SURFACE ROUGHNESS EFFECTS ON SQUEEZE FILM BEHAVIOR IN POROUS TRANSVERSELY TRIANGULAR PLATES WITH...

8/2/2019 SURFACE ROUGHNESS EFFECTS ON SQUEEZE FILM BEHAVIOR IN POROUS TRANSVERSELY TRIANGULAR PLATES WIT

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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976

6340(Print), ISSN 0976 6359(Online) Volume 3, Issue 2, May-August (2012), IAEME

1

SURFACE ROUGHNESS EFFECTS ON SQUEEZE FILM

BEHAVIOR IN POROUS TRANSVERSELY TRIANGULAR PLATES

WITH COUPLE STRESS FLUID

Sundarammal Kesavan & Santhana Krishnan N

Department of Mathematics, SRM University, Kattankulathur, Tamilnadu, IndiaEmail : [email protected]

ABSTRACT

The effect of surface roughness on squeeze film behavior between two transversely

triangular plates with couple stress lubricant is analyzed when the upper plate has porousfacing which approaches the lower plate with uniform velocity. The modified Stochastic

Reynolds equation is derived on the basis of Stokes micro-continuum theory for couplestress fluid and Christensen Stochastic theory for the rough surfaces. Closed formsolution of the Stochastic Reynolds equation is obtained in terms of Fourier-Bessel series.

It is observed that, effect of couple stress fluid and surface roughness is more pronounced

compared to classical case.

Keywords: Couple stress fluid, Rough surface, Squeeze film, Triangular plates, Fourier

Bessel series

Nomenclature

a Length of an equilateral triangular plate

E Expectancy operator

f Probability density function

h Nominal film height

hs Deviation of film height from nominal level

H Film thickness

H* Film thickness of the porous layer

Jo Bessel function of first kind of zeroth order

l Couple stress parameter ( =

INTERNATIONAL JOURNAL OF MECHANICAL

ENGINEERING AND TECHNOLOGY (IJMET)

ISSN 0976 6340 (Print)ISSN 0976 6359 (Online)

Volume 3, Issue 2, May-August (2012), pp. 01-12

IAEME: www.iaeme.com/ijmet.html

Journal Impact Factor (2011):1.2083 (Calculated by GISI)www.jifactor.com

IJMET

I A E M E


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p Pressure in the film region

p* Pressure in the porous region

T Non-dimensional squeeze film time

u,w Velocity components in r and z-directions respectively

u*,w

* Darcys velocity components in the porous region

Modified form of the Darcys law

Greek symbols

Nth

eigen value

Ratio of microstructure size of polar additives to the pore size

Viscosity of the lubricant

Material constant

Non-dimensional couple stress parameter

Random variable

Angular co-ordinatePermeability parameter

Non-dimensional permeability parameter

1 INTRODUCTION

The determination of squeeze film characteristics has attracted the attention of several

investigators due to its importance in the practical problems of improving the

performance of hydraulic machine elements. This discovery, that a film of lubrication

could completely separate two surfaces resulted in a complete reversal in understandingthe limitation of bearing design and has had a profound effect on the design, operation

and life expectancy of machinery. The squeeze film phenomenon is found when theupper surface approaches the lower surface with a normal velocity. The viscous lubricantpresent in the film region cannot be squeezed out instantaneously, so it offers resistance

to extrusion. This results in build-up of pressure, which support load. Self-lubricating

porous bearings are widely used in industry due to their self-contained oil reservoir inaddition to their low cost and other aspects concerned with lubrication mechanism.

Porous bearing are extensively used in brakes, clutches, etc. due to their self-contained oil

reservoir and favorable low friction characteristics. Besides engineering applications,their detailed study is of immense use in understanding lubrication aspects of synovial

joints.

The analysis of squeeze film bearings is made by many researchers for Newtonianlubricants based on the assumption of perfectly smooth bearing surfaces. However, it is

well known in the tribology literature that the bearing surfaces develop roughness after

having some run-in and wear. The chemical degradation of lubricants leading to thecontamination of lubricants is also one of the plausible reasons for developing the

roughness on bearing surfaces in some cases. Since the surface roughness distribution is

random in nature, a stochastic approach to model the surface roughness mathematicallyhas to be adopted. Several approaches have been proposed to study the effect of


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roughness of the bearing surfaces on the performance of hydrodynamics bearings.

Ramanaiah [1] analyzed the Squeeze films between finite plates lubricated by fluids with

couple stresses. Ramanaiah and Sarkar [2] studied the performance between squeeze filmplates and proved that the couple stress effects increase the load-carrying capacity and the

time of approach of the plates. Ramanaiah and Sundarammal Kesavan [3] studied the

effects of bearing deformation on the characteristics of a slider bearings and Ramanaiahand Sundarammal Kesavan [4] investigated squeeze films between circular andrectangular plates.

All the investigations mentioned above are confined to the study of surface roughness onporous bearing with Newtonian and non-Newtonian fluid as lubricant. However, with the

development of modern industry, the importance of non-Newtonian fluids as lubricant

has been emphasized, as Newtonian constitutive approximation is not a satisfactory

engineering approach for most of the lubrication problems. Hence, the effect of non-Newtonian property of lubricant must be taken into account in the realistic study of these

bearing. The common lubricants exhibiting non-Newtonian behavior are polymer-

thickened oils, greases, and natural lubricating fluids, which appear in animal joints.

Recently, the effect of transverse surface roughness on the performance of hydromagnetic

squeeze film between conducting rough porous plates of different geometricalconfigurations was analyzed by many researchers. Sudha and Sundarammal Kesavan [5]

analysed the squeeze film characteristics of couple stress fluid between the porous

triangular plates. Bujurke et al. [6] have investigated the surface roughness effects on

squeeze film behavior in porous circular disks with couple stress fluid. Here, it has beensought to analyze the performance of a couple stress fluid based squeeze film behavior

between transversely rough porous triangular plates.

2 MATHEMATICAL FORMULATION OF THE PROBLEM

A schematic diagram of squeeze film geometry of the problem considered is shown infigure 1. The squeeze film characteristics between two triangular plates are analyzed

when one triangular plate has a porous facing which approaches the other rough

triangular plate with uniform velocity. The lubrication in the film region is assumed to bea stokes couple stress fluid. It is also assumed that, the body forces and body couples are

absent. The bearing surfaces are assumed to be transversely rough.

Figure 1 Configuration of the Squeeze Film bearing system


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Under the usual assumption of hydrodynamic lubrication, applicable to thin films, the

equation of motion for couple stress fluid [7] takes the forms

(1)

(2)

(3)

where u and w are the velocity components in r and z directions respectively, p is the

pressure, is the Newtonian viscosity, is the new material constant characterizing the

couple stress and is of dimension of momentum. The ratio has the dimension of

length squared and hence characterizes the material length of the fluid. The film thickness

of the lubricant film geometry is

(4)

where h denotes the nominal smooth part of the film geometry, while is the part due to

the surface asperities measured from the nominal level and is regarded as a randomly

varying quantity of zero mean, rand are the radial and angular coordinates and is the

index parameter determining a definite roughness arrangement.

The relevant boundary conditions for the velocity components are

u=0, w=0, atz = 0 (5)

u=0, w = + , atz = H (6)

Here is the constant velocity of the upper porous triangular plate approaching the

lower impermeable triangular plate. Last conditions in equation (5) and (6) are due to thevanishing of the couple stresses atz = 0 and Hrespectively.

Since, p is independent ofz, the solution of equation (1), subject to the boundaryconditions (5) and (6) is


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Taking , , , we get

(7)

where

Integration of equation (3) across the fluid film and using the boundary conditions (5) and

(6) and the expression (7) for u, gives the modified Reynolds equation

On simplifying, we get

(8)

where

The flow of couple stress fluid in a porous medium is governed by the modified form ofthe Darcys law, which accounts for the [8] polar effects

(9)

where are the Darcys velocity components along r and z

directions respectively, is the pressure in the porous region, and is the

permeability of the porous medium. The parameter represents the ratio of

microstructure size of polar additives to the pore size of the porous medium. If

then the microstructure additives present in the Newtonian fluid

block the pores of the porous region and this reduces the Darcy flow through the porous

matrix. When microstructure is very small compared to the porous size, i.e., the

polar additives percolate in to the porous matrix.

The pressurep* in the porous region satisfies the Laplace equation


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(10)

Using equation (9) in (8), we get

(11)

Taking the expected values of both sides of equation (11), we get

(12)

where expectancy operatorE(.) is defined by

(13)

andfis the probability density function of the stochastic film thickness hs. In many real

engineering problems, sliding surfaces show a roughness in height distribution which isGaussian in nature. Therefore, a polynomial form which approximates the Gaussian is

chosen in the analysis. Such a probability density function [9] is given by

(14)

where c is the half total range of random film thickness variable and function terminates

at c = with being the standard deviation.

In the context of stochastic theory, the following type of directional roughness structuresis of special interest.

2.1 Sides of one-dimensional roughness

In this model, the roughness is assumed to have the form of long narrow ridges and

valleys in r-direction. The film thickness assumes the form

(15)

and the equation (12) reduces to

(16)


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The relevant boundary conditions for pressure in the film region are

(17)

(18)

(19)

(20)

(21)

and the interface condition is

(22)

where a is the side of an equilateral triangle and H* is the thickness of the porous layer.

Condition (18) and (19) show that both the film region and the porous facing are open to

ambient pressure. There is no flow through the impervious boundary at the top of the

porous medium (condition 20). Pressure continuity at the film plate interface requirescondition (22). The problem is thus reduced to solving equation (16) and (10) with

boundary conditions (17) and (18).

Using equation (13) for the distribution function, we have

(23)

The cylindrical coordinates are appropriate to the present problem, with the pole at the

centroid of the triangular plates and the z-axis along the axis of a triangular plates. The

pressure p* in the porous region does not depend on which satisfies the Laplace

Equation

i.e.,

The solution of the above equation of the form u = RZ, whereR is a function ofronly

andZis a function ofz only is

(24)

Putting each side equal to we have


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(25)

(or) the equivalent form

In each case, A,B,C,D denote arbitrary constants. We must suppose for

to be real when r>0. are Bessel functions of first kind and zeroth order and is

the nth

eigen value which satisfies

(26)

Applying boundary condition (19) at r = a, the most general solution is

(27)

Also,

Applying condition (20) and (21),

, (0


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(30)

Again Integrating equation (30) with respect to r and making use of the boundary

condition (18), the mean pressure in the film region is

(31)

Substituting equations (28) and (31) in the interface condition (22) and using

orthogonality of the eigen function , we get

(32)

3 SOLUTION OF THE PROBLEM

Let the vertices of the equilateral triangular plates be (-2a, 0), (a, -a ) and (a, a )

where 2 a > 0 is the length of each side of the equilateral triangle. The equation

governing the film pressure is obtained as

(33)

where in , k being suitably chosen

constant, from dimensionless point of view and

and is the permeability of the

free space. Integration of equation (33) with respect to the appropriate boundary

conditions

(34)

where in and the introduction of the

following non-dimensional quantities

and

leads to the non-dimensional pressure distribution


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(35)

Then the load carrying capacity given by is obtained in

dimensionless form as

(36)

The dimensionless time T for a given load taken by the upper plate to reach a film

thickness from an initial film thickness can be obtained as

(37)

where

4 RESULTS AND DISCUSSIONS

Figures 2-5 shows the variation of non-dimensional squeeze film pressure as a function

of dimensionless coordinates X(r) and Y(z) for different values of a with =0.9, =0.9,

=2.9. This reveal that for a given value of the dimensionless height a, the dimensionlesspressure for a couple stress fluid increases.

Figure 2. Variation of distribution of pressure for

=0.9, =0.9, =2.9, a=0.5

Figure 3. Variation of distribution of pressure

for =0.9, =0.9, =2.9, a=0.55


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Figure 4. Variation of distribution of pressure for=0.9, =0.9, =2.9, a=0.7

Figure 5. Variation of distribution of pressure for=0.9, =0.9, =2.9, a=1.0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

Dimensionless Height H

DimensionlessTimeT

A. = 0.2

A. = 0.3

A. = 0.4

A. = 0.5

A. = 0.6

A. = 0.7

Behaviour of Couple Stress Fluid

with different parameters of A.

A. = 0.7

A. = 0.2

Figure 6. Behavior of couple stress fluid with

different parameters of

Behaviour of Newtonian Fluid

0

0.5

1

1.5

2

0.20 0.25 0.30 0.35 0.40 0.45 0.50

Dimensionless Height (H)

DimensionlessTime(T)

Newtonian Fluid

Figure 7. Behavior of Newtonian fluid with

parameters of

The values of the dimensionless Time Tare compared for the couple stress fluid and theNewtonian fluid. Figure 6 and Figure 7 reveal that for a given value of the

dimensionless height H, the dimensionless time Tfor a couple stress fluid is larger than

the Newtonian fluid. The dimensionless time T increases as the couple stress parameterincreases for any given value ofH.


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5 CONCLUSION

From the above discussions it is observed that for a given value of the dimensionlessheight a, the dimensionless pressure for a couple stress fluid increases. Also, the squeeze

in couple stress fluid is slower than in Newtonian fluid films. The time of approach T

increases with the dimensionless parameter showing that couple stress fluids are

better lubricants than Newtonian fluids. The presence of microstructures in the fluid film

causes an enhancement of the squeeze film characteristics.

ACKNOWLEDGEMENTS

Authors would like to thank the anonymous referees for their comments.

REFERENCES

1. Ramanaiah, G., Squeeze films between finite plates lubricated by fluids with couplestresses, Wear, 54, (1979), 315

2. Ramanaiah, G. and Sarkar P., Squeeze films and thrust bearing lubricated by fluids

with couple stress, Wear, 48, (1978), 309-316

3. Ramanaiah, G. and Sundarammal Kesavan., Effect of bearing deformation on thecharacteristics of a slider bearing, Wear, 82, (1982), 273-278

4. Ramanaiah, G. and Sundarammal Kesavan., Effect of bearing deformation on the

characteristics of squeeze film between circular and rectangular plates, Wear, 82, (1982),49-55

5. Sudha V., Sundarammal Kesavan., Ramamurthy V., Squeeze film characteristics ofcouple stress fluid between porous triangular plates, Journal of Manufacturing

Engineering, (2008), 36. Bujurke N.M., Basti D.P Ramesh B. Kudenatti, Surface roughness effects on squeeze

flm behavior in porous circular disks with couple stress fluid, Transp Porous Med, 71,

(2008), 185-1977. Stokes V.K., Couple stress in Fluids, Phys Fluids, 9, (1966), 1709-1715

8. Naduvinamani N.B., Hiermath P.S., Gurubasawaraj G., Surface roughness effects in a

short porous journal bearing with a couple stress fluids, Fluids Dyn. Res., 31, (2002) 3339. Christensen H., Stochastic models for hydrodynamic lubrication of rough surfaces,

Proc. Inst. Mech. Eng. (Part J), 185(55), (1969), 1013

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