Performance of Magnetic-Fluid-Based Squeeze Film between Longitudinally Rough Elliptical Plates

An attempt has been made to analyze the performance of a magnetic �uid-based-squeeze �lm between longitudinally rough elliptical plates. A magnetic �uid is used as a lubricant while axially symmetric �ow of the magnetic �uid between the elliptical plates is taken into consideration under an oblique magnetic �eld. Bearing surfaces are assumed to be longitudinally rough. e roughness of the bearing surface is characterized by stochastic random variable with nonzero mean, variance, and skewness. e 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. e results are presented graphically. It is clearly seen that the magnetic �uid lubricant improves the performance of the bearing system. It is interesting to note that the increased load carrying capacity due to magnetic �uid lubricant gets considerably increased due to the combined e�ect of standard deviation and negatively skewed roughness. is performance is further enhanced especially when negative variance is involved. is 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 �ow.


Introduction
e transient load carrying capacity of a �uid �lm 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 �lm between various geometrical con�gurations.Subsequently, Wu [2,3] investigated the squeeze �lm 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 �lm 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 �lm 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.
e above studies dealt with conventional lubricant.Verma [5] considered the application of magnetic �uid as a lubricant.e magnetic �uid consisted of �ne magnetic grains coated with a surfactant and dispersed in a nonconducting magnetically passive solvent.Later on, Bhat and Deheri [6] discussed the squeeze �lm behaviour between porous annular disks using a magnetic �uid lubricant with the external magnetic �eld, oblique to the lower disk.is analysis was improved further by Bhat and Deheri [7] to deal with the performance of a magnetic �uid based squeeze �lm in curved circular plates.�urthermore, Patel and Deheri [8] studied the behaviour of a magnetic �uid based squeeze �lm between porous conical plates.All these above studies established that the performance of the bearing system was modi�ed and enhanced owing to the magnetic �uid lubricant.In all the above analyses bearing surfaces were assumed to be smooth.However, the bearing surfaces aer having some run-in and wear develop roughness.In fact, due to elastic, thermal, and uneven wear effects, the con�gurations encountered in practice are usually far from smooth.Sometimes the contamination of the lubricant and chemical degradation of the surfaces result in roughness.e 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][10][11][12][13].Christensen and Tonder [14][15][16] mathematically modelled the random roughness and suggested a comprehensive general analysis for investigating the effect of transverse as well as longitudinal surface roughness.is approach of Christensen and Tonder was the basis for investigating the effect of surface roughness in a number of investigations [17][18][19][20][21][22][23][24].
Recently, Andharia and Deheri [25] discussed the performance of a magnetic �uid based s�ueeze �lm in longitudinally rough conical plates.Here it has been proposed to study and analyze the performance of a s�ueeze �lm between longitudinally rough elliptical plates under the presence of a magnetic �uid lubricant.

Analysis
e con�guration of the bearing system shown in Figure 1 consists of two elliptical plates.e upper plate moves normally towards the lower plate with uniform velocity ḣ 0 (= ℎ 0 / where ℎ 0 is the central �lm thickness.e assumptions of usual hydrodynamic lubrication theory are taken into consideration in the analysis.e lubricant �lm is considered to be isoviscous and incompressible and the �ow is laminar.
e bearing surfaces are assumed to be longitudinally rough.e thickness ℎ of the lubricant �lm is where ℎ is the mean �lm thickness and ℎ  is the deviation from the mean �lm thickness characterizing the random roughness of the bearing surfaces.ℎ  is considered to be stochastic in nature and governed by the probability density function (ℎ  , −  ℎ   , where  is the maximum deviation from the mean �lm thickness.e mean , the standard deviation  and the parameter  which is the measure of symmetry, of the random variable ℎ  , are de�ned by the relationships: where  denotes the expected value de�ned by Axially symmetric �ow of magnetic �uid between the elliptical plates is taken into consideration under an oblique magnetic �eld  whose magnitude  is expressed as where  is semimajor axis and  is semiminor axis.e direction of the magnetic �eld is signi�cant since  needs to satisfy the equations erefore,  arises out of a potential function and the inclination  of the magnetic �eld  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 �lm pressure  in the present case is obtained as where  is �uid 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 presents (10) in the form e associated boundary conditions are Solving ( 13) using boundary condition given in (14), one obtains the dimensionless pressure distribution: where e load carrying capacity of the bearing in nondimensional form can be expressed as

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 ( * /2)(1 + 1/ 2 ) while load carrying capacity is increased by ( * /4)(1 + 1/ 2 ) due to magnetic �uid lubricant.In the absence of roughness this study reduces to the performance of a magnetic �uid based squeeze �lm 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 �gures 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 �gures one can easily observe that the increasing values of the aspect ratio cause increased load  carrying capacity.Figures 9 and 10 give the pro�le 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.

Conclusion
A close look at the �gures 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 ℎ 0 : Film thickness : Aspect ratio  : Magnetic �eld  2 : Magnitude of magnetic �eld : �ressure in the �lm region : Expected value of the pressure : Non�dimensional �lm pressure : Load capacity : Nondimensional load capacity   : Cartesian coordinates : Mean of the stochastic �lm thickness : Standard deviation of the stochastic �lm thickness : Measure of symmetry of the stochastic �lm thickness

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Variation of load carrying capacity with respect to  for different .Variation of load carrying capacity with respect to  for different .

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Variation of load carrying capacity with respect to  for different .

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Variation of load carrying capacity with respect to  for different .

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Variation of load carrying capacity with respect to  for different  * .

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Variation of load carrying capacity with respect to  for different .
Variation of load carrying capacity with respect to  * for different .
LoadF 3: Variation of load carrying capacity with respect to  * for different .