Squeeze film lubrication between circular stepped plates: Rabinowitsch fluid model

Squeeze film lubrication between circular stepped plates: Rabinowitsch fluid model

Tribology International 73 (2014) 78–82 Contents lists available at ScienceDirect Tribology International journal homepage: www.elsevier.com/locate/...

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Tribology International 73 (2014) 78–82

Contents lists available at ScienceDirect

Tribology International journal homepage: www.elsevier.com/locate/triboint

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

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