A Study of Hydromagnetic Longitudinal Rough Circular Step Bearing.
It is documented that gears, braking units, hydraulic dampers, skeletal bearings, and synovial joints make use of squeeze film mechanism. Generally an electrically conducting fluid with high thermal and electrical conductivity is applied as a lubricant for squeeze film to work under such extreme circumstances. Also, a use of external magnetic field then advances the performance of lubrication.
The performance of an oil lubricated circular step bearing is analyzed by Majumdar ; this study underlines the importance of radii ratio. Lin  discussed the couple stress effect on the performance of an externally pressurized circular step bearing. It was established that the couple stress fluid modified the performance of the bearing system. Deheri et al.  reformed and developed the study of Majumdar  to use the magnetization effect by considering a magnetic fluid as the lubricant, the flow being regulated by Neuringer-Rosensweig's model .
It is well known that the effect of surface roughness is quite important in different types of bearing systems. Most of the discussions have considered the stochastic averaging method of Christensen and Tonder [5-7] who presented a stochastic model to evaluate the effect of rough surfaces. (Ting , Prakash and Tiwari , Prajapati , and Andharia et al. ).
The hydromagnetic squeeze films between two conducting rough circular plates are studied by Vadher et al. . It was investigated that the negative effect of roughness got minimized up to some extent by applying magnetization. Patel and Deheri  evaluated the performance of ferrofluid squeeze film in rotating curved circular plates by using Shliomis model. Here, the outcomes of this article suggested that Shliomis model based ferrofluid lubrication was relatively better than the model of Neuringer-Rosensweig . Patel et al.  extended the study of Deheri et al.  to study the effect of transverse surface roughness. It was indicated that the bearing performance was adversely affected by the roughness of the bearing surfaces. But the situation was found to be a little enhanced in case of variance (-ve) was considered. A modified lubrication equation for hydromagnetic non-Newtonian cylindrical squeeze films was presented by Lin et al. . The hydromagnetic non-Newtonian effects offered quite better performance for circular squeeze films in comparison with the case of a nonconducting Newtonian lubricant.
Patel and Deheri  studied the magnetic fluid lubrication of a squeeze film in transversely rough porous circular plates with concentric circular pocket. Longitudinal roughness effect has been a matter of discussions in Andharia and Deheri , Andharia and Deheri , Andharia and Deheri , and Shimpi and Deheri . Recently, Lin  examined the ferrofluid lubrication of a longitudinally rough journal bearing.
Patel et al.  discussed the hydromagnetic lubrication of a rough porous circular step bearing. Hence, it has been sought to analyze the effect of longitudinal roughness pattern on behavior of a hydromagnetic squeeze film in circular step bearings.
The configuration of the bearing system is presented in Figure 1.
For the computation of different performance characteristics, generally the following assumptions are considered:
(1) The recess is deep enough for the pressure in it to be uniform.
(2) The bearing has very low rotational velocity and its effect is neglected for the pressure generation.
As shown in Figure 1 a thrust load w is applied and the bearing supports the load without metal to metal contact. The load w is supported by the fluid within the pocket and land. The fluid escapes in the radial direction through the restrictions by a land or sill around the recess.
In view of the discussions of Christensen and Tonder [5-7] the surfaces of bearing are taken to be longitudinally rough. The thickness h(x) of the lubricant film is given by
h (x) = [bar.h] (x) + hs (1)
where [bar.h] (x) is the mean film thickness and [h.sub.s] is the deviation from the mean film thickness characterizing the random roughness surface. [h.sub.s] is considered in nature and governed by the probability density function f ([h.sub.s]), - c [less than or equal to] f ([h.sub.s]) [less than or equal to] c, where c is the maximum deviation from the mean film thickness. The mean [alpha] and standard deviation [sigma] and the measure of symmetry [epsilon] of the random variable [h.sub.s] are defined by the relationships.
[alpha] = E ([h.sub.s]) [[sigma].sup.2] = E [[([h.sub.s] - [alpha]).sup.2]] (2)
[epsilon] = E [[([h.sub.s] - [alpha]).sup.3]] (3)
where E denotes the expected value defined by
[mathematical expression not reproducible] (4)
The concerned Reynolds' type equation gives the pressure induced flow for a circular step bearing as (Majumdar (1985), Patel, Deheri, and Vadher (2015))
[mathematical expression not reproducible] (5)
[mathematical expression not reproducible] (6)
Using Reynolds' boundary conditions
r = [r.sub.i]; p = [p.sub.s]; (7)
the governing equation for the film pressure p is given by
[mathematical expression not reproducible] (8)
[mathematical expression not reproducible] (9)
Introduction of the dimensionless terms
[mathematical expression not reproducible] (10)
paves the way for the expression of pressure distribution in nondimensional form as
P = [P.sub.s] x ln (r/[r.sub.0])/ln ([r.sub.i]/[r.sub.0]) (11)
The load bearing capacity w is calculated by integrating the pressure which takes the dimensionless form
W = [P.sub.s.sup.*] ln (1 - [k.sup.2]/ln (1/k) (12)
3. Results and Discussions
The representation for dimensionless pressure profile is given in (11) while the nondimensional load bearing capacity is governed by (12). From these equations it is clearly observed that the load carrying capacity
W [infinity] [P.sub.s.sup.*] (13)
[mathematical expression not reproducible] (14)
This suggests that for a constant flow rate the load increases as stochastically averaged squeeze film thickness decreases. Therefore, the bearing turns out to be self-compensating provided the flow rate is considered as constant. It is depicted from (11) and (12) that the effect of conductivity parameters on the distribution of pressure and load bearing capacity is determined by
[[phi].sub.0] + [[phi].sub.1] + (tanh (M/2))/(M/2)/ [[phi].sub.0] + [[phi].sub.1] + 1 (15)
which turns to
[[phi].sub.0] + [[phi].sub.1]/ [[phi].sub.0] + [[phi].sub.1] + 1 (16)
for large values of M, because tanh(M) [congruent to] 1 and (2/M) [congruent to] 0. Further, it is noticed that the pressure and load carrying capacity increases with increases in [[phi].sub.0] + [[phi].sub.1] because both the functions are increasing functions of [[phi].sub.0] + [[phi].sub.1].
It is clearly depicted from Figures 2-6 that the hydromagnetization parameter induces a sharp increase in the load bearing capacity while Figures 7-10 gives the profile of load with respect to [[phi].sub.0] + [[phi].sub.1] for different values of standard deviation, variance, skewness, and radii ratio, respectively. Here, also the load carrying capacity increases with an increase in conductivity parameter. Figures 7-10 also tells that the load increases sharply up to some extent ([[phi].sub.0] + [[phi].sub.1] [congruent to] 1.32) and then this rate of increase gets slower.
It is seen from Figures 11-13 that the standard deviation associated with longitudinal roughness causes increased load bearing capacity with respect to various parameters such as variance, skewness, and radii ratio. Besides, variation of load carrying capacity for different values of variance is described by Figures 14 and 15. From these figures it is noticed that the load decreases with increase in positive variance while the variance (- ve) increases it. The trends of load bearing capacity with respect to skewness are similar to that of variance which is observed from Figure 16.
Form Figures 5 and 9 it is clearly observed that the effect of [[epsilon].sup.*] on the variation of load carrying capacity with respect to magnetization and conductivity, respectively, remains negligible while this effect of [[epsilon].sup.*] with regards to variance is registered to be nominal as suggested by Figure 14.
This means that this type of bearing system the suitable combination of negatively skewed roughness and negative variance may go a long way for better performance of the bearing system.
It is seen that for small as well as large values of M the performance of the bearing suffers when the plates are considered to be electrically conducting, in matching with the hydromagnetic case, when the plates are taken to be nonconducting. Further, this can physically be clarified by fringing phenomena which happens when the plates are electrically conducting. Finally, the load is acquired to be more in comparison with the Neuringer Rosensweig model based ferrofluid squeeze film.
The trio of hydromagnetization, supply pressure, and standard deviation counters fruitfully the negative effect induced by positively skewed roughness and variance (+ve) in the case of negatively skewed roughness when variance comes out to be negative. Of course, in bettering the bearing performance the plate conductivities and radii ratio may play an important role. Needless to say, the longitudinal roughness parameter behaves in a little better way as compared to the transverse roughness. Therefore, it becomes very much essential that the longitudinal roughness aspect must be given due consideration from longevity point of view while designing this type of bearing system.
Nomenclature r: Radial coordinate [r.sub.o]: Outerradius [r.sub.i]: Inner radius k: Radii ratio ([r.sub.i]/[r.sub.o]) h: Lubricant film thickness H: Thickness of solid housing s: Electrical conductivity of the lubricant [mu]: Viscosity of the lubricant [B.sub.o]: Uniform transverse magnetic field applied between the plates M: Hartmann Number [[B.sub.0]h[(s/[mu]).sup.1/2]] [p.sub.s]: Supply Pressure Q: Flow rate [P.sub.s.sup.*] Dimensionless supply pressure p: Lubricant pressure P: Nondimensional pressure w: Load carrying capacity W: Dimensionless load carrying capacity [h.sub.0]': Surface width of lower plate [h.sub.1]': Surface width of upper plate [s.sub.0]: Electrical conductivity of lower surface [s.sub.1]: Electrical conductivity of upper surface [[phi].sub.0](h): Electrical permeability of lower surface ([s.sub.0][h.sub.0]'/sh) [[phi].sub.1] (h): Electrical permeability of upper surface ([s.sub.1][h.sub.1]'/sh) [[sigma].sup.*]: Nondimensional standard deviation [[alpha].sup.*]: Dimensionless variance [[epsilon] .sup.*]: Nondimensional skewness.
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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Jatinkumar V. Adeshara, (1) M. B. Prajapati, (2) G. M. Deheri (iD), (3) and R. M. Patel (iD) (4)
(1) Research Scholar, Department of Mathematics, H. N. G. University, Patan, 384 265, Gujarat State, India
(2) Head, Department of Mathematics, HNG University, Patan 384 265, Gujarat, India
(3) Department of Mathematics, SP University, Vallabh Vidyanagar 388120, Gujarat, India
(4) Department of Mathematics, Gujarat Arts and Science College, Ahmedabad 380 006 Gujarat, India
Correspondence should be addressed to R. M. Patel; firstname.lastname@example.org
Received 13 May 2018; Accepted 10 July 2018; Published 2 September 2018
Academic Editor: Huseyin Cimenoglu
Caption: Figure 1: Configuration of bearing system.
Caption: Figure 2: Variation of load carrying capacity with respect to M and [[phi].sub.0] + [[phi].sub.i].
Caption: Figure 3: Distribution of load bearing capacity with respect to M and [[sigma].sup.*].
Caption: Figure 4: Profile of load with respect to M and [[alpha].sup.*].
Caption: Figure 5: Variation of load carrying capacity with respect to M and [[epsilon].sup.*]
Caption: Figure 6: Distribution of load bearing capacity with respect to M and k.
Caption: Figure 7: Profile of load with respect to [[phi].sub.0] + [[phi].sub.1] and [[sigma].sup.*].
Caption: Figure 8: Variation of load carrying capacity with respect to [[phi].sub.0] + [[phi].sub.1] and [[alpha].sup.*].
Caption: Figure 9: Distribution of load bearing capacity with respect to [[phi].sub.0] + [[phi].sub.1] and [[epsilon].sup.*]
Caption: Figure 10: Profile of load with respect to [[phi].sub.0] + [[phi].sub.1] and k.
Caption: Figure 11: Variation of load carrying capacity with respect to [[sigma].sup.*] and [[alpha].sup.*].
Caption: Figure 12: Distribution of load bearing capacity with respect to [[sigma].sup.*] and [[epsilon].sup.*].
Caption: Figure 13: Profile of load with respect to [[sigma].sup.*] and k.
Caption: Figure 14: Variation of load carrying capacity with respect to [[sigma].sup.*] and [[epsilon].sup.*].
Figure 15: Distribution of load bearing capacity with respect to [[sigma].sup.*] and k.
Caption: Figure 16: Profile of load with respect to [[epsilon].sup.*] and k.
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|Title Annotation:||Research Article|
|Author:||Adeshara, Jatinkumar V.; Prajapati, M.B.; Deheri, G.M.; Patel, R.M.|
|Publication:||Advances in Tribology|
|Date:||Jan 1, 2018|
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