Acta Mechanica 98, 2 7 - 3 8 (1993)
ACTA MECHANICA
9 Springer-Verlag 1993
Ferrofluid lubrication
of cylindrical rollers with cavitation
P. Sinha a n d P. Chandra, K a n p u r , a n d D. Kumar, Bharuch, I n d i a
(Received July 24, 1991; revised March 18, 1992)
Summary. This paper analyses the ferrofiuid lubrication of cylindrical rollers under combined rolling and normal motion. The analysis, which takes into account the rotation of magnetic particles, has been made for general cases where the magnetization vectors need not be parallel to the applied magnetic field. Cavitation boundary conditions are used and the applied magnetic field is assumed to be imposed in a direction transverse to the fluid motion. A perturbation scheme in terms of non-dimensional Brownian time relaxation parameter has been used and the effects of various parameters on bearing characteristics have been studied.
List of symbols
A b F h ho h2
H
Ho I M M0 M1, M2 N P Pi q R
bt~ ?) 1,1 i
Uo, Vo W
X~ Z X 2
ti 12o
"C
non-dimensional relaxation time parameter length of the cylindrical roller [m]
frictional force [N]
film thickness [m]
minimum film thickness [m]
film thickness at the point of cavitation [m]
applied magnetic field [A/m]
constant applied magnetic field strength [A/m]
sum of moments of inertia of particles per unit volume [kg/m]
magnetization vector (M1, 0, M2) [A/m]
equilibrium magnetization [A/m]
components of magnetization vector [A/m]
dimensionless parameter pressure [(N/m;)]
pressure of i-th order, i = 0, 1, 2, 3 [(N/mZ)]
nondimensional velocity parameter equivalent radius of the cylinders [m]
velocity components [m/s]
velocity component of i-th order, i = 0, 1, 2, 3 [m/s]
reference velocity components of moving surface [m/s]
load capacity [N]
coordinates cavitation point Ira]
viscosity coefficient [Ns/m 2]
permeability of free space [Kg ms 2 A - 2]
Brownian relaxation parameter [s]
relaxation time parameter due to rotations [s]
non-dimensional ~B
A bar above a variable denotes a corresponding dimensionless quantity.
28 R Sinha et al.
1 Introduction
In recent years, an increasing attention has been focused on the study of ferrofluids. These fluids are stable colloidal suspensions of very fine magnetic partMes in a carrier fluid. They exhibit unusual properties under an externally applied magnetic field, that is, they can be confined, positioned or controlled at desired places. This has led to important applications of such fluids in the lubrication of liquid seals, journal bearing, thrust bearing etc. Several investigators have theoretically analysed ferrofluid lubrication in different bearing configurations, like thrust bearings (Walker and Buckmaster [1]), short bearings (Tipei [2]), seals (Kamiyama et al. [3]), sliding bearing (Miyaki et al. [4]), squeeze film (Verma [5]), finite journal bearing (Sorge [6]), etc.
However no attention seems to have been paid to the analysis of roller bearing with ferrofluids.
The rolling contact bearings utilize the rolling action of balls and/or rollers having point or line contacts, and hence exercise minimum rolling frictional resistance. Such bearings are generally used in situations where either rotatory motion, relative motion or a combination of both types of motions is required. Since 1960, increasing emphasis has been laid on the study of mechanism of roller bearing lubrication. The advantages and disadvantages of roller bearing over other types of bearings have been discussed in detail by Wilcock and Booser [7].
Pioneering work in this direction was initiated by Floberg [8] who analyzed the hydrodyna- mic lubrication of roller bearings, considering the rupture of fluid film at the outlet. Dowson et al. [9] presented a generalization of this work by considering rolling, sliding and normal motion of the rollers.
It must be emphasized here that the performance of rollers which operate with thinner oil films, could be enhanced in the presence of a magnetic fluid, since the flow flux, for a ferrofluid lubricant, would reduce. It is also to be noted that in all the ferro lubrication studies of bearings, conducted so far, the magnetization vector is taken to be parallel to the applied magnetic field.
This assumption reduces the complexity of the equations to be solved drastically, but nevertheless imposes a physical restriction which may not be always valid.
In view of this, in the present study offerro lubrication of rollers, under combined rolling and normal motion, no such restriction is imposed on the magnetization vector. Cavitation boundary conditions are used and an applied magnetic field is assumed to be imposed in a direction transverse to the fluid motion. Various bearing characteristics are obtained through perturbation analysis and elaborated through graphs.
2 Mathematical formulation
We consider here the axisymmetric flow of ferrofluid between identical rollers moving with equal uniform tangential velocity Uo and normal velocity Vo (Fig. 1). A constant magnetic field is applied tranverse to the direction of fluid motion. Further, we make the following simplifying assumptions (Walker and Buckmaster [1], Tipei [2]):
(i) The plates are non-magnetic and non-conducting so that the applied field is not modified.
(ii) Magnetization in the ferrofluid is negligible compared to the applied magnetic field.
(iii) The ferrofluid is saturated so that Mo does not depend upon the applied magnetic field.
(iv) Thin film lubrication approximations are valid so that the derivatives of velocity, magnetization etc. along the film are negligible in comparison to their derivatives across the film.
Ferrofluid lubrication of cylindrical rollers 29
Vo Z
-.oo ( o ~,o) x = x 2
H o
Fig. 1. Cylindrical rollers
Thus, for the steady, two-dimensional, viscous flow of a ferrofluid the equations of motions (Shliomis [10], K a m i y a m a et al. [11]) under the above assumptions reduce to the following:
@ OZu #oHo aM1
- ? ~ + t/Oz 2 + 2 0z 0 (1)
1 ? u kZoZ~rs
M1 = 2 ZBM2 Oz l M 1 M 2 H ~ (2)
1 c~u /t0"CBZs M12Ho (3)
M2 = Mo - ~ znM, ~zz + I
~ u cqv
a x + ~zz = 0. (4)
B o u n d a r y conditions for velocity c o m p o n e n t s are dh
u = Uo, v = Vo + UO dx at z = h - 0 , v = 0 at z = 0
0z
(5)
and for pressure the cavitation b o u n d a r y conditions are assumed, i.e.,
p = 0 a t x = - - ~ dp
p = 0 = - - a t x = x2
dx
(6)
where x 2 is the cavitation point.
Film thickness h is given as
h = h o + x 2 / 2 R (7)
where ho is the m i n i m u m film thickness and R the equivalent radius of the cylinders.
Now, introducing the non-dimensional variables
pho 2 u v x z
P - qUo ]/2Rho
'
a -Vo'
~= ~ '
Yc -21/2R~o'
e --ho'
- -/~ = h ]1~/1 - M 1 ]~[2 - M 2
ho' M o ' Mo
30 R Sinha et al.
and n o n - d i m e n s i o n a l p a r a m e t e r s
go ( 2 R ' ] 1/2 #oMoHoho v~Vo 7~vs
q -- Uoo \ ~o J ' N - tlUo ' ~ - ho ' A = - I '
Eqs. (1) to (4) in n o n - d i m e n s i o n a l form can be written as follows:
Off 02~ N ~J~fl
- ~ x + ae ~ + 2 a~ - ~ (8)
f/Is - N A ' c M l f 4 2 (9)
2 0~
J~/I 2 = 1 -~ N A z M 1 2 (10)
2 ~
a~
+ q ~ = 0.(11)
The b o u n d a r y conditions b e c o m e : 1 d/~
~ : 1 , ~ = 1 + - - -
q d2 at ~ = / ~ = I A c X 2 0~
- - = 0 , ~ = 0 at ~ = 0
/3 = 0 at s = - o o
p = 0 = __dP at x = )~2.
d2
(12)
(13)
3 A n a l y s i s
E q u a t i o n s (8) to (10) are to be solved s i m u l t a n e o u s l y to o b t a i n the velocity and pressure distribution. These equations are coupled n o n - l i n e a r equations a n d are not easily a m e n a b l e to analytical solution. Therefore, we use p e r t u r b a t i o n technique to solve them. Considering the representative values of various m a g n e t i c fluid p a r a m e t e r s ( K a m i y a m a et al. [11], Rosensweig [12]), e.g.,
Ho ~ 105 A m -1 ' Mo ~ 104 A m -1 , #o ~ 10- v K g m s - 2 A - 2 , -CB ~ 1 0 - 6 S,
Z s ~ 10 - ~ I S, I ~ 10 -14 K g m - ~ ,
we find t h a t the n o n - d i m e n s i o n a l quantities z, N a n d A are of the o r d e r 1 0 - 2, 1 a n d i respectively.
So we use z as a p e r t u r b a t i o n parameter. Thus, the following p e r t u r b a t i o n scheme is a d o p t e d :
f = f o + z f l + "t-*f2 + ~ f3 + ... 3 (14)
w h e r e f stands for the variable quantities p, fi, ~ r t , ~r2- Using the a b o v e p e r t u r b a t i o n scheme and eliminating m a g n e t i z a t i o n vector c o m p o n e n t s of different order, we o b t a i n the equations
Ferrofluid lubrication of cylindrical rollers 31 governing io, 3z, i~, i3 as follows:
d/5o 8~io - d ~ + ~ 2 - = 0
-d~- + ~ + ] - ~ LOeJ = ~
N o O o]
]~- + a T + 7 ~ Las - N A 8sj = 0
dp3 82u3 N 8 [8/2 2 ~31 { 1 (~30~2~ 8b~0]
dx + 8~ ~ + -4 ~
L az
- NA -- + NZA 2 -- =- - - - -
~= ~ \ o ~ ) j
o ~ j o .(15)
(16)
(17)
(18)
On perturbing the boundary conditions (12) and (13), we obtain
3o = 1, 3j = 0 ]
1 d/~ l at 5 =/~
~ o = 1
+ - - - ~ j = 0 q dS'
(19)
p ~ = 0 at 2 = - 0 % P i = 0 = d~ at 2 = 2 2 (20)
where i = 0, 1, 2, 3, a n d j = 1, 2, 3.
Solving Eqs. (15) to (18) along with the boundary conditions (19), we get the expressions for 3o, 31, 32 and 33 as follows:
3o
kd~j
+ 1=r,, N+0t( )
31 [_d2 4 d2 J
32
kd2 ~ (d2 -- N\~+
+d2JJ
N (dfio~ 3
A
(21)
(22)
(23)
(24)
Integrating the equation of continuity (11) over the film thickness and using boundary conditions (19) along with the perturbation scheme, we get
0 ~ ~o d~ = - q (25)
0
82 3j d5 = 0; j = 1, 2, 3. (26)
0
32 R Sinha et al.
Using the expressions offii from Eqs. (21) to (24) in the Eqs. (25) and (26) we get equations for pressure distribution, as follows:
~ - \ d e J + ~ = - q
[~3 ~ d~I N dpo~]
~ ( d e 4 ~2 JA = o
~x ~3 (d2 4 \d2 - N +A d2]jJ=O
0 Ih_~d~3 N(dp2 ( 1 ) @ 1 N 2 ( 1 A )dpo~
+
W \ ~ )
j(27)
(28)
(29)
(30)
By solving Eqs. (27) to (30) along with the associated boundary conditions the expressions for pressure of different orders are obtained, which finally give the pressure distribution as follows:
I N'c {1 / 5 = 3 1 + ~ -
81Nz380 f
-co
-NA~+N2A2z z} . ~ [ q ( e - 2 2 ) + ( h - h 2 ) ] d 2
--o9
1 [ q ( 2 - - 2 2 ) ~- (h - - ]~2)] 3 d e ~- 0 ( ' c 4 ) . ( 3 1 )
The cavitation point 22 is determined by the boundary condition at e = 22. Thus Eq. (31) gives the equation determining the cavitation point e = 22 as
X2
3 i + T ~ [q(X - x2) + (h - h2)] d e
81N~ 3 f
80 ~ [q(2 - 22) + (h -1
h2)] 3 d 2 = 0 . -co(32)
From Eq. (32) the cavitation point is calculated numerically by using bisection rule.
Non-dimensional load capacity is given by
~2
W- Wh _ 2 f p d2 btlUoR
oo
(33)
and the force of friction is given by
F _
~2
bqUo -oo
(34)
Ferrofluid lubrication of cylindrical rollers 33 where
a~_ e=fi 3 27Nz ~
= ~7 [q(2 - s + (h - h;)] + ~ [q(2 - :~2) + (h - h2)] 3.
(35)
It m a y be mentioned here that the term due to the cavitation region as obtained by Floberg [8]
does not a p p e a r in Eq. (34) because the cylinders rotate with the same velocity.
F r o m Eq. (35), it is obvious that the effect of magnetic fluid parameters on the frictional force P is appearing only through the terms of third order of z. Thus their effect on P will not be appreciable.
4 Results and discussion
The non-dimensional parameters appearing in the expressions of pressure and load are (a) Magnetic parameters N, z and A,
N -
# o M o H oho r B Vo ~IZ s, ~:-- , A -
~IUo ho I
A : 0 - O [- : 0 - 0 2 N = O . 0 N : 5 - 0
q=-l.0 11"0
:7 \'t
q=0"0
q=0"5
q=l.0
-3"0 -2-0 -I'0 " 0
Fig. 2. Variation of non-dimensional pressure (p) vs.
0"6 1"2
34 R Sinha et al.
A = 0 " 0 N = 5 " 0 - - C = 0 " 0 2 . . . . E, = 0 ' 0 8
q= -1"0
/
/ ~
/
I I I I II q = - 0 5
/
1.0
q-- 0 . 0
q= 0-5
q=1"O
-215 21)!0 -1"5 -1"0 -0'5 Fig. 3. Variation of non-dimensional pressure (/3) vs.
0 0-5 1.0
(b) Velocity parameter q,
Vo (2R'~ 1/2
q = Uoo \ - ~ o / I "
Non-dimensional pressure distribution (/5) vs. 2 has been plotted in Fig. 2 for various values ofq and N keeping A and ~ fixed and in Fig. 3 for various values ofq and 9 keeping A and N fixed, whereas the effect of A on/5 is shown in Fig. 4, for fixed N and r. The qualitative behaviour of pressure for the magnetic fluid remains similar to that of Newtonian case, only the quantitative changes are noticed. It is noted from these figures that as q decreases peak pressure shifts towards the centre line. This behaviour with respect to q is similar to that observed by D o w s o n et al. [9].
Further, it is seen from Figs. 2 and 3 that the effect of N or ~ is to increase pressure. It m a y be noted from Figs. 2 and 3 that the pressure enhancement with N and ~ is more for small values of q.
In particular for squeezing (q < 0) the variation is more in comparison to the case of separation (q > 0). However, Fig. 4 shows very small variation of pressure with A. It is also observed that the effect of magnetic fluid parameters on the cavitation point is not significant. In Table 1 we see the effect of magnetic fluid parameters on peak pressure for q = - 1 (this value has been chosen because the variation is more as compared to other values). F o r small values of r (~ = 0.02, N = 5), the peak pressure decreases as A increases from 0 to 3 while for 9 = 0.08, peak pressure
Ferrofluid lubrication of cylindrical rollers
~ =0-02
N = 5 " 0
- - . - - A = O ' O
. . . . A = I " O
- - A = 2 " O
q =-I-0 I'0
35
q = o - o
q=l'O
-2"8 -2.0 -I-0
Fig. 4. Variation of non-dimensional pressure (p) vs.
0 0 6 1.2
Table 1. Dimensionless peak pressure (/5) for q = - 1.0
A N = 5 N = 10
= 0.02 r = 0.04 r = 0.08 z = 0.02 z = 0.04 9 = 0.08 0.0 0.92699025 0.94958953 0.99472614 0.94959836 0.99479686 1.0850701 0.2 0.92654710 0.94785311 0.98806991 0.94786194 0.98814061 1.0607604 0.4 0.92612204 0.94626141 0.98257128 0.94627023 0.98264197 1.0457114 0.6 0.92571507 0.94481438 0.97823027 0.94482321 0.97830095 1.0399234 0.8 0.92532618 0.94351208 0.97504685 0.94352091 0.97511751 1,043 3962 1.0 0.92495539 0.94235456 0.97302103 0.94236329 0.97309169 1.0561299 1.2 0.92460267 0.94134156 0.97215282 0.94135039 0.97222348 1.0781245 1A 0.924 268 06 0.940 473 36 0.972 442 22 0.940 48218 0.972 512 88 1.109 379 9 1.6 0.923 951 52 0.939 749 87 0.973 889 22 0.939 758 69 0.973 959 89 1.149 896 3 1.8 0.92365307 0.93917105 0.97649386 0.93917987 0.97656453 1.1996735 2.0 0.92337272 0.93873695 0.98025608 0.93874577 0.98032676 1.2587115 2.2 0.923 11046 0.93844754 0.985 17591 0,93845636 0.98524660 1,3270105 2.4 0.92286628 0.93830285 0.99125332 0.93831167 0.99132403 1.4045703 2.6 0.92264016 0.93830285 0.99848843 0.93831167 0.99855909 1.4913910 2.8 0.92243217 0.93844754 1.00688110 0.93845636 1.00695180 1.5874726 3.0 0.92224223 0.93873695 1.0164313 0.93874577 1.01650200 1.6928150
36 R Sinha et al.
v~
1,8'
1.2
0"6
J
A=O.0 N : 0 . 0 N : 5 O N : I 0 . 0
9 I J J
J J
[ I I 1 I , I I 1 I I 1
0-01 0.05 0-1
C
Fig. 5. Variation of non-dimensional load (l~) vs.
q : -1'0
q=O'O
q=l'O
decreases as A increases from 0 to 1.2, and then shows an increasing trend as the value of A is increased beyond 1.2. The critical value of A beyond which inversion of peak pressure takes place depends upon the combination of N and ~.
Dimensionless load capacity vs. 9 has been shown in Fig. 5 for A = 0 and various values of N, q, and in Fig. 6 for N = 5.0 and various values of A, q. F r o m Fig. 5 it m a y be noted that the load increases linearly with respect to z. It is seen from these figures that an increase in q decreases load capacity, whereas an increase in applied magnetic field enhances load capacity. It is observed that as the applied magnetic field increases such that N is doubled from 5 to 10, for given z = 0.04, the increase in load is 4.5% and for v = 0.08, the increase is 8.3%. Similarly when z is doubled from 0.04 to 0.08, the increase for N = 5 is 4.5% and for N - 10 is 8.3%. It m a y also be noted from Fig. 5 that the effect of magnetic fluid parameters N and r is higher in case of squeezing then in case of separation. Figure 6 shows that for a fixed q the load capacity decreases with increase in A for small values of z, however an increase is observed with respect to A for higher values of r.
This behaviour of A for the case of squeezing is further elaborated in Table 2, which shows that the load capacity attains a minimum with respect to A and then increases. This behaviour is more evident for higher values of N and ~.
Ferrofluid lubrication of cylindrical rollers 37
1'03
O,B7
0-ll
0.55
N = 5 " 0 A = 0 . 0 . . . A : I - 0
--.-- A = 2 " 0
- , A = 3 " O
q = - 0 " ~
~ ~ : ; ~ ~ , ~ ~ q:O'O
q = 0.5 0"39
o t o : ' ' ' o & ' o C
Fig. 6. Variation of non-dimensional load (14/) vs. z
Table 2. Dimensionless load (17r for q = - 1.0
A N = 5 N = 1 0
r = 0.02 r = 0.04 ~ = 0.08 ~ = 0.02 ~ = 0.04 z = 0.08 0.0 1.4318549 1.4667652 1.5365065 1.4667764 1.5365967 1.6760795 0.2 1.4311703 1.4640830 1.5262251 1.4640943 1.5263153 1.6385299 0.4 1.4305138 1.4616245 1.5177319 1.4616357 1.5178220 1,6152851
0.6 1.4298852 1.4593893 1.5110267 1.4594006 1.511 1168 1.6063446
0,8 1,4292845 1.4573778 1.5061095 1.4573890 1.506199 5 1,6117090 1.0 1.4287118 1.4555897 1.5029802 1.4556009 1.5030704 1.6313777 1.2 1.428166 9 1.454025 1 1.5016392 1.4540364 1.501729 3 1.6653512 1.4 1.4276501 1.4526841 1.5020862 1.4526953 1.5021763 1.7136291 1.6 1.4271612 1.4515665 1.5043213 1.4515778 1.5044114 1.7762118 1.8 1.4267002 1.4506725 1.5083445 1.4506837 1.5084346 1.8530991 2.0 1.4262671 1.4500020 1.5141557 1.4500132 1.5142459 1.9442909 2.2 1.425862 0 1.449 559 4 1,521755 0 1.449 566 2 1.521845 2 2.049 787 3 2.4 1.4254848 1.4493314 1.5311423 1.4493427 1.5312325 2.1695881 2.6 1.4251356 1.4433140 1.5423180 1.4493427 1.5424080 2.3036937 2.8 1.4248143 1.4495594 1.5552814 1.4495662 1.5553716 2.4521041 3.0 1.4245210 1.4500020 1.5700330 1.4500132 1,5701232 2.6148189
38 R Sinha et al.: Ferrofluid lubrication of cylindrical rollers
5 Conclusions
Ferrofluid lubrication, in the presence of a transversely applied magnetic field, of cylindrical rollers with cavitation has been analyzed. In contrast to the earlier analysis, this study relaxes the assumption of the magnetization vector being parallel to the magnetic field. In addition, the rotation of magnetic particles has also been accounted for. A perturbation scheme in terms of non-dimensional Brownian time relaxation parameter has been used and expressions for various bearing characteristics are obtained.
The qualitative behaviour of pressure and load-carrying capacity for a ferrofluid remains similar to that of a Newtonian fluid, only quantitative changes are noticed. Pressure enhancement is observed as N and ~ increase, more so in the case of squeezing. The impact of the ferrofluid parameters on the cavitation point is negligible. It is further noted that an increase in the applied magnetic field increases the load-carrying capacity. This increase is more significant in the case of squeezing than in the case of separation.
It is thus seen that the application of the magnetic field to the ferrofluid lubrication increases the load-carrying capacity without really affecting the point of cavitation, which in turn depends upon the fluid flowing out of the bearing. One could thus interpret that the performance of rollers which operate with thinner oil film could be enhanced in the presence of a magnetic field.
References
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[2] Tipei, N. : Theory of lubrication with ferrofluids: application to short bearings. J. Lub. Tech. (ASME) 104, 510-515 (1982).
[3] Kamiyama, S., Oyama, T., Htwe, J.: Basic study on the performance of magnetic fuid seals. Proc. of JSLE International Tribology Conf., pp. 985-990, Tokyo 1985.
[4] Miyake, S., Sadao, S., Tanahashi, T.: Sliding bearing lubricated with ferromagnetic fluid. ASLE Trans.
28, 461 (1985).
[5] Verma, R D. S.: Magnetic fluid-based squeeze film. Int. J. Engng. Sci. 24, 395-401 (1986).
[6] Sorge, F.: A numerical approach to finite journal bearings lubricated with ferrofluids. J. Trib. (Trans.
ASME) 109, 7 7 - 8 2 (1987).
[7] Witcock, D. F., Booser, E. R.: Bearing design and applications. New York: McGraw-Hill 1957.
[8] Floberg, L.: Lubrication of two cylindrical surfaces considering cavitation. Trans. Chalmers Univ. of Tech. 14 (Inst. of Machine Elements) 234 (1961).
[9] Dowson, D., Markho, R H., Jones, D. A.: The lubrication of lightly loaded cylinders in combined rolling, sliding and normal motion. I. J. Lub. Tech. (ASME) 98, 509 (1976).
[10] Shliomis, M. I.: Effective viscosity of magnetic suspensions. Soy. Phys. JETP 34, 1291-1294 (1972).
[11] Kamiyama, S., Koike, K., Iizuka, N.: On the flow of a ferromagnetic fluid in a circular pipe (Report 1, Flow in a uniform magnetic field). Bull. JSME 22, 1205-1211 (1979).
[12] Rosensweig, R. E.: Ferrohydrodynamics. Cambridge: University Press 1987.
Authors' addresses: R Sinha and R Chandra, Department of Mathematics, Indian Institute of Technology Kanpur, Kanpur-208016, D. Kumar, Department of Mathematics, Narmada College of Science and Technology, Bharuch, Gujrat, India
