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Squeeze Film Lubrication between Parallel Stepped Plates with Couplestress Fluids

Biradar Kashinath

Department of Mathematics, Government First Grade College, Shorapur, Dist: Yadgir, INDIA.

Corresponding Address:

[email protected]

Research Article

Abstract: In this paper, a theoretical analysis of the effects of couple stresses on the squeeze film lubrication between parallel stepped plates is presented. The modified Reynolds type equation is derived on the basis of Stokes microcontinuum theory of couplestress fluids. The closed form solution is obtained. According to the results obtained, the influence of couple stresses enhances the squeeze film pressure, load carrying capacity and decreases the response time as compared to the classical Newtonian-lubricant case. The load carrying capacity decreases as the step height increases.

Keywords: Squeeze film, Parallel stepped plates, couple stresses.

1 Introduction

The squeeze film lubrication phenomenon is widely observed in several applications such as gears, bearings, machine tools, rolling elements and automotive engines. The squeeze film action is also seen during approach of faces of disc clutches under lubricated condition. The squeeze film phenomenon arises when the two lubricating surfaces move towards each other in the normal direction and generates a positive pressure and hence supports a load. This phenomenon arises from the fact that a viscous lubricant present between the two surfaces cannot be instantaneously squeezed out when the two surfaces moving towards each other and this action provides a cushioning effect in bearings. The squeeze film lubrication between two infinitely long parallel plates is studied by Cameron (1981). The flow of an incompressible fluid between two parallel plates due to normal motion of the plates is investigated by Bujurke. et al (1995). The unsteady flow between two parallel discs with arbitrary varying gap width was studied by Ishizawa (1996). The squeeze film with Newtonian lubricants has been studied by several investigators (Jackson, 1963; Burbidge and Colin servais, 2004; Gupta, 1977). The Rayleigh step-bearings with non-Newtonian fluids has been studied by many researchers (Hughes, 1963; Bujurke et al., 1987; Maiti, 1973; Elkouh and Yang, 1991).

                          The use of different liquids as lubricants under different circumstances has gained its importance with a development of modern machines. In most of these lubricating oils the additives of high molecular weight polymers are present as a kind of viscosity index improvers. The presence of these additives in the lubricant prevents the viscosity variation of the lubricants with a change in temperature. The lubricants with additives causes the non-Newtonian behavior of the lubricating oils since the classical continuum mechanics of fluids neglects the size of fluid particles in the flow of fluids and hence several microcontinuum theories has been proposed to take into account of the intrinsic motion of material constituents(Ariman et al., 1973,1974). The Stokes (1966) microcontinuum theory of couplestress fluid accounts for the polar effects such as the couple stresses, body couples and asymmetric tensor. During last few decades Stokes microcontinuum theory has been extensively used to study the effect of couple stresses on the performance of various bearing systems viz: the slider bearings (Bujurke et al., 1990; Naduvinamani et al.,2003), Journal bearings (Mak and Conway, 1978; Guha, 2004), Squeeze film bearings (Lin, 1998; Naduvinamani et al., 2001) and thrust bearings (Ramesh and Dubey, 1975). These studies predicted the advantages of couple stress fluid lubricants over the Newtonian lubricants such as the increased load carrying capacity, decreased coefficient of friction in the slider bearings and delayed time of approach in squeeze film bearings.

          So far no attempt has been made to study the squeeze film characteristics with couplestress lubricants between flat stepped plates. Hence, in this chapter an attempt has been made to analyze the effect of couple stresses on these bearings.

2 Basic Equations

The constitutive equations for force and couple stresses proposed by Stokes are

1       (2.1)

2                     (2.2)

and      3       (2.3)




and      6

The dimensions of 7and 8are those of Newtonian viscosity and the dimensions of 9and 10 are those of momentum and the ratio 11 has a dimension of length square. For the incompressible fluids when the body forces and body moments are absent the equations of motion derived by Stokes are

12           (2.4)

13                                      (2.5)


3 Mathematical formulation of the problem

            The squeeze film between parallel stepped plates approaching each other with a normal velocity V is shown in figure 1. The lubricant in the film region is considered to be an incompressible Stokes couple stress fluid. When body forces and body couples are absent, under the usual assumptions of hydrodynamic lubrication applicable to thin films the equations of motion (2.4) and (2.5) for the couple stress fluids in Cartesian coordinates take the form

14,             (3.1)

15,            (3.2)

16.        (3.3)

The relevant boundary conditions are

(i)         At the upper surface y = h

17                   19     (3.4a)

20                               (3.4b)

(ii)        At the bearing surface y = 0

18        21       (3.5a)

22.             (3.5b)



 4 Solution of the problem

Solution of equation (3.1) subject to the boundary conditions (3.4a) and (3.5a) is

24         (4.1)

Where 25  is the couplestress parameter.

The volume flux of the lubricant is given by

26                    (4.2)

where b is the width of the bearing.

On using equation (4.1) in (4.2) gives

27       (4.3)



Integration of the continuity equation (4.3) over the film thickness and the use of boundary conditions (3.4a) and (3.5a) gives

29.                    (4.4)

Integration of equation (4.4) w.r.t x and using the condition 30 at x = 0 gives

31.                 (4.5)

The modified Reynolds type equation for determining the pressure is obtained from equations (4.3) and (4.5) in the form

32       (4.6)



34  .

The relevant boundary conditions for the pressure are

35   at     36 ,        (4.7a)

37    at       38          (4.7b)

Solution of equation (4.6) subject to the boundary conditions (4.7a) and (4.7b) is

39     (4.8)  and

40            (4.9)

The load carrying capacity, W is obtained in the form

41        (4.10)

Which in nondimensional form

42     (4.11)

Where  43     and    44



Writing 47 in equation (4.11), the squeezing time for reducing the initial film thickness 48 of 49 to a final thickness  50  of   52   is given by

54      (4.12)

Which in nondimensional form

55    (4.13)



57 ,

58    59      60    61


5 Results and Discussion

            This paper predicts the influence of couple stresses on the squeeze film characteristics of step bearings on the basis of Stokes couplestress fluid theory. The effect of couple stresses can be observed with the aid of nondimensional couplestress parameter62, where 63  which has the dimension of length and this length can be identified as the chain length of the polar additives in a non-polar lubricant. Hence the parameter, 64 provides the mechanism of the interaction of the lubricant with the bearing geometry.

5.1 Load carrying capacity

                  The variation of nondimensional load carrying capacity, 73 as a function of 80 for different values of couple stress parameter65 with K = 0.7 is as shown in the Fig. 2. The dashed curve in the graph corresponds to the Newtonian case. Compared with the Newtonian lubricant case, the effect of couple stresses increase the load carrying capacity and this increase in 74  is more accentuated for larger values of66. An increase of nearly 40% in 75  observed in the present study when 83 and85.  Figure 3 depicts the variation of nondimensional load carrying capacity, 76 as a function of 81 for different values of K for both Newtonian lubricants 87 and couplestress lubricants88. It is observed that, 77 increases for decreasing value of K and this increase in 78  is more pronounced for larger values of82. The relative percentage increase in79, 8990 for different values of K is given in the table 1

5.2 Time-height relationship

                      The most important characteristics of the squeeze film bearings is the squeeze film time i.e. the time required for reducing the initial film thickness 53 of 91 to a final value 51. Figure 4   shows the variation of the nondimensional time of approach 92 as a function of 99 for different values of67   with101,102. It is observed that, the presence of couple stresses provides an increase in the response time as compared to the Newtonian lubricant case. The relative increase in93103  108 is given in the Table 1 for various values of 68 and109. It is found that, an increase of nearly 20% in 94for 84 and86. The variation of  95 with  100 for different values of K for both Newtonian  111 and couplestress lubricants 112 is depicted in the Fig. 5. It is observed that,  96 increases for decreasing values of K. The variation of  97 with  113 for different values of the step height 114 with 116 is shown in the Fig. 6. It is observed that,  98 decreases for increasing values of nondimensional step height115.

Table 1: Values of 117 and 104 for different values of 69 and K with 120 121.






































6 Conclusions

              The squeeze film lubrication between parallel stepped bearings with couplestress fluid as lubricant is studied on the basis of Stokes microcontinuum theory for couplestress fluids. On the basis of the numerical computations of the results presented the following conclusions are drawn

  1. The effect of couple stresses enhances the load carrying capacity significantly.
  2. The relative increase in the load carrying capacity 119 is found to be a function of K.
  3. The relative squeeze film time 106 is found to be a function of 71 and K and 107 increases for increasing values of72  and decreases for increasing values of K.

            In view of the above the squeeze film characteristics between parallel stepped plates can be improved by the use of lubricants with microstructure additives.






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