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Optimum slider profile of a slider bearing lubricated with a micropolar fluid
Wear, 42 (1977) 1 - 7 0 Elsevier Sequoia S. A., Lausanne
1 - Printed
in the Netherlands
OPTIMUM SLIDER PROFILE OF A SLIDER WITH A MICROPOLAR FLUID
G. RAMANAIAH Department (Received
BEARING
LUBRICATED
and J. N. DUBEY
of Mathematics, January
Indian Institute
of Technology,
Rharagpur
(India)
1, 1976; in final form May 16, 1976)
Summary The slider profile which gives the maximum load capacity for a slider bearing lubricated with a micropolar fluid is considered. The optimum profile is a step function with a riser location and a step height ratio that depend on the two basic dimensionless parameters of micropolar fluid dynamics. The results reduce to those of the classical Rayleigh problem when the coupling parameter tends to zero. The optimum load capacity increases with the coupling parameter.
1. Introduction The theory of micropolar fluids introduced by Eringen [ 1] deals with a class of fluids which exhibit a microrotational effect and microrotational inertia and support couple stress and body couples. Physically, some polymeric fluids and fluids containing a rigid substructure may be represented by the mathematics model covering micropolar fluids. The use of micropolar fluids as lubricants was suggested by Cowin [2]. Allen and Kline [3] studied the inclined slider bearing with a micropolar fluid lubricant and concluded that micropolar fluids are superior to Newtonian lubric~~. Similar conclusions were reported by Khader and Vachon [4] for hydrostatic bearings, by Prakash and Sinha [ 51 for journal bearings and by Ramanaiah and Dubey [6] for squeeze film bearings. Datta [ 71, Agrawal et al. [8] and Maiti [9, lo] investigated micropolar lubricated bearings by neglecting the term 2kv in the basic equation of motion (eqn. (6)). The present paper extends the classical Rayleigh [ 111 problem of the optimum slider to slider bearings lubricated by a micropolar fluid.
2. Basic equations The two-dimensional bearing geometry shown in Fig. 1 consists of two rigid surfaces in relative motion. The lower surface moves with a constant
2
velocity u. in the x direction while the upper surface (y = h(x)) is stationary. The lubricant between the surfaces is an incompressible micropolar fluid. The body forces and body couples are assumed to be absent. b
Fig. 1. Geometry
of the slider bearing.
Taking the velocity vector ; = (u, u, 0) and the microrotation t = (0, 0, v), the equations of motion of the fluid reduce to au au _+-_=O ax ay
(1)
ap pri =x+
vector
(2) (3)
(4)
where p is the density, p is the pressure and ~1is the shear viscosity. k and y are the new material constants peculiar to micropolar fluids, j is the moment of microinertia and the superposed dot denotes the material derivative. In deriving the Reynolds equation the usual assumptions of lubrication are made [12] : (1) the fluid film thickness h is very much less than b where b is the breadth of the bearing; (2) the inertia terms are negligible; (3) the variation of pressure across the fluid film is negligible, i.e. aplay = 0; and (4) the x derivatives of u and v are much smaller than the corresponding y derivatives. Thus eqns. (2) - (4) simplify to (5)
(6)
3
Solving the coupled
eqns. (5) and (6) with the no-slip boundary
condi-
tions u = ue
v=o
at y=O
(7)
U=O
v=o
at y=h
(8)
gives h-y-k/sinh$--sinhm(y-:)I
i(Zil+k)mcosh$/-’
u = 7.40 h - (2k tanh f)
1(2~ + k)m/-r
cosh$-coshm 1 dp
+-2P *
(2~ + k) sinh F
I
and
v=- uo 2
h - 2y + h cosech $
sinh m
(10)
where
Integration per unit length:
of eqn. (9) across the fluid film yields the volume flow rate
h 4'
s udy 0
1 =P”k--Integrating conditions
1 dp 12p dx
(12)
eqn. (1) across the fluid film using eqn. (12) and the boundary
Z!=O gives Reynolds’
aty=Oandy=h equation
(13)
mh
= 6/u,,
coth~~~--l
introducing
the dimensionless
(14)
quantities
3KH(MH
coth MH - 1)
(2 + K)M2
where ho is the minimum
;:
film thickness,
-I (15)
I
eqn. (14) takes the form
(16)
The boundary form
p = 0 at x = 0 and x = b take the dimensionless
conditions
P=OatX=O
and X=1
Solving eqns. (16) and (17), the dimensionless P=
s
x(H-Q)FdX=
(17) pressure is obtained
](Q-H)FdX
as (IS)
X
0
where Q= j,FdX i)
The load capacity
(19)
/FdX I
0
of the bearing per unit length
h w=
p&x I 0
takes the dimensionless W=-----_
wh’ Guob2
form
- ]PdX= o
jX(Q-H)FdX 0
(20)
3. Optimum
slider profile
The load capacity of the slider bearing depends on the shape of the fluid. It is necessary to determine the H(X) which results in the maximum load capacity W. Rayleigh [ll] considered this problem for Newtonian lubricants. Using the calculus of variations, he found that the optimum profile (the profile which makes W a maximum) was of the stepped form shown in Fig. 2.
Fig. 2. Rayleigh step.
Following Rayleigh [ 111, let H in eqns. (19) and (20) become H + 6H where 6 H is an admissible variation in H. Then Q and W become, respectively, &+6QandW+6W. Thus, 11 F+(H-Q)g 0
aQjFdX= 0
6HdX I
&‘W=6QjXFdX-f/F+(H-Q)d&5HdX 0 0 Elimination of 6 Q from eqns. (21) and (22) yields the variation the variation of H :
(21)
(22) of W due to
1
6W= 0J1
F+(H-Q)~@XFdX/%‘dX)-X/SHdX
(23)
From this equation it is possible to satisfy the optimum condition by a profile of stepped form (Fig. 2) in the present case also. If H = 1 on the lower step and if F + (H - Q)(dF/dH) = 0 on the upper step, every admissible H from this stepped form diminishes W. No matter how 6 H varies along the upper step, there is no contribution to 6 W. In contrast, an admissible 6 H can only be non-negative along the lower step since H must not be less than 1 and thus only a non-positive contribution to 6 W can result. Therefore the optimum film profile must be of the stepped form. The step height ratio r and riser location s will depend on the parameters K and M. Indeed, r and s are obtained from eqn. (22) and F(r) + (r-Q)
0 dF (XHElzr =
FdX
(24)
(25)
6
K
0
K-O
2-
r: K-1
Fig. 3. Plots of step height
ratio r, riser location
Thus, r is a root of the transcendental
s and maximum
load W, us. M.
equation
ml] F+(H-1)sI+F3’2(H)=0
(26)
s = dm{#(l)
(27)
and the optimum
+ m}-’ load W, is given by
(28)
4. Discussion
and conclusions
The optimum slider profile is of the stepped form even for micropolar fluid lubricants. However the step height ratio r and riser location s depend on the parameters K and M. From eqn. (15), F increases monotonically with the coupling parameter K and decreases monotonically with M. Also F = T3 whenK=O or A4 = 00 and F = {l + (K/2)}W3 when M = 0. The transcendental eqn. (26) takes the simple form (2H- 3)a= 1 when K = 0 orM=OorM=m. Hence the classical results r = 1.866 and s = 0.718 are obtained. However the optimum load capacity depends on M; W, = 0.0343 for K = 0 or M = 00and W, = 0.0343 (1 + K/2) for M = 0.
7
As the parameter M is the ratio of the characteristic linear dimension of the flow to the particle size. of a micropolar fluid, the classical results of Stokesian fluid theory are recovered from the present theory as it4 + ~0, i.e. as the particle size becomes negligible compared with the characteristic linear dimension of the fI ow. However in lubrication theory dealing with flow through narrow recesses, M- ’ may not be negligible and the effect of M can be considerable on the bearing characteristics. For example, with K = 1 and M = 5, W, = 0.0357 which is an increase of more than 4% over the classical value W, = 0.0343. The plots of r, s and W, versus M for K = 0 and 1 are shown in Fig. 3; r and s are virtually insensitive to M but the optimum load increases with K and decreases with M.
References 1 2 3 4 5 6 7 8 9 10 11 12
A. C. Eringen, J. Math. Mech., 16 (1966) 1. S. C. Cowin, Phys. Fluids, 11 (1968) 1919. S. J. Allen and K. A. Kline, Trans. ASME (E), (1971) 646. M. S. Khader and R. I. Vachon, Trans. ASME (F), (1973) 104. J. Prakash and Prawal Sinha, Int. J. Eng. Sci., (1975) 217. G. Ramanaiah and J. N. Dubey, Wear, 32 (1975) 343. A. B. Datta, Jpn. J. Appl. Phys., 11 (1972) 98. V. K. Agrawal, K. L. Ganju and S. C. Jethi, Wear, 19 (1972) 259. G. Maiti, Jpn. J. Appl. Phys.. 12 (1973) 1052. G. Maiti, Jpn. J. Appl. Phys., 13 (1974) 1440. Lord Rayleigh, Phitos. Mag., 35 (1918) 1. 0. Pinkus and B. Sternlicht, Theory of Hydrodynamic Lubrication, McGraw-Hi& New York, 1961.
1 - Printed
in the Netherlands
OPTIMUM SLIDER PROFILE OF A SLIDER WITH A MICROPOLAR FLUID
G. RAMANAIAH Department (Received
BEARING
LUBRICATED
and J. N. DUBEY
of Mathematics, January
Indian Institute
of Technology,
Rharagpur
(India)
1, 1976; in final form May 16, 1976)
Summary The slider profile which gives the maximum load capacity for a slider bearing lubricated with a micropolar fluid is considered. The optimum profile is a step function with a riser location and a step height ratio that depend on the two basic dimensionless parameters of micropolar fluid dynamics. The results reduce to those of the classical Rayleigh problem when the coupling parameter tends to zero. The optimum load capacity increases with the coupling parameter.
1. Introduction The theory of micropolar fluids introduced by Eringen [ 1] deals with a class of fluids which exhibit a microrotational effect and microrotational inertia and support couple stress and body couples. Physically, some polymeric fluids and fluids containing a rigid substructure may be represented by the mathematics model covering micropolar fluids. The use of micropolar fluids as lubricants was suggested by Cowin [2]. Allen and Kline [3] studied the inclined slider bearing with a micropolar fluid lubricant and concluded that micropolar fluids are superior to Newtonian lubric~~. Similar conclusions were reported by Khader and Vachon [4] for hydrostatic bearings, by Prakash and Sinha [ 51 for journal bearings and by Ramanaiah and Dubey [6] for squeeze film bearings. Datta [ 71, Agrawal et al. [8] and Maiti [9, lo] investigated micropolar lubricated bearings by neglecting the term 2kv in the basic equation of motion (eqn. (6)). The present paper extends the classical Rayleigh [ 111 problem of the optimum slider to slider bearings lubricated by a micropolar fluid.
2. Basic equations The two-dimensional bearing geometry shown in Fig. 1 consists of two rigid surfaces in relative motion. The lower surface moves with a constant
2
velocity u. in the x direction while the upper surface (y = h(x)) is stationary. The lubricant between the surfaces is an incompressible micropolar fluid. The body forces and body couples are assumed to be absent. b
Fig. 1. Geometry
of the slider bearing.
Taking the velocity vector ; = (u, u, 0) and the microrotation t = (0, 0, v), the equations of motion of the fluid reduce to au au _+-_=O ax ay
(1)
ap pri =x+
vector
(2) (3)
(4)
where p is the density, p is the pressure and ~1is the shear viscosity. k and y are the new material constants peculiar to micropolar fluids, j is the moment of microinertia and the superposed dot denotes the material derivative. In deriving the Reynolds equation the usual assumptions of lubrication are made [12] : (1) the fluid film thickness h is very much less than b where b is the breadth of the bearing; (2) the inertia terms are negligible; (3) the variation of pressure across the fluid film is negligible, i.e. aplay = 0; and (4) the x derivatives of u and v are much smaller than the corresponding y derivatives. Thus eqns. (2) - (4) simplify to (5)
(6)
3
Solving the coupled
eqns. (5) and (6) with the no-slip boundary
condi-
tions u = ue
v=o
at y=O
(7)
U=O
v=o
at y=h
(8)
gives h-y-k/sinh$--sinhm(y-:)I
i(Zil+k)mcosh$/-’
u = 7.40 h - (2k tanh f)
1(2~ + k)m/-r
cosh$-coshm 1 dp
+-2P *
(2~ + k) sinh F
I
and
v=- uo 2
h - 2y + h cosech $
sinh m
(10)
where
Integration per unit length:
of eqn. (9) across the fluid film yields the volume flow rate
h 4'
s udy 0
1 =P”k--Integrating conditions
1 dp 12p dx
(12)
eqn. (1) across the fluid film using eqn. (12) and the boundary
Z!=O gives Reynolds’
aty=Oandy=h equation
(13)
mh
= 6/u,,
coth~~~--l
introducing
the dimensionless
(14)
quantities
3KH(MH
coth MH - 1)
(2 + K)M2
where ho is the minimum
;:
film thickness,
-I (15)
I
eqn. (14) takes the form
(16)
The boundary form
p = 0 at x = 0 and x = b take the dimensionless
conditions
P=OatX=O
and X=1
Solving eqns. (16) and (17), the dimensionless P=
s
x(H-Q)FdX=
(17) pressure is obtained
](Q-H)FdX
as (IS)
X
0
where Q= j,FdX i)
The load capacity
(19)
/FdX I
0
of the bearing per unit length
h w=
p&x I 0
takes the dimensionless W=-----_
wh’ Guob2
form
- ]PdX= o
jX(Q-H)FdX 0
(20)
3. Optimum
slider profile
The load capacity of the slider bearing depends on the shape of the fluid. It is necessary to determine the H(X) which results in the maximum load capacity W. Rayleigh [ll] considered this problem for Newtonian lubricants. Using the calculus of variations, he found that the optimum profile (the profile which makes W a maximum) was of the stepped form shown in Fig. 2.
Fig. 2. Rayleigh step.
Following Rayleigh [ 111, let H in eqns. (19) and (20) become H + 6H where 6 H is an admissible variation in H. Then Q and W become, respectively, &+6QandW+6W. Thus, 11 F+(H-Q)g 0
aQjFdX= 0
6HdX I
&‘W=6QjXFdX-f/F+(H-Q)d&5HdX 0 0 Elimination of 6 Q from eqns. (21) and (22) yields the variation the variation of H :
(21)
(22) of W due to
1
6W= 0J1
F+(H-Q)~@XFdX/%‘dX)-X/SHdX
(23)
From this equation it is possible to satisfy the optimum condition by a profile of stepped form (Fig. 2) in the present case also. If H = 1 on the lower step and if F + (H - Q)(dF/dH) = 0 on the upper step, every admissible H from this stepped form diminishes W. No matter how 6 H varies along the upper step, there is no contribution to 6 W. In contrast, an admissible 6 H can only be non-negative along the lower step since H must not be less than 1 and thus only a non-positive contribution to 6 W can result. Therefore the optimum film profile must be of the stepped form. The step height ratio r and riser location s will depend on the parameters K and M. Indeed, r and s are obtained from eqn. (22) and F(r) + (r-Q)
0 dF (XHElzr =
FdX
(24)
(25)
6
K
0
K-O
2-
r: K-1
Fig. 3. Plots of step height
ratio r, riser location
Thus, r is a root of the transcendental
s and maximum
load W, us. M.
equation
ml] F+(H-1)sI+F3’2(H)=0
(26)
s = dm{#(l)
(27)
and the optimum
+ m}-’ load W, is given by
(28)
4. Discussion
and conclusions
The optimum slider profile is of the stepped form even for micropolar fluid lubricants. However the step height ratio r and riser location s depend on the parameters K and M. From eqn. (15), F increases monotonically with the coupling parameter K and decreases monotonically with M. Also F = T3 whenK=O or A4 = 00 and F = {l + (K/2)}W3 when M = 0. The transcendental eqn. (26) takes the simple form (2H- 3)a= 1 when K = 0 orM=OorM=m. Hence the classical results r = 1.866 and s = 0.718 are obtained. However the optimum load capacity depends on M; W, = 0.0343 for K = 0 or M = 00and W, = 0.0343 (1 + K/2) for M = 0.
7
As the parameter M is the ratio of the characteristic linear dimension of the flow to the particle size. of a micropolar fluid, the classical results of Stokesian fluid theory are recovered from the present theory as it4 + ~0, i.e. as the particle size becomes negligible compared with the characteristic linear dimension of the fI ow. However in lubrication theory dealing with flow through narrow recesses, M- ’ may not be negligible and the effect of M can be considerable on the bearing characteristics. For example, with K = 1 and M = 5, W, = 0.0357 which is an increase of more than 4% over the classical value W, = 0.0343. The plots of r, s and W, versus M for K = 0 and 1 are shown in Fig. 3; r and s are virtually insensitive to M but the optimum load increases with K and decreases with M.
References 1 2 3 4 5 6 7 8 9 10 11 12
A. C. Eringen, J. Math. Mech., 16 (1966) 1. S. C. Cowin, Phys. Fluids, 11 (1968) 1919. S. J. Allen and K. A. Kline, Trans. ASME (E), (1971) 646. M. S. Khader and R. I. Vachon, Trans. ASME (F), (1973) 104. J. Prakash and Prawal Sinha, Int. J. Eng. Sci., (1975) 217. G. Ramanaiah and J. N. Dubey, Wear, 32 (1975) 343. A. B. Datta, Jpn. J. Appl. Phys., 11 (1972) 98. V. K. Agrawal, K. L. Ganju and S. C. Jethi, Wear, 19 (1972) 259. G. Maiti, Jpn. J. Appl. Phys.. 12 (1973) 1052. G. Maiti, Jpn. J. Appl. Phys., 13 (1974) 1440. Lord Rayleigh, Phitos. Mag., 35 (1918) 1. 0. Pinkus and B. Sternlicht, Theory of Hydrodynamic Lubrication, McGraw-Hi& New York, 1961.