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Magnetohydrostatics equation and its application

Abstract: magnetohydrostatics equation is one of the basic equations of magnetic fluid and the basis of the application of magnetic fluid. Based on the force of magnetic fluid in the magnetic field, this paper deduces the magnetohydrostatics equation, and uses the basic equation to analyze the effect of long straight wire on magnetic fluid and the change of apparent density of magnetic fluid under the action of magnetic field

key words: magnetohydrostatics equation magnetic field

introduction

magnetohydrostatics equation is the basic equation of magnetic fluid. Because the shrinkage of the same material produced by different factories is different, it is the mechanical theoretical basis of studying magnetic fluid in the equilibrium state of static or uniform motion

in this paper, the static equation of magnetic fluid is derived from the force of magnetic fluid in the magnetic field, and the basic equation is used to analyze the effect of long straight wire magnetic field on magnetic fluid and the density characteristics of magnetic fluid under the action of magnetic field

1. Magnetohydrostatics equation

where, ρ Is the density of the magnetic fluid, h is the distance from any point in the magnetic fluid to the reference point, and C is the integral constant, which is determined by the boundary conditions

equation (4) is the basic form of magnetohydrostatics equation [2] and the theoretical basis for the application of magnetofluids

2. Application example of magnetohydrostatics equation

(1) effect of electrified long straight wire on magnetic fluid

Figure 1 shows the shape of magnetic fluid level under the action of vertical long straight wire

since the pressure on the liquid surface of the magnetic fluid is constant, it can be obtained from formula (4)

if the directivity of H is not considered, formula (5) can be written as

generally, within a certain radius with the center line of the long straight wire as the axis, that is, when it is small, it can be considered that the magnetic field is strong, the magnetic fluid is in a saturated magnetization state, and its magnetization is approximately equal to the saturated magnetization. Formula (6) can be rewritten as

, C 'is a constant

it can be seen from formula (9) that when the current is constant, the liquid level of the magnetic fluid decreases with the increase of the distance r from the center of the wire, and its shape is hyperbola [3]

formula (9) is the result of derivation under the premise of strong magnetic field, but in practice, with the increase of distance r, the strength of the magnetic field will weaken. In this way, the magnetic fluid far away from the center line will be in an unsaturated state. At this time, the nonlinear problem of magnetic fluid must be considered

the magnetism in magnetic fluid comes from the magnetic particles in it. The magnetic field has a certain influence on the shape and distribution of the liquid surface of magnetic fluid, and the shape of the liquid surface of magnetic fluid will also have an influence on the magnetic field in turn. Therefore, there is the coupling problem of magnetic field and force field. The cross-section shape of magnetic fluid under the action of long straight wire can be obtained by decoupling calculation. Figure 2 shows the numerical calculation results of the radial r change of the magnetic fluid level in the current carrying conductor magnetic field when the current is constant

Figure 3 shows the photographic results of the liquid surface shape of the magnetic fluid under the action of the magnetic field of the electrified long straight wire, and the calculation is consistent with the experimental results. It can be seen that the liquid level of the magnetic fluid will rise with the increase of the current of the wire. Based on this characteristic, magnetic fluid can be used for current sensing [4]

(2) change of density of magnetic fluid under the action of magnetic field

according to the magnetohydrostatics equation, it can be concluded that the density of magnetic fluid changes after the application of magnetic field. The stronger the magnetic field, the greater the change of density. The density of magnetic fluid after applying magnetic field is called apparent density

the effect of external magnetic field on the density of magnetic fluid can be verified by experiments. Figure 4 shows the schematic diagram of the apparent density test of magnetic fluid

through the finite element method, the automobile industry and aircraft manufacturing have always been one of the most important industrial sectors in the UK. Calculate the magnetic field generated by the coil shown in Figure 4. Figure 5 shows the calculation results of the magnetic field distribution, in which figure (a) the magnetic field distribution results of the energized coil; (b) Distribution results of magnetic induction intensity on the central axis of the energized coil. Since the applied magnetic field is proportional to the coil current, the relationship between magnetic field and current can be obtained by calculating the magnetic field generated by a certain current. In the experiment, the non-magnetic weight with known density is placed in the magnetic fluid, and the coil is connected with current to keep the position of the weight in the magnetic fluid unchanged. By reading the value of the spring scale, the buoyancy of the magnetic fluid can be measured, so as to measure the density of the magnetic fluid

Figure 6 shows the relationship between magnetic fluid density and magnetic field strength. The saturation magnetization of magnetic fluid is 259gs, and its own density is 1.35g/mm3. It can be seen from the changing trend of the curve that with the continuous increase of the magnetic field strength, the density of the magnetic fluid increases in proportion. Using this characteristic, magnetic fluid can be used for the separation of materials with different densities, such as beneficiation

3. Conclusion

① magnetohydrostatics equation is the basic equation of magnetic fluid and the basis of the application of magnetohydrostatics

② when the magnetic field formed by the long straight wire acts on the magnetic fluid, the rise of the magnetic fluid level is directly proportional to the square of the energized current and inversely proportional to the distance from the liquid level to the center line of the long straight wire; Its liquid surface shape is hyperbolic

③ under the action of magnetic field, the density of magnetic fluid increases with the increase of magnetic field strength, and the density of magnetic fluid is in direct proportion to the change of magnetic field

References:

[1] zoujibin et al Principle and design of magnetic fluid seal National Defense Industry Press [m] 2000

Zou Jibin. Fundametals and Design Method of Magnetic Fluid Seals. [M] National Defence Industry Press. 2000

[2] iedrichs, atics and dynamics of a Single Freefluid-pesk. The European Physical Journal B. 2000, (18):

[3] Victor Bashtovoi, etc. Magnetic Field Effect on Capillary Rise of Magnetic Fluids. JMMM. 2005, (289):

[4] so. Applications of magnetic fluids for other inertia sensors JMMM. 1999, (201): (end)

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