Ferrofluid lubrication in porous inclined slider bearing with velocity slip

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International Journal of Mechanical Sciences 44 (2002) 2495 – 2502

Ferrouid lubrication in porous inclined slider bearing with velocity slip Rajesh C. Shaha;∗ , M.V. Bhatb a

Department of Mathematics, Nirma Institute of Technology, Gandhinagar-Serkhej Highway, Post. Chandlodia, Ahmedabad-382 481, Gujarat State, India b E-202, Riddhi Complex, Near Jodhpur Village, Ahmedabad-380 015, Gujarat State, India

Received 7 March 2002; received in revised form 14 November 2002; accepted 26 November 2002

Abstract The e1ects of slip velocity and the material constant in a porous inclined slider bearing lubricated with a ferrouid were theoretically studied by using Jenkins model. Expressions were obtained for pressure, load capacity, friction on the slider, coe5cient of friction and position of the centre of pressure. The increase in slip parameter caused decrease in load capacity as well as friction and increase in the coe5cient of friction without altering the centre of pressure much. As the material constant increased, the load capacity decreased, friction and coe5cient of friction increased and the position of the centre of pressure shifted slightly towards the inlet of the bearing. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Ferrouid lubrication; Porous; Inclined slider; Jenkins model; Slip

1. Introduction Prakash and Vij [1] investigated a porous inclined slider bearing and found that porosity caused decrease in the load capacity and friction, while it increased the coe5cient of friction. Gupta and Bhat [2] found that the load capacity and friction could be increased by using a transverse magnetic =eld on the bearing and a conducting lubricant. With the advent of ferrouid lubricant, Agrawal [3] studied its e1ects on a porous inclined bearing and found that the magnetization of the magnetic particles in the lubricant increased its load capacity without a1ecting the friction on the moving slider. Recently, Ram and Verma [4] extended the analysis [3] using Jenkins model. They studied only the load capacity which increased with increasing magnetization of the uid and the material constant of the model. ∗

Corresponding author.

0020-7403/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0020-7403(02)00187-X

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Nomenclature a A B C E f F G h h1 h2 H∗ k K p P Q s u U W x; y; z X  0 ∗ fG FG hG HG PG sG uG wG WG XG YG H G 

h2 =h1 bearing length bearing breadth constant of integration de=ned in Eq. (3) coe5cient of friction friction on the slider de=ned in Eq. (2) =lm thickness outlet =lm thickness inlet =lm thickness thickness of porous matrix uid permeability de=ned in Eq. (A.13) =lm pressure =lm pressure in porous matrix constant of integration slip constant x-component of the velocity of =lm uid velocity of slider load capacity coordinates x=A uid viscosity free space permeability de=ned in Eq. (A.17) de=ned in Eq. (A.17) de=ned in Eq. (11) de=ned in Eq. (10) de=ned in Eq. (7) applied magnetic =eld de=ned in Eq. (A.17) sh1 x—component of the velocity of uid in porous matrix z—component of the velocity of uid in porous matrix de=ned in Eq. (9) x coordinate of the centre of pressure XG =A magnitude of HG magnetic susceptibility inclination of HG with the x-axis

R.C. Shah, M.V. Bhat / International Journal of Mechanical Sciences 44 (2002) 2495 – 2502 2

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de=ned in Eq. (A.17) z-component of the velocity of =lm uid value of w when z = 0 value of w when z = h material constant of Jenkins model de=ned in Eq. (A.17) uid density

w w0 wh !2 "∗ #

All the above investigations assumed that there was no slip at the interface of =lm and the porous matrix. Beavers et al. [5] found that the no slip condition was not true at the boundary of a naturally permeable material. The no slip condition might fail when the bearing material was made of foam or soft metal. In the present paper we consider the lubrication of porous inclined slider bearing with a ferrouid lubricant considering slip velocity at the interface of porous and =lm regions using Jenkins ow behaviour which takes into account a material constant. 2. Analysis The bearing shown in Fig. 1 consists of a ferrouid =lm of thickness h within an inclined slider bearing of length A in the x-direction and width B in the y-direction, AB. The value of h is h2 at the inlet and h1 at the outlet. The slider is impermeable and moves with a uniform velocity U in the x-direction. The stator has a porous matrix of uniform thickness H ∗ and k is the uid permeability there. The porous matrix is backed by a solid wall. The Eq. (A.18) giving the dimensionless =lm pressure pG can be written as
  d 1 d dE G pG − ∗ X (1 − X ) = ; (1) dX dX 2 dX where

 G − "∗ 2 sGhG2 X (1 − X ) hG3 (4 + sGh)  G = 12$ + G (1 + sGh)(1 − 2 X (1 − X ))

(2)

Z U h2

h

H h1

Y O

H*

Porous matrix A

Fig. 1. Porous inclined slider bearing.

X

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and

 G + sGh) G − 2 2 s" 6h(2 G ∗ X (1 − X ) E= : 1 + sGhG Solving Eq. (1) under the boundary conditions p(0) G = p(1) G = 0; we obtain 1 pG = ∗ X (1 − X ) + 2 where

(3)

(4)  1

X

E−Q dX G

(5)

1

(E=G) dX : Q = 01 (1=G) dX 0

(6)

For an inclined slider bearing h = h2 − (h2 − h1 )x=A;

hG = a − (a − 1)X;

(7)

where a = h2 =h1 :

(8)

The load carrying capacity W , friction on the moving slider F, coe5cient of friction f and position of the centre of pressure XG of the bearing can be expressed in dimensionless forms as  1 E−Q h21 W ∗ G X W= = − dX; (9) 2 UA B 12 G 0  1 G + sGh)(E G s G h(2 − Q) F h 1  = + dX; (10) FG = G UAB 1 + sGhG 2G(1 + sGh)(1 − 2 X (1 − X )) 0 Af FG fG = = ; h1 WG
  XG 1 ∗ 1 1 2 E − Q YG = = X − dX : A G WG 24 2 0

(11) (12)

3. Results and discussion G fG and YG are computed using Simpson’s one third rule Values of the bearing characteristics WG , F, with step size 0.1 with an accuracy of three signi=cant digits. They are displayed in Tables 1–4. It is seen from the Tables that WG as well as FG decrease, fG increases and YG is not altered much when the slip parameter 1= sG increases. But, WG decreases, FG and fG increase and the position YG of the centre of pressure shifts towards the inlet of the bearing when the material constant 2 increases. The present analysis reduces to the no slip case [4] which gives the value of WG only, when 1= sG → 0.

R.C. Shah, M.V. Bhat / International Journal of Mechanical Sciences 44 (2002) 2495 – 2502 Table 1 Values of dimensionless load capacity WG for various values of the slip parameter 1= sG and the material constant 2 2

1= sG

0.02

0.2

0.4

0.8

1.6

0.02 0.03 0.04

0.2346 0.2318 0.2291

0.2275 0.2248 0.2223

0.2195 0.2169 0.2145

0.2026 0.2003 0.1982

0.1646 0.1631 0.1616

Table 2 Values of dimensionless friction FG for various values of 1= sG and

2

2

1= sG

0.02

0.2

0.4

0.8

1.6

0.02 0.03 0.04

0.7599 0.7536 0.7476

0.7649 0.7586 0.7524

0.7713 0.7649 0.7586

0.7874 0.7806 0.7740

0.8494 0.8415 0.8338

Table 3 Values of dimensionless coe5cient of friction fG for various values of 1= sG and

2

2

1= sG

0.02

0.2

0.4

0.8

1.6

0.02 0.03 0.04

3.239 3.252 3.263

3.362 3.374 3.385

3.514 3.526 3.537

3.886 3.897 3.906

5.160 5.160 5.159

Table 4 Values of dimensionless position of centre of pressure YG for various values of 1= sG and 2 2

1= sG

0.02

0.2

0.4

0.8

1.6

0.02 0.03 0.04

0.5437 0.5432 0.5427

0.5426 0.5421 0.5416

0.5412 0.5407 0.5402

0.5378 0.5373 0.5368

0.5254 0.5250 0.5247

= 0:001, ∗ = 1:0, "∗ = 0:3, a = 2.

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4. Conclusions Magnetization of the uid particles increased the load capacity without altering the friction, but the slip velocity decreased them. However, the load capacity decreased and friction increased owing to the material constant of Jenkins model. We think that the above model is more realistic because we consider the material used for bearing design in this model. Acknowledgements The authors are grateful to the referees for their valuable suggestions. Appendix Following Ram and Verma [4] the governing Eq. is

 @2 u @ 0 H G 2 1 p− ; = @z 2 (1 − #!2 H=2) G @x 2

(A.1)

where u is the velocity of the uid in the =lm region in the x-direction,  is the uid viscosity, # is the uid density, !2 is the material constant, G is the magnetic susceptibility, H is the magnitude of the applied =eld, p is the pressure of the uid in the =lm region and 0 is the free space permeability. Solving Eq. (A.1) under the slip boundary conditions of Sparrow et al. [6]

 1 @u u = U when z = h; u = when z = 0; (A.2) s @z z=0 1=s being the slip parameter, we obtain 1 + sz {z(1 + sh) + h}(z − h) @ u= U+ 1 + sh (1 + sh)2(1 − #!2 H=2) G @x



0 H G 2 p− 2

 :

Substituting the above value of u in the integral form of the continuity equation  h @ u d z + wh − w0 = 0; @x 0 w being the axial component of the uid velocity in the =lm, we have


 h3 (4 + sh) d 0 H d Uh(2 + sh) G 2 − p− = w0 d x 2(1 + sh) 12(1 + sh)(1 − #!2 H=2) G dx 2

(A.3)

(A.4)

(A.5)

remembering that wh = 0 since the slider is non-porous. The velocity components of the uid in the porous matrix are given by Ram and Verma [4]




 0 H #!2 @ @u k @ G 2 P− + G H ; (A.6) uG = −  @x 2 2 @z @z

R.C. Shah, M.V. Bhat / International Journal of Mechanical Sciences 44 (2002) 2495 – 2502






 0 H #!2 @ @u k @ G 2 P− − H ; wG = − G  @z 2 2 @x @z

2501

(A.7)

k being the uid permeability and P the pressure there. The continuity equation of the uid in the porous region yields  



@2 @2 0 H 0 H G 2 G 2 + 2 P− =0 (A.8) P− @x2 2 @z 2 whose integration across the porous matrix (−H ∗ ; 0) gives

 

2 0 H @ 0 H G 2 G 2 ∗ d P− ; = −H d x2 p − 2 @z 2 z=0

(A.9)

using Morgan–Cameron approximation [7] that when H ∗ is small, the pressure in the porous region can be replaced by the average pressure with respect to the bearing wall thickness and which was used by Prakash and Vij [1] and remembering that the surface z = −H ∗ is non-porous. Owing to the continuity of normal velocity components across the porous-=lm interface


    @ 1 @u k 1 2 @ 2 P − 0 H #! G H w0 = wG 0 = − ; (A.10) G −  @z 2 2 @x @z z=0 z=0 using Eq. (A.7). Using Eqs. (A.3), (A.9) and (A.10), Eq. (A.5) yields
 

 1 G 2 H )= d h3 (4 + sh) − (3#!2 ksh d ∗ 2 12kH + p − 0 H G dx (1 + sh)(1 − #!2 H=2) G dx 2
 d Uh(2 + sh) − (#!2 ksUH G )= = 6 dx 1 + sh

(A.11)

which is the Reynolds-type equation in this case. We take a magnetic =eld vanishing at the inlet and outlet of the bearing and inclined at an angle  with the x-axis. Thus HG = H (x)(cos ; 0; sin );

 = (x; z)

(A.12)

and H 2 = Kx(A − x);

(A.13)

K being a quantity chosen to suit the dimensions of both sides of Eq. (A.13). Such a magnetic =eld attains a maximum at the middle of the bearing producing magnetic pressure. On the other hand a constant =eld cannot produce magnetic pressure because (d=d x)H 2 = 0. The direction of the magnetic =eld is signi=cant since HG has to satisfy the equations ∇ • HG = 0;

∇ × HG = 0;

(A.14)

so HG arises out of a potential and  satis=es the equation cot 

A − 2x @ @ + =− @x @z 2(Ax − x2 )

(A.15)

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whose solution is determined from equations cosec2  = C 2 (Ax − x2 ); C(2x − A) = [C 2 A2 − 4 sin(Cz)]1=2 ; C being an arbitrary constant. Introducing the dimensionless quantities x kH ∗ X = ; $= 3 ; A h1 2 KAh G  0 1 ; "∗ = ∗ = U

h hG = ; h1 6k h21

sG = sh1 ;

2

√ #!2 G KA = ; 2

(A.16)

pG =

h21 p ; UA

Eq. (A.11) transforms to     G − "∗ 2 sGhG2 X (1 − X ) d  hG3 (4 + sGh) 1 ∗ d  12$ + pG −  X (1 − X ) G dX dx 2 (1 + sGh)(1 − 2 X (1 − X ))   G + sGh) G − 2 2 s" G ∗ X (1 − X ) d 6h(2 : = dX 1 + sGhG

(A.17)

(A.18)

References [1] Prakash J, Vij SK. Hydrodynamic lubrication of a porous slider. Journal of Mechanical Engineering Science 1973; 15:232–4. [2] Gupta JL, Bhat MV. An inclined porous slider bearing with a transverse magnetic =eld. Wear 1979;55:359–67. [3] Agrawal VK. Magnetic uid based porous inclined slider bearing. Wear 1986;107:133–9. [4] Ram P, Verma PDS. Ferrouid lubrication in porous inclined slider bearing. Indian Journal of Pure and Applied Mathematics 1999;30(12):1273–81. [5] Beavers GS, Joseph DD. Boundary conditions at a naturally permeable wall. Journal of Fluid Mechanics 1967;30: 197–207. [6] Sparrow EM, Beavers GS, Hwang IT. E1ect of velocity slip on porous walled squeeze =lms. Journal of Lubrication Technology 1972;94:260–5. [7] Morgan VT, Cameron A. Mechanism of lubrication in porous metal bearing. Proceedings of the Conference of Lubrication and Wear, Institute of Mechanical Engineers, London, Paper 89, 1957. p. 151–7.