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EFFECTS OF SURFACE ROUGHNESS ON POROUS INCLINED R BEARINGS

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					278                                                                 Journal of Marine Science and Technology, Vol. 15, No. 4, pp. 278-286 (2007)




           EFFECTS OF SURFACE ROUGHNESS ON
            POROUS INCLINED SLIDER BEARINGS
           LUBRICATED WITH MICROPOLAR FLUIDS

                                         N.B. Naduvinamani* and T.V. Biradar**

Key words: surface roughness, micropolar fluids, porous, slider bearings.      Reynolds type equation which is applicable to any
                                                                               general surface roughness structure. Chritensen [3]
                                                                               proposed a new stochastic averaging approach for the
                            ABSTRACT                                           study of roughness effects on the hydrodynamic lubri-
                                                                               cation of bearings. Christensen and Tonder [4-6] pre-
      This paper describes a theoretical analysis of the effect of surface     sented a comprehensive general analysis for the two
roughness on the hydrodynamic lubrication of one-dimensional po-
                                                                               types of one dimensional surface roughness patterns
rous inclined slider bearings with micropolar fluids. To account for
more realistic situations of the non-uniform rubbing of the bearing            Viz. transverse and longitudinal, based on the general
surfaces, it is assumed that, the probability density function for the         probability density function and this approach formed
random variable characterizing the surface roughness is asymmetrical           the basis for the study of surface roughness effects by
with non-zero mean. The averaged modified Reynolds type equation               several researchers [9, 11, 12, 14]. In all these studies
is derived and the closed form expressions for the bearing character-
                                                                               it is assumed that, the probability density function for
istics are obtained. The numerical computations of the results show
that, the performance of the porous inclined slider bearing is im-             the random variable characterizing the surface rough-
proved for the micorpolar lubricants as compared to the correspond-            ness is symmetric with zero mean. However, in general
ing Newtonian lubricants. Further, it is found that, the negatively            due to non-uniform rubbing of the surfaces, especially
skewed surface roughness improves the porous bearing performance               in slider bearings the distribution of surface roughness
whereas, the bearing performance suffers due to the presence of
                                                                               may be asymmetrical. In view of this, Andharia et al.
positively skewed surface roughness.
                                                                               [2] studied the effect of surface roughness on the perfor-
                        INTRODUCTION                                           mance characteristics of one-dimensional slider bear-
                                                                               ings with an assumption of the probability density func-
      The study of the effects of surface roughness on                         tion for the random variable characterizing the surface
the hydrodynamic lubrication of various bearing sys-                           roughness is asymmetrical with a non-zero mean. All
tems has been a subject of growing interest. This is                           these studies are limited to the study of surface rough-
mainly because of the reason that, in practice all bearing                     ness effects on bearing performance with Newtonian
surfaces are rough. The study of the effect of surface                         lubricants.
roughness has a greater importance in the study of                                    Most of the modern lubricants are no longer
porous bearings as the surface roughness is inherent to                        Newtonian fluids, since the use of the lubricant addi-
the process used in their manufacture. In general, the                         tives in lubricants has become a common practice in
roughness asperity height is of the same order as the                          order to improve the performance of lubricants.
mean separation between the lubricated contacts. In                            Therefore, several microcontinuum theories [7] have
such situations, surface roughness affects the perfor-                         been proposed to account for the effects of additives.
mance of the bearing system. The stochastic study of                           Eringen [8] micropolar fluid theory is a subclass of
Tzeng and Saibel [21] has fascinated several investiga-                        microfluids that ignores the deformation of the micro-
tors in the field of tribology. Patir and Cheng [15, 16]                       elements and allows for the particle micromotion to take
proposed an average flow model for deriving the                                place. Allen and Kline [1] presented the lubrication
                                                                               theory for micropolar fluids in which the order-of-
                                                                               magnitude arguments are used to simplify the govern-
Paper Submitted 08/04/06, Accepted 12/14/06. Author for Correspondence:        ing equations to a system of coupled, linear , ordinary
N.B. Naduvinamani. E-mail: naduvinamaninb@yahoo.co.in.                         differential equations. The lubrication theory for
*Department of Mathematics, Gulbarga University, Gulbarga-585 106,
                                                                               micropolar fluids and its application to a journal bear-
 INDIA.
**Department of Mathematics, Appa Institute of Engineering and                 ing is presented by Prakash and Sinha [17]. Shukla and
  Technology, Gulbarga-585 103, INDIA.                                         Isa [20] derived a generalized Reynolds equation for
                    N.B. Naduvinamani & T.V. Biradar: Effects of Surface Roughness on Porous Inclined Slider Bearings                   279




micropolar lubricants and its application to optimum                           y
one-dimensional slider bearings. Ramanaiah and Dubey
[19] presented the analysis for optimum slider profile of
a slider bearing lubricated with a micropolar fluid. The
micropolarity-roughness interaction in hydrodynamic                                                                U
lubrication is studied by Praksh et al. [18]. This is based
on the Chritensen’s [3] stochastic model for hydrody-
namic lubrication of rough surfaces. The porous in-                            h1                  C
clined slider bearing lubricated with micropolar fluid is
studied by Verma et al. [22] with an assumption of                                                                     h0
perfectly smooth bearing surfaces. The effect of three-                                                                          x
dimensional irregularities on the hydrodynamic lubri-                      δ                                                Porous bearing
                                                                                                                            Solid backing
cation of journal bearings lubricated with micropolar                                                   L
lubricants is studied by Lin [13].
      In this paper an attempt has been made to study the                            Fig. 1. Porous plane inclined slider bearing.
effect of micropolarity-roughness interaction on the
performance of one-dimensional porous inclined slider
bearings. It is assumed that, the probability density                    random variable h s are defined as
function for the random variable characterizing the
surface roughness is asymmetrical with non-zero mean.                          α * = E(h s)                                             (2)

        FORMULATION OF THE PROBLEM                                             σ * = E[(h s – α *) 2]                                   (3)

      Figure 1 shows the schematic diagram of the po-                          ε * = E[(h s – α *) 3]                                   (4)
rous inclined slider bearing. It consists of two surfaces
separated by a lubricant film. The lower surface of the                  where E is the expectation operator defined by
porous bearing is at rest and the upper solid surface is
moving in its own plane with a constant velocity U. It                                    ∞
is assumed that the bearing surfaces are rough and                             E( • ) =        ( • ) f (h s)dh s                        (5)
infinitely wide in the z-direction. The lubricant in the                                  –∞
film region as well as in the porous region is assumed to
be micropolar fluid.                                                     It is to be noted that, the parameters α *, σ * and ε * are
      To mathematically model the surface roughness,                     independent of x, the mean α* and the parameter ε * can
expression for the film thickness is considered to be                    assume both positive and negative values however σ *
consisting of two parts.                                                 can always assume positive values.
                                                                                The field equations for micropolar fluids proposed
     H = h(x) + h s                                            (1)       by Eringen [8] simplify considerably under the usual
                                                                         assumptions of hydrodynamic lubrication i.e. laminar,
where                                                                    incompressible fluid with negligible body forces and
                                                                         negligible inertia forces. The resulting equations under
                     h1 – h0                                             steady state conditions given by [22] are;
     h(x) = h 1 –            x                                           Conservation of mass:
                        L
                                                                                ∂u ∂v
                                                                                  +   =0                                                (6)
                                                                                ∂x ∂y
is the mean film thickness, h s is a randomly varying
quantity measured from the mean level and thus charac-                   Conservation of momentum:
terizes the surface roughness and L is the length of the
bearing. The stochastic part h s is assumed to have the                                   ∂ 2u          ∂ν ∂p
probability density function f(h s) defined over the do-                       (µ + K)             +K     –   =0                        (7)
                                                                                          ∂y   2        ∂y ∂x
main –C ≤ h s ≤ C where C is the maximum deviation
from the mean level. For a continuous non-broken fluid
                                                                                ∂p
film thickness it is assumed that C < < h(x).                                      =0                                                   (8)
      The mean α * , the standard deviation σ * and the                         ∂y
parameter ε *, which is the measure of symmetry of the                   Conservation of angular momentum:
280                                         Journal of Marine Science and Technology, Vol. 15, No. 4 (2007)




          ∂ 2ν          ∂u                                                       h            Mh
      γ            –K      – 2Kν = 0                                  (9)    +      1 – tanh2                       Mα * M3
          ∂y   2        ∂y                                                       h0           h0                        –   A
                                                                                                                     h0   3

where γ and K are the viscosity coefficients for                                   2
                                                                                 α*      Mh               Mh
micropolar fluids.                                                           +      tanh    + M 1 – tanh2    B
The flow of micropolar lubricants in the porous matrix                           h0      h0               h0
is governed by the modified Darcy’s law
                                                                                                      h3      Mh
          –k                                                                 + 2M2(1 + λ)                tanh
      q=     ∇p *                                                   (10)
                                                                                                       3
                                                                                                      h0      h0
         µ+K


where q = (u*, v*) is the modified Darcy velocity vector,                        h3          Mh                     Mα * M3
                                                                             +      1 – tanh                            –   A
k is the permeability of the porous matrix and p * is the                         3
                                                                                 h0          h0                      h0   3
pressure in the porous matrix.
      The modified Reynolds equation for smooth po-
rous inclined slider bearing lubricated with micropolar                                 Mh    3h 2α * Mh  3h 2        Mh
fluid was obtained by [22] in the form                                       + tanh        A + 3 tanh    + 2 1 – tanh    MB
                                                                                        h0     h0     h0  h0          h0


       ∂                         kδ ∂p     U ∂H
                   f (H, M) +            =                          (11)         3hB      Mh                       3h           Mh
       ∂x                       µ + K ∂x   2 ∂x                              +       tanh                      +      1 – tanh2    MA – 3λ MB
                                                                                  h0      h0                       h0           h0

where

                h3
                                2                   3
                      – 3λ H M + 3λ H tanh(MH ) + 2H M2(1 + λ)tanh(MH )                                                        1
f (H, M) =
                 0                                                           *
             µ(2 + λ)               12M 2(1 + λ)tanh(M H )
                                                                                                                   Mh             Mh          Mα * M3
                                                                                 12M2(1 + λ ) tanh                    + 1 – tanh2                 –   A
                                                                                                                   h0             h0           h0   3
                                                                    (12)

where                                                                                                                                                      (14)
                                                     1
   µ     H     mh      K(2µ + K)                         2                   with
λ = , H = , M = 0, m =                                       and δ is the
   K     h0     2      γ (µ + K)                                                                           3              2                   2            2
                                                                                             ε*       α*             σ*       α*         α*           σ*
porous layer thickness.                                                                A=         +            +3                and B =          +
                                                                                             h3
                                                                                              0       h3
                                                                                                       0             h2
                                                                                                                      0
                                                                                                                              h0         h0           h2
                                                                                                                                                       0
      Multiplying both sides of Eq.(11) by h s and inte-
grating with respect to h s over the interval –C to C and
using Eqs. (2)-(4), gives the averaged Reynolds type                         The relevant boundary conditions for the pressure field
equation in the form                                                         are
                                                                                       –
                                                                                       p=0            at           x = 0, L                                (15)
       ∂                    *       *   * kδ ∂p           U ∂h
                   G(h, M, α , σ , ε ) +                =           (13)
       ∂x                                µ + K ∂x         2 ∂x               Introducing the non-dimensional scheme
      –
where p is the expected value of p and                                                       x      h      hs     p h20      kδ
                                                                                       x =     , h = , hs = , P =       , ψ= 3
                                                                                             L      h0     h0     µUL       h 0
                               h3
                                0       h       Mh                                       K    α*    σ*     ε*
G(h, M, α *, σ *, ε *) =             3λ    tanh                                        λ= , α= , σ= 2 , ε= 3
                            µ(2 + λ)    h0      h0                                       µ    h0                                                           (16)
                                                                                                   h      h               0         0
                           N.B. Naduvinamani & T.V. Biradar: Effects of Surface Roughness on Porous Inclined Slider Bearings                             281




into the Eqs. (13) and (15) gives the non-dimensional                                                                        –1
                                                                                                                      ψ
averaged Reynolds type equation and boundary condi-                                + D G(h , M, α, σ, ε) +                        dx                   (22)
tions in the form                                                                                                    1+λ

                                                                                The component of stress tensor required for computing
      ∂                                        ψ ∂P     1 ∂h
                G(h , M, α, σ, ε) +                   =                 (17)    frictional force is
     ∂x                                      1 + λ ∂x   2 ∂x

                                – = 0, 1                                                     1        ∂u
     P=0                   at   x                                       (18)            τyx = (2µ + K) – Kv                                            (23)
                                                                                             2        ∂y


                BEARING CHARACTERISTICS                                         The frictional force F per unit width on the sliding
                                                                                surface at y = H is defined as
      The solution of Eq. (17) subject to boundary con-
                                                                                              L
ditions (18) is obtained in the form
                                                                                        F=           (τyx)y = H dx                                     (24)
     P = I 1 + DI 2                                                     (19)                 0

                                                                                where
where
            x                                               –1                                                  K ∂u
                     h                      ψ                                           (τyx)y = H = µ +                                               (25)
     I1 =              G(h , M, α, σ, ε) +                       dx ,                                           2 ∂y
                     2                     1+λ                                                                         y=H
            0

                                                                                since    v = 0 at y = H
            x                                          –1
                                                  ψ                                  Using the solution of Eq. (7) in Eq. (25) and then
     I2 =             G(h , M, α, σ, ε) +                   dx ,                in Eq. (24) and the use of non-dimensional scheme
                                                 1+λ
            0                                                                   gives the non-dimensional mean frictional force in the
                    I3                                                          form
     D=–               ,
                    I4
                                                                                                 1
            1
                                                                                                         h + α dP    M(1 + λ)(2 + λ)
                                                            –1                          F =                       +                       dx
                     h                      ψ                                                              2   dx 4Mh (h + α)(1 + λ) + λg
     I3 =              G(h , M, α, σ, ε) +                    dx and                          0
                     2                     1+λ
            0                                                                                                                                          (26)
            1                                          –1
                                                  ψ                             where
     I4 =             G(h , M, α, σ, ε) +               dx .            (20)
                                                 1+λ
            0

                                                                                                                                       M3
The non-dimensional mean load carrying capacity is                              g = tanh(Mh ) + 1 – tanh2(Mh ) Mα –                       (ε + α 3 + α 2σ)
                                                                                                                                       3
obtained in the form [22];
                1                1
                                         dP                                                                                                            (27)
     W =             Pdx = –         x      dx                          (21)
                                         dx
                0                0                                              The non-dimensional coefficient of friction is then ob-
                                                                                tained by
Use of Eq. (19) in (21) gives
                                                                                                  F
                                                                                        Cf =                                                           (28)
        1                                                   –1                                    W
                     h                      ψ
W = –       x          G(h , M, α, σ, ε) +
                     2                     1+λ
        0                                                                                            RESULTS AND DISCUSSION

                                                                                        The Effect of surface roughness and micropolar
282                                     Journal of Marine Science and Technology, Vol. 15, No. 4 (2007)




lubricants on the performance characteristics of the                     parameter ψ . The integrations I1, I2 and I 3 involved in
porous inclined slider bearings is studied through the                   the calculation of non-dimensional pressure P given in
dimensionless parameters α , ε , σ , M, ψ and λ . As the                 Eq. (17) are calculated numerically [10]. It is observed
roughness parameters tends to zero the results obtained                  that P increases for increasing values of M and for
in this paper reduce to the smooth case studied by                       decreasing value of ψ . Further, it is observed that the
Verma et al. [22]. For the numerical computations of                     point of maximum pressure is a function of the perme-
the slider characteristics, the following set of values are              ability parameter ψ .
used for various non-dimensional parameters α = -0.1 –                         The effect of roughness parameter α , ε and σ on
0.1; ε = -0.1 – 0.1; σ = 0.0 – 0.4; λ = 0.0 – 0.4; M = 4,                                         –
                                                                         the variations of P with x is shown in the Figures 3, 4
6, 8, 10; and ψ = 0.1, 0.2. The numerical values for the                 and 5 respectively for two values of the parameter M. It
roughness parameters α , ε, σ are also so chosen that the                is observed from Figures 3 and 4 that P increases for
corresponding film shapes are feasible.                                  negatively increasing values of α and ε whereas P
                                                                         decreases for positively increasing values of α and ε .
1. Pressure                                                              The increasing values of σ decreases the fluid film
                                                                         pressure P (Figure 5).
     The effect of micropolar lubricants on the varia-
tion of the non-dimensional pressure P with – is de-
                                              x                          2. Load carrying capacity
picted in Figure 2 for two values of the permeability
                                                                                    Figure 6 shows the variation of non-dimensional
      0.5
                ψ = 0.1 ψ = 0.2
          M=4                                                                 0.5
      0.4 M = 6
          M=8
          M = 10                                                              0.4
      0.3
 P
                                                                              0.3
      0.2

                                                                              0.2                                 M = 4 M = 10
      0.1                                                                 P                           ε = -0.1
                                                                                                      ε = -0.05
      0.0                                                                     0.1                     ε = 0.0
        0.0        0.2        0.4       0.6         0.8        1.0                                    ε = 0.05
                                –
                                x                                                                     ε = 0.1
                                                                              0.0
                                                         –
Fig. 2. Variation of non-dimensional pressure P with x for different                        0.2      0.4          0.6        0.8        1.0
        values of M with α = -0.1, σ = 0.1, ε = -0.1 and λ = 2.                                         –
                                                                                                        x
                                                                                                                                   –
                                                                         Fig. 4. Variation of non-dimensional pressure P with x for different
                                                                                 values of ε with α = -0.1, σ = 0.1, ψ = 0.1 and λ = 2.
     0.5
                  M = 4 M = 10
         α = -0.1
         α = -0.05                                                            0.5
     0.4 α = 0.0
         α = 0.05
         α = 0.1                                                              0.4
     0.3

P                                                                             0.3
     0.2                                                                                                       M = 4 M = 10
                                                                         P 0.2                       σ = 0.0
                                                                                                     σ = 0.1
     0.1                                                                                             σ = 0.2
                                                                              0.1                    σ = 0.3
                                                                                                     σ = 0.4
     0.0                                                                      0.0
       0.0       0.2         0.4       0.6         0.8        1.0                           0.2      0.4          0.6        0.8        1.0
                               –                                                                       –
                                                                                                       x
                               x
                                                          –
Fig. 3. Variation of non-dimensional pressure P with x for different                                                               –
                                                                         Fig. 5. Variation of non-dimensional pressure P with x for different
        values of α with σ = 0.1, ε = -0.1, ψ = 0.1 and λ = 2.                   values of σ with α = -0.1, ε = -0.1, ψ = 0.1 and λ = 2.
                     N.B. Naduvinamani & T.V. Biradar: Effects of Surface Roughness on Porous Inclined Slider Bearings                       283




load carrying capacity W with λ for different values of                       where ψ = 0.2.
                                                                                                                                       –
M. The numerical value of λ = 0 corresponds to the                                  Figures 11, 12 and 13 shows the variation of F
Newtonian case. It is observed that W increases for                           with λ for different values of α, ε and σ respectively for
increasing values of λ and M. The effect of roughness                         two values of M. It is observed that negatively skewed
                                                                                                                       –
parameter α , ε and σ on the variation of W with λ is                         surface roughness pattern decreases F whereas posi-
depicted in the Figures 7, 8 and 9 respectively for two                       tively skewed surface roughness increases the frictional
values of M. It is observed that, negatively skewed                           force on the sliding surface.
surface roughness increases W whereas positively
skewed surface roughness decreases W . Further, the                           4. Coefficient of friction
significant increase in W is observed for larger values
of λ as compared to the Newtonian case ( λ = 0).                                    The variation of non-dimensional coefficient of
                                                                              friction C f with the parameter λ for different values of
3. Frictional force                                                           M is shown in the Figure 14 for two values of permeabil-
                                                                              ity parameter ψ . It is observed that C f decreases
      Figure 10 shows the variation of non-dimensional                        rapidly for increasing values of λ as compared to the
                 –
frictional force F with λ for different values of M. It is                    Newtonian case ( λ = 0). However, the marginal in-
                            –
observed that, for ψ = 0.1 F decreases up to the value of                     crease in C f is observed for increasing values of M.
                    –
λ = 0.5 and then F increases for further increasing                                 The effect of roughness parameters α , ε and σ on
                          –
values of λ . However, F increases for all values of λ


                                                                                                 M = 10 M = 4
                 ψ = 0.1 ψ = 0.2                                                 1.4
                                                                                     ε = -0.1
   1.4 M = 4                                                                                                                  M = 10
                                                                                     ε = -0.05
       M=6                                                                       1.2 ε = 0.0
   1.2 M = 8                                  ψ = 0.1                                ε = 0.05                                            M=4
       M = 10                                                                        ε = 0.1
                                                              ψ = 0.2            1.0
   1.0
                                                                               W 0.8
W 0.8

                                                                                 0.6
   0.6
                                                                                 0.4
   0.4                                                                             0.0    0.5    1.0    1.5       2.0   2.5   3.0      3.5   4.0
      0                1                2               3               4                                     λ
                                 λ                                            Fig. 8. Variation of non-dimensional load W with λ for different
Fig. 6. Variation of non-dimensional load W with λ for different                      values of ε with α = -0.1, σ = 0.1 and ψ = 0.1.
        values of M with α = -0.1, σ = 0.1 and ε = -0.1.

                                                                                                M = 4 M = 10
                                                                                 1.4 σ = 0.0
     1.4 α = -0.1 Μ = 4 Μ = 10                                                       σ = 0.1
                                                                                     σ = 0.2
         α = -0.05                                                               1.2 σ = 0.3
     1.2 α = 0.0                                                                     σ = 0.4
         α = 0.05
     1.0 α = 0.1                                                                 1.0

     0.8                                                                      W 0.8
 W
     0.6
                                                                                 0.6
     0.4
                                                                                 0.4
         0.0   0.5     1.0    1.5       2.0     2.5     3.0    3.5      4.0        0.0    0.5    1.0    1.5       2.0   2.5   3.0      3.5   4.0
                                    λ                                                                     λ
Fig. 7. Variation of non-dimensional load W with λ for different              Fig. 9. Variation of non-dimensional load W with λ for different
        values of α with σ = 0.1, ε = -0.1 and ψ = 0.1.                               values of σ with α = -0.1, ε = -0.1 and ψ = 0.1.
284                                          Journal of Marine Science and Technology, Vol. 15, No. 4 (2007)




                                                                                   1.05
      1.00           ψ = 0.1 ψ = 0.1                                                                       M = 4 M = 10
      0.95 M = 4
                                                                                   1.00          σ = 0.0
           M=6                                                                     0.95          σ = 0.1
      0.90 M = 8                                                                                 σ = 0.2
           M = 10                                                                  0.90          σ = 0.3
      0.85                                                                                       σ = 0.4
      0.80                                                                         0.85
      0.75                                                                       0.80
                                                                               F 0.75                            M = 10
F 0.70
      0.65                                                                         0.70
      0.60                                                                         0.65                                                       M=4
      0.55                                                                         0.60
      0.50                                                                         0.55
         0.0   0.5     1.0   1.5       2.0    2.5    3.0    3.5    4.0                   0.0      0.5      1.0    1.5          2.0      2.5      3.0   3.5   4.0
                                   λ                                                                                       λ
                                                           –                                                                              –
Fig. 10. Variation of non-dimensional frictional force F with λ for           Fig. 13. Variation of non-dimensional frictional force F with λ for
         different values of M with α = -0.1, σ = 0.1 and ε = -0.1.                    different values of σ with α = -0.1, ε = -0.1 and ψ = 0.1.




                                                                                   1.0
   1.10            M = 4 M = 10                                                                                                           ψ = 0.1 ψ = 0.2
   1.05 α = -0.1                                                                   0.9                                          M=4
   1.00 α = -0.05                                                                  0.8
                                                                                                                                M=6
                                                                                                                                M=8
   0.95 α = 0.0                                                                                                                 M = 10
          α = 0.05                                                                 0.7
   0.90 α = 0.1
   0.85                                                                            0.6                           ψ = 0.2
                                                                                               ψ = 0.1
 F 0.80                                                                        C
                                                                                   0.5
   0.75                                                                            0.4
   0.70
   0.65                                                                            0.3
   0.60                                                                            0.2
   0.55                                                                            0.1
       0.0 0.5 1.0 1.5 2.0                     2.5    3.0    3.5   4.0                   0.0      0.5
                                                                                                    .      1.0   1.5 2.0                2.5      3.0   3.5   4.0
                           λ                                                                                        λ
                                                            –
Fig. 11. Variation of non-dimensional frictional force F with λ for           Fig. 14. Variation of non-dimensional Coefficient of friction C f with λ
         different values of α with σ = 0.1, ε = -0.1 and ψ = 0.1.                     for different values of M with α = -0.1, σ = 0.1 and ε = -0.1



                                                                                   1.0
      1.00            M = 4 M = 10                                                                                                               M = 4 M = 10
        ε = -0.1                                                                   0.9                                               α = -0.1
   0.95 ε = -0.05                                                                                                                    α = -0.05
   0.90 ε = 0.0                                                                    0.8                                               α = 0.0
        ε = 0.05                                                                                                                     α = 0.05
   0.85 ε = 0.1                                                                    0.7                                               α = 0.1
   0.80                                                                            0.6
 F 0.75                      M = 10
                                                                               C 0.5
   0.70                                                     M=4                  0.4
   0.65
   0.60                                                                            0.3
   0.55                                                                            0.2
         0.0   0.5     1.0   1.5       2.0     2.5    3.0    3.5   4.0             0.1
                                   λ                                                  0.0        0.5       1.0   1.5           2.0      2.5      3.0   3.5   4.0
                                                                                                                       λ
                                                            –
Fig. 12. Variation of non-dimensional frictional force F with λ for           Fig. 15. Variation of non-dimensional Coefficient of friction C f with λ
         different values of ε with α = -0.1, σ = 0.1 and ψ = 0.1.                     for different values of α with σ = 0.1, ε = -0.1 and ψ = 0.1.
                    N.B. Naduvinamani & T.V. Biradar: Effects of Surface Roughness on Porous Inclined Slider Bearings                            285




                                                                              1.0
    0.9                                                  M = 4 M = 10                                                             M = 4 M = 10
                                             ε = -0.1                         0.9                                       σ = 0.0
    0.8                                      ε = -0.05                        0.8                                       σ = 0.1
                                             ε = 0.0                                                                    σ = 0.2
    0.7                                      ε = 0.05                         0.7                                       σ = 0.3
                                             ε = 0.1                                                                    σ = 0.4
    0.6                                                                       0.6
                                                                              0.5
 C 0.5                                                                      C
   0.4                                                                        0.4
                                                                              0.3
    0.3
                                                                              0.2
    0.2
                                                                              0.1
    0.1                                                                          0.0      0.5   1.0     1.5       2.0    2.5      3.0   3.5    4.0
       0.0    0.5     1.0    1.5       2.0      2.5      3.0   3.5   4.0
                                                                                                              λ
                                   λ
Fig. 16. Variation of non-dimensional Coefficient of friction C f with λ   Fig. 17. Variation of non-dimensional Coefficient of friction C f with λ
         for different values of ε with α = -0.1, σ = 0.1 and ψ = 0.1.              for different values of σ with α = -0.1, ε = -0.1 and ψ = 0.1.




the variation of C f with λ is depicted in Figures 15, 16                  Sharnbasawappa Appa, President , Sharnbasveshwar
and 17. It is observed that C f decreases for negatively                   Vidya Vardhak Sangha, Gulbarga for his encour-
skewed surface roughness pattern whereas it increases                      agements. This work is supported by the UGC New
for positively skewed surfaces roughness pattern.                          Delhi under the Project No.F-31-84/2005(SR).
Further, the increasing values of σ increases the coeffi-
cient of friction C f marginally. These increase/de-                                                NOMENCLATURE
crease in C f are more pronounced for larger values of λ.
                                                                           C           maximum asperity height from the mean level
                        CONCLUSIONS
                                                                           Cf          non-dimensional frictional coefficient,
       The performance characteristics of the rough po-
rous inclined slider bearings lubricated with micropolar                                Cf = F W
fluids is analysed on the basis of Eringen [8] constitu-
tive equations for micropolar fluids and the stochastic                    E           expectancy operator defined by equation (5)
random variable to represent the surface roughness of                      F           frictional force
the bearing with non-zero mean variance and skewness.                      –                                                 Fh o
On the basis of the numerical computations performed,                      F           non-dimensional frictional force, F =
                                                                                                                             µUL
the following conclusions can be drawn.                                    H           fluid film thickness, H = h + h s
1. The significant increase in the load carrying capacity
   and decrease in the coefficient of friction can be                                                                              h1 – h0
   attained by the use of micropolar lubricants.                           h           mean film thickness, h(x) = h 1 –                   x
                                                                                                                                      L
2. The porous facing on the slider bearing affects its
   performance.
                                                                           –
3. The presence of negatively skewed surface roughness                     h           non-dimensional mean film thickness, h = h h
                                                                                                                                    o
   pattern on the bearing surface improves its
                                                                           –
   performance.                                                            H           non-dimensional fluid film thickness, H = H h
                                                                                                                                                     o
4. The positively skewed surface roughness pattern on
   the porous slider bearing affects its performance.                      ho          outlet film thickness
5. The above effects are more accentuated for larger                       h1          inlet film thickness
   values of the parameter λ .                                             hs          random variable
                                                                           K           viscosity coefficient for micropolar fluid.
                    ACKNOWLEDGEMENTS                                       k           permeability of the porous matrix.
                                                                           L           bearing length
     The authors are grateful to the referees for their                    M           non-dimensional micropolar parameter,
valuable comments on the earlier draft of the paper.
One of the authors (TVB) is thankful to Poojya Dr.                                     M = mh o 2
286                                 Journal of Marine Science and Technology, Vol. 15, No. 4 (2007)




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