EFFECT OF SLIP VELOCITY ON THE PERFORMANCE OF A SHORT BEARING LUBRICATED WITH A MAGNETIC FLUID

This paper aims at analyzing the effect of velocity slip on the behavior of a magnetic fluid based infinitely short hydrodynamic slider bearing. Solving the Reynolds’ equation, the expression for pressure distribution is obtained. In turn, this leads to the calculation of the load carrying capacity. Further, the friction is also computed. It is observed that the magnetization paves the way for an overall improved performance of the bearing system. However the magnetic fluid lubricant fails to alter the friction. It is established that the slip parameter needs to be kept at minimum to achieve better performance of the bearing system, although the effect of the slip parameter on the load carrying capacity is in most situations, negligible. It is found that for large values of the aspect ratio, the effect of slip is increasingly significant. Of course, the aspect ratio plays a crucial role in this improved performance. Lastly, it is established that the bearing can support a load even in the absence of flow, which does not happen in the case of a conventional lubricant.

All these above studies considered conventional lubricants.Agrawal [12] dealt with the configuration of Prakash and Vij [10] with a magnetic fluid lubricant, and found that the performance was better than with a conventional lubricant.Bhat and Deheri [13] modified and extended the analysis of Agrawal [12] by considering a magnetic fluid based porous composite slider bearing with its slider consisting of an inclined pad and a flat pad.Bhat and Deheri established that the magnetic fluid increased the load carrying capacity, did not affect the friction, decreased the coefficient of friction, and shifted the centre of pressure towards the inlet.Patel et al. [14] analyzed the performance of a magnetic fluid based infinitely short bearing.It was shown that the magnetization sharply increased the load carrying capacity.The friction remained unchanged due to magnetization.Prajapati [15] investigated the performance of a magnetic fluid based porous inclined slider bearing with velocity slip, and concluded that the magnetic fluid lubricant minimized the negative effect of the velocity slip.Recently, hydrodynamic lubrication of Short bearings have been subjected to investigations in Patel et al. [23], Vakis and Polycarpous [24] and Patel and Deheri [25].
The present study discusses the performance of a magnetic fluid based short bearing system with slip effect while the magnitude of the magnetic field is represented by a cosine function.

Analysis
Figure 1 consists of the configuration of the bearing system, which is infinitely short in the Z-direction.The slider runs with uniform velocity u in the Xdirection.The length of the bearing is L and the breadth B is in the Z-direction, where B L. The pressure gradient ∂p ∂x can be neglected because the pressure gradient ∂p ∂z is much larger as a consequence of B being very small.The magnetic fluid is a suspension of solid magnetic particles approximately 3-10 nanometers in diameter stabilized by a surfactant in a liquid carrier.With the help of an external magnetic field these fluids can be confined, positioned, shaped and controlled as desired.For details, see Bhat [22].The magnetic field is taken to be oblique to the stator, as in Agrawal [12].Following Bhat [22] and Prajapati [16], the magnetic field is taken as where the inclination angle of the magnetic field is described from the partial differential equation In view of the deliberation carried out in Prajapati [16], Verma [17] and Bhat and Deheri [18]the magnitude of the magnetic field is assumed to be of the form where k is chosen to suit the dimensions of both sides and the strength of the magnetic field.Under the usual assumptions of hydrodynamic lubrication, and employing the Beavers and Joseph [19] model for slip, the governing Reynolds' equation (Agrawal [12], Prajapati [16], Patel et al. [14]) turns out to be where µ 0 is the magnetic susceptibility, µ is free space permeability, µ is lubricant viscosity and an m is the aspect ratio.The associated boundary conditions are The expression for pressure distribution is obtained by integrating Equation (3) with respect to the boundary condition (4), as where Introduction of the dimensionless quantities leads to the expression for the non-dimensional pressure distribution, obtained as Then the load carrying capacity per unit width is determined from . (7) Thus, the dimensionless load carrying capacity of the bearing system comes out to be . ( The frictional force F per unit width of the lower plane of the moving plate is obtained as where is the non-dimensional shearing stress, while A little computation indicates that where At Y = 0 (moving plate), one computes that  Therefore, the friction force in non-dimensional form at the moving plate is calculated as Next, at Y = 1 (fixed plate), one concludes that which transforms to the non dimensional form as It is clearly seen from Equations ( 14) and ( 16) that

Results and Discussion
Equations ( 6) and ( 8), respectively, present the variation of non-dimensional pressure distribution and load carrying capacity, while the frictional force is determined from Equation (9).Comparison with the conventional lubricant indicates that the non-dimensional  pressure increases by while the load carrying capacity enhances by For lower values of the slip parameter, the load carrying capacity estimated here is approximately three times more than the load calculated from the investigation of Patel [20].It is interesting to note that the friction remain unchanged in spite of the presence of slip, which is clear from Equation (17).However, for large values of the aspect ratio, the effect of slip on friction is significant.
The distribution of load carrying capacity with respect to magnetization µ * for various values of m, s, L/h 2 and h 2 /B is presented in Figures 2-5.All these figures make it clear that the load carrying capacity increases due to magnetization.Further, the load carrying capacity increases for increasing values of m, L/h 2 and h 2 /B while it decreases with increasing slip velocity values.However, the effect of m and s on µ * is negligible so far as the load carrying capacity is concerned.(Figures 2 and 3).
Figures 6-8 show the variation of load carrying capacity with respect to slip velocity s for different values of m, L/h 2 and h 2 /B, respectively.It is clearly  seen from these figures that the load carrying capacity decreases with increasing slip velocity values.However, the decrease remains nominal, as can be seen from Figures 6-8.
Figures 9 and 10 deal with the distribution of load carrying capacity with respect to m.The load carrying capacity increases with increasing values of m. Figure 11 confirms that the rate of increase in load carrying capacity with respect to L/h2 increases with increasing values of h 2 /B.Thus, the combined effect of the two ratios L/h 2 and h 2 /B is significantly positive.From Figure 12, it is found that the friction decreases with respect to the aspect ratio, whereas it increases with increasing slip parameter values.

Conclusions
This paper underlines that from the point of view life time of bearing, the slip parameter needs to be put at a minimum value.A comparison of our paper with the discussions of Patel et al. [21] indicates that the load carrying capacity remains almost identical for lower aspect ratio values.The industrial importance of this work is that it offers an additional degree of freedom from the design point of view, in terms of the form of the magnitude of the magnetic field.It is suggested that the adverse effect of slip velocity can be compensated to a large extent by the magnetic fluid lubricant, when a suitable aspect ratio value is chosen.

Figure 1 .
Figure 1.Configuration of the bearing system.

Figure 2 .
Figure 2. Variation of load carrying capacity with respect to µ * and m.

Figure 3 .
Figure 3. Variation of load carrying capacity with respect to µ * and s.

Figure 4 .
Figure 4. Variation of load carrying capacity with respect to µ * and L/h2.

Figure 5 .
Figure 5. Variation of load carrying capacity with respect to µ * and h2/B.

Figure 6 .
Figure 6.Variation of load carrying capacity with respect to s and m.

Figure 7 .
Figure 7. Variation of load carrying capacity with respect to s and L/h2.

Figure 8 .
Figure 8. Variation of load carrying capacity with respect to s and h2/B.

Figure 9 .
Figure 9. Variation of load carrying capacity with respect to m and L/h2.

Figure 10 .
Figure 10.Variation of load carrying capacity with respect to m and h2/B.

Figure 11 .
Figure 11.Variation of load carrying capacity with respect to L/h2 and h2/B.