1. Introduction

Ferrofluid or magnetic fluid[1] are stable colloidal suspensions containing fine ferromagnetic particles dispersing in a liquid, called carrier liquid, in which a surfactant is added to generate a coating layer preventing the flocculation of the particles. When an external magnetic field is applied, ferrofluids experience magnetic body forces depends upon the magnetization of ferromagnetic particles. Owing to these features ferrofluids are useful in many applications like in sensors, sealing devices, filtering apparatus, etc.[2,3].

Wu[4], in an innovative analysis , dealt with the case of squeeze film behavior for porous annular disks in which he showed that owing to the fact that fluid can flow through the porous material as well as through the space between the bounding surfaces , the performance of a porous walled squeeze film can differ substantially from that of a solid walled squeeze film. Murti[5] analyzed the squeeze film behavior between porous circular discs.

With the advent of ferrofluid, Agrawal[6] studied its effects on a porous inclined bearing and found that the magnetization of the magnetic particles in the lubricant increased its load capacity without affecting the friction on the moving slider. Later Verma[7] and Shah and Bhat[8,9] investigated the squeeze film between porous plates and found that its performance with magnetic fluid lubricant was better than with conventional lubricant.

In all above investigations, none of the authors considered various porous structure for porous plates in their study. The porous layer in the bearing is considered because its advantages property of self lubrication. With this motivation, we recapitulate the study of the paper of Shah and Bhat[10] for various porous structure without the effect of rotation, and with the effect of ferrofluid as lubricant under a magnetic field oblique to the lower plate. The problem also include the effect of squeeze velocity and the curvature of the upper disc. The ferrofluid flow model considered here is due to R. E. Rosensweig[1].

A ferrofluid lubrication equation is derived for the above problem and the various porous structure is considered for computation of bearing characteristics like pressure and load capacity. The ferrofluid used in computations are water based.

2. The Mathematical Model

The configuration of the bearing, shown in Figure 1, consists of two circular plates each of radius a (metre). The upper plate has a porous facing of thickness l₁(metre) which is backed by a solid wall. It moves normally towards an impermeable and flat lower plate with uniform velocity where (metre) is the central film thickness and t is time in seconds. The film thickness h (metre) is given by

(1)

where is the curvature of the upper plate and r (metre) is the radial coordinate.

Assuming axially symmetric flow of the ferrofluid between the plates under an oblique magnetic field whose magnitude H vanishes at the modified Reynolds’s equation governing the film pressure p is[10]

(2)

where is the permeability of the porous region, is the permeability of the free space, is the magnetic susceptibility and is the fluid viscosity.

Figure 1. Configuration of the problem

3. Solution

CASE I (a globular sphere model as shown in Figure 2). A porous material is filled with globular particles ( a mean particle size

Figure 2. Structure model of porous sheets given by Kozeny-Carman

In this case Kozeny-Carman formula[11] gives

(3)

where is the porosity.

Let us introduce the dimensionless quantities

(4)

and take, for example,

(5)

where is chosen so as to have a magnetic field of strength between the order of 10⁵ to 10⁶.

The angle ,since the magnetic field arise out of a potential, can be determined from

(6)

with the help of elliptic integrals depending on the value of

Using equations (1) - (5), we see that the dimensionless film pressure P satisfies the equation

(7)

Solving equation (7) under the boundary conditions

(8)

we obtain

(9)

The load capacity W of the bearing can be expressed in dimensionless form as

(10)

CASE II (a capillary fissures model as shown Figure 3). The model is composed of three sets of mutually orthogonal fissures (a mean solid size and assuming no loss of hydraulic gradient at the junctions, Irmay[12] derived the permeability

Figure 3. Structure model of porous sheets given by Irmay

(11)

where and is the porosity

Let us introduce the dimensionless quantities

(12)

Using equations(1), (2), (5), (11) and (12), the dimensionless film pressure P satisfies the equation

(13)

Solving equation (13) under the boundary conditions (8), we obtain

(14)

The load capacity W of the bearing can be expressed in dimensionless form as

(15)

4. Results and Discussion

The dimensionless pressure P and load capacity are given, by equations (9), (10) for globular sphere model given by Kozeny-Carman and by equations (14), (15) for capillary fissures model by Irmay.

Setting the magnetization parameter and without considering the two cases of the present analysis reduces to nonmagnetic case[13].

Also, setting in equations (9), (10) and (14), (15), we obtain the results for flat upper plate for two cases.

The computed values of the dimensionless load capacity are displayed in figure 4 and figure 5 for two different cases and for the following value of the parameters:

Figure 4 and figure 5 show that increase with and The increases are substantial in the case of concave plates for case I.

Figure 4. Values of the dimensionless load capacity for different values of and for case I

Figure 5. Values of the dimensionless load capacity for different values of and for case II

The load capacity also increases more in case I, this may be because of the slow coming out of the ferrofluid from the porous region.

5. Conclusions

Based upon the results and discussion it can be concluded that the better load carrying capacity is obtained for globular sphere model with the effect of ferrofluid under an oblique magnetic field to the lower plate.

Thus, it is suggested to have a design of porous squeeze bearing with globular sphere in the porous region.