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Short Communication

Squeeze ﬁlm lubrication between circular stepped plates: Rabinowitsch ﬂuid model N.B. Naduvinamani a,n, M. Rajashekar b, A.K. Kadadi a,b a b

Department of Mathematics, Gulbarga University, Gulbarga-585 106, Karnataka, India Department of Mathematics, Govt. PU. College for Girls, Raichur-584 101, Karnataka, India

art ic l e i nf o

a b s t r a c t

Article history: Received 24 September 2013 Received in revised form 31 December 2013 Accepted 5 January 2014 Available online 11 January 2014

In this paper, a theoretical analysis on the squeeze ﬁlm characteristics between circular stepped plates lubricated with Rabinowitsch ﬂuid is presented. By using Rabinowitsch ﬂuid model, the modiﬁed Reynolds type equation is derived to study the dilatant and pseudoplastic nature of the ﬂuid in comparison with Newtonian ﬂuid. The closed form solution is obtained by using perturbation method. According to the results obtained, the load-carrying capacity and squeeze ﬁlm time increases for dilatant ﬂuids as compared to the corresponding Newtonian ﬂuids whereas the reverse trend is observed for pseudoplastic ﬂuids. Further, it is observed that the response time decreases as the step height increases. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Rabinowitsch ﬂuid Squeeze-ﬁlm Circular stepped plates

1. Introduction The use of squeeze ﬁlm lubrication is observed in several applications such as clutches, breaks, gears, machine tools, etc. The squeeze ﬁlm bearings are found to be very useful because of squeezing action between the surfaces due to which high pressure and more load can be generated. Earlier, several researchers studied the squeeze ﬁlm bearings lubricated with Newtonian ﬂuids. Cameron [1] analyzed the squeeze ﬁlm lubrication between two inﬁnitely long parallel plates. The squeeze ﬁlm with Newtonian lubricants has been studied by Jackson [2]. Gupta and Gupta [3] investigated the squeezing ﬂow between parallel plates. The squeeze ﬂow of apparently lubricated thin ﬁlms was studied by Burbidge and Servais [4]. Bujurke et al. [5] analyzed the ﬂow of an incompressible ﬂuid between two parallel plates due to the normal motion of the plate. Nowadays, the use of Newtonian ﬂuids blended with various additives increases due to their effective improvement in the bearing characteristics as compared to the Newtonian lubricants. Due to the addition of additives to the Newtonian lubricant, the non-linear relationship exists between the shear stress and shear strain rate. There are several ﬂuid models to study the nonNewtonian properties of the lubricants such as power law, couple stress and micropolar ﬂuid model. The squeeze ﬁlm between ﬁnite plates lubricated by ﬂuids with couple stress was studied by Ramanaiah [6]. It is observed that, the squeeze time increases

n

Corresponding author. Tel.: þ 91 8472263296. E-mail address: [email protected] (N.B. Naduvinamani).

0301-679X/$ - see front matter & 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.triboint.2014.01.004

if a ﬂuid with couple stress is used as the lubricant. The squeeze ﬁlm lubrication between circular stepped plates of couple stress ﬂuids was studied by Naduvinamani [7]. It is found that the inﬂuence of couple stresses enhances the squeeze ﬁlm pressure, load-carrying capacity and decreases the response time as compared to classical Newtonian lubricant case. The squeeze ﬁlm lubrication between parallel stepped plates with couplestress ﬂuids was studied by Kashinath [8]. It is observed that, the loadcarrying capacity decreases as the step height increases. Shukla and Isa [9] investigated the characteristics of non-Newtonian power law lubricants in step bearings and hydrostatic step seals. Apart from these, Rabinowitsch ﬂuid model is also one such ﬂuid model which can be applied to analyze the non-linear behavior of non-Newtonian lubricants. In the Rabinowitsch ﬂuid model, the non-linear relationship between shear stress and shear strain rate can be described for one-dimensional ﬂuid ﬂow as follows τrz þ κτ3rz ¼ μ

∂u ∂z

ð1Þ

where μ denotes the zero shear rate viscosity and κ denotes the non-linear factor which describes the non-Newtonian effects of the lubricant. By using this ﬂuid model, one can analyze the Newtonian ﬂuids ðκ ¼ 0Þ, dilatant ﬂuids ðκ o 0Þ and pseudoplastic ﬂuids ðκ 4 0Þ. The theoretical analysis of this ﬂuid model agrees well with the experimental results conducted by Wada and Hayashi [10]. Determination of the load capacity of ﬁnite width journal bearing by ﬁnite element method in the case of a nonNewtonian lubricant was analyzed by Bouring and Gay [11]. Hashmimoto and Wada [12] studied the effects of ﬂuid inertia

N.B. Naduvinamani et al. / Tribology International 73 (2014) 78–82

R r; z t tn u; w

Nomenclature b H* h0 h1 h2 hs n hs KR p p1 p2

bearing width non-dimensional ﬁlm thickness (¼h1/h2) minimum ﬁlm thickness at time t¼0 maximum ﬁlm thickness minimum ﬁlm thickness step height step height ratio ( ¼hs/h0) the position of the step 0 o K o 1 pressure in the ﬁlm region ﬂuid ﬁlm pressure in the region 0 r r r KR ﬂuid ﬁlm pressure in the region KR rr r R

2. Mathematical formulation of the problem The physical conﬁguration of circular stepped plates approaching each other with a normal velocity Vð ¼ dh=dtÞ is as shown in Fig. 1. The lubricant in the ﬁlm region is considered as Rabinowitsch ﬂuid. The basic equations governing the ﬂow of an incompressible non-Newtonian Rabinowitsch ﬂuid under the assumptions of hydrodynamic lubrication for thin ﬁlm is given by 1 ∂ðruÞ ∂w þ ¼0 r ∂r ∂z

ð2Þ

V

∂p ∂τrz ¼ ∂r ∂z

ð3Þ

∂p ¼0 ∂z

ð4Þ

The relevant boundary conditions for velocity components are (i) At the upper surface z¼ h u¼0

and

w ¼ V

ð5aÞ

(ii) At the lower surface z ¼0 u¼0

and

w¼0

ð5bÞ

Integrating Eq. (3) with respect to z subject to the boundary conditions (5a) and (5b) and using constitutive Eq. (1), the expression for velocity component can be obtained as 1 1 gF 1 þ κg 3 F 2 u¼ ð6Þ μ 2 where 1 1 3 1 3 ∂p 2 F 1 ¼ zðz hÞ; F 2 ¼ z4 z3 h þ z2 h zh and g ¼ 4 2 8 8 ∂r Using Eq. (6) in continuity Eq. (2) and integrating with respect to z under the relevant boundary conditions (5a) and (5b) for z, the Reynolds type equation for non-Newtonian Rabinowitsch ﬂuid is obtained in the form 1∂ 3 5 ∂h 3 r h g þ κh g 3 ð7Þ ¼ 12μ r ∂r 20 ∂t As Eq. (7) is a non-linear equation in p, it is not easy to solve by using analytical methods. Therefore, the classical perturbation method is used to solve it. The perturbation series for p can be expressed in the form

R KR hs

p ¼ p00 þεp01 þ ε2 p02 þ ⋯ð*Þ Z

h2

radius of the circular plate radial and axial coordinates time of approach non-dimensional time of approach velocity components of lubricant in the r and zdirections, respectively velocity of approach load-carrying capacity non-dimensional load-carrying capacity dimensionless non-linear factor of lubricants initial viscosity of a Newtonian ﬂuid non-linear factor of lubricants shear stress component

V W Wn α μ κ τrz

forces in parallel circular squeeze ﬁlm bearing lubricated with pseudoplastic ﬂuids. Recently, several researchers have investigated the non-Newtonian effects of Rabinowitsch lubricants on various types of bearings. Lin et al. studied the non-Newtonian effects of Rabinowitsch ﬂuid model on the slider bearings [13], parallel annular disks [14] and parallel rectangular squeeze-ﬁlm plates [15], respectively. Variational principal for non-Newtonian lubrication of Rabinowitsch ﬂuid model analyzed by He [16]. Singh et al. investigated the effects of Rabinowitsch ﬂuid model on the hydrostatic thrust bearing [17], annular ring hydrostatic thrust bearing [18], a squeeze ﬁlm characteristics between a long cylinder and a ﬂat plate [19] and a sphere and a ﬂat plate [20]. In this paper, an attempt has been made to analyze the squeeze ﬁlm characteristics between circular stepped plates lubricated with Rabinowitsch ﬂuid which has not been studied so far.

79

h1

Film Region r

For ε 5 1, it is sufﬁcient, for analysis, to consider the ﬁrst order term in ε as follows: p ¼ p00 þεp01

ð8Þ

Solid Backing Fig. 1. Squeeze ﬁlm between circular stepped plates.

Substituting into the Reynolds type Eq. (7) and the two separated equation governing the squeeze ﬁlm pressure p00 and

80

N.B. Naduvinamani et al. / Tribology International 73 (2014) 78–82

p01 can be derived, respectively. 1d dh 3 dp00 rh ¼ 12μ r dr dt dr

ð9Þ

" 3 # 1 d 3 5 dp00 3 dp01 h ¼0 þh r dr 20 dr dr

ð10Þ

The modiﬁed Reynolds type equation for determining the squeeze ﬁlm pressure is obtained from Eqs. (9) and (10) in the form ð11Þ

dp01i 162 μV r 3 ¼ 5 h7 dr

ð12Þ

i

where pji ¼ p001 ¼ p011 ¼ p1 ;

hi ¼ h1

pji ¼ p002 ¼ p012 ¼ p2 ;

0 r r r KR

for

hi ¼ h2

for

KR r r r R

p1 and p2 being the pressure in the region-I ð0 r r r KRÞ and in the region-II ðKR rr r RÞ, respectively. The relevant boundary conditions for the pressure are

p2 ¼ 0

at at

3. Results and discussion The non-Newtonian effects on squeeze ﬁlm characteristics between circular stepped plates are investigated using a Rabinowitsch ﬂuid model. 3.1. Load carrying capacity

dp00i 6μVr ¼ 3 dr hi

p1 ¼ p2

In the limiting case of ðα ¼ 0Þ, Eqs. (17) and (19) reduce to their corresponding Newtonian case presented by Naduvinamani and Siddanagouda [7] (when the couplestress parameter tends to zero).

r ¼ KR

Fig. 2 shows the variation of non-dimensional load-carrying capacity W n with H n for different values of α. It is observed that, the maximum load is delivered for dilatant ﬂuids at small value of H n . As the height of the ﬂuid ﬁlm thickness decreases the loadcarrying capacity increases. The dotted curve represents the Newtonian case. The increase in W n is observed more for pseudoplastic ﬂuids as compared to the Newtonian ﬂuids whereas the reverse trend is observed for dilatant lubricants. An increase of nearly 4% is observed for dilatant ﬂuids than the Newtonian ﬂuids.

1.05

ð13aÞ

r¼R

= = = = =

1.00

ð13bÞ

0.95

The solution of Eqs. (11) and (12) subject to the boundary conditions (13a) and (13b) is " # " # K 2 R2 r 2 R2 ð1 K 2 Þ 81 r 4 K 4 R4 R4 ð1 K 4 Þ κμV p1 ¼ 3μV þ þ þ 7 7 3 3 10 h2 h1 h1 h2 " p2 ¼ 3μV

R2 r 2

#

3 h2

"

þ

81 R4 r 4 κμV 7 10 h2

0.90

W

*

0.85

ð14Þ

0.80

ð15Þ

0.75

#

0.70 1.0

The load carrying capacity W is obtained in the form Z R Z KR rp1 dr þ 2π rp2 dr W ¼ 2π 0

KR

which in the non-dimensional for is ( ) ( ) 3 2Wh2 K4 1 K4 18α K 6 1 K 6 ¼ þ þ Wn ¼ þ 7 3 5 1 1 3πμVR3 Hn Hn n

ð16Þ

1.6

1.8

2.0

2.2

2.4

2.6

2.8

Fig. 2. Variation of non-dimensional load-carrying capacity W n with H n for different values of α with K ¼ 0:7.

ð17Þ

1.0

2

0.9 0.8 0.7

W

= -0.005

*

= 0.0 [Newtonian]

0.6

= 0.005 K = 0.5 K = 0.7 K = 0.9

0.5

which in non-dimensional form is 2 Wh0 t 3

0.4

8μbL ) ( )# Z 1 "( K4 1 K4 18 K6 1 K6 n ¼ þ n3 þ n7 þ α dh2 n n n n n 5 ðh2 þ hs Þ7 ðh2 þhs Þ3 hf h2 h2

0.3 1.0

n

h2 ¼ h2 =h0 ;

n

hs ¼ hs =h0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

*

ð19Þ where hf ¼ hf =ho ;

1.4

*

ð18Þ

n

1.2

H

where H ¼ ðh1 =h2 Þ and α ¼ kðR=h2 Þ . Writing V ¼ ðdh2 =dtÞ in Eq. (17), the squeezing time for reducing the ﬁlm thickness from an initial value h0 of h2 to a ﬁnal value hf is given by ) ( )# Z h f "( 4 3μπR4 K 1 K4 18 2 K 6 1 K 6 κR ¼ t¼ þ þ þ dh2 7 7 3 3 5 2W h0 h1 h2 h1 h2

tn ¼

-0.01 -0.005 0.0 [Newtonian] 0.005 0.01

H

Fig. 3. Variation of non-dimensional load-carrying capacity W n with H n for different values of K.

N.B. Naduvinamani et al. / Tribology International 73 (2014) 78–82

5

3.5

= = = = =

4

= -0.005

-0.01 -0.005 0.0 0.005 0.01

= 0.0 [Newtonian]

= 0.005 K = 0.3 K = 0.5 K = 0.7

3.0 2.5

3

t

81

2.0

t

*

*

1.5

2

1.0

1

0 0.4

0.5

0.5

0.6

0.7

0.8

0.9

0.0 0.4

1.0

0.5

0.6

0.9

1.0

hf

*

n

n

Fig. 4. Variation of non-dimensional response time t n with hf for different values n of α with hs ¼ 0:15.

Fig. 5. Variation of non-dimensional response time t n with hf for different values n of K with hs ¼ 0:15.

3.5

Table 1 The variation of relative load (Rwn) and relative time (Rtn) for different values of α and K.

= -0.005

= 0.0 [Newtonian]

= 0.005

3.0

h = 0.0 h = 0.15

K

α

R wn

Rtn

0.5

0.01 0.005 0.005 0.01 0.01 0.005 0.005 0.01 0.01 0.005 0.005 0.01

3.715 1.858 1.868 3.726 3.897 1.955 1.943 3.897 3.491 1.746 1.746 3.491

25.628 12.811 12.818 25.628 25.698 12.849 12.849 25.698 20.887 10.448 10.439 20.877

0.9

0.8

*

hf

0.7

0.7

h = 0.2

2.5 2.0

t

*

1.5 1.0 0.5 0.0 0.4

0.5

0.6

The variation of non-dimensional load-carrying capacity W with H n for different values of K for α o 0; α ¼ 0 & α 4 0 is plotted in Fig. 3. It is found that, W n increases for decreasing values of K and H n , that is, for the ﬁxed R the increasing values of K leads to increase in the step region. This leads to increase in the over all ﬂuid ﬁlm region. Since the same amount of lubricant has to occupy the larger area leading to reduction in the load carrying capacity. For smaller values of K, the large amount of load is delivered in the circular step bearing for dilatant ﬂuids as compared to the Newtonian and pseudoplastic ﬂuids. 3.2. Time-height relationship The squeeze ﬁlm time is an important characteristics of any squeeze ﬁlm bearing. The variation of non-dimensional squeezen ﬁlm time t n as a function of hf for different values of α is depicted in Fig. 4. It is observed that, the response time t n increases for n decreasing ﬁlm thickness hf . An increase in t n is more accentuated for dilatant ﬂuids than the corresponding Newtonian ﬂuids whereas the reverse trend is observed for pseudoplastic ﬂuids. A relative increase in t n , Rtn ¼ ððt nRabinof luid 7t nNewtonian =t nNewtonian Þ 100Þ is given in Table 1. It is found that, as compared to the Newtonian ﬂuids, an increase of 25% is observed for dilatant ﬂuids. The n variation of t n as a function of hf for different values of K is plotted in Fig. 5. It is found that, as the ﬂuid ﬁlm thickness decreases, the squeeze ﬁlm time of circular stepped plates approaching each other increases. Also, the response time t n

0.7

0.8

0.9

1.0

*

hf

n

n

Fig. 6. Variation of non-dimensional response time t n with hf for different values n of hs with K ¼ 0:7.

increases for decreasing values of K. Fig. 6 shows the variation of n non-dimensional response time t n as a function of hf for different n values of step height ratiohs . It is observed that, the approaching time of plate-plate contact can be increased by decreasing the ratio of step height and minimum ﬁlm thickness at initial time. The response time t n increases as the step height ratio of the ﬂuid n ﬁlm thickness hs decreases. The increase in t n is observed more for dilatant ﬂuids then Newtonian ﬂuids whereas the reverse trend is observed for pseudoplastic ﬂuids. 4. Design example For the illustration of engineering design application, the following numerical example is considered. Physical parameter

Notation Range of values chosen

Radius of the circular plate Initial viscosity of a Newtonian ﬂuid

R

10 mm

μ

500 cp

82

Minimum ﬁlm thickness at time t¼0 Minimum ﬁlm thickness Step height

N.B. Naduvinamani et al. / Tribology International 73 (2014) 78–82

h0

0.05 mm

h2

0.05 mm

hs

(Archana K.K) sincerely acknowledges the ﬁnancial assistance from the University Grants Commission, New Delhi under UGCBSR Fellowship for meritorious students.

0 mm, 7:5 10 3 mm, 10 10 3 mm

References

2:5 10 7 ,

Non-linear factor of lubricants

κ

Dimensionless nonlinear factor of lubricants

α

1:25 10 7 ,0, 2:5 10

7

, 1:25 10

7

0.01, 0.005, 0, 0.005, 0.01

By using the deﬁnitions of various non-dimensional parameters one can obtain all the parametric values used in calculations of the squeeze ﬁlm characteristics and presented in Figs. 2–6. 5. Conclusions On the basis of Rabinowitsch ﬂuid model, this paper predicts the non-Newtonian effects on squeeze ﬁlm lubrication between circular stepped plates using a dimensionless parameter α, which accounts for the non-Newtonian nature of the lubricant. The nature of the lubricant is dilatant for α o 0, Newtonian for the parameter α ¼ 0 and pseudoplastic for α 4 0. According to the results obtained, the following conclusions can be drawn: 1. Dilatant ﬂuids signiﬁcantly enhance the load-carrying capacity and response time as compared to the Newtonian ﬂuids, whereas the reverse trend is observed for pseudoplastic ﬂuids. 2. On comparing with Newtonian ﬂuids, the response time increases for dilatant ﬂuids whereas it decreases for pseudoplastic ﬂuids. 3. The relative load-carrying capacity, RW n , and the relative response time, Rtn , are found to be a function of K and α. 4. The relative load-carrying capacity, RW n , and the relative response time, Rtn , increases for decreasing values of α and K. For dilatant ﬂuids, 4% increase in RW n and 25% increase in Rt n is observed in comparison with corresponding pseudoplastic ﬂuids.

Acknowledgements Authors are thankful to the reviewers for their valuable comments on the earlier draft of the paper. One of the authors

[1] Cameron A. Basic lubrication theory. New York: Wiley; 1981. [2] Jackson JD. A study of squeezing ﬂow. Appl Sci Res Sec A 1963;11:148–52. [3] Gupta PS, Gupta AS. Squeezing ﬂow between parallel plates. Wear 1977;45 (2):177–85. [4] Burbidge AS, Servais C. Squeeze ﬂows of apparently lubricated thin ﬁlms. J Non-Newtonian Fluid Mech 2004;124(1–3):115–27. [5] Bujurke NM, Achar PK, Pai NP. Computer extended series for squeezing ﬂow between plates. Fluid Dyn Res 1995;16(2–3):173–87. [6] Ramanaiah G. Squeeze ﬁlms between ﬁnite plates lubricated by ﬂuids with couple stress. Wear 1979;54(2):315–20. [7] Naduvinamani NB, Siddanagouda A. Squeeze ﬁlm lubrication between circular stepped plates of couple stress ﬂuids. J Braz Soc Mech Sci Eng 2009;XXXI (1):21–6. [8] Kashinath B. Squeeze ﬁlm lubrication between parallel stepped plates with couplestress ﬂuids. Int J Stat Math 2012;3(2):65–9. [9] Shukla JB, Isa M. Characteristics of non-Newtonian power law lubricants in step bearings and hydrostatic step seals. Wear 1974;30:51–71. [10] Wada S, Hayashi H. Hydrodynamic lubrication of journal bearings by pseudoplastic lubricants, (Part-2: Experimental studies),. Bull JSME 1971;14:279–86. [11] Bourging P, Gay B. Determination of the load capacity of ﬁnite width journal bearings by ﬁnite element method in the case of a non-Newtonian lubricant. ASME J Tribol 1984;106:285–90. [12] Hashimoto H, Wada S. The effects of ﬂuid inertia forces in parallel circular squeeze ﬁlm bearings lubricated with pseudoplastic ﬂuids. J Tribol 1986;108:282–7. [13] Lin JR. Non-Newtonian effects on the dynamic characteristics of one-dimensional slider bearings: Rabinowitsch ﬂuid model. Tribol Lett 2001;10:237–43. [14] Lin JR. Non-Newtonian squeeze ﬁlm characteristics between parallel annular disks: Rabinowitsch ﬂuid model. Tribol Int 2012;52:190–4. [15] Lin JR, Hung CR, Chu LM, Liaw WL, Lee PH. Effects of non-Newtonian Rabinowitsch ﬂuids in wide parallel rectangular squeeze-ﬁlm plates. Ind Lubr Tribol 2013;65 (Online Date 27/6/2012). [16] He JH. Variational principle for non-Newtonian lubrication: Rabinowitsch ﬂuid model. Appl Math Comput 2004;157:281–6. [17] Singh UP, Gupta RS, Kapur VK. On the steady performance hydrostatic thrust bearing: Rabinowitsch ﬂuid model. Tribol Trans 2011;54:723–9. [18] Singh UP, Gupta RS, Kapur VK. On the application of Rabinowitsch ﬂuid model on an annular ring hydrostatic thrust bearing. Tribol Int 2013;58:65–70. [19] Singh UP, Gupta RS, Kapur VK. On the squeeze ﬁlm characteristics between a long cylinder and a ﬂat plate: Rabinowitsch ﬂuid model. Proc Inst Mech Eng Part J J Eng Tribol 2012. [20] Singh UP, Gupta RS. Non-Newtonian effects on the squeeze ﬁlm characteristics between a sphere and a ﬂat plate: Rabinowitsch ﬂuid model. Adv Tribol 2012;2012:1–7.

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