Lubrication theory for couple stress fluids and its application to short bearings

Wear, 80 (1982) 281- 290 281

LUBRICATION THEORY FOR COUPLE STRESS FLUIDS AND ITS APPLICATION TO SHORT BEARINGS

CHANDAN SINGH

Department of Mathematics, Indian Institute of Technology, Kanpur, Kanpur 208016, Uf far Pradesh {India)

(Received August 5,1981; in revised form December 21,198l)

Summary

Couple stresses may have a significant effect on the behaviour of bear- ings and a generalized Reynolds equation is derived to include these effects. The problem of the short journal bearing is analysed. An increase in the effective viscosity increases the load-carrying capacity and frictional drag and decreases the coefficient of friction. The longer the chain length of the lubricant or of the additive molecule is, the greater are the effects due to couple stresses.

1. Introduction

Lubricant films in which the properties are different from those of the bulk lubricant are of great rheological importance because most machine elements in practice owe their performance to them, especially under condi- tions of thin film lubrication. The creation of such a film requires the addi- tion of small amounts of additives; these additives are generally long-chain organic compounds. Rheolo~c~ abnorm~ities can therefore have a considerable effect on the minimum film thickness, which in turn strongly influences wear and fatigue failure in service.

Various theories have been postulated to describe these rheological ab- normalities in thin films. The theory of Stokes [l] includes polar effects such as the presence of couple stresses and body couples. Couple stresses occur in a liquid containing long-chain molecules. The couple stresses are particularly significant in lubrication problems, concerning thin films and they affect the bearing characteristics.

Couple stress theory [l] has been applied to various lubrication problems [ 2 - 71. Couple stresses in lubricants increase load-carrying capacity [ 2 - 71 and frictional drag [ 51 and reduce the coefficient of friction [ 51. The inlet film thickness of elastohydrodynamic films is increased [ 71. These results are, however, either for infinitely long geometries [ 2, 3, 5 - 7 ] or for a finite geometry [4] for squeezing only. No attempt has been made

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282

to study the effects of couple stresses in short bearings or for finite bearings which operate under sliding. In this paper a generalized Reynolds equation is derived for couple stress fluids and the characteristics of short journal bear- ings are analysed.

The si~ifi~ant feature of this study is a new dimensionless parameter L = Ifc where 1 = (q/p) l/2 1 has the dimensions of length and may be . considered to be some property of the fluid that depends on the molecular dimension of the lubricant or the additive molecule.

2. The generalized Reynolds equation

The equation of continuity and the equation of motion for an incom- pressible fluid with a couple stress are [I]

V-V=0 (1)

where V, A, F* and T are the velocity, acceleration, body force per unit mass and body couple per unit mass respectively, p is the density, p is the hydro- dynamic pressure, /J is the newtonian viscosity and 71 is a new material constant defining the couple stress property.

If the usual assumptions of lubrication theory are imposed [ 81 and the couple stress parameter q is retained, the equations of motion, in the absence of body forces and body couples, are obtained:

(3)

()A&

a4w a% ap Qay4-p

_=-- ay2 a.2

(4)

(5)

and the equation of continuity is

au au aw +--+-=o

iii ay a.2

The boundary conditions for the velocity distributions are

(6)

u = Ul, w = WI aty=O (7)

u = U.& w = wz aty=h (8)

The solution of eqns. (3) and (5) with the boundary conditions (7) and (8) yields

283

zl = 2.41 + (u, 1 3P -u,,r+--

h 2/l ax

x jcosh( Xl 1 (9)

w = WI + (wz 1 aP -wl)Y+ - -

h 2pa2 ( y2

(lo)

where

The flow fluxes qw and qE along the x axis and the z axis are given by

h ufl +w2

Qz = w dy = 0

2

where

(12)

(13)

By integrating the equation of continuity (eqn. (6)) across the film, the generalized Reynolds equation governing the pressure distribution in the film is obtained:

(14)

where the material coefficients JI and g may vary along the x and z directions.

3. The lubrication of short journal bearings

For short journal bearings the circumferential variation in the pressure is negligible compared with the axial pressure variation. The bearing is sta- tionary and the journal rotates with a uniform velocity U. Therefore eqn. (14) reduces to

284

d2p 6j.dJ 1 ah -=_-- dz2 R h3f(l, h) a0

(15)

where x = Rf3 and the film thickness is given by

h = Cfi + f COS 8) (16)

Integration of eqn. (15) using the boundary conditions

p = 0 at 2 = *B/2 (17)

where B is the axial dimension of the bearing, gives the pressure distribution:

3j.f ue

‘= Rc2 (18)

The load components along and perpendicul~ to respectively the line of centres are

Wa = W cos $9

n B/2 = 2

ss pcost’Rd6d.a

0 0

and

w lr/2 = W sin q

TI B/2

=2JJ P sin 8 R d6 dz 0 0

or

1 ,uUeB3 li Wo=_-

s

sin8 cos@ d8

2 Rc’ o (1 + E cosQ3f(l, h)

W 1 j.iUeB3 n

J

sin20 d0 ai2 = 2 RC2

o (1 + E coseyf(l, h)

If we non-dimensionalize by using

h H=-= 1+&cost? L=C

c c

w= W

(1/2)~.(UB~lRc~

the non-Dimensions load components are 7r

W, = s

sin 0 cos 8 d0

~ (1 + E cos f3)2F(L, H)

(19)

(20)

(21)

(22)

(23)

(24)

sin28 dtI

where

F(L,H)=l-12$+24

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(25)

W-9

The total load capacity of the bearing is

w = ( w02 + w~,2z)1/z (27)

The attitude angle cp which the load line makes with the line of centres is given by

R/2 lp=tan-l T

i 1 wo

The shear stress along the 8 direction is [l]

au a3u re=Ccay-77ay3

After simplification, eqn. (29) becomes

h a~ NJ 70 = y-2 Rye-y

i 1 The shear stress on the journal is

(r,,,_o=-;$-P+

Similarly the shear stress on the bearing surface is

(28)

W-9

(39)

(31)

(32)

The frictional drag F1 is obtained by integrating the shearing stress over the bearing surface. Thus the frictional drag on the journal surface is

2n B/2

F, = 2 J-I (T~)~=~R de dz (33) 0 0

Because the pressure and the pressure gradient are taken to be zero after 8 = T, eqn. (33) after simplification becomes

/.dJBR B ’ n 48 F, = -

sin 8 de +2 -

2c 10 D s o (1 + E cos tQ3F(L, H) + (1 - c’)l” (34)

where D = 2R. The non-dimensional frictional drag is

286

F, = Fl &lUBR/2C

=2; aj 0 sin 0 de 4?r

o (1 + E coseyF(L, H) + (1 - t?y

The coefficient of friction is obtained from

(35)

The integrations in eqns. (24), (25) and (35) were carried out numerically by the Romberg integration method.

4. Results and discussion

4.1. The couple stress parameter In addition to the well-known parameters for newtonian theory, one

more parameter is used in the present study, i.e. the couple stress parameter L which is defined by

A=!_ (37) c

where t = (~/~)l”. The parameter I has the dimensions of length and can be identified with

a property which depends on the size of the fluid molecule, e.g. the chain length of the molecule of a polar additive in a non-polar lubricant. Thus the parameter L can be considered as a characteristic of the interaction of the fluid with the bearing geometry. Therefore the parameter L provides a mechanism which might be helpful in explaining some of the rheological ab- normalities that are commonly observed in certain long-chain additives when the flows are confined to narrow passages.

It is expected that the effects of couple stresses on various bearing characteristics would be prominent when L is large: a large value of L corre- sponds either to an additive or lubricant molecule with a large chain length or to a small clearance width. The second case is of importance in lubrication theory where the bearing clearance is small. Thus, the larger L is, the more pronounced are the effects due to couple stresses.

Couple stresses which arise as a consequence of the intrinsic motion of the lubricant or additive molecule when confined to narrow passages are not important when L is small. A small value of L corresponds to a short molec- ular length and a large clearance width. When L + 0, the bearing character- istics reduce to their equivalents in newtonian theory. This would happen when 7) -+ 0, which in turn implies that the lubricant or the additive molecule has a vanishing chain length.

The parameter L can be determined once 1 is known. For different fluids the parameter Z can be obtained experiment~ly as suggested by Stokes [l] .

287

4.2. Bearing characteristics The load-carrying capacity, attitude angle, frictional drag and friction

coefficient (B/c)p, are given in Tables 1 - 4 for a wide range of values of the eccentricity ratio and the length ratio parameter L. The value of B/D is taken as 0.2 throughout the numerical calculations.

TABLE 1

Variation in the non-dimensional load capacity with the eccentricity ratio for various values of the couple stress parameter L

E w

0.1 1.60765 1.65372 1.78400 2.28862 3.12326 5.79144

0.2 1.72546 1.77990 1.93399 2.53341 3.52820 6.71470

0.3 1.94916 2.02133 2.22593 3.02700 4.36189 8.64577

0.4 2.33418 2.44250 2.75040 3.96513 5.99615 12.51903

0.5 3.00153 3.18935 3.72582 5.85807 9.42895 20.89163

0.6 4.24212 4.63471 5.76450 10.28221 17.84499 42.09228

0.7 6.89720 7.97756 11.12147 23.73314 44.80739 112.29706

0.8 14.32575 19.14868 33.38009 90.44298 185.62200 490.25689

0.9 53.31126 122.42383 328.79002 1154.54270 2531.16590 6932.30570 0.95 207.58573 1289.13930 4528.97810 17488.02900 39103.47000 108232.68000

L = 0.0 L = 0.05 (newtonian)

L = 0.1 L = 0.2 L = 0.3 L = 0.5

BID = 0.2.

TABLE 2

Attitude angle q versus the eccentricity ratio for various values of the couple stress parameter L

E @J (deg)

L = 0.0 (newtonian)

L = 0.05 L = 0.1 L = 0.2 L = 0.3 L = 0.5 ! /

0.0 90 90 90 90 90 90

0.1 82.70 82.57 82.25 81.32 80.42 79.29 0.2 75.43 75.16 74.50 72.65 70.95 68.90 0.3 68.17 67.76 66.73 64.01 61.68 59.06 0.4 60.93 60.34 58.91 55.39 52.69 49.94 0.5 53.68 52.85 50.94 46.80 44.05 41.58 0.6 46.32 45.16 42.70 38.25 35.84 33.96 0.7 38.70 37.02 33.96 29.83 28.11 26.95 0.8 30.50 27.87 24.51 21.65 20.79 20.28 0.9 20.80 16.51 14.41 13.52 13.33 13.23 0.95 14.32 9.79 9.16 8.97 8.93 8.92

BID = 0.2.

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TABLE 3

Variation in the non-dimensional frictional drag with the eccentricity ratio for various values of

the couple stress parameter L

L =o.o L = 0.05 L = 0.1 L = 0.2 L = 0.3 L = 0.5 (ne wtonian)

0.1 12.79292 12.79759 12.81077 12.86178 12.94608 13.21537

0.2 12.99911 13.00452 13.01982 13.07916 13.17731 13.49089

0.3 13.36635 13.37333 13.39309 13.46999 13.59734 14.00437

0.4 13.93779 13.94795 13.97672 14.08920 14.27580 14.87238

0.5 14.79484 14.81188 14.86029 15.05064 15.36690 16.37839

0.6 16.09859 16.13308 16.23166 16.62157 17.27029 19.34557

0.7 18.21157 18.30372 18.56978 19.62773 21.38964 27.02716

0.8 22.17831 22.57890 23.75240 28.43697 36.24297 61.22278

0.9 33.25917 38.87038 55.56985 122.35174 233.67760 589.58999

0.95 37.17291 144.15635 404.53461 1445.98850 3183.10490 8738.64510

B/D = 0.2.

TABLE 4

Coefficient (B/c)pf of friction Versus the eccentricity ratio for various values of the couple stress parameter L

E (Bk)i.tf

L = 0.0 L = 0.05 I, = 0.1 L = 0.2 L = 0.3 L = 0.5 (newtonian)

0.1 19.89375 19.34658 17.95224 14.04971 10.36264 5.70469

0.2 18.83423 18.26573 16.83026 12.90664 9.33711 5.02289 0.3 17.14370 16.54020 15.04213 11.12485 7.79326 4.04948

0.4 14.92789 14.27630 12.70422 8.88318 5.95207 2.96996 0.5 12.32274 11.61039 9.97115 6.42304 4.07439 1.95992

0.6 9.48733 8.70231 7.03949 4.04134 2.41949 1.14900

0.7 6.60108 5.73600 4.17431 2.06754 1.19342 0.60169 0.8 3.87036 2.94784 1.77894 0.78605 0.48813 0.31220

0.9 1.55967 0.79377 0.42253 0.26494 0.23080 0.21262 0.95 0.68855 0.27956 0.22330 0.20671 0.20350 0.20186

BID = 0.2.

Table 1 shows the variation in the load-carrying capacity with the eccentricity ratio for various values of the length ratio parameter L. The load capacity increases with increasing couple stress. At lower eccentricities the increase in load capacity has little significance compared with higher eccen- tricities .

Table 2 shows the variation in the attitude angle as a function of the eccentricity ratio for various values of the couple stress parameter L. At higher values of L the attitude angle is lower for a couple stress fluid com-

289

pared with a newtonian fluid, showing that the resultant load is nearer to the line of centres for a couple stress fluid.

Table 3 shows the variation in the frictional drag as a function of the eccentricity ratio. The trend is similar to that for the load capacity; however, the increase in frictional drag is not as pronounced as that for load capacity.

Table 4 shows the variation in the coefficient (B/c)pr of friction with the eccentricity ratio for different values of L. As L increases, the coefficient of friction first decreases rapidly and then becomes constant for higher values of L. The value at L = 0.0 gives values for the coefficient of friction for the newtonian case for different values of e. The coefficient of friction is smaller for large eccentricity ratios. This result is similar to the result for newtonian fluids.

Acknowledgments

The author thanks Dr. Prawal Sinha, Department of Mathematics, Indian Institute of Technology, Kanpur, for useful discussions.

The financial assistance by the Council of Scientific and Industrial Research, New Delhi, is sincerely acknowledged.

Nomenclature

A B

i&h)

FI QL, H) h H 1 L

P

4x. qz R T 4 v, w Ul. u2 u

V WI? WP W

Wo, wlr,, x9 Y, 2 E

21

acceleration vector axial length of the hearing radial clearance function defined by eqn. (13) body force per unit mass vector frictional force on the journal function defined by eqn. (26) oil film thickness h/c, dimensionless film thickness (v//A)~/~, characteristic material length Z/c, length ratio hydrodynamic pressure fluxes along the x and z directions respectively journal radius body couple per unit mass vector velocity components tangential velocity components along the x axis velocity of the journal velocity vector tangential velocity components along 2 axis resultant load-carrying capacity of the bearing load components along and perpendicular to the line of centres Cartesian coordinates eccentricity ratio couple stress characteristic angular coordinate (x = R6)

290

P newtonian viscosity coefficient Pf coefficient of friction 7 shear stress p attitude angle

Superscript corresponding dimensionless quantity

References

1 V. K. Stokes, Couple stresses in fluids, Phys. Fluids, 9 (1966) 1709. 2 G. Ramanaiah and P. Sarkar, Squeeze films and thrust bearings lubricated by fluids

with couple stress, Wear, 48 (1978) 309. 3 G. Ramanaiah and P. Sarkar, Optimum load capacity of a slider bearing lubricated by

a fluid with couple stress, Wear, 49 (1978) 61. 4 R. Ramanaiah, Squeeze films between finite plates lubricated by fluids with couple

stress, Wear, 54 (1979) 315. 5 P. Sinha and C. Singh, Couple stresses in journal bearing lubricants and the effect of

cavitation, Wear, 67 (1981) 15. 6 P. Sinha and C. Singh, Couple stresses in the lubrication of rolling contact bearings

considering cavitation, Wear, 67 (1981) 85. 7 P. Sinha and C. Singh, Couple stresses in elastohydrodynamic film roller bearings,

Wear, 71 (1981) 129. 8 0. Pinkus and B. Sternlicht, Theory of Hydrodynamic Lubrication, McGraw-Hill,

New York, 1961.