{0}. 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 ( ${R}_{0}\ll {R}_{\text{1}}$ ).

2) The effect of magnetic field is limited within the distance 50R_{2}.

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

4) The magnetic fluid is incompressible fluid.

Figure 1. The analytical region.

2.2. Governing Equations of an Incompressible Magnetic Fluid

The equation of continuity is

$\nabla \cdot \nu =\text{0}$ , (1)

The momentum equation neglecting gravity force is

$\rho \frac{D\nu}{Dt}=-\nabla p+\eta {\nabla}^{2}v+\nabla \cdot \left(BH\right)+\frac{I}{{t}_{s}}\nabla \times \left(\Omega -\omega \right),$ (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
$\equiv \left(1/2\right)\nabla \times \nu $ .

The internal angular momentum equation is

$I\frac{D\Omega}{Dt}=\gamma I\text{}{\nabla}^{2}\Omega -\frac{I}{{t}_{s}}\left(\Omega -\omega \right)+{\mu}_{0}M\times H,$ (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

$\frac{DM}{Dt}=\Omega \times M-\frac{1}{{t}_{b}}\left(M-{M}_{0}\right)$ , (4)

where t_{b} is a relaxation time of the particle rotation by Brownian rotation motion, and M_{0} is the equilibrium magnetization of the magnetic fluid, and magnitude is given by

$\left|{M}_{0}\right|\left(\left|H\right|,T\right)=\varphi {M}_{s}\left(T\right)L\left(\frac{\text{\pi}{\mu}_{0}{M}_{s}\left(T\right){d}^{3}\left|H\right|}{6kT}\right),$ (5a)

where
$L\left(x\right)=\mathrm{coth}x-1/x$ is a Langevin function, T: temperature, k: Boltzmann constant, d: diameter of the particle,
$\varphi $ : volume fraction of ferromagnetic particles, M_{s}: magnetic moment of a particle, if T ≤ 0.8T_{c} (T_{c}: temperature of Curie point),

${M}_{s}\left(T\right)\approx {M}_{s}\left(0\right)\left(1-\frac{kT}{n{M}_{B}{H}_{M}}\right),$ (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

$\rho {C}_{m}\frac{DT}{Dt}-{\mu}_{0}\rho T{\rm K}\left(\frac{D\left|H\right|}{Dt}\right)={k}_{1}{\nabla}^{2}T+\Phi ,$ (6)

where C_{m} is the heat capacity at constant volume with a magnetic field, K: pyromagnetic coefficient of magnetic fluid, k_{1}: the thermal conductivity, Φ: a viscous dissipation term usually negligible. The Maxwell’s equation for a non-conducting fluid with no displacement current becomes

$\nabla \times H=\text{0,}\nabla \cdot B=0,$ (7)

$B={\mu}_{0}\left(H+M\right).$ (8)

2.3. Coordinate System

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

$x+iy=R{\text{e}}^{i\theta},$ (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.

2.4. Equation of the Distribution of Magnetic Field

Introduced are a magnetic induction stream functions, ${\psi}_{i}^{m}$ :

${B}_{ix}=\frac{\partial {\psi}_{i}^{m}}{\partial y}\text{,}{B}_{iy}=-\frac{\partial {\psi}_{i}^{m}}{\partial x},$ (10)

${\psi}_{i}^{m}\equiv \stackrel{\xaf}{{\psi}_{i}^{m}}-\frac{{\mu}_{0}{I}_{0}}{2\text{\pi}}\mathrm{ln}R,$ (11)

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

・ R_{0} ≤ R ≤ R_{1}:

${\nabla}^{2}\stackrel{\xaf}{{\psi}_{1}^{m}}=0,$ (12a)

・ R_{1} ≤ R ≤ R_{2}:

$\frac{1}{{\mu}_{0}}{\nabla}^{2}\stackrel{\xaf}{{\psi}_{2}^{m}}+\left(\frac{{M}_{\theta}}{R}+\frac{\partial {M}_{\theta}}{\partial R}-\frac{1}{R}\frac{\partial {M}_{R}}{\partial \theta}\right)=0,$ (12b)

・ R_{2} ≤ R ≤ 50R_{2}:

${\nabla}^{2}\stackrel{\xaf}{{\psi}_{3}^{m}}=0.$ (12c)

where M_{R} and M_{θ} are R- and θ-component of magnetization of the magnetic fluid, M.

2.5. Boundary Conditions

The boundary conditions used are as follows:

・ At R = R_{1}, no-slip, no-spin, and thermal insulation:

${\left[v\right]}_{R={R}_{1}}=\text{0,}{\left[\Omega \right]}_{R={R}_{1}}=\text{0,}{\left(\frac{\partial T}{\partial R}\right)}_{R={R}_{1}}=0\text{,}$ (13)

・ At R=R_{2}, no-slip, no-spin, and Dirichlet temperature condition:

${\left[v\right]}_{R={R}_{2}}=\text{0},\text{}{\left[\Omega \right]}_{R={R}_{2}}=\text{0},$ (14)

$T\left(\theta ,{R}_{2}\right)={T}_{h}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\theta \in \left(\frac{\text{\pi}}{2},\frac{3}{2}\text{\pi}\right),$ (15a)

$T\left(\theta ,{R}_{2}\right)={T}_{l}\text{for}\text{\hspace{0.17em}}\theta \in \left[0,\frac{\text{\pi}}{2}\right]\cup \left[\frac{3}{2}\text{\pi},2\text{\pi}\right].$ (15b)

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

$\text{at}\text{\hspace{0.17em}}R={R}_{\text{1}}:{\left[{B}_{\text{1}}\right]}_{n}={\left[{B}_{\text{2}}\right]}_{n}\text{,}{\left[{H}_{\text{1}}\right]}_{t}={\left[{H}_{\text{2}}\right]}_{t},$ (16a)

$\text{at}\text{\hspace{0.17em}}R={R}_{\text{2}}:{\left[{B}_{\text{2}}\right]}_{n}={\left[{B}_{\text{3}}\right]}_{n}\text{,}{\left[{H}_{\text{2}}\right]}_{t}={\left[{H}_{\text{3}}\right]}_{t},$ (16b)

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

$\stackrel{\xaf}{{\phi}_{1}^{m}}=0\text{,}\frac{\partial \stackrel{\xaf}{{\phi}_{1}^{m}}}{\partial R}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(R=0\right),$ (17a)

$\frac{\partial \stackrel{\xaf}{{\phi}_{1}^{m}}}{\partial R}=\frac{\partial \stackrel{\xaf}{{\phi}_{2}^{m}}}{\partial R}+{\mu}_{0}{M}_{\theta}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(R={R}_{\text{1}}\right),$ (17b)

$\frac{\partial \stackrel{\xaf}{{\phi}_{3}^{m}}}{\partial R}=\frac{\partial \stackrel{\xaf}{{\phi}_{2}^{m}}}{\partial R}+{\mu}_{0}{M}_{\theta}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(R={R}_{\text{2}}\right),$ (17c)

$\stackrel{\xaf}{{\phi}_{3}^{m}}=\text{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(R=\text{5}0{R}_{\text{2}}\right).$ (17d)

3. Results

Assuming that the magnetic particle of magnetic fluid is iron oxide, and that the carrier liquid is hydrocarbon, the following constants and parameters of magnetic fluid are used

${C}_{m}=2.09\times 1{\text{0}}^{\text{3}}\text{J}\cdot {\text{K}}^{-1}\cdot {\text{kg}}^{-1}$ , $d=1.0\times {10}^{-8}\text{\hspace{0.17em}}\text{m}$ , ${H}_{M}=1.7\times {10}^{9}\text{\hspace{0.17em}}\text{A}\cdot {\text{m}}^{-1}$ , $I=1.0\times {10}^{-4}\text{kg}\cdot {\text{m}}^{-1}$ , $k=1.38\times {10}^{-23}\text{\hspace{0.17em}}\text{J}\cdot {\text{K}}^{-1}$ , ${k}_{1}=0.22\text{\hspace{0.17em}}\text{J}\cdot {\text{K}}^{-1}\cdot {\text{m}}^{-1}\cdot {\text{s}}^{-1}$ , $K=220\text{\hspace{0.17em}}\text{A}\cdot {\text{m}}^{-\text{1}}\cdot {\text{K}}^{-1}$ , ${M}_{B}=1.17\times {10}^{-29}\text{\hspace{0.17em}}\text{A}\cdot {\text{m}}^{-1}$ , ${M}_{s}\left(0\right)=3.18\times {10}^{4}\text{\hspace{0.17em}}\text{A}\cdot {\text{m}}^{-1}$ , $n=2.2$ , $\varphi =0.\text{041}$ , $\eta =0.2\text{\hspace{0.17em}}\text{Pa}\cdot \text{s}$ , ${t}_{b}=7.\text{6}\times {\text{10}}^{-\text{7}}\text{\hspace{0.17em}}\text{s}$ , ${t}_{s}=4.17\times {\text{10}}^{-\text{11}}\text{\hspace{0.17em}}\text{s}$ , ${\mu}_{0}=4\text{\pi}\times 1{\text{0}}^{-7}\text{\hspace{0.17em}}\text{H}\cdot {\text{m}}^{-1}$ , $\rho =1.\text{22}\times {\text{10}}^{\text{3}}\text{\hspace{0.17em}}\text{kg}\cdot {\text{m}}^{-3}$ .

3.1. Magnetic Body Force of Magnetic Fluid

Since the relaxation time t_{s} (~10^{−11} s) of internal spin rotation and the dissipation coefficient γ (
$\approx {t}_{b}/2~{10}^{-\text{7}}\text{\hspace{0.17em}}{\text{s}}^{-\text{1}}$ ) of inertia spin moment is very small, Equation (3) can be reduced as

$\frac{I}{{t}_{s}}\left(\Omega -\omega \right)={\mu}_{0}M\times H,$ (18)

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

$\rho \frac{Dv}{Dt}=-\nabla p+\eta {\nabla}^{2}v+{\mu}_{0}\left(M\cdot \nabla \right)H,$ (19)

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

$\nabla \times {\mu}_{0}\left(M\cdot \nabla \right)H={\mu}_{0}\frac{\partial \chi}{\partial T}\left[\nabla T\times \left(H\cdot \nabla \right)H\right],$ (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 $\nabla T\times \nabla \left|H\right|\ne \text{0}$ .

3.2. Numerical Results

In the case of I_{0} =10 A, T_{h} − T_{l} = 20 K and R_{1}/R_{2} = 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 L_{0} (=R_{2}), a temperature T_{0} (=T_{h} − T_{l}),

and reference velocity V_{0} (
$={H}_{0}\sqrt{{\mu}_{0}/\rho}$ ), H_{0} (
$=\frac{{I}_{0}}{2\text{\pi}{L}_{0}}$ ) 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.

4. Conclusions

1) For producing thermomagnetic convection of a magnetic fluid, a spatially non-uniform of temperature and a spatially non-uniform of external magnetic field are required so that $\nabla T\times \nabla \left|H\right|\ne \text{0}$ .

Figure 2. Isotherms (at an interval of 0.1), streamlines(at an interval of 1.0 × 10^{−}^{7} stream function) and dimensionless magnetization field strengths (at an interval of 1.0 × 10^{−}^{3} with from 0.001 to 0.007) in the case of the electric current wire fixed at (x, y) = (0, 0), where the direction of velocity along the center is leftward.

Figure 3. Isotherms (at an interval of 0.1), streamlines (at an interval of 2.0 × 10^{−}^{8} stream function) and dimensionless magnetization field strengths (at an interval of 2.0 × 10^{−}^{3} with from 0.002 to 0.02) in the case of the electric current wire fixed at (x, y) = (0.15, 0), where the direction of velocity along the center is leftward.

Figure 4. Isotherms (at an interval of 0.1), streamlines (at an interval of 1.0 × 10^{−}^{7} stream function) and dimensionless magnetization field strengths (at an interval of 2.0 × 10^{−}^{3} with from 0.002 to 0.02) in the case of the electric current wire fixed at (x, y) = (0, 0.15), where the direction of flow is clockwise.

2) In the case that the electric current wire is fixed in center of annular pipe, two symmetrical circulating flows are produced.

3) If the position of electric current wire is different from the center, the center of the circulating flows moves to the side of lower or higher temperature depending on whether the location of electric current wire is to the lower or higher temperature side. Especially in the case of the position of electric current wire deviates greatly from the median line of temperature, only one circulating flow zone is produced.

Nomencalature

B: magnetic induction

C_{m}: heat capacity at constant volume with a magnetic field

d: diameter of the particle

H: magnetic field

H_{0}: external magnetic field

H_{M}: molecular field

I: average inertia moments of particles per unit volume

I_{0:} a direct current

k: Boltzmann constant

k_{1}: thermal conductivity

K: pyromagnetic coefficient of magnetic fluid

M: magnetization of the magnetic fluid

M_{0}: equilibrium magnetization of the magnetic fluid

M_{B}: Bohr magneton

M_{s}: magnetic moment of a particle

n: number of spin per unit atom of ferromagntic metal

p: pressure

R_{0}: radius of conductive wire

R, θ: cylindrical coordinate system

t: time

T: temperature

T_{c}: temperature of Curie point

t_{b}: relaxation time of the particle rotation by Brownian rotation motion

t_{s}: relaxation time of internal spin rotation

v: velocity

x, y: Cartesian coordinate system

γ: dissipation coefficient of inertia spin moment

η: viscosity of magnetic fluid in absence of a magnetic field

μ_{0}: permeability of the magnetic fluid

ρ: mass density of magnetic fluid

$\psi $ : stream function

${\psi}_{i}^{m}$ : a magnetic induction stream functions

$\varphi $ : volume fraction of ferromagnetic particles

Φ: viscous dissipation term

χ: magnetic susceptibility of magnetic fluid

ω: effective rate of rotation of a fluid element

Ω: internal spin rate

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Applied Mathematics*,

**8**, 655-662. doi: 10.4236/am.2017.85052.

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