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ANALYSIS OF GREASE LUBRICATED ISOVISCOUS-ELASTIC POINT CONTACTS-2-3-4

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ANALYSIS OF GREASE LUBRICATED ISOVISCOUS-ELASTIC POINT CONTACTS-2-3-4 Powered By Docstoc
					    INTERNATIONAL JOURNAL OF ADVANCED RESEARCH ISSN
International Journal of Advanced Research in Engineering and Technology (IJARET),IN
0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME
               ENGINEERING AND TECHNOLOGY (IJARET)

ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
                                                                            IJARET
Volume 4, Issue 5, July – August 2013, pp. 207-217
© IAEME: www.iaeme.com/ijaret.asp                                           ©IAEME
Journal Impact Factor (2013): 5.8376 (Calculated by GISI)
www.jifactor.com



     ANALYSIS OF GREASE LUBRICATED ISOVISCOUS-ELASTIC POINT
                           CONTACTS

                                       Vivek Chacko1
        Department of Mechanical Engineering, Saintgits College of Engineering, Kottayam, Kerala,
                                             India
                                  Bindu Kumar Karthikeyan2
                 Department of Mechanical Engineering, Government Engineering College,
                                Thiruvananthapuram, Kerala, India



ABSTRACT

       Greases generally behave as shear-thinning or pseudo-plastic fluids- their viscosity reduces
under shear i.e. with sufficient shear the viscosity of grease approaches that of the base lubricant. By
this behaviour grease may be considered as a plastic fluid. Numerical solution of the modified
Reynolds equation for grease for point contacts remains challenging, despite the advent of powerful
computational techniques and platforms. Grease lubricated concentrated point contacts are analysed
under isothermal conditions. Grease is seen to provide better lubrication in comparison with oil
lubricant, thus protracting the life of the contact conjunction. Operating conditions have been defined
by dimensionless control parameters. The asymptotic behaviour of grease allows for the
development of film thickness equations. From the results of the numerical method, extrapolated
equations for grease film thickness in iso-viscous elastic regime lubrication is developed using the
dimensionless parameters.

Keywords: Grease lubrication, Hydrodynamic, Herschel-Bulkley Model, Moes Parameters,
Iso-viscous elastic

1.    INTRODUCTION

        Lubrication theories for oil lubricated contacts have been well documented; however the
theory for grease lubrication lags considerably behind because of the complexity of its rheological
properties. In practice, approximately 80–90% of rolling element bearings is lubricated with grease.
The understanding of the rheological behaviour of lubricating greases is nowadays a decisive factor
in the design and optimisation of the tribological systems as well as in the control of their processing.


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Greases are two-phase lubricants composed of a thickener dispersed in base oil. Due to the effects of
the thickener, greases are often modelled as a plastic solid. Unlike oil, grease can withstand shear
and will not flow until a critical yield stress is reached. Traditionally, these properties of grease have
been related to the so-called ‘yield state’ at low strain rates and a shear-thinning behaviour at
medium and high strain rates. Balan& Franco (2001) recorded that the typical grease flow curve
exhibits constant values of shear stress at low strain rates.
        The behaviour of grease inside a contact conjunction differs from that for oil so the Reynolds
equation used for oil lubricated contacts cannot be used directly for grease lubricated contacts. The
flow of grease follows a non-Newtonian behaviour. The numerical solution of the modified Reynolds
equation for grease remains challenging despite the advent of powerful computational techniques and
platforms. Herschel and Bulkley [1] presented the Herschel–Bulkley (HB) equation as a realistic
constitutive model for grease behaviour. Kauzlarich and Greenwood [2] found that most grease
behave pseudo-plastically.Kauzlarich and Greenwood [2], Balan& Franco [3], Jonkisz and Freda [4],
Zhu and Neng[5],Yoo and Kim [8] have carried out numerical method research in grease lubricated
using HB Model. But all the studies described are limited to line contact conjunctions.
        Numerical model for grease lubrication capable of solving EHL of circular point contacts for
isoviscous elastic regime of lubrication has been developed herein. For this a modified Reynolds
equation is derived considering grease as a Bingham solid incorporating Herschel Bulkley flow
model. Results from the numerical method, with input values taken from Williamson [17] are found
to be well comparable to experimental film thickness. The generic nature of the numerical method
developed is confirmed. A range of load, speed, material and lubricant viscosity parameters have
then been analysed.
        Extrapolated film thickness equation, incorporating asymptotic behaviour of grease in the iso-
viscous elastic (IE) regime of lubrication, is developed herein. The film thickness and corresponding
dimensionless parameters are tabulated from the results of the numerical method.
The equation for grease lubricant film thickness in IE regime in terms of dimensionless Moes’
Parameters [18] is reported in this paper.

2. MATHEMATICAL MODEL

        The mathematical model is composed of the Modified Reynolds’ equation for grease flow
[9and10], the film thickness equation, the load balance equation, together with the expression of
viscosity and density of lubricant. Details of the model and the numerical methods can be seen in
refs. [9], [10]. For the numerical analysis grease flow Herschel–Bulkley flow equation is selected,
which is described by a three-parameter rheological model as [1]:

                                         nh
               τ = τ    0   +η   p   D                                                      (1)

  Assuming no side leakage, modified Reynolds’ equation for grease flow the equation can be
simplified to [10]:



  ∂  ρ ha 3 ∂p  ∂  ρ ha 3 ∂p       ∂                    ∂ ( hρ ) 
               +             = 12  ρ ( ub h − uh p ) +          
  ∂x  η p ∂x  ∂y  η p ∂y 
                                     ∂x                     ∂t 
                                                                                             (2)




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For oil lubricated contacts with no core formation and plug flow: h p = 0 and the plug flow velocities
equate to that of the base oil, thus: u p = u b and v p = v b . This yields the usual Reynolds’ equation.

a. LUBRICANT RHEOLOGY
       The density dependence on pressure follows the usual relationship given by Dowson and
Higginson [11], in dimensionless form, is considered here:

                                        α . Ph ⋅ Pi , j
                ρ   i, j    = 1+                                                             (3)
                                      1 + β . P h ⋅ Pi , j

                                 −10                                −9
The values are α = 5 . 8 3 × 1 0     ,           β = 1 .6 8 × 1 0

        In general, grease is assumed to be fully shear-degraded by its passage through the contact
inlet when its structure becomes a large scale tri-dimensional network of discrete spherical soap
particles, dispersed in the base oil [12]. In this case the viscosity of the grease would conform to:

     η   p   = η    bo     (1   + BΦ       )                                                 (4)

where η p is the plastic viscosity of grease, η b o is the base oil viscosity, B ≈ 0 . 2 5 is a constant
and Φ is the volume fraction of soap in oil.

        The region of contact in rolling element bearings is subjected to high pressures. Therefore,
the viscosity of the grease inside the rolling element contact cannot remain constant, and changes
with the pressure increase. In order to calculate the variation of viscosity owing to pressure, the
Reolands’ equation[9]is used and is expressed as:

                                                             R
                                                   P 
      ln φ + 1 .2 =        (ln φ 0   + 1 .2 ) 1 +                                          (5)
                                                  2000 

where P is the gauge pressure and R is the Reolands’ viscosity-pressure index.
Elastohydrodynamic lubrication has a characteristically flat film shape, with parallel surfaces at
contact.

b. THICKNESS OF PLUG
       To find a solution the thickness of grease plug must be determined at any position x, y in the
conjunction,

                      2τ 0
      hp =
              ∂p 2  ∂p 2 
                                                                                           (6)
                    +      
              ∂x 
                       ∂y  
                              


c. ELASTIC FILM SHAPE
       The elastohydrodynamic film shape is given by the approximate parabolic shape of the
contact of a ellipsoidal solid near a flat semi-infinite elastic half-space subjected to a localised


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Hertzian deformation. With an initial gap of h0 the elastic film shape for a spherical solid such as a
ball bearing is [13,14]:

                                                x2    y2
        h      (x, y ) =          h0 +             +     + δ          (x, y )                                      (7)
                                               2Rx   2Ry


δ     ( x , y ) is the local deformation
d. METHOD OF SOLUTION
The value of pressure at iteration k within a time step is obtained as:

         p ik, j = p ik, − 1 + Ω ∆ p ik, j
                         j                                                                                         (8)


where Ω               is an under-relaxation factor.

       In each small step of time, two convergence criteria are required: one for pressures pi , j and
the other for the contact load, which should attain the value determined by a previous dynamic
analysis, for instance load per ball-to-race contact in a ball bearing.
For pressure convergence:

 nx     ny

∑∑
 i =1   j =1
                p ik, j − p ik, − 1
                                j

          nx     ny
                                           ≤ ε   p                                                                 (9)
        ∑∑
        i =1    j =1
                       p   k
                           i, j



                                                                                  −5          −4
The error tolerance                ε   p   is usually in the range of 5 X 1 0          − 10        .

For load convergence the following criterion is used:

                                  *
                            W          − π           ≤ ε      w                                                    (10)

The error tolerance ε                      w   ≈ 1 0 − 2 .If the criterion in (9) is not satisfied the film thickness is relaxed
as:

                                  *k                 * k −1           *
                           H       0       = H         0      + ξ W       − π                                      (11)

                                  −7
where        ξ ≈ 10                        is referred to as a damping factor.




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e. ALGORITHM FOR COMPUTER PROGRAM




                             Figure 1 Algorithm for computer program


f. VALIDATION OF NUMERICAL METHOD
        The computer program for the numerical method is executed for values of input conditions in
Williamsons’ experimental analysis [16].The entrainment velocity vs. film thickness plot is shown
below in Figure 2. Comparing the results of film thickness from the numerical method with those
from experiment, a good match is obtained. The numerical results show a slightly higherpredicted
film thickness over experimental results at higher entrainment velocities. A maximum difference of
10% in the film thickness is found at aninlet linear velocityof about 1.1 m/s. Viscosity of the
lubricant is sensitive to temperature variations. Experimental investigations are prone to temperature
variations, in contrast to the isothermal method assumed for numerical analysis. Also squeeze
phenomenon is not factored in the current study. The slight difference in film thickness may be
attributed to these the differences in considerations.



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             Figure 2 Overlay of Grease film thickness over experimental OFT (Williamson)


  g. DIMENSIONLESS PARAMETERS
          If lubricant is assumed to be incompressible and Roelands’ equation is followed for viscosity
  calculation inside contact conjunction, then three parameters can be defined to fully express the
  contact parameters in dimensionless from [17].These are the operating parametersM, L and the
  curvature ratio,D expressed as:

                       η
                                                                                           (12)
               ′           ′


                   η
             α ′       ′
                                                                                           (13)

  For circular point contacts, the ratio of reduced radii,

  D = Rx/Ry =1. The advantage over the conventional dimensionless parameters of speed, load and
  material is that the number of control parameters defining the contact conditions can be reduced for
  the given equation[18].

          The load, speed and material parameters and the corresponding Moes’ Parameters have been
  tabulated in table 1. Regression analysis was used to determine the relation between dimensionless
  central film thickness and Moes’ parameters.

3. RESULTS AND DISCUSSION

          Grease lubrication studies for numerical solution of IE regime have not been reported hitherto
  in open literature. The analysis method is generic and may be valid across applications. Figure shows
  three dimensional pressure profile for IE regime of grease lubrication in point contact conjunctions.
  The maximum dimensionless load acting on the ball is W*=7.0088E-06. Centre line film thicknesses
  are assumed for further calculation since the maximum pressure is known to lie on this axis.



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Figure 3: Pressure profiles for different speeds    Figure 3: 3D Pressure Profile for W*=7
                                                                                      W*=7.0088E-06




       Figure 5: Corresponding film thicknesses                Figure 6: W*=2.89743E
                                                                         W*=2.89743E-06




         Figure 7: Film thickness contour for             Figure 8: W*=7.0088E-06 (for 168.5 rpm)
                                                                               06
                  W*=6.14241E-07  07


                         s
      Moes Dimensionless Parameters are introduced for the formation of film thickness equation.
Table 1 shows the input parameters along with the corresponding minimum film thicknesses from
                                        Iso viscous                                    p
the results of the numerical method for Iso-viscous rigid regime lubrication and Moes’ parameters.


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The variation of material parameters is expected to bring slight variations to the coefficients in the
below equation. Using suitable regression analysis, the film thickness equation is arrived at:

                                                                                                                                  (14)


                            Table 1 Contact characteristics in terms of Moes' Parameters
           Sl.   Load    W (N)     Rx (m)     E'( Gpa)    η0       us                   M        L      hcen(m)     h*       h* from     Difference
           No    Point                                                                                                       equation     in film
                                                                                                                                         thickness
                                                                                                                                            (%)
  n=500    1      1      4.30143   6.70E-03   1.56E+11   0.0202   2.42     1.33E-08    34.28    5.42   2.18E 06   3.25E-04   4.05E-04       24.7
           2      11     7.30756   6.70E-03   1.56E+11   0.0202   2.424    1.33E-08    58.23    5.42   1.97E-06   2.94E-04   3.10E-04       5.4
           3      21     11.0558   6.70E-03   1.56E+11   0.0202   2.424    1.33E-08    88.10    5.42   1.77E-06   2.65E-04   2.51E-04      -5.2
           4      31     15.4373   6.70E-03   1.56E+11   0.0202   2.424    1.33E-08   123.01    5.42   1.59E-06   2.37E-04   2.12E-04      -10.5
           5      41     20.2902   6.70E-03   1.56E+11   0.0202   2.424    1.33E-08   161.68    5.42   1.41E-06   2.11E-04   1.85E-04      -12.6
           6      51     25.4101   6.70E-03   1.56E+11   0.0202   2.424    1.33E-08   202.48    5.42   1.25E-06   1.87E-04   1.65E-04      -11.9
           7      61     30.5624   6.70E-03   1.56E+11   0.0202   2.424    1.33E-08   243.53    5.42   1.11E-06   1.66E-04   1.50E-04      -9.7
           8      71     35.4969   6.70E-03   1.56E+11   0.0202   2.424    1.33E-08   282.85    5.42   9.96E-07   1.49E-04   1.39E-04      -6.5
           9      81     39.9641   6.70E-03   1.56E+11   0.0202   2.424    1.33E-08   318.45    5.42   8.99E-07   1.34E-04   1.31E-04      -2.5
           10     91     43.7322   6.70E-03   1.56E+11   0.0202   2.424    1.33E-08   348.47    5.42   8.25E-07   1.23E-04   1.25E-04       1.5
           11    101     46.6012   6.70E-03   1.56E+11   0.0202   2.424    1.33E-08   371.33    5.42   7.72E-07   1.15E-04   1.21E-04       5.0
           12    111     48.4172   6.70E-03   1.56E+11   0.0202   2.424    1.33E-08   385.80    5.42   7.40E-07   1.11E-04   1.19E-04       7.5
           13    121     49.0815   6.70E-03   1.56E+11   0.0202   2.424    1.33E-08   391.10    5.42   7.28E-07   1.09E-04   1.18E-04       8.5
  n=250    1      1      4.30143   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   114.61    3.62   2.05E-06   3.06E-04   3.54E-04      15.9
           2      11     7.30756   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   194.70    3.62   1.86E-06   2.77E-04   2.71E-04      -2.2
           3      21     11.0558   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   294.57    3.62   1.66E-06   2.48E-04   2.20E-04      -11.6
           4      31     15.4373   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   411.31    3.62   1.48E-06   2.20E-04   1.85E-04      -15.8
           5      41     20.2902   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   540.61    3.62   1.31E-06   1.95E-04   1.61E-04      -17.2
           6      51     25.4101   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   677.02    3.62   1.15E-06   1.72E-04   1.44E-04      -16.3
           7      61     30.5624   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   814.30    3.62   1.02E-06   1.52E-04   1.31E-04      -13.6
           8      71     35.4969   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   945.77    3.62   9.01E-07   1.35E-04   1.22E-04      -9.6
           9      81     39.9641   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   1064.79   3.62   8.06E-07   1.20E-04   1.14E-04      -4.8
           10     91     43.7322   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   1165.19   3.62   7.32E-07   1.09E-04   1.09E-04       0.1
           11    101     46.6012   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   1241.63   3.62   6.79E-07   1.01E-04   1.06E-04       4.5
           12    111     48.4172   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   1290.01   3.62   6.47E-07   9.65E-05   1.04E-04       7.6
           13    121     49.0815   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   1307.71   3.62   6.36E-07   9.49E-05   1.03E-04       8.7
 n=168.5   1      1      4.30143   6.70E-03   1.56E+11   0.0202   0.792    1.33E-08    79.31    4.10   2.00E-06   2.98E-04   3.69E-04      23.7
           2      11     7.30756   6.70E-03   1.56E+11   0.0202   0.792    1.33E-08   134.74    4.10   1.81E-06   2.70E-04   2.82E-04       4.5
           3      21     11.0558   6.70E-03   1.56E+11   0.0202   0.792    1.33E-08   203.85    4.10   1.61E-06   2.40E-04   2.29E-04      -4.9
           4      31     15.4373   6.70E-03   1.56E+11   0.0202   0.792    1.33E-08   284.64    4.10   1.43E-06   2.14E-04   1.93E-04      -9.7
           5      41     20.2902   6.70E-03   1.56E+11   0.0202   0.792    1.33E-08   374.12    4.10   1.27E-06   1.89E-04   1.68E-04      -11.0
           6      51     25.4101   6.70E-03   1.56E+11   0.0202   0.792    1.33E-08   468.52    4.10   1.11E-06   1.66E-04   1.50E-04      -9.7
           7      61     30.5624   6.70E-03   1.56E+11   0.0202   0.792    1.33E-08   563.52    4.10   9.78E-07   1.46E-04   1.37E-04      -6.4
           8      71     35.4969   6.70E-03   1.56E+11   0.0202   0.792    1.33E-08   654.51    4.10   8.63E-07   1.29E-04   1.27E-04      -1.7
           9      81     39.9641   6.70E-03   1.56E+11   0.0202   0.792    1.33E-08   736.87    4.10   7.69E-07   1.15E-04   1.19E-04       3.9
           10     91     43.7322   6.70E-03   1.56E+11   0.0202   0.792    1.33E-08   806.35    4.10   6.95E-07   1.04E-04   1.14E-04       9.8
           11    101     46.6012   6.70E-03   1.56E+11   0.0202   0.792    1.33E-08   859.25    4.10   6.43E-07   9.60E-05   1.10E-04      15.0
           12    111     48.4172   6.70E-03   1.56E+11   0.0202   0.792    1.33E-08   892.74    4.10   6.12E-07   9.13E-05   1.08E-04      18.5
           13    121     49.0815   6.70E-03   1.56E+11   0.0202   0.792    1.33E-08   904.98    4.10   6.01E-07   8.97E-05   1.07E-04      19.8
  n=100    1      1      4.30143   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   114.61    3.62   1.96E-06   2.92E-04   3.54E-04      21.2
           2      11     7.30756   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   194.70    3.62   1.77E-06   2.64E-04   2.71E-04       2.6
           3      21     11.0558   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   294.57    3.62   1.57E-06   2.35E-04   2.20E-04      -6.6
           4      31     15.4373   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   411.31    3.62   1.40E-06   2.09E-04   1.85E-04      -11.3
           5      41     20.2902   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   540.61    3.62   1.23E-06   1.84E-04   1.61E-04      -12.3
           6      51     25.4101   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   677.02    3.62   1.08E-06   1.61E-04   1.44E-04      -10.8
           7      61     30.5624   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   814.30    3.62   9.45E-07   1.41E-04   1.31E-04      -7.1
           8      71     35.4969   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   945.77    3.62   8.32E-07   1.24E-04   1.22E-04      -2.1
           9      81     39.9641   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   1064.79   3.62   7.38E-07   1.10E-04   1.14E-04       4.0
           10     91     43.7322   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   1165.19   3.62   6.65E-07   9.93E-05   1.09E-04      10.1
           11    101     46.6012   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   1241.63   3.62   6.14E-07   9.17E-05   1.06E-04      15.6
           12    111     48.4172   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   1290.01   3.62   5.83E-07   8.69E-05   1.04E-04      19.5
           13    121     49.0815   6.70E-03   1.56E+11   0.0202   0.485    1.33E-08   1307.71   3.62   5.71E-07   8.53E-05   1.03E-04      21.0




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4. CONCLUSION

        Grease is the preferred lubricant in most rolling-element bearings. The study of grease is
desired in optimizing bearing performance. Grease behaves as pseudo-plastic fluid which makes it
different from oil, which is a fluid.
        This necessitates an original study of the performance of grease starting from modified
Reynolds equation, considering it as a Bingham plastic, based on which the above numerical model
is developed. The analysis based on the numerical model is considered, and the above assumptions
yield results which are well-comparable to established experimental data. Grease is found to offer
better film thickness under same conditions than base oil.
        Also grease lubricant has a longer service life in comparison to oil lubricants. Better lubricant
film thickness would in turn translate to extended life of the contact conjunction.
The result from the numerical method is consolidated in terms of the dimensionless Moes
Parameters. By method of regression analysis, the equation for film thickness of grease lubricated
circular point contact under iso-viscous elastic regimes of EHL lubrication has been presented.

                                             APPENDIX

                            Table 2 Lubricant and Geometrical Parameters




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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME

NOMENCLATURE

E’         Effective Elastic Modulus of contacting material      GPa
ho         Initial film thickness (un-deformed)                  m
H          Film thickness                                        m
h een      Centre film thickness of contact                      m
    *                                            ’   2
h          Dimensionless Film thickness (=W/E Rx )               -
hp         Plug flow thickness                                   m
nh         Herschel Bulkley index                                -
P          Pressure                                              Pa
Rx         Radius of ball in entrainment direction               m
U          Speed of entraining motion in x direction             m/s
ub         Speed of entraining motion of the base oil            m/s
up         Speed of entraining of the plug flow in x direction   m/s
us         Speed parameter used in Moes Parameter                m/s
v          Velocity of side-leakage in y direction               m/s
vb         Base oil side-leakage velocity                        m/s
vp         Plug flow velocity in y direction                     m/s
W          Applied load                                          N
                       th
Wi         Load on 'i' ball                                      N
W0         Initial load                                          N
Z0         Initial axial deflection                              m
                                                                     -1
           Pressure- Viscosity coefficient of lubricant          Pa s
δ0         Initial deflection                                    m
η0         Lubricant viscosity before entrainment                Pa.s
ηp         Plastic viscosity                                     Pa.s
                                                                          3
ρ          Lubricant density                                     kg/m
                                                                          2
τ          Shear Stress                                          N/m
                                                                          2
τ0         Yield shear stress                                    N/m


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