#### Abstract

An attempt has been made to analyze the performance of a magnetic fluid-based-squeeze film between longitudinally rough elliptical plates. A magnetic fluid is used as a lubricant while axially symmetric flow of the magnetic fluid between the elliptical plates is taken into consideration under an oblique magnetic field. Bearing surfaces are assumed to be longitudinally rough. The roughness of the bearing surface is characterized by stochastic random variable with nonzero mean, variance, and skewness. The associated averaged Reynolds’ equation is solved with appropriate boundary conditions in dimensionless form to obtain the pressure distribution leading to the calculation of the load-carrying capacity. The results are presented graphically. It is clearly seen that the magnetic fluid lubricant improves the performance of the bearing system. It is interesting to note that the increased load carrying capacity due to magnetic fluid lubricant gets considerably increased due to the combined effect of standard deviation and negatively skewed roughness. This performance is further enhanced especially when negative variance is involved. This paper makes it clear that the aspect ratio plays a prominent role in improving the performance of the bearing system. Besides, the bearing can support a load even when there is no flow.

#### 1. Introduction

The transient load carrying capacity of a fluid film between two surfaces having a relative normal velocity plays a crucial role in frictional devices such as clutch plates in automatic transmissions. Archibald [1] studied the behaviour of squeeze film between various geometrical configurations. Subsequently, Wu [2, 3] investigated the squeeze film performance mainly for two types of geometries, namely, annular and rectangular when one of the surfaces was porous faced. Prakash and Vij [4] discussed the load carrying capacity and time height relations for squeeze film performance between porous plates. In that study various geometries such as circular, annular, elliptical, rectangular, conical, and truncated conical were considered. Besides, a comparison was made between the squeeze film performance of various geometries of equivalent surface area and it was established that the circular geometry registered the highest transient load carrying capacity, other parameters remaining same.

The above studies dealt with conventional lubricant. Verma [5] considered the application of magnetic fluid as a lubricant. The magnetic fluid consisted of fine magnetic grains coated with a surfactant and dispersed in a non-conducting magnetically passive solvent. Later on, Bhat and Deheri [6] discussed the squeeze film behaviour between porous annular disks using a magnetic fluid lubricant with the external magnetic field, oblique to the lower disk. This analysis was improved further by Bhat and Deheri [7] to deal with the performance of a magnetic fluid based squeeze film in curved circular plates. Furthermore, Patel and Deheri [8] studied the behaviour of a magnetic fluid based squeeze film between porous conical plates. All these above studies established that the performance of the bearing system was modified and enhanced owing to the magnetic fluid lubricant.

In all the above analyses bearing surfaces were assumed to be smooth. However, the bearing surfaces after having some run-in and wear develop roughness. In fact, due to elastic, thermal, and uneven wear effects, the configurations encountered in practice are usually far from smooth. Sometimes the contamination of the lubricant and chemical degradation of the surfaces result in roughness. The roughness appears to be random in character, which was recognized by many investigators who analyzed the effect of surface roughness resorting to a stochastic method [9–13]. Christensen and Tonder [14–16] mathematically modelled the random roughness and suggested a comprehensive general analysis for investigating the effect of transverse as well as longitudinal surface roughness. This approach of Christensen and Tonder was the basis for investigating the effect of surface roughness in a number of investigations [17–24].

Recently, Andharia and Deheri [25] discussed the performance of a magnetic fluid based squeeze film in longitudinally rough conical plates. Here it has been proposed to study and analyze the performance of a squeeze film between longitudinally rough elliptical plates under the presence of a magnetic fluid lubricant.

#### 2. Analysis

The configuration of the bearing system shown in Figure 1 consists of two elliptical plates. The upper plate moves normally towards the lower plate with uniform velocity where is the central film thickness. The assumptions of usual hydrodynamic lubrication theory are taken into consideration in the analysis. The lubricant film is considered to be isoviscous and incompressible and the flow is laminar.

The bearing surfaces are assumed to be longitudinally rough. The thickness of the lubricant film is where is the mean film thickness and is the deviation from the mean film thickness characterizing the random roughness of the bearing surfaces. is considered to be stochastic in nature and governed by the probability density function , , where is the maximum deviation from the mean film thickness. The mean , the standard deviation and the parameter which is the measure of symmetry, of the random variable , are defined by the relationships: where denotes the expected value defined by wherein

Axially symmetric flow of magnetic fluid between the elliptical plates is taken into consideration under an oblique magnetic field whose magnitude is expressed as where is semimajor axis and is semiminor axis. The direction of the magnetic field is significant since needs to satisfy the equations Therefore, arises out of a potential function and the inclination of the magnetic field with the lower plate is determined from

Following Prakash and Vij [4], Bhat and Deheri [6], and Andharia and Deheri [25], the Reynolds’ equation governing the film pressure in the present case is obtained as where is fluid viscosity, is the magnetic susceptibility, and stands for permeability of the free space.

It is easily observed that , , and are all independent of and while and can assume both positive and negative values, is always positive. Following the averaging process discussed by Andharia et al. [24] and using (6), (9) takes the form where is the expected value of the lubricant pressure while

Introduction of dimensionless quantities: presents (10) in the form

The associated boundary conditions are Solving (13) using boundary condition given in (14), one obtains the dimensionless pressure distribution: where

The load carrying capacity of the bearing in non-dimensional form can be expressed as

#### 3. Results and Discussion

It is easily observed that the non-dimensional pressure is determined from (15) while (17) presents the distribution of load carrying capacity in dimensionless form. It is clearly seen from these two equations that the dimensionless pressure increased by while load carrying capacity is increased by due to magnetic fluid lubricant. In the absence of roughness this study reduces to the performance of a magnetic fluid based squeeze film in elliptical plates.

To analyze the quantitative effect of various parameters such as the magnetization parameter , the aspect ratio and roughness parameters , , and on the performance of the bearing, dimensionless load carrying capacity is computed numerically for different values of these parameters. Results are presented graphically in Figures 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15.

In Figures 2, 3, 4, and 5 one can have the variation of load carrying capacity with respect to magnetization parameter for different values of , , , and , respectively. All these figures suggest that the load carrying capacity increases sharply with respect to the magnetization parameter. Figures 6, 7 and 8 depict the variation of load carrying capacity with respect to the aspect ratio for different values of , , and , respectively. From these figures one can easily observe that the increasing values of the aspect ratio cause increased load carrying capacity. Figures 9 and 10 give the profile for load carrying capacity with respect to the skewness for different values of and respectively. It is clearly seen from these graphs that the positively skewed roughness decreases the load carrying capacity while the negatively skewed roughness increases the load carrying capacity. From Figure 11 it is observed that the trend of variance is quite similar to that of skewness so far as the distribution of load carrying capacity is concerned. It is also interesting to note that the effect of standard deviation is negligible beyond the variance’s value 0.1 with regards to the distribution of load carrying capacity. Lastly, Figures 12, 13, 14, and 15 describe the variation of the load carrying capacity with respect to the standard deviation for various values of , , , and , respectively. Unlike the case of transverse roughness, here it is established that the standard deviation tends to increase the load carrying capacity. However, the increase at the initial stage is relatively less.

#### 4. Conclusion

A close look at the figures reveals that the negative effect of positive can be compensated to a considerable extent by the positive effect of the magnetization parameter by choosing a suitable aspect ratio in the case of negatively skewed roughness. In the similar way the negative effect of positive can be compensated by choosing suitable combination of magnetization parameter and aspect ratio especially when negative variance occurs.

Hence, this study makes it mandatory that the roughness must be given due consideration while designing the bearing system from bearing’s life period point of view.

#### Nomenclature

: | Dimensions of the bearing |

: | Film thickness |

: | Aspect ratio |

: | Magnetic field |

: | Magnitude of magnetic field |

: | Pressure in the film region |

: | Expected value of the pressure |

: | Non-dimensional film pressure |

: | Load capacity |

: | Nondimensional load capacity |

: | Cartesian coordinates |

: | Mean of the stochastic film thickness |

: | Standard deviation of the stochastic film thickness |

: | Measure of symmetry of the stochastic film thickness |

: | Variance |

: | Fluid viscosity |

: | Magnetic susceptibility |

: | Dimensionless magnetization parameter |

: | Permeability of the free space. |

#### Acknowledgment

The authors are grateful to the referees for their valuable comments on the earlier version of the paper.