Shliomis Model Based Ferrofluid Lubrication of Squeeze Film in Rotating Rough Curved Circular Disks with Assorted Porous Structures

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Employing the discussion for the stochastic modeling of transverse roughness of ^{[23, 24, 25]}, the expression for the film thickness h(x) of the lubricant film is considered to be,

(1)

where denote the mean film thickness and is the deviation from the mean film thickness characterizing the random roughness of the bearing surfaces. The deviation s derived by the probability density function

where c is the maximum deviation from the mean film thickness. The details of the mean , the standard deviation and the parameter which is the measure of symmetry of the random variable are considered from the discussion of ^{[23, 24, 25]}.

It is assumed that the rotating upper disk lying along the surface determined by the relation

approaches with normal velocity to the rotating lower plate lying along the surface given by

where and are the curvature parameters of the corresponding plates and is the central film thickness. The film thickness then, is defined by (^{[22, 26]})

^[27] investigated that magnetic particles of a magnetic fluid can relax in two ways when the applied magnetic field changes. One is by the rotation of magnetic particles in the fluid and the other one by rotation of the magnetic moment with in the particles. Brownian relaxation time parameter gives particle rotation while the relaxation time parameter describes the intrinsic rotational process. Assuming steady flow, neglecting inertial and second derivatives of , the equations governing the flow become,

(2)

(3)

(4)

Where in is the internal angular momentum, is the sum of moments of inertia of the particles per unit volume, together with

^[26], is the fluid viscosity in the film region, is external magnetic field, is magnetic susceptibility of the magnetic field, is the film pressure, is the fluid viscosity and is the permeability of the free space.

By making use of equation (3), in equation (2) and (4), one finds that,

(5)

and

(6)

Neglecting terms, substitution of in above equation, leads to

(7)

From equation (6), it is easily observed that an initial approximation to is

Substituting the value of on the right side of equation (6), a second approximation to is easily seen to be

Again, substituting this value of on the right side of equation (6), third approximation to is found as

(8)

In view of the equations of Shliomis model in cylindrical polar coordinates with uniform magnetic field, when both surfaces are solid and upper one rotates, the governing equation for the film pressure is obtained from (^[26]),

(9)

with being layer thickness.

Neglecting term, following the discussions of ^{[23, 24, 25]} regarding the modelling of roughness and under the usual assumptions of hydro-magnetic lubrication (^{[26, 28, 29]}) the modified Reynolds equation when both plates rotate, takes the form

(10)

where

the following non dimensional quantities are introduced

(11)

The relevant boundary conditions for the pressure distribution are

(12)

2.1. A Globular Sphere Model

In Figure 2, a porous material is filled by globular spherical particles (a mean particle size ).

Figure 2. Configuration of porous structure model sheets

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The Kozeny-Carman equation is well known in fluid dynamics. Relatively better results for pressure drop are obtained when this model is applied to laminar flow. ^[30] suggested that the use of Kozeny-Carman formula turned in the relation

where is the porosity and is the length ratio. From experimental investigations the length ratio was proposed to be around 2.5 under suitable situations. Thus, the Kozeny-Carman formula takes the form

By making use of the boundary conditions (12) and non dimensional quantities (11), one gets the dimensionless form of the pressure distribution in the case of Kozeny- Carman, in the following form

(13)

where

and the dimensionless load carrying capacity of the bearing system is found from

(14)

Therefore, the non dimensional load carrying capacity is calculated as

(15)

2.2. A Capillary Fissures Model

The model of porous sheets, investigated by Irmay and displayed in Figure 3, consists of three sets of mutually orthogonal fissures (a mean solid size ).

Figure 3. Geometry of porous Structure model sheets

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Considering no loss of hydraulic gradient at the junctions, ^[31] presented the permeability,

where e is the porosity.

Resorting to the boundary conditions (12) and non dimensional quantities (11), the non dimensional pressure distribution for Irmay model is derived as

(16)

In view of equation (14), the dimensionless form of load carrying capacity is calculated from

(17)

3. Results and Discussions

It is easily noticed that equation (13) and equation (16) determine the non dimensional pressure distribution while the load carrying capacity in dimensionless form is given by equation (15) and equation (17) with respect to Kozeny- Carman’s model and Irmay model respectively.

It is observed that the expressions for non dimensional load carrying capacity are linear with respect to the magnetization parameter . Accordingly, an increase in magnetization will result in increased load carrying capacity in case of both the models. Probably, this may be due to the fact that magnetization increases the effective viscosity of the lubricant. Further a close glance at these two equations trends to suggest that the bearing can support certain amount of load even in the absence of flow unlike the case of traditional lubricants.

Setting the roughness parameters to be zero, this discussion reduces to the performance analysis of a Shliomis model based rotating porous squeeze film in smooth circular plates. In the absence of the magnetization this turns to the rotating porous squeeze film performance in curved circular plates. Further, this gives the study of ^[32] when there is no rotation. Lastly, taking curvature parameter to be zero, this is the effect of rotation on squeeze film performance as discussed in ^[1].

Figure 4 - Figure 20 and Table 1 deal with the graphical representation of Kozeny- Carman’s model based porous structure while Figures 21-32 and Table 2, Table 3, Table 4 account for the case of Irmay’s model based porous structure. A noticeable difference between the above two models is found from the performance points of view as can be seen from the discussions outlined below.

The Variation of the load carrying capacity presented in Figure 4 and Table 1 makes it clear that the load carrying capacity increases sharply due to the effect of magnetization.

Figure 4. Variation of Load carrying capacity with respect to

and e

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Table 1. Variation of Load carrying capacity with respect to and

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The effect of porous structures on the load distribution is presented in Figures 5-7, 20 and Tables 2-4. It is clearly seen that the porous structure effect is relatively significant in the case of Kozeny –Carman’s model.

Figure 5. Variation of Load carrying capacity with respect to

and e

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Figure 6. Variation of Load carrying capacity with respect to

and C

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Figure 7. Variation of Load carrying capacity with respect to

and

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Table 2. Variation of Load carrying capacity with respect to and B

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Table 3. Variation of Load carrying capacity with respect to and

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Table 4. Variation of Load carrying capacity with respect to and

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Figure 8. Variation of Load carrying capacity with respect to e and C

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Figure 9. Variation of Load carrying capacity with respect to e and

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Figure 8 - Figure 11 and Figure 21, Figure 22 depict the variation of load carrying capacity with respect to the porosity parameter. It is noticed that the decrease in load carrying capacity is comparatively less in the case of Kozeny –Carman’s model.

Figure 10. Variation of Load carrying capacity with respect to e and

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Figure 11. Variation of Load carrying capacity with respect to e and S

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The effects of curvature parameters for both models are provided in Figure 12 - Figure 15 and Figure 23 - Figure 25. These Figures show that a proper choice of the ratio of the curvature parameters may go some way in improving the performance of the bearing system.

Figure 12. Variation of Load carrying capacity with respect to B and

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Figure 13. Variation of Load carrying capacity with respect to B and

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Figure 14. Variation of Load carrying capacity with respect to C and

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Figure 15. Variation of Load carrying capacity with respect to C and S

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Figure 16. Variation of Load carrying capacity with respect to

and

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The adverse effect of the standard deviation associated with roughness is shown in Figure 16 - Figure 17 and Figure 26. The rate of decrease in the load carrying capacity is more in the case of Irmay’s model.

Figure 17. Variation of Load carrying capacity with respect to

and S

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The fact that negatively skewed roughness tends to increase the load carrying capacity is exhibited in Figure 18 and Figure 27. At the same time positively skewed roughness induces a decrease in the load carrying capacity. Furthermore, the trends of load carrying capacity with respect to variance remain identical to that of the skewness as can be seen from Figures 19 and 28.

Figure 18. Variation of Load carrying capacity with respect to

and S

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Figure 19. Variation of Load carrying capacity with respect to

and S

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Lastly, the effect of rotation is presented in Figure 11, Figure 15, Figure 17, Figure 18, Figure 19, Figure 20 and Figure 28. It is needless to say that for an overall improved performance, a suitable choice of rotational parameter is required.

A noticeable fact is that the compensation of the negative influence of standard deviation and porosity by the magnetization is relatively more in the case of Kozeny- Carman model when negatively skewed roughness surfaces. Of course, in this compensation the rotational inertia and curvature parameters are required to be suitably chosen.

Figure 20. Variation of Load carrying capacity with respect to

and

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Figure 21. Variation of Load carrying capacity with respect to

and

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Figure 22.Variation of Load carrying capacity with respect to

and

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Figure 23. Variation of Load carrying capacity with respect to

and

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Figure 24. Variation of Load carrying capacity with respect to

and

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Figure 25. Variation of Load carrying capacity with respect to

and

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Figure 26. Variation of Load carrying capacity with respect to

and

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Figure 27. Variation of Load carrying capacity with respect to

and

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Figure 28. Variation of Load carrying capacity with respect to

and

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Figure 29. Variation of Load carrying capacity with respect to

and porosity parameters of both the structures

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Figure 30. Variation of Load carrying capacity with respect to

and porosity parameters of both the structures

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A comparison between the performance analyses of both the models can be had from the Figure 29 - Figure 30 and Table 5 - Table 6. It is easily observed from these figures and tables that the Kozeny- Carman’s model is a bit superior.

Here is the porosity parameter of Irmay’s model.

Table 5. Variation of Load carrying capacity with respect to and porous structure parameters of both the models

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Table 6. Variation of Load carrying capacity with respect to and porous structure parameters of both the models

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