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Effect of velocity slip on the squeeze film between rotating porous annular discs
73
Wear, 38 (1976) 73 - 85 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
EFFECT OF VELOCITY SLIP ON THE SQUEEZE ROTATING POROUS ANNULAR DISCS
J. PRAKASH* Department (India)
FILM BETWEEN
and S. K. VIJ
of Mathematics, Indian Institute of Technology,
Powai, Bombay 400076
(Received February 17, 1975; in final form October 2, 1975)
Summary The analysis of the squeeze film between two rotating annular discs, one with a porous facing, is extended to include the effect of velocity slip at the porous surface through the Beavers-Joseph slip model. The problem is solved analytically using the separation of variables method. The effect of slip is to reduce the load capacity and the response time of the squeeze film.
Introduction Squeeze films between porous plates have been analysed by many authors [ 1 -81. Wu [ 1, 21, Prakash and Vij [ 31 and Murti [4] considered the normal approach of non-rotating parallel plates using the conventional no-slip condition at the porous surface. The effect of velocity slip at the porous interface was taken into account in squeeze film studies by Sparrow et al. [ 51, Wu [ 61 and Prakash and Vij [ 71. None of these analyses considered relative sliding between the two surfaces and the inertia effects. Recently, Wu [ 81 analysed squeeze films between rotating discs for annular geometry and considered inertia effects due to rotation. The usual no-slip condition at the porous surface was assumed, which does not comply with the actual physical situation [9, lo]. The present investigation is concerned with squeeze films between rotating porous discs and is aimed at generalizing Wu’s analysis [8] to account for the effect of velocity slip at the surface of the porous material. The problem is formulated and solved analytically. Results for dimensionless load capacity for various sets of values of operating parameters are computed and illustrated. The effect of rotation is to reduce load capacity and time of approach of the discs. Consideration of velocity slip at the porous surface further diminishes load capacity and hence the response time of thesqueeze film. *Present address: Department of Machine Design, NTH-SINTEF, The Engineering Research Foundation at The Technical University of Norway, Trondheim, Norway.
NON-POROUS HOUSING POROUS FILM
DISC REBION
NON-POROUS
Fig. 1. Geometry
Mathematical
and coordinates
DISC
of the problem.
formulation
Figure 1 is a schematic diagram of the squeeze film between rotating discs. The upper disc, made of porous material, rotates with an angular velocity & and the non-porous lower disc rotates with an angular velocity Ri. The usual assumptions pertaining to thin film lubrication are made. It is assumed that all the inertia terms except the centrifugal force term can be neglected; flow in the film and the porous region is axisymmetric. On the basis of these assumptions, the Navier-Stokes equations and the continuity equation reduce respectively to
a2u 1 dp _-=.----~ az2 P dr
pv2
(1)
v
a2v/az2= 0 apia2 1 r
a ar
Boundary
(2)
= 0 (ru)
(31 +
z=
conditions
0
for the velocity
(41
components
in the film region are
75
u(r,d, 0)
=0
u(r,8, 0)
=rSZ,
w(r,0, 0)
=0
j
(5)
brr
(6)
o
(7)
atthe non-porous disc (z = 0). At the interface of the porous disc (z = h), taking account of the velocity slip through the BeaversJoseph slip model [ 91, the velocity components are given by
u(r, 8, h) = r& zu(r, 8, h) = w’(r, 8, h) where @ is the permeability of the porous material and (IL,a dimensionless quantity, is the slip coefficient, which depends on the characteristics of the porous material and is independent of gap height and lubricant properties [9]. u(r, 8, h) is the slip velocity in the radial direction at the porous interface, which was assumed to be zero in Wu’s analysis [8]. u’, u’ and W’ are velocity components in the porous region obtained by Darcy’s law. Assuming that the porous matrix is homogeneous and isotropic and that the fluid in the porous region rotates at the same velocity as the porous disc, Darcy’s law may be stated as u’ =--_
G P
ap
(ar
1
----pra2Q
v’ = r-flu
(11) (12)
dh wI -__-!-_!.!_ dt
(13)
P a2
where P is the pressure in the porous region and is given by Poisson’s equation (obtained from eqns. (11) - (13) and the continuity equation):
ia
_~
r ar
ap
( 1 r_
ar
2
+ $=
2p@
(14)
Integrating eqn. (2) twice with respect to z and using the boundary conditions (6) and (9), (15) Substituting for u from eqn. (15), on integration and using the boundary conditions (5) and (8) eqn. (1) yields
‘76
z3 -- h3 - 3h3& --_“_____ 12h2
.z2 - h2 --- 2h2 Co fin,2+ -.-___%a 3h
+
(16)
where &
=s s+h
c, =
3s(k
(17) + 2a2s)
(18)
h(s + I-Y) ($112
SE-
&!?
(19)
aho
ho
Integrating the continuity eqn. (4) with respect to z after substituting for u from eqn. (16) and using the boundary conditions (7) and (lo), the equation for determining the film pressure is (l+
Z1)‘%
= 2p fJ;2,2(X1-3Zo)
r> (
1
+ (1’: 4 &J&Qi
+ 2P
;
(1+ 5 c&2,2 +
I
+ (1 + 3 X0) n,”
+ I (20)
a=h
The boundary conditions region are [8]
for pressure in the porous region and in the film
P(a, 2) = 0
(21)
P(b, 2) = 0
(22)
= 0
(23)
r=h+H
P(r, h) =p(r)
(24)
Using eqn. (24), eqns. (21) and (22) reduce to p(u) = P(a, h) = 0
(25)
= P(b, h) = 0
(26)
p(6)
77
It is required either to solve eqns. (14) and (20) simultaneously with boundary conditions (21) - (26) or to solve eqn. (14) with conditions (20) (23) and then use condition (24) to obtain the film pressure. Solutions Equation with boundary
(14) is solved, using the separation conditions (21) - (23) to yield
of variables technique, m
Pa2
r2 - b2 - (a2 -- b”)3 W/b)
p*=L2
I
+ 1
C,exp(a,z)
x
PI=1
&(a,r)
X [l + exp {2a,(h + H-z)}]
(27)
where &(a&
= K&MJ&kr)
The (IL,are eigenvalues U,(ct,b) = 0
-
(28)
J0(fwWhw-)
and are given by n = 1, 2, 3, . . .
(2%
The (zonstants C, are determined using eqn. (20). The film pressure is obtained from eqns. (24) and (27) as P=-
PJX
r2-b2-(a2-b2)-
2
ln(r/b) ln(alb) I
_
+ C
G ew(d)
X
n=l
(30)
X 11+ exp (%fW-40 (w) Substituting for P from eqn. (27) and for p from eqn. (30) into eqn. (20), after rearrangement .azexp(a,h){l CC ?I=1
+ exp(2a,H))(l-
[,)UO(a,r) = -
12/A (1 - PA) P(l + Xc,) (31)
where &I =
m3
1 - exp(2cw,H)
(32)
(Y, h3 (1+ X1) 1 + exp(%lJi)
and
x= By using the orthogonality are given by
(33) of the eigenfunctions
Uo(a,r), the constants
C,,
Substituting eqn. (34) into eqn. (30), the non-dimensional distribution in the film region is ~=.
j-_l_(k2-l)ln~
+12n(l--‘A) I
In k
I
%f
(34)
x C,)
Jo(G) Mom=) Cl-- tn) {Jo(&) + Jd&k)~
XC n=l
7i3 (1+
pressure
(35)
where
(36) F = r/b
k = a/b
ii = H/b
12 \I’0 &I-,=[” =
& h3(l
&- = h/h,
I
1 -- exp( 2&H) (37)
+ Zr) i + exp(2&H)
and i&O= $ b/h:
(38)
The load-carrying capacity the bearing area. Thus
is obtained
by integrating
the film pressure
(I
w= 271
(39)
s p(r) r dr b
Substituting given by
for p and integrating,
W@ = ___ /J
jjF=
--
715
b4A
the non-dimensional
load capacity -is
k4_1-!k2-1)2 In k
2
where 48~
Jo(&) - Jo(& k) --
%I
Jo(G)
A,, =- -4
The film thickness eqn. (40) in the form
+
Jdcu,
(41)
k)
and the time relation
can be obtained
by rearranging
79
c L-)-l n=I 1-L
M
IK3=o
(42)
where L=
np b4
4w
Ci:
1 k4
_
1
_
(k2- 1J2 In k
I
(43)
(44) and integrating with respect to t for given initial conditions (h(t,) h( t2) = h2 ) and given load. Thus
= hl and
In eqn. (45), C1,
80
Fig. 2. Non-dimensional load capacity k us. permeability parameter *g for various values of the slip coefficient 01; Al, = al.
decreases with increasing \ko for both slip and no-slip cases. The effect of slip is to decrease the load capacity. The relative decrease in load capacity reduces on increasing cy (for a fixed qo). The curves also show that at some value of *o the load capacity becomes zero, i.e. there is no squeeze action and the upper plate falls down instantly. If q. is increased further, the load capacity becomes negative, i.e. instead of restricting the falling ofthe upper plate, the squeeze actions hastens the downward motion. The reason for the load capacity becoming zero or negative may be explained by eqn. (40). The contribution of the first term, which contains the rotation parameter but not the permeability parameter or the slip parameter, is negative, while the contribution of the second term, which for the case of 52, = 52, contains the permeability parameter and the slip parameter but not the rotation parameter, is positive. As the permeability parameter Q. increases, the positive contribution of the second term diminishes until this contribution becomes equal to or less than the negative contribution of the first term, so that the load capacity becomes zero or negative. Figure 3 shows load capacity uersus \ko for various values of the rotation parameter CJ.Increase in u decreases the load capacity linearly for both the slip and no-slip cases. For this case (8, = a,, i.e. h = 0), the effect of slip as well as that of permeability is independent of the rotation parameter CJ(eqn. (40) or Fig. 3). However, the relative significance of the inertia effect increases with increasing \ko because the effect of increasing q. is to decrease the load capacity.
81 l0.C
r
I ---------
--
A
WITH
- ‘“?O
- -_--_-_
--
\
\
. 8.0
I
\
2.0
x.0.0
NO-SLIP
H-0.01 M-
\
k=
i.1.0,
SLIP
0.1,1,/b
= O.ool
I
\ ‘1 \ \
6.C
\ \ *(-
\
b_ 0.1 31
I 0.1
I
1.c
la0
Fig. 3. Non-dimensional load capacity %’ vs. permeability parameter q. for various values of the rotation parameter u ;
CZ,= al.
Figure 4 shows that load capacity decreases with increasing fi for all \ko in both the slip and no-slip cases. For low values of q. (\ko < O.Ol), the decrease in load capacity is insignificant; however, for higher values of \ko it is considerable and must be taken into account in the design of porous b_earings. The dependence of Bon the load capacity is insignificant when H> 0.4 (q. G 0.1) or H> 0.5 (e. > 0.1). This critical value of His the value beyond which flow in the porous wall is governed only by the permeability of the material and not by the wall thickness. For a constant normal velocity, the instantaneous non-dimensional load capacity w is plotted against dimensionless film thickness h for various values of \ko in Fig. 5. The load capacity increases as the plates approach each other for both the slip and no-slip cases. Thus as 6 decreases, the resistance encountered by fluid flowing in the gap increases and hence the time of approach of the plates increases. The effect of slip is to decrease the load capacity for all &, implying thereby that the response time of the squeeze film is reduced on considering the_veJocity slip at the porous surface. Slip effects are not significant at low h (h < 0.04) for \ko > 0.01 since, at low
82
7.0
6.0
\
5.0 G
4.0
3.C
2.C
,
1.c
OS
‘t bi
0.1
a2
0.3
0.4
0.5
0.6
0.7
0.8
Fi
Fig. 4. Non-dimensional load capacity ~VS. dimensionless porous disc thickness fi for various values of the permeability parameter qo; 52, = Q.
h, dp/dh increases rapidly while dp/dh decreases; therefore, the fluid flow into the porous disc dominates the radial flow in the film. For cases where s2, + (nl, Fig. 6 shows the load capacity as a function of, the permeability parameter \ko for various sets of values of 52, and Q. Corresponding no-slip curves are also shown. The following observations are made. (i) The curves corresponding to 52, = RI = 0 and S’12,= Q = 7000 are the upper bound and lower bound, respectively, of the set of curves in Fig. 6 for both the slip and no-slip cases. Thus the effect of inertia is greatest for S2,, = I& (= Sl, say) and the curves of Figs. 2 - 5 are lower bounds of load capacity for all cases, provided that the maximum value of (a,,, Q,) does not exceed a. (ii) For a constant value of &, slip effects are more prominent for Q, > S’&than for QU < S& since the (radial) slip velocity at the porous disc surface increases on increasing QU. (iii) With rotating non-porous discs, the load capacity of the squeeze film is the same whether the upper or the lower disc rotates. This is evident from Fig. 6, where the curves corresponding to 52, = 7000, Szr = 0 and fiU = 0, Q 7 7000 almost coincide in the domain q. < 0.05. However, with rotating
a3
000
Fig. 5. Non-dimensional film thickness /i us. dimensionless load capacity values of the permeability parameter qo; a, = al.
w for
various
porous discs, if only the porous disc rotates, the fluid in both the film and the porous regions rotates so that the loss of fluid due to centrifugal force increases. Thus the load capacity of the squeeze film when only the upper (porous) disc rotates is less than that when only the lower (non-porous) disc rotates (Fig. 6). (iv) Comparing the curves corresponding to a2, = 7000, SZr= 0 and Sl, = 1000, QI = 8000, the load capacity in the former is greater than that in the latter up to a certain \ko;after this value of \k,, it is lower because the loss of fluid due to centrifugal force is less in the film region and greater in the porous region for the former than for the latter. Conclusions A more rational analysis of the squeeze film between rotating annular discs, one of which is porous, has been presented. The following points may be useful for the design of such bearings. (1) The response time of squeeze films is less than that predicted by the no-slip analysis. (2) As fi decreases, the response time of the squeeze film increases; however, this decrease in if results in a reduction in the strength of the porous disc. Therefore, a compromise between the response time and the disc strength should be made depending on the type of duty such bearings are required to perform. (3) Where the two discs rotate at different velocities, that rotating at a lower velocity should be made of porous material.
84
i=l.o,k=
2.0
ti = o-01 --NO-SLIP
d = 0.1,
&./b=O.OOl
4
-
2 \’
\’
‘\’
‘\\ ‘\ \ ‘\\
’
\’
+
\
1”
\\’
\
\
\\’ \\
k\
\
’ \ \ \
I.0 Fig. 6. Non-dimensional load capacity tus. of values of a,, and ~21.
100.0
permeability parameter *O for various sets
Nomenclature a
An, G b h h-0 h
L h2
ri H
R 4, Yo
outer radius of disc constants defined in eqns. (41) and (34), respectively inner radius of disc instantaneous film thickness initial film thickness h/h, dimensionless film thickness at time tI and t2, respectively dh/dt porous disc thickness H/b
Bessel functions
of order zero
k
a/b
-5 M P
defined in eqns. (43) and (44), respectively instantaneous film pressure
P
-ph;/pb2h
85
instantaneous pressure in the porous region space coordinates r/b (@)“‘/aho time t2 - tl, the time required to reduce the film thickness from hl to k? radial, tangential and axial components, respectively, of the fluid velocity in the film region radial, tangential and axial components, respectively, of the fluid velocity in the porous region eigenfunction defined in eqn. (28) instantaneous load capacity -Wh$pb41i slip coefficient nth eigenvalue defined in eqn. (29) ba, azimuthal coordinate defined in eqn. (33) absolute viscosity of the fluid defined in eqns. (32) and (37), respectively fluid density -p Ci: h$/2 /.di defined in eqns. (17) and (18), respectively permeability of the porous facing permeability parameter #b/h: angular speeds of the upper and lower discs, respectively %I -
s-4
References 1 H. Wu, Squeeze film behaviour for porous annular disks, Trans. ASME, F92 (1970) 593 - 596. 2 H. Wu, Analysis of the squeeze film between porous rectangular plates, Trans. ASME, F94 (1972) 64 - 68. 3 J. Prakash and S. K. Vij, Load capacity and time-height relations for squeeze films between porous plates, Wear, 24 (1973) 309 - 322. 4 P. R. K. Murti, Squeeze films in porous circular discs, Wear, 23 (1973) 283 - 289. 5 E. M. Sparrow, G. S. Beavers and I. T. Hwang, Effect of velocity slip on porouswalled squeeze films, Trans. ASME, F94 (1972) 260 - 265. 6 H. Wu, Effect of velocity slip on the squeeze film between porous rectangular plates, Wear, 20 (1972) 67 - 71. 7 J. Prakash and S. K. Vij, Effect of velocity slip on porous-walled squeeze films, Wear, 29 (1974) 363 - 372. 8 H. Wu, The squeeze film between rotating porous annular disks, Wear, 18 (1971) 461- 470. 9 G. S. Beavers and D. D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967) 197 - 207. 10 G. S. Beavers, E. M. Sparrow and R. A. Magnuson, Experiments on coupled parallel flows in a channel and a bounding porous medium, Trans. ASME, D92 (1970) 843 848.
Wear, 38 (1976) 73 - 85 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
EFFECT OF VELOCITY SLIP ON THE SQUEEZE ROTATING POROUS ANNULAR DISCS
J. PRAKASH* Department (India)
FILM BETWEEN
and S. K. VIJ
of Mathematics, Indian Institute of Technology,
Powai, Bombay 400076
(Received February 17, 1975; in final form October 2, 1975)
Summary The analysis of the squeeze film between two rotating annular discs, one with a porous facing, is extended to include the effect of velocity slip at the porous surface through the Beavers-Joseph slip model. The problem is solved analytically using the separation of variables method. The effect of slip is to reduce the load capacity and the response time of the squeeze film.
Introduction Squeeze films between porous plates have been analysed by many authors [ 1 -81. Wu [ 1, 21, Prakash and Vij [ 31 and Murti [4] considered the normal approach of non-rotating parallel plates using the conventional no-slip condition at the porous surface. The effect of velocity slip at the porous interface was taken into account in squeeze film studies by Sparrow et al. [ 51, Wu [ 61 and Prakash and Vij [ 71. None of these analyses considered relative sliding between the two surfaces and the inertia effects. Recently, Wu [ 81 analysed squeeze films between rotating discs for annular geometry and considered inertia effects due to rotation. The usual no-slip condition at the porous surface was assumed, which does not comply with the actual physical situation [9, lo]. The present investigation is concerned with squeeze films between rotating porous discs and is aimed at generalizing Wu’s analysis [8] to account for the effect of velocity slip at the surface of the porous material. The problem is formulated and solved analytically. Results for dimensionless load capacity for various sets of values of operating parameters are computed and illustrated. The effect of rotation is to reduce load capacity and time of approach of the discs. Consideration of velocity slip at the porous surface further diminishes load capacity and hence the response time of thesqueeze film. *Present address: Department of Machine Design, NTH-SINTEF, The Engineering Research Foundation at The Technical University of Norway, Trondheim, Norway.
NON-POROUS HOUSING POROUS FILM
DISC REBION
NON-POROUS
Fig. 1. Geometry
Mathematical
and coordinates
DISC
of the problem.
formulation
Figure 1 is a schematic diagram of the squeeze film between rotating discs. The upper disc, made of porous material, rotates with an angular velocity & and the non-porous lower disc rotates with an angular velocity Ri. The usual assumptions pertaining to thin film lubrication are made. It is assumed that all the inertia terms except the centrifugal force term can be neglected; flow in the film and the porous region is axisymmetric. On the basis of these assumptions, the Navier-Stokes equations and the continuity equation reduce respectively to
a2u 1 dp _-=.----~ az2 P dr
pv2
(1)
v
a2v/az2= 0 apia2 1 r
a ar
Boundary
(2)
= 0 (ru)
(31 +
z=
conditions
0
for the velocity
(41
components
in the film region are
75
u(r,d, 0)
=0
u(r,8, 0)
=rSZ,
w(r,0, 0)
=0
j
(5)
brr
(6)
o
(7)
atthe non-porous disc (z = 0). At the interface of the porous disc (z = h), taking account of the velocity slip through the BeaversJoseph slip model [ 91, the velocity components are given by
u(r, 8, h) = r& zu(r, 8, h) = w’(r, 8, h) where @ is the permeability of the porous material and (IL,a dimensionless quantity, is the slip coefficient, which depends on the characteristics of the porous material and is independent of gap height and lubricant properties [9]. u(r, 8, h) is the slip velocity in the radial direction at the porous interface, which was assumed to be zero in Wu’s analysis [8]. u’, u’ and W’ are velocity components in the porous region obtained by Darcy’s law. Assuming that the porous matrix is homogeneous and isotropic and that the fluid in the porous region rotates at the same velocity as the porous disc, Darcy’s law may be stated as u’ =--_
G P
ap
(ar
1
----pra2Q
v’ = r-flu
(11) (12)
dh wI -__-!-_!.!_ dt
(13)
P a2
where P is the pressure in the porous region and is given by Poisson’s equation (obtained from eqns. (11) - (13) and the continuity equation):
ia
_~
r ar
ap
( 1 r_
ar
2
+ $=
2p@
(14)
Integrating eqn. (2) twice with respect to z and using the boundary conditions (6) and (9), (15) Substituting for u from eqn. (15), on integration and using the boundary conditions (5) and (8) eqn. (1) yields
‘76
z3 -- h3 - 3h3& --_“_____ 12h2
.z2 - h2 --- 2h2 Co fin,2+ -.-___%a 3h
+
(16)
where &
=s s+h
c, =
3s(k
(17) + 2a2s)
(18)
h(s + I-Y) ($112
SE-
&!?
(19)
aho
ho
Integrating the continuity eqn. (4) with respect to z after substituting for u from eqn. (16) and using the boundary conditions (7) and (lo), the equation for determining the film pressure is (l+
Z1)‘%
= 2p fJ;2,2(X1-3Zo)
r> (
1
+ (1’: 4 &J&Qi
+ 2P
;
(1+ 5 c&2,2 +
I
+ (1 + 3 X0) n,”
+ I (20)
a=h
The boundary conditions region are [8]
for pressure in the porous region and in the film
P(a, 2) = 0
(21)
P(b, 2) = 0
(22)
= 0
(23)
r=h+H
P(r, h) =p(r)
(24)
Using eqn. (24), eqns. (21) and (22) reduce to p(u) = P(a, h) = 0
(25)
= P(b, h) = 0
(26)
p(6)
77
It is required either to solve eqns. (14) and (20) simultaneously with boundary conditions (21) - (26) or to solve eqn. (14) with conditions (20) (23) and then use condition (24) to obtain the film pressure. Solutions Equation with boundary
(14) is solved, using the separation conditions (21) - (23) to yield
of variables technique, m
Pa2
r2 - b2 - (a2 -- b”)3 W/b)
p*=L2
I
+ 1
C,exp(a,z)
x
PI=1
&(a,r)
X [l + exp {2a,(h + H-z)}]
(27)
where &(a&
= K&MJ&kr)
The (IL,are eigenvalues U,(ct,b) = 0
-
(28)
J0(fwWhw-)
and are given by n = 1, 2, 3, . . .
(2%
The (zonstants C, are determined using eqn. (20). The film pressure is obtained from eqns. (24) and (27) as P=-
PJX
r2-b2-(a2-b2)-
2
ln(r/b) ln(alb) I
_
+ C
G ew(d)
X
n=l
(30)
X 11+ exp (%fW-40 (w) Substituting for P from eqn. (27) and for p from eqn. (30) into eqn. (20), after rearrangement .azexp(a,h){l CC ?I=1
+ exp(2a,H))(l-
[,)UO(a,r) = -
12/A (1 - PA) P(l + Xc,) (31)
where &I =
m3
1 - exp(2cw,H)
(32)
(Y, h3 (1+ X1) 1 + exp(%lJi)
and
x= By using the orthogonality are given by
(33) of the eigenfunctions
Uo(a,r), the constants
C,,
Substituting eqn. (34) into eqn. (30), the non-dimensional distribution in the film region is ~=.
j-_l_(k2-l)ln~
+12n(l--‘A) I
In k
I
%f
(34)
x C,)
Jo(G) Mom=) Cl-- tn) {Jo(&) + Jd&k)~
XC n=l
7i3 (1+
pressure
(35)
where
(36) F = r/b
k = a/b
ii = H/b
12 \I’0 &I-,=[” =
& h3(l
&- = h/h,
I
1 -- exp( 2&H) (37)
+ Zr) i + exp(2&H)
and i&O= $ b/h:
(38)
The load-carrying capacity the bearing area. Thus
is obtained
by integrating
the film pressure
(I
w= 271
(39)
s p(r) r dr b
Substituting given by
for p and integrating,
W@ = ___ /J
jjF=
--
715
b4A
the non-dimensional
load capacity -is
k4_1-!k2-1)2 In k
2
where 48~
Jo(&) - Jo(& k) --
%I
Jo(G)
A,, =- -4
The film thickness eqn. (40) in the form
+
Jdcu,
(41)
k)
and the time relation
can be obtained
by rearranging
79
c L-)-l n=I 1-L
M
IK3=o
(42)
where L=
np b4
4w
Ci:
1 k4
_
1
_
(k2- 1J2 In k
I
(43)
(44) and integrating with respect to t for given initial conditions (h(t,) h( t2) = h2 ) and given load. Thus
= hl and
In eqn. (45), C1,
80
Fig. 2. Non-dimensional load capacity k us. permeability parameter *g for various values of the slip coefficient 01; Al, = al.
decreases with increasing \ko for both slip and no-slip cases. The effect of slip is to decrease the load capacity. The relative decrease in load capacity reduces on increasing cy (for a fixed qo). The curves also show that at some value of *o the load capacity becomes zero, i.e. there is no squeeze action and the upper plate falls down instantly. If q. is increased further, the load capacity becomes negative, i.e. instead of restricting the falling ofthe upper plate, the squeeze actions hastens the downward motion. The reason for the load capacity becoming zero or negative may be explained by eqn. (40). The contribution of the first term, which contains the rotation parameter but not the permeability parameter or the slip parameter, is negative, while the contribution of the second term, which for the case of 52, = 52, contains the permeability parameter and the slip parameter but not the rotation parameter, is positive. As the permeability parameter Q. increases, the positive contribution of the second term diminishes until this contribution becomes equal to or less than the negative contribution of the first term, so that the load capacity becomes zero or negative. Figure 3 shows load capacity uersus \ko for various values of the rotation parameter CJ.Increase in u decreases the load capacity linearly for both the slip and no-slip cases. For this case (8, = a,, i.e. h = 0), the effect of slip as well as that of permeability is independent of the rotation parameter CJ(eqn. (40) or Fig. 3). However, the relative significance of the inertia effect increases with increasing \ko because the effect of increasing q. is to decrease the load capacity.
81 l0.C
r
I ---------
--
A
WITH
- ‘“?O
- -_--_-_
--
\
\
. 8.0
I
\
2.0
x.0.0
NO-SLIP
H-0.01 M-
\
k=
i.1.0,
SLIP
0.1,1,/b
= O.ool
I
\ ‘1 \ \
6.C
\ \ *(-
\
b_ 0.1 31
I 0.1
I
1.c
la0
Fig. 3. Non-dimensional load capacity %’ vs. permeability parameter q. for various values of the rotation parameter u ;
CZ,= al.
Figure 4 shows that load capacity decreases with increasing fi for all \ko in both the slip and no-slip cases. For low values of q. (\ko < O.Ol), the decrease in load capacity is insignificant; however, for higher values of \ko it is considerable and must be taken into account in the design of porous b_earings. The dependence of Bon the load capacity is insignificant when H> 0.4 (q. G 0.1) or H> 0.5 (e. > 0.1). This critical value of His the value beyond which flow in the porous wall is governed only by the permeability of the material and not by the wall thickness. For a constant normal velocity, the instantaneous non-dimensional load capacity w is plotted against dimensionless film thickness h for various values of \ko in Fig. 5. The load capacity increases as the plates approach each other for both the slip and no-slip cases. Thus as 6 decreases, the resistance encountered by fluid flowing in the gap increases and hence the time of approach of the plates increases. The effect of slip is to decrease the load capacity for all &, implying thereby that the response time of the squeeze film is reduced on considering the_veJocity slip at the porous surface. Slip effects are not significant at low h (h < 0.04) for \ko > 0.01 since, at low
82
7.0
6.0
\
5.0 G
4.0
3.C
2.C
,
1.c
OS
‘t bi
0.1
a2
0.3
0.4
0.5
0.6
0.7
0.8
Fi
Fig. 4. Non-dimensional load capacity ~VS. dimensionless porous disc thickness fi for various values of the permeability parameter qo; 52, = Q.
h, dp/dh increases rapidly while dp/dh decreases; therefore, the fluid flow into the porous disc dominates the radial flow in the film. For cases where s2, + (nl, Fig. 6 shows the load capacity as a function of, the permeability parameter \ko for various sets of values of 52, and Q. Corresponding no-slip curves are also shown. The following observations are made. (i) The curves corresponding to 52, = RI = 0 and S’12,= Q = 7000 are the upper bound and lower bound, respectively, of the set of curves in Fig. 6 for both the slip and no-slip cases. Thus the effect of inertia is greatest for S2,, = I& (= Sl, say) and the curves of Figs. 2 - 5 are lower bounds of load capacity for all cases, provided that the maximum value of (a,,, Q,) does not exceed a. (ii) For a constant value of &, slip effects are more prominent for Q, > S’&than for QU < S& since the (radial) slip velocity at the porous disc surface increases on increasing QU. (iii) With rotating non-porous discs, the load capacity of the squeeze film is the same whether the upper or the lower disc rotates. This is evident from Fig. 6, where the curves corresponding to 52, = 7000, Szr = 0 and fiU = 0, Q 7 7000 almost coincide in the domain q. < 0.05. However, with rotating
a3
000
Fig. 5. Non-dimensional film thickness /i us. dimensionless load capacity values of the permeability parameter qo; a, = al.
w for
various
porous discs, if only the porous disc rotates, the fluid in both the film and the porous regions rotates so that the loss of fluid due to centrifugal force increases. Thus the load capacity of the squeeze film when only the upper (porous) disc rotates is less than that when only the lower (non-porous) disc rotates (Fig. 6). (iv) Comparing the curves corresponding to a2, = 7000, SZr= 0 and Sl, = 1000, QI = 8000, the load capacity in the former is greater than that in the latter up to a certain \ko;after this value of \k,, it is lower because the loss of fluid due to centrifugal force is less in the film region and greater in the porous region for the former than for the latter. Conclusions A more rational analysis of the squeeze film between rotating annular discs, one of which is porous, has been presented. The following points may be useful for the design of such bearings. (1) The response time of squeeze films is less than that predicted by the no-slip analysis. (2) As fi decreases, the response time of the squeeze film increases; however, this decrease in if results in a reduction in the strength of the porous disc. Therefore, a compromise between the response time and the disc strength should be made depending on the type of duty such bearings are required to perform. (3) Where the two discs rotate at different velocities, that rotating at a lower velocity should be made of porous material.
84
i=l.o,k=
2.0
ti = o-01 --NO-SLIP
d = 0.1,
&./b=O.OOl
4
-
2 \’
\’
‘\’
‘\\ ‘\ \ ‘\\
’
\’
+
\
1”
\\’
\
\
\\’ \\
k\
\
’ \ \ \
I.0 Fig. 6. Non-dimensional load capacity tus. of values of a,, and ~21.
100.0
permeability parameter *O for various sets
Nomenclature a
An, G b h h-0 h
L h2
ri H
R 4, Yo
outer radius of disc constants defined in eqns. (41) and (34), respectively inner radius of disc instantaneous film thickness initial film thickness h/h, dimensionless film thickness at time tI and t2, respectively dh/dt porous disc thickness H/b
Bessel functions
of order zero
k
a/b
-5 M P
defined in eqns. (43) and (44), respectively instantaneous film pressure
P
-ph;/pb2h
85
instantaneous pressure in the porous region space coordinates r/b (@)“‘/aho time t2 - tl, the time required to reduce the film thickness from hl to k? radial, tangential and axial components, respectively, of the fluid velocity in the film region radial, tangential and axial components, respectively, of the fluid velocity in the porous region eigenfunction defined in eqn. (28) instantaneous load capacity -Wh$pb41i slip coefficient nth eigenvalue defined in eqn. (29) ba, azimuthal coordinate defined in eqn. (33) absolute viscosity of the fluid defined in eqns. (32) and (37), respectively fluid density -p Ci: h$/2 /.di defined in eqns. (17) and (18), respectively permeability of the porous facing permeability parameter #b/h: angular speeds of the upper and lower discs, respectively %I -
s-4
References 1 H. Wu, Squeeze film behaviour for porous annular disks, Trans. ASME, F92 (1970) 593 - 596. 2 H. Wu, Analysis of the squeeze film between porous rectangular plates, Trans. ASME, F94 (1972) 64 - 68. 3 J. Prakash and S. K. Vij, Load capacity and time-height relations for squeeze films between porous plates, Wear, 24 (1973) 309 - 322. 4 P. R. K. Murti, Squeeze films in porous circular discs, Wear, 23 (1973) 283 - 289. 5 E. M. Sparrow, G. S. Beavers and I. T. Hwang, Effect of velocity slip on porouswalled squeeze films, Trans. ASME, F94 (1972) 260 - 265. 6 H. Wu, Effect of velocity slip on the squeeze film between porous rectangular plates, Wear, 20 (1972) 67 - 71. 7 J. Prakash and S. K. Vij, Effect of velocity slip on porous-walled squeeze films, Wear, 29 (1974) 363 - 372. 8 H. Wu, The squeeze film between rotating porous annular disks, Wear, 18 (1971) 461- 470. 9 G. S. Beavers and D. D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967) 197 - 207. 10 G. S. Beavers, E. M. Sparrow and R. A. Magnuson, Experiments on coupled parallel flows in a channel and a bounding porous medium, Trans. ASME, D92 (1970) 843 848.