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Viscous shear effects on the squeeze-film behavior in porous circular disks
Pergamon
Int. J. Mech. Sci Vol. 38, No. 4, pp. 373 384, 1996
Copyright ~ff 1996 Elsevier Science Lid Printed in Great Britain. All rights reserved 0020-.7403/96 $15.00 + 0.00
0020-7403(95)00066-6
VISCOUS SHEAR EFFECTS ON THE SQUEEZE-FILM BEHAVIOR IN POROUS CIRCULAR DISKS JAW-REN LIN Department of Mechanical Engineering, Nanya Junior College, 414, Sec. 3, Chung-Shang East Road, Chung-Li 320-34, Taiwan, R.O.C. {Received 10 November 1994; and in revisedform 22 June 1995) Abstract--On the basis of the Brinkman model (BM), the main objective of this paper is to predict the viscous shear effects on the squeeze-filmcharacteristics betweenporous circular disks. According to the results obtained, the influences of viscous shear stresses on the squeeze-film behavior are significantand not negligible.It is found that the BM predicts quite different squeeze-filmaction to those derived by using the Slip-flowmodel (SFM) and Darcy model (DM). When comparing with that of the SFM, the viscous shear effects of the BM provide an enhancement in the load-carrying capacity and, then, increase the response time of the squeeze-film behavior; but these trends are reversed as compared to that derived by using the DM.
NOTATION h film thickness ho filmthickness at t = 0 nondimensional film thickness, h/ho H thicknessof porous facing /~ facing-radius ratio, H/R K permeability of porous facing material p film pressure /~ nondimensional film pressure, -ph3/l~R2h r, z radial coordinate and axial coordinate R radius of circular disk 7 nondimensional radial coordinate, r/R t time u, w radial and axial velocity components in the film region W load capacity 1~ nondimensional load capacity, - Wh3/pR4h 6o /z ~bo r ( ") ( )*
height-permeability ratio, ho/K t/2 lubricant viscosity permeability parameter, KH/h3o nondimensional time, Wh~t/#R 4 indicates the derivative with respect to time, t indicates the quantity within the porous medium 1. I N T R O D U C T I O N
Squeeze-film characteristics play a n i m p o r t a n t role in m a n y applications, such as lubrication of m a c h i n e elements, a u t o m a t i c transmissions, a n d artificial joints. The squeeze-film action comes from the b e h a v i o r of two lubricated surfaces a p p r o a c h i n g each other with a n o r m a l velocity. Since the viscous l u b r i c a n t c o n t a i n e d between these surfaces has a resistance to extrusion a n d c a n n o t be i n s t a n t a n e o u s l y squeezed out, a certain time is needed for these surfaces to come into contact. D u r i n g that interval, a pressure is built up a n d the load is, then, s u p p o r t e d by the squeeze-film. Traditionally, the study of the squeeze-film b e h a v i o r between two circular disks when one disk has a porous facing was based o n the D a r c y model (DM) where viscous shearing stresses are neglected [ 1 - 3 ] . It is f o u n d that the permeability p a r a m e t e r of p o r o u s facing adversely affects the load capacity a n d the response time needed for the circular disk to a t t a i n a given film height. In this D M , the fluid b e h a v i o r in the p o r o u s facing was a s s u m e d to obey Darcy's law a n d the no-slip c o n d i t i o n 373
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Jaw-Ren Lin
was used at the porous facing/lubricant film interface. However, Beavers and Joseph [4] found the experimental evidence of a tangential velocity slip of fluid on the porous wall and defined a dimensionless slip coefficient, 2, to modify the no-slip condition at the interface. According to their results, the value of ~ = 0.1 is used for the aloxite porous specimens, and the value of ~ = 0.78-4.01 is used for the foametal porous specimens. Beavers et al. [5] confirmed that the value of ~ = 0.1 should be used in the slip-flow formulation. But, Taylor [6] proposed that 2 = 1 should be used in the slip-flow model (SFM). Based on the SFM, many authors investigated the affect of velocity slip on the squeeze-films between porous disks [7-9]. According to their results, the effect of slip reduces the load capacity and the response time of the squeeze-films compared to that derived by using the DM. However, the SFM is valid only in a dense porous medium of large thickness so that the variation of velocity in it can be neglected. But, many porous squeeze-films involve porous layers of shallow depth. In these porous facings the velocity is no longer uniform and the distortion of velocity yields the viscous shearing stresses. Although the SFM introduces the effect of velocity slip at the interface, it yields a discontinuity of the tangential velocity component across the permeable surface. Therefore, the Darcy equation has to be modified to better describe the flow phenomena within the porous matrix. The Brinkman equation [10] accounting for the viscous shear effects and the viscous damping effects (Darcy resistance) can get around the above obstacles. By virtue of the viscous shear term contained in the Brinkman equation, the Brinkman model (BM) is mathematically and physically compatible with the a c t u a l velocity profile within the porous medium. Since the DM fails to predict the true characteristics of porous squeeze-films and the SFM cannot give an a c t u a l velocity profile within the porous medium, the BM seems to be appropriate to predict the more realistical performance of porous squeeze-films. Recently, many studies have employed the BM to study the fluid flow and heat transfer problems involving the porous media [11 15]. Based on this BM, Lin and Hwang [16 191 have examined the effects of viscous shear stresses on the performances of porous journal bearings. According to their results, the viscous shear effects improve the static and dynamic characteristics of porous journal bearings. However, we have no idea how the viscous shear stresses of the BM affect the squeeze-film characteristics for porous circular disks. Therefore, the study is needed. On the basis of the BM, the main objective of this paper is to predict the influences of the viscous shear effects on the squeeze-film behavior between porous circular disks. The modified Reynolds equation is derived by using the Brinkman equations to account for the viscous shear effects. The film pressure is solved and, then, applied to predict the loadcarrying capacity and the time-height relationship. To show the influences of the viscous shear effects on the squeeze-film performance, the results are compared with those derived by using the DM and SFM. The results of the limiting case (i.e., the value permeability parameter approaches zero) obtained from the BM give a good agreement with a nonporous disk and support the present prediction.
2. A N A L Y S I S
Consider the physical configuration of two circular disks shown as in Fig. 1, in which the upper disk has a porous facing. Assuming no variation of pressure across the fluid film, no external forces acting on the film, fluid inertia being small compared to the viscous shear stresses, and an incompressible Newtonian lubricant having constant properties, then the reduced Navier-Stokes equations and the continuity equation in the film region are given by the following ¢~p
c~Z u
& - ~ ~z z
(1)
nc'~= 0. #z
(2)
Viscous shear effectson the squeeze-filmbehavior
375
/
/
/
/
'\ \
~- FiLm
(
\
~Z T
!:i : !:i : !:; : !::l: i:i': ]
P°r°us e'ecing
!:; : !!i ~.
h
Fig. 1. Configurationof porous circular disks. For the porous region, it is assumed that the porous material is homogeneous and isotropic, the flow of lubricant in the porous facing is laminar, and the axial pressure gradient across the porous facing is a function of the radial coordinate, then the Brinkman equations gives ~:p*
?r
# -
K
?'u* u * + t~*
?z 2
(3)
?p*
(z
= f(r)
(4)
where ()* indicates the quantity within the porous medium; /~* denotes the effective viscosity of the fluid in the porous matrix which is assumed to be different in value from/~, the viscosity of the fluid layer; a n d f ( r ) is an unknown function to be determined. Since the pressure is continuous at the interface z = h, p(r) = p*(r, h)
(5)
integrating Eqn (4) with respect to z yields p*(r, z) = p(r) + (z - h)f(r).
(6)
To solve for the velocities u and u* of the lubricant in the film region and the porous region, we have the boundary conditions for the velocities, and the continuity conditions for the velocities and the viscous shear stresses u(r, 0) = 0
(7)
u*(r, h + H) = 0
(8)
utr, h) = u*(r, h) ~' ~i~u z=h = ~ , ~OU* - z z=h "
(9) (10)
Integrating the r-momentum equation with respect to z, we can obtain the expression of radial velocity in the film region, u, containing two integration constants. Substituting
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Jaw-Ren Lin
Eqn (6) into the r-Brinkman equation and solving the equation, also we have the expression of radial velocity within the porous medium, u*, containing the other two integration constants. Applying the boundary conditions (7) and (8) at z = 0 and z = h + H, the continuity condition of velocities (9), and the continuity condition of viscous shear stresses (10) at the interface z = h, the four integration constants can be obtained, Finally, we have the expressions of velocities u and u*, respectively, 4, xp
-~t(2+f2)[l+exp(-2--~)]-26II-exp(--2-~a~)l z} (--~),~6[1 + exp(-~) 1 + [1 - exp(--~)] K'/z
xK3"Zdffa I l - e x p ( - ~ ) l - 2 t r [ e x p ( - ~ ) l
+m__
p
dr l~t6[' + e x p ( - - ~ ) ] + [ 1 - e x p ( - ~ ) ]
z (11) K1/2
and
o.
['z/K
K dp 2(1 + ~o)exp~ u*(r,
z) = - - -
2p dr
+exp(
E, oxp(
(2_6Z)exp(6-~K'"2)_2(l_ot6)exp(6-tr~z/K'/2) + :t6 [1 + e x p ( - - ~ ) 1 + I1 - e x p ( - - - ~ ) 1
-2t
+---K3/2df dr { Cr(l gE;:e--~p(-_~m)]Til--e~p~-_~)]
~tZfexp(f---~-lJ2)+a(l-~tf)exp(6~a~z/K'/2)
} (12)
where (13)
H tr - K 1/2
(14)
6 = Kh1;----~"
(15)
Viscoussheareffectsonthesqueeze-filmbehavior
377
2.1. Modified Reynolds equation The continuity equations in the film region and the porous medium are, respectively, 1 ,~(ru) Ow + ~r ~r cz
....
=0
(16)
and 1 ~(ru*)
r
?~w*
~r + ~ = 0 .
(17)
The boundary conditions for the axial velocities at the lower and upper disks are w(r, 0) = w*(r,
0
(18)
dh h + H) = - - . dt
(19)
Integrating the continuity Eqn (16) in the film region with respect to z with the boundary condition (18), the axial velocity at the interface, w(r, h), can be obtained.
12~ur dr r
w(r,h)-
f
12c~exp(-~)-3~(2+62)I1 +exp(-~)l-6~II-exp(--~)l}
x. 26+
:~c~[1 + exp--~ - ~-)--+---~ ( ] [
ex--p-----~---( )]
2
~ [1 - exp ( i : ~ 0 - 2aexp ( - ~) l +2K ~/ddr(r~fr) ~6f, ÷exp([1 ~ e x p ( ~ ) ] I '
(20)
Integrating the continuity Eqn (17) within the porous medium with respect to z with the boundary condition (19) also we have the axial velocity at the interface w*(r, h).
w*(r,h)=dt
×
2# rdr rdrr (4 + 276
2~r-~ 1-exp
-
-(4-2~6-62)exp(--~)
z
K 2 1r ddr
(21) Since the axial velocities are continuous at the interface, z = h, w(r, h) = w*(r, h)
(22)
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the modified Reynolds equation can be derived as
rdr
dt
rTrkT
/
0
(23)
where the expressions of A and/~ are 3(8at + 4ot26 -- 4~62 -- 2,53 -- ct64)
= 120- - 263 +
ct6 I1 + e x p ( - ~ ) 1 24~t(1- 62)exp ( - ~ ) -
+ I1 - exp ( - ~ ) 1
3 ( 8 c t - 4 ~ 2 6 - 4~t62 + 2 6 3 - ~tf4)exp (-- - ~ )
-~
(24) Ct6[1 + e x p ( - - ~ ) ]
+ [1- exp(---~) 1
6ct(2o + 2eta6 -- 2~t26 -- ct62)
/~ -~- 60-2
+ exp(- )1 + [1- exp(ot6[1 + exp ( - - ~ ) ] + I , - e x p ( - ~ ) ] (25) Introducing the nondimensional variables and parameters, ph 3 -
fh 3
p R 2 dh/dt'
y-
r
#Rdh/dt'
= ~,
H
H = -R'
~o =
KH h~ '
60 =
ho
(26)
the modified Reynolds equations can be written in a nondimensional form as • o6o ~
f
+ 12~°6°6h-3 +
fdf f
= 0.
(27)
2.2. Film pressure Observing the modified Reynolds equation (27), two unknowns p and f appear in this equation. To solve for the two unknowns, we need to derive another complementary equation. It is noted that the examination of the effect of velocity slip on porous-walled squeeze-films by Prakash and Vii [8] was based on this thin-walled facing assumption. On the other hand, according to the discussions of the lubrication of porous bearings by using the BM, Lin and Hwang [20] have proposed that the zero pressure gradient assumption across the porous wall (i.e., f(7) = 0) can be directly applied to the thin-walled bearings for simplifying the problem. Without loss of generality, we still retain the function f(f); but, a simple form is assumed for the function f(7): f(/7) = /72 + a l f + a2"
(28)
For the present problem of porous circular disks, the nondimensional pressure boundary conditions for the film region,/5, and for the porous region,/~* = - p * h a / # R 2 ( d h / d t ) , are /5=15"=0
d/5 ~/~* ----0 dr dr
atf=l ate=0.
(29) (30)
Equations (29) show that the pressure at the position of 7 = 1 is equal to the ambient pressure. Equations (30) indicate that the pressure is symmetry at f = 0.
Viscous shear effects on the squeeze-film behavior
379
Utilizing the relation (6) and the above boundary conditions, the function f(?) can be obtained as f(~) = ~2 _ 1.
(31)
Substituting f(?) into Eqn (27) the modified Reynolds equation can be, then, rearranged as 1 d { d/~= rd~\ d?]
4(3"o6o6/~ 3 +/4/~) ~o63A
(32)
Integrating the modified Reynolds equation twice with respect to f by applying the pressure boundary conditions, the film pressure is obtained as p(e) =
3(I)o66/~3 + B/~ ~o6o3~ (1 - f2).
3. S Q U E E Z E - F I L M
(33)
CHARACTERISTICS
3.1. Load capacity The load capacity is calculated by integrating the pressure force acting over the disk surface W = 27r
=0 fr="
p(r)rdr.
(34)
Expressing in terms of the nondimensional form, we have
2~o63d
05)
3.2. Time-height relation For constant load W the film thickness h at any given time t can be found from Eqn (35). If the initial film height is ho at time t = 0, we introduce the nondimensional time r and the nondimensional film thickness h: Wh z r = /~R~-t,
--.h h=ho
(36)
Then, the nondimensional time can be determined by integrating Eqn (35) r :~
rt f~ = ~ 3@066 t73 +/4/3 d/~ =~ ~o63~ 3 .
(37)
Once the facing-radius ratio /4, the permeability parameter ~o, and the heightpermeablity ratio 6o are specified, then the nondimensional time r required for the circular disk to reach a given height /~ can be numerically obtained by the method of Gaussian Quadrature. 4. R E S U L T S
AND DISCUSSION
Based on the BM, this paper predicts the influences of viscous shear stresses on the squeeze-film behavior between porous circular disks. The modified Reynolds equation is derived by using the Brinkman equations to account for the viscous shear effects. The film pressure is solved and, then, applied to calculate the load-carrying capacity and the time-height relationship, To reveal how the viscous shear effects of the BM affect the squeeze-film characteristics the results are compared to those obtained from the SFM and DM. The solutions of a nonporous facing disk are also included. The performances for the SFM, DM, and that of a solid disk are calculated from the formulas derived by Prakash and Vij [8-1. Since the comparison should be under the same parameter, their equations have been nondimensionalized in the same manner as the present paper.
380
Jaw-Ren Lin lOa_-
B--
21 ;-_6~W
42-
W
0.1 s6-
o
o
o
o
o
o
o
o
o
o 0
....... ___ _ _ _ _ .....
0
001
8~
BM, BM, BM, BM, BM,
W~, W, W, W, Ew
a=0.78 a=l a=1.45 a=4.01
~=0.01 0.001
~
, ,.,~.i.
0.0001
~
, ;,~,~,~
0001
~
, ,,~,~.
0.01
~
, ;.,~,~
0.1
~o
Fig. 2. Effectof :~on I~.
4.1. Effect of ~ The existence of a ( = [/~,//~]u2) provides the viscous shear effects of the BM on the squeeze-film characteristics of porous circular disks. The value of the parameter • depends upon the variation of the porosity. In order to show the variation of ~ on the squeeze-film characteristics, Fig. 2 depicts the affect of ~ on the dimensionless load capacity if" vs permeability parameter Oo at facing-radius r a t i o / t = 0.01. This figure further shows the relative error in load capacity calculations for ~ = 0.78 and 4.01. The relative error Ew is defined as [vV~=4.01 Ew
~
--
[/~z = 0 . 7 8
[/~'~ = 4 . 0 1
It is found that a larger value of ~ results in a higher load capacity. But the maximum of the relative error is within about 0.1. Totally, the effect of the variation of ~ for different porous media on the squeeze-film performance between porous circular disks is found to be small and negligible. On the other hand, the accurate prediction of a for any given porous medium does not yet appear so far. Moreover, Neale and Nader [11] have proposed that assigning /~* the value of/~ (i.e., ~ = 1) provides satisfactory correlation of experimental data and increases substantially the effectiveness of the BM. Without loss of generality, the value of is chosen to be 1 in the following prediction. 4.2. Accuracy of the model used When the value of permeability parameter @o approaches zero, the porous solutions should approach the nonporous solutions. In order to verify the accuracy of the model used, the results are compared with those of nonporous circular disks. Figure 3 presents the nondimensional film thickness h vs nondimensional time z for the squeeze-film with -Q = 0.01 and ¢I)o = 0.00001. It is seen that the results of porous solutions predicted by using the BM for a small value of O ( = 0.00001) approach the nonporous solutions. In other words, as the value of@ approaches zero, the squeeze-film characteristics predicted by using the BM approach that of a solid solution. This close agreement provides a support to the present prediction. 4.3. Load capacity and time-height relation Figure 4 shows the comparison of dimensionless load capacity if" vs permeability parameter @ at facing-radius ratio H = 0.01 predicted by using the BM, DM and SFM. It is found that the presence of the porous facing reduces the load-carrying capacity of the squeeze-film. Moreover, the decrement is enlarged as the value of ~o increases. This result
Viscous shear effects on the squeeze-film behavior l.O
-
381
-
0,9 0.8
_ _ ......
Solid DM SFM
0.7
.....
BM H=O.01
0.6
¢0=0.00001
vl 0.5 0.4 0.3 0,2 0.1
0.0
t~o
2~o
3~o
4~o
5~o
600
T
Fig. 3. Compared to that of a nonporous solution.
10---
9-
8-"
.... __
DM _
SFM
_ _
BM
6~
~=oo~
4321
01 o.oool
~ ~ ' ~ I~$~Jl
0.001
~
' I ,~,~,l
~
0.01
@o
T ~ I~1~11
2
T ~ ' ~
0,1
Fig, 4. Load capacity I~ vs Do f o r / ~ = 0.01 using the BM, D M and SFM.
can be explained from the fact that the presence of the porous facing provides a path for radial flow toward the environments. For the porous squeeze-film, only part of the lubricant will be squeezed out as two circular disks approach each other; the remaining part will simply flow out through the porous medium. With the porous facing in place, it becomes easier to flow out of the squeeze-film. In this way, the presence of the porous facing reduces the resistance to radial flow. As a consequence, it makes the film pressure evenly distributed and the load capacity is, then, reduced. By observing the results predicted from the three models, it is found that the viscous shear effects of the BM is to reduce the load capacity of the squeeze-film as compared to that of the DM; but the viscous shear effects provide an enhancement in value of the load capacity when comparing with that derived by the SFM. Since the introduction of the viscous shear effects by the velocity continuity and viscous shear continuity at the interface is to reduce the resistance encountered by the fluid flowing toward the environments, it then diminishes the load-carrying capacity of the squeeze-film
382
J a w - R e n Lin 1.0
-
-
0.9
.....
Solid
. . . .
DM 5FM
0.8 __
_
DM
0.7 0.6 h- 0.5 0.4 ,
0.3
¢0 =o.ol
0.2 0.1 0.0 I00
200
300
400
500
6()0
700
T Fig. 5. Time--height relation f o r / 1 = 0.01 using the BM, D M a n d SFM.
as compared to that of the DM. But for the artificial SFM the slip velocity may be over introduced. An increase in value of slip velocity reduces the radial flow resistence. As a result, the SFM predicts a lower load-carrying capacity. In this sense, the SFM seems to require a suitable value of slip velocity for the present problem to derive the velocity profile; but the BM better describes the flow phenomena of the squeeze-film behavior and, thus, provides an accurate prediction of the load-carrying capacity of porous circular disks. Figure 5 displays the comparison of time-height relation at H = 0.01 for different @o predicted by using the BM, SFM and DM. Comparing with that of the solid case, the presence of the porous facing reduces the time required to attain a given value of/Y, and the viscous shear effects of the BM are to bring about a still further reduction in value of the time. However, the viscous shear effects of the BM provide an increase in the response time as compared to that derived by the SFM. These results can also be explained from the same physical considerations that are already discussed in connection with the results of loadcarrying capacity. Since the BM results in a higher load capacity as compared to that of the SFM, a higher/~ is attained for the same time to be taken. Accordingly, the BM predicts quite different squeeze-film action to those derived by the SFM and DM. From above we can say that, comparing with that of the SFM, the viscous shear effects of the BM enhance the load capacity and lengthen the response time of the squeeze-film action; but these trends reverse as compared to that derived by the DM. 4.4. Designexample Let us consider a porous facing disk with thickness H for various permeabilities K. The data of the disk are: h0 = 0.005 cm H = 0.05 cm R=5cm K = 2.5 x 1 0 - t l ; 2.5 × 10-t°; 2.5 x 10-9; 2.5 x 10-8; 2.5 x 10 -7 cm 2. From these data we find that the facing-radius r a t i o / 4 H/R the permeability parameter ~o = =
=
0.01
KH/h3 =
0.0001; 0.001; 0.1; 1.
Viscous shear effects on the squeeze-film behavior
383
Table 1. Dimensionless load capacity ff vs ~o at /4 = 0.01 predicted by using the BM with c~ --- 1
0.0001 0.0002 0.0003 0.0006 0.0010 0.0018 0.0032 0.0056 0.0100 0.0178 0.0316 0.0562 0.1000 0.1778 0.3162 0.5623 1.000(3
4.6622 4.6430 4.6157 4.5763 4.5187 4.4337 4.3077 4.1218 3.8533 3.4807 2.9979 2.4299 1.8381 1.2979 0.8633 0.5489 0.3390
Table 2. Time height relation for the squeeze-film at/4 = 0.01 for various ~0 predicted by using the BM with ~ = 1 ~o=0.0001
1,00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00
0.0000 0.2517 0.5464 0.8942 1.3089 1.8086 2.4184 3.1729 4.1217 5.3376 6.9306 9.0732 12.0483 16.3436 22.8542 33.3473 51.6202 86.5530 159.5828 309.9997 522.6009
~o=0.001
0.0000 02437 0.5280 0.8624 1.2593 1.7351 2.3118 3.0194 3.8998 5.0120 6.4417 8.3146 10.8190 14.2395 19.0030 25,7229 35.1741 48.0357 64.3020 83.2144 110.1631
~o=001
0.0000 0.2060 0.4418 0.7131 1.0265 1.3903 18141 2.3098 2.8911 3,5735 43743 5.3111 6.3999 7,6530 9.(1752 1(/.6629 12.4(154 14.2974 16,3895 19.1162 290771
~o=0.1
0.0000 0.0953 0.1975 0.3071 0,4242 0.5492 0.6820 0.8229 0.9717 1.1283 1.2926 1.4645 1.6438 1.8308 2.0265 2.2336 2.4595 2.7267 3.1181 4.0999 12.2980
~o=1
0,0000 0,0171 0,0347 0.0526 0,0710 0.0898 0.1091 0.1288 0.1492 0.1702 0.1921 0.2151 0.2397 0.2668 0.2980 0.3368 0.3914 0.4847 0.7004 1.5050 9.5252
With the values of ho and K given, the height-permeability ratio 60 can be specified. The results of nondimensional load capacity if" and time-height relation are presented in Tables 1 and 2. It is hoped that this design example can demonstrate how the results could be used by a design engineer. 5. C O N C L U S I O N
On the basis of the BM, this paper predicts the influences of the viscous shear effects on the squeeze-film behavior between porous circular disks. The modified Reynolds equation is derived by using the Brinkman equations to account for the viscous shear effects. The film pressure is solved and, then, applied to predict the load-carrying capacity and the
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Jaw-Ren Lin
time-height relationship. As the value of permeability parameter approaches zero, the squeeze-film characteristics predicted by using the BM approach that of a nonporous solution. This close agreement supports the present prediction. According to the results calculated, the effect of the variation of c~ for different porous media on the squeeze-film behavior is slight and negligible; but, the influences of viscous shear stresses on the squeeze-film characteristics are significant and not negligible. It is also found that the BM predicts quite different squeeze action to those derived by using the SFM and DM. When comparing with that of the SFM, the viscous shear effects of the BM provide an increase in value of the load-carrying capacity and, then, lengthen the response time of the squeeze-film action; but these trends are reversed as compared to that derived by using the DM. REFERENCES 1. H. Wu, Squeeze-film behavior for porous annular disks. ASME J. Lubr. Techno. 92, 593-596 (1970). 2. P. R. K. Murti, Squeeze-film behavior in porous circular disks. ASME J. Lubr. Techno. 96, 206-209 (1974). 3. J. Prakash and S. K. Vij, Load capacity and time-height relations for porous squeeze-films. Wear 24, 309-322 (1973). 4. G. S. Beavers and D. D. Joseph, Boundary conditions at a naturally permeable wall, Part 1. J. Fluid Mech. 30, 191 207 (1967). 5. G. S. Beavers, E. M. Sparrow and R. A. Magnuson, Experiments on coupled parallel flows in a channel and a bounding porous medium. J. Basic En#ng, Trans~ ASME, Series S, 92, 843-848 (1970). 6. G. I. Taylor, A model for the boundary condition of a porous material, Part 1. J. Fluid Mech. 49, 319-326 (1971). 7. E. M. Sparrow, G. S. Beavers and 1. T. Hwang, Effect of velocity slip on porous-walled squeeze-films. A S M E J . Lubr. Techno. 260-265 (1972). 8. J. Prakash and S. K. Vij, Effect of velocity porous-walled squeeze-films. Wear 29, 363 372 (1974). 9. H. Wu, A review of porous squeeze-films. Wear 47, 371-385 (1978). 10. H. C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. AppL Sci. Res. A1, 27-34 (1947). 11. G. Neale and W. Nader, Practical significance of Brinkman's extension of Darcy's law: coupled parallel flows within a channel and a bounding porous medium. Canadian J. Chem. Engng 52, 475-478 (1974). 12. D. Poulikakos and M. Kazmierczak, Forced convection in a duct partially filled with a porous material. Trans. ASME, J. Heat Transfer 109, 653-662 (1986). 13. C. T. Hsu and P. Cheng, A singular perturbation solution for Couette flow over a semi-infinite porous bed. Trans. J. Fluids Enong 113, 113-142 (1991). 14. P. Vasseur and U Robillard, The Brinkman model for natural convection in a porous layer: effects of nonuniform thermal gradient. Int. J. Heat Mass Transfer 36, 4199-4206 (1993). 15. B. S. Bhatt and N. C. Sacheti, Flow past a porous spherical shell using the Brinkman model. J. Physics D: Appl. Phys 27, 37 41 (1994). 16. J. R. Lin and C. C. Hwang, Lubrication of short porous journal bearings--use of the Brinkman-extended Darcy model. Wear 161, 93-104 (1993). 17. J. R. Lin and C. C. Hwang, Static and dynamic characteristics of long porous journal bearings--use of the Brinkman-extended Darcy model. J. Phys. D: Appl. Phys. 27, 634-643 (1994). 18. J. R. Lin and C. C. Hwang, Hydrodynamic lubrication of finite porous journal bearings--use of the Brinkman-extended Darcy model. Int. J. Mech. Sci. 36, 631-644 (1994). 19. J. R. Lin and C. C. Hwang, Linear stability analysis of finite porous journal bearings--use of the Brinkmanextended Darcy model. Int. J. Mech. Sci. 36, 645-658 (1994). 20. J. R. Lin and C. C. Hwang, On the lubrication of short porous journal bearings--use of the Brinkman model. Trans. ASME, J. Tribology 117, 196-199 (1995).
Int. J. Mech. Sci Vol. 38, No. 4, pp. 373 384, 1996
Copyright ~ff 1996 Elsevier Science Lid Printed in Great Britain. All rights reserved 0020-.7403/96 $15.00 + 0.00
0020-7403(95)00066-6
VISCOUS SHEAR EFFECTS ON THE SQUEEZE-FILM BEHAVIOR IN POROUS CIRCULAR DISKS JAW-REN LIN Department of Mechanical Engineering, Nanya Junior College, 414, Sec. 3, Chung-Shang East Road, Chung-Li 320-34, Taiwan, R.O.C. {Received 10 November 1994; and in revisedform 22 June 1995) Abstract--On the basis of the Brinkman model (BM), the main objective of this paper is to predict the viscous shear effects on the squeeze-filmcharacteristics betweenporous circular disks. According to the results obtained, the influences of viscous shear stresses on the squeeze-film behavior are significantand not negligible.It is found that the BM predicts quite different squeeze-filmaction to those derived by using the Slip-flowmodel (SFM) and Darcy model (DM). When comparing with that of the SFM, the viscous shear effects of the BM provide an enhancement in the load-carrying capacity and, then, increase the response time of the squeeze-film behavior; but these trends are reversed as compared to that derived by using the DM.
NOTATION h film thickness ho filmthickness at t = 0 nondimensional film thickness, h/ho H thicknessof porous facing /~ facing-radius ratio, H/R K permeability of porous facing material p film pressure /~ nondimensional film pressure, -ph3/l~R2h r, z radial coordinate and axial coordinate R radius of circular disk 7 nondimensional radial coordinate, r/R t time u, w radial and axial velocity components in the film region W load capacity 1~ nondimensional load capacity, - Wh3/pR4h 6o /z ~bo r ( ") ( )*
height-permeability ratio, ho/K t/2 lubricant viscosity permeability parameter, KH/h3o nondimensional time, Wh~t/#R 4 indicates the derivative with respect to time, t indicates the quantity within the porous medium 1. I N T R O D U C T I O N
Squeeze-film characteristics play a n i m p o r t a n t role in m a n y applications, such as lubrication of m a c h i n e elements, a u t o m a t i c transmissions, a n d artificial joints. The squeeze-film action comes from the b e h a v i o r of two lubricated surfaces a p p r o a c h i n g each other with a n o r m a l velocity. Since the viscous l u b r i c a n t c o n t a i n e d between these surfaces has a resistance to extrusion a n d c a n n o t be i n s t a n t a n e o u s l y squeezed out, a certain time is needed for these surfaces to come into contact. D u r i n g that interval, a pressure is built up a n d the load is, then, s u p p o r t e d by the squeeze-film. Traditionally, the study of the squeeze-film b e h a v i o r between two circular disks when one disk has a porous facing was based o n the D a r c y model (DM) where viscous shearing stresses are neglected [ 1 - 3 ] . It is f o u n d that the permeability p a r a m e t e r of p o r o u s facing adversely affects the load capacity a n d the response time needed for the circular disk to a t t a i n a given film height. In this D M , the fluid b e h a v i o r in the p o r o u s facing was a s s u m e d to obey Darcy's law a n d the no-slip c o n d i t i o n 373
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Jaw-Ren Lin
was used at the porous facing/lubricant film interface. However, Beavers and Joseph [4] found the experimental evidence of a tangential velocity slip of fluid on the porous wall and defined a dimensionless slip coefficient, 2, to modify the no-slip condition at the interface. According to their results, the value of ~ = 0.1 is used for the aloxite porous specimens, and the value of ~ = 0.78-4.01 is used for the foametal porous specimens. Beavers et al. [5] confirmed that the value of ~ = 0.1 should be used in the slip-flow formulation. But, Taylor [6] proposed that 2 = 1 should be used in the slip-flow model (SFM). Based on the SFM, many authors investigated the affect of velocity slip on the squeeze-films between porous disks [7-9]. According to their results, the effect of slip reduces the load capacity and the response time of the squeeze-films compared to that derived by using the DM. However, the SFM is valid only in a dense porous medium of large thickness so that the variation of velocity in it can be neglected. But, many porous squeeze-films involve porous layers of shallow depth. In these porous facings the velocity is no longer uniform and the distortion of velocity yields the viscous shearing stresses. Although the SFM introduces the effect of velocity slip at the interface, it yields a discontinuity of the tangential velocity component across the permeable surface. Therefore, the Darcy equation has to be modified to better describe the flow phenomena within the porous matrix. The Brinkman equation [10] accounting for the viscous shear effects and the viscous damping effects (Darcy resistance) can get around the above obstacles. By virtue of the viscous shear term contained in the Brinkman equation, the Brinkman model (BM) is mathematically and physically compatible with the a c t u a l velocity profile within the porous medium. Since the DM fails to predict the true characteristics of porous squeeze-films and the SFM cannot give an a c t u a l velocity profile within the porous medium, the BM seems to be appropriate to predict the more realistical performance of porous squeeze-films. Recently, many studies have employed the BM to study the fluid flow and heat transfer problems involving the porous media [11 15]. Based on this BM, Lin and Hwang [16 191 have examined the effects of viscous shear stresses on the performances of porous journal bearings. According to their results, the viscous shear effects improve the static and dynamic characteristics of porous journal bearings. However, we have no idea how the viscous shear stresses of the BM affect the squeeze-film characteristics for porous circular disks. Therefore, the study is needed. On the basis of the BM, the main objective of this paper is to predict the influences of the viscous shear effects on the squeeze-film behavior between porous circular disks. The modified Reynolds equation is derived by using the Brinkman equations to account for the viscous shear effects. The film pressure is solved and, then, applied to predict the loadcarrying capacity and the time-height relationship. To show the influences of the viscous shear effects on the squeeze-film performance, the results are compared with those derived by using the DM and SFM. The results of the limiting case (i.e., the value permeability parameter approaches zero) obtained from the BM give a good agreement with a nonporous disk and support the present prediction.
2. A N A L Y S I S
Consider the physical configuration of two circular disks shown as in Fig. 1, in which the upper disk has a porous facing. Assuming no variation of pressure across the fluid film, no external forces acting on the film, fluid inertia being small compared to the viscous shear stresses, and an incompressible Newtonian lubricant having constant properties, then the reduced Navier-Stokes equations and the continuity equation in the film region are given by the following ¢~p
c~Z u
& - ~ ~z z
(1)
nc'~= 0. #z
(2)
Viscous shear effectson the squeeze-filmbehavior
375
/
/
/
/
'\ \
~- FiLm
(
\
~Z T
!:i : !:i : !:; : !::l: i:i': ]
P°r°us e'ecing
!:; : !!i ~.
h
Fig. 1. Configurationof porous circular disks. For the porous region, it is assumed that the porous material is homogeneous and isotropic, the flow of lubricant in the porous facing is laminar, and the axial pressure gradient across the porous facing is a function of the radial coordinate, then the Brinkman equations gives ~:p*
?r
# -
K
?'u* u * + t~*
?z 2
(3)
?p*
(z
= f(r)
(4)
where ()* indicates the quantity within the porous medium; /~* denotes the effective viscosity of the fluid in the porous matrix which is assumed to be different in value from/~, the viscosity of the fluid layer; a n d f ( r ) is an unknown function to be determined. Since the pressure is continuous at the interface z = h, p(r) = p*(r, h)
(5)
integrating Eqn (4) with respect to z yields p*(r, z) = p(r) + (z - h)f(r).
(6)
To solve for the velocities u and u* of the lubricant in the film region and the porous region, we have the boundary conditions for the velocities, and the continuity conditions for the velocities and the viscous shear stresses u(r, 0) = 0
(7)
u*(r, h + H) = 0
(8)
utr, h) = u*(r, h) ~' ~i~u z=h = ~ , ~OU* - z z=h "
(9) (10)
Integrating the r-momentum equation with respect to z, we can obtain the expression of radial velocity in the film region, u, containing two integration constants. Substituting
376
Jaw-Ren Lin
Eqn (6) into the r-Brinkman equation and solving the equation, also we have the expression of radial velocity within the porous medium, u*, containing the other two integration constants. Applying the boundary conditions (7) and (8) at z = 0 and z = h + H, the continuity condition of velocities (9), and the continuity condition of viscous shear stresses (10) at the interface z = h, the four integration constants can be obtained, Finally, we have the expressions of velocities u and u*, respectively, 4, xp
-~t(2+f2)[l+exp(-2--~)]-26II-exp(--2-~a~)l z} (--~),~6[1 + exp(-~) 1 + [1 - exp(--~)] K'/z
xK3"Zdffa I l - e x p ( - ~ ) l - 2 t r [ e x p ( - ~ ) l
+m__
p
dr l~t6[' + e x p ( - - ~ ) ] + [ 1 - e x p ( - ~ ) ]
z (11) K1/2
and
o.
['z/K
K dp 2(1 + ~o)exp~ u*(r,
z) = - - -
2p dr
+exp(
E, oxp(
(2_6Z)exp(6-~K'"2)_2(l_ot6)exp(6-tr~z/K'/2) + :t6 [1 + e x p ( - - ~ ) 1 + I1 - e x p ( - - - ~ ) 1
-2t
+---K3/2df dr { Cr(l gE;:e--~p(-_~m)]Til--e~p~-_~)]
~tZfexp(f---~-lJ2)+a(l-~tf)exp(6~a~z/K'/2)
} (12)
where (13)
H tr - K 1/2
(14)
6 = Kh1;----~"
(15)
Viscoussheareffectsonthesqueeze-filmbehavior
377
2.1. Modified Reynolds equation The continuity equations in the film region and the porous medium are, respectively, 1 ,~(ru) Ow + ~r ~r cz
....
=0
(16)
and 1 ~(ru*)
r
?~w*
~r + ~ = 0 .
(17)
The boundary conditions for the axial velocities at the lower and upper disks are w(r, 0) = w*(r,
0
(18)
dh h + H) = - - . dt
(19)
Integrating the continuity Eqn (16) in the film region with respect to z with the boundary condition (18), the axial velocity at the interface, w(r, h), can be obtained.
12~ur dr r
w(r,h)-
f
12c~exp(-~)-3~(2+62)I1 +exp(-~)l-6~II-exp(--~)l}
x. 26+
:~c~[1 + exp--~ - ~-)--+---~ ( ] [
ex--p-----~---( )]
2
~ [1 - exp ( i : ~ 0 - 2aexp ( - ~) l +2K ~/ddr(r~fr) ~6f, ÷exp([1 ~ e x p ( ~ ) ] I '
(20)
Integrating the continuity Eqn (17) within the porous medium with respect to z with the boundary condition (19) also we have the axial velocity at the interface w*(r, h).
w*(r,h)=dt
×
2# rdr rdrr (4 + 276
2~r-~ 1-exp
-
-(4-2~6-62)exp(--~)
z
K 2 1r ddr
(21) Since the axial velocities are continuous at the interface, z = h, w(r, h) = w*(r, h)
(22)
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Jaw-Ren Lin
the modified Reynolds equation can be derived as
rdr
dt
rTrkT
/
0
(23)
where the expressions of A and/~ are 3(8at + 4ot26 -- 4~62 -- 2,53 -- ct64)
= 120- - 263 +
ct6 I1 + e x p ( - ~ ) 1 24~t(1- 62)exp ( - ~ ) -
+ I1 - exp ( - ~ ) 1
3 ( 8 c t - 4 ~ 2 6 - 4~t62 + 2 6 3 - ~tf4)exp (-- - ~ )
-~
(24) Ct6[1 + e x p ( - - ~ ) ]
+ [1- exp(---~) 1
6ct(2o + 2eta6 -- 2~t26 -- ct62)
/~ -~- 60-2
+ exp(- )1 + [1- exp(ot6[1 + exp ( - - ~ ) ] + I , - e x p ( - ~ ) ] (25) Introducing the nondimensional variables and parameters, ph 3 -
fh 3
p R 2 dh/dt'
y-
r
#Rdh/dt'
= ~,
H
H = -R'
~o =
KH h~ '
60 =
ho
(26)
the modified Reynolds equations can be written in a nondimensional form as • o6o ~
f
+ 12~°6°6h-3 +
fdf f
= 0.
(27)
2.2. Film pressure Observing the modified Reynolds equation (27), two unknowns p and f appear in this equation. To solve for the two unknowns, we need to derive another complementary equation. It is noted that the examination of the effect of velocity slip on porous-walled squeeze-films by Prakash and Vii [8] was based on this thin-walled facing assumption. On the other hand, according to the discussions of the lubrication of porous bearings by using the BM, Lin and Hwang [20] have proposed that the zero pressure gradient assumption across the porous wall (i.e., f(7) = 0) can be directly applied to the thin-walled bearings for simplifying the problem. Without loss of generality, we still retain the function f(f); but, a simple form is assumed for the function f(7): f(/7) = /72 + a l f + a2"
(28)
For the present problem of porous circular disks, the nondimensional pressure boundary conditions for the film region,/5, and for the porous region,/~* = - p * h a / # R 2 ( d h / d t ) , are /5=15"=0
d/5 ~/~* ----0 dr dr
atf=l ate=0.
(29) (30)
Equations (29) show that the pressure at the position of 7 = 1 is equal to the ambient pressure. Equations (30) indicate that the pressure is symmetry at f = 0.
Viscous shear effects on the squeeze-film behavior
379
Utilizing the relation (6) and the above boundary conditions, the function f(?) can be obtained as f(~) = ~2 _ 1.
(31)
Substituting f(?) into Eqn (27) the modified Reynolds equation can be, then, rearranged as 1 d { d/~= rd~\ d?]
4(3"o6o6/~ 3 +/4/~) ~o63A
(32)
Integrating the modified Reynolds equation twice with respect to f by applying the pressure boundary conditions, the film pressure is obtained as p(e) =
3(I)o66/~3 + B/~ ~o6o3~ (1 - f2).
3. S Q U E E Z E - F I L M
(33)
CHARACTERISTICS
3.1. Load capacity The load capacity is calculated by integrating the pressure force acting over the disk surface W = 27r
=0 fr="
p(r)rdr.
(34)
Expressing in terms of the nondimensional form, we have
2~o63d
05)
3.2. Time-height relation For constant load W the film thickness h at any given time t can be found from Eqn (35). If the initial film height is ho at time t = 0, we introduce the nondimensional time r and the nondimensional film thickness h: Wh z r = /~R~-t,
--.h h=ho
(36)
Then, the nondimensional time can be determined by integrating Eqn (35) r :~
rt f~ = ~ 3@066 t73 +/4/3 d/~ =~ ~o63~ 3 .
(37)
Once the facing-radius ratio /4, the permeability parameter ~o, and the heightpermeablity ratio 6o are specified, then the nondimensional time r required for the circular disk to reach a given height /~ can be numerically obtained by the method of Gaussian Quadrature. 4. R E S U L T S
AND DISCUSSION
Based on the BM, this paper predicts the influences of viscous shear stresses on the squeeze-film behavior between porous circular disks. The modified Reynolds equation is derived by using the Brinkman equations to account for the viscous shear effects. The film pressure is solved and, then, applied to calculate the load-carrying capacity and the time-height relationship, To reveal how the viscous shear effects of the BM affect the squeeze-film characteristics the results are compared to those obtained from the SFM and DM. The solutions of a nonporous facing disk are also included. The performances for the SFM, DM, and that of a solid disk are calculated from the formulas derived by Prakash and Vij [8-1. Since the comparison should be under the same parameter, their equations have been nondimensionalized in the same manner as the present paper.
380
Jaw-Ren Lin lOa_-
B--
21 ;-_6~W
42-
W
0.1 s6-
o
o
o
o
o
o
o
o
o
o 0
....... ___ _ _ _ _ .....
0
001
8~
BM, BM, BM, BM, BM,
W~, W, W, W, Ew
a=0.78 a=l a=1.45 a=4.01
~=0.01 0.001
~
, ,.,~.i.
0.0001
~
, ;,~,~,~
0001
~
, ,,~,~.
0.01
~
, ;.,~,~
0.1
~o
Fig. 2. Effectof :~on I~.
4.1. Effect of ~ The existence of a ( = [/~,//~]u2) provides the viscous shear effects of the BM on the squeeze-film characteristics of porous circular disks. The value of the parameter • depends upon the variation of the porosity. In order to show the variation of ~ on the squeeze-film characteristics, Fig. 2 depicts the affect of ~ on the dimensionless load capacity if" vs permeability parameter Oo at facing-radius r a t i o / t = 0.01. This figure further shows the relative error in load capacity calculations for ~ = 0.78 and 4.01. The relative error Ew is defined as [vV~=4.01 Ew
~
--
[/~z = 0 . 7 8
[/~'~ = 4 . 0 1
It is found that a larger value of ~ results in a higher load capacity. But the maximum of the relative error is within about 0.1. Totally, the effect of the variation of ~ for different porous media on the squeeze-film performance between porous circular disks is found to be small and negligible. On the other hand, the accurate prediction of a for any given porous medium does not yet appear so far. Moreover, Neale and Nader [11] have proposed that assigning /~* the value of/~ (i.e., ~ = 1) provides satisfactory correlation of experimental data and increases substantially the effectiveness of the BM. Without loss of generality, the value of is chosen to be 1 in the following prediction. 4.2. Accuracy of the model used When the value of permeability parameter @o approaches zero, the porous solutions should approach the nonporous solutions. In order to verify the accuracy of the model used, the results are compared with those of nonporous circular disks. Figure 3 presents the nondimensional film thickness h vs nondimensional time z for the squeeze-film with -Q = 0.01 and ¢I)o = 0.00001. It is seen that the results of porous solutions predicted by using the BM for a small value of O ( = 0.00001) approach the nonporous solutions. In other words, as the value of@ approaches zero, the squeeze-film characteristics predicted by using the BM approach that of a solid solution. This close agreement provides a support to the present prediction. 4.3. Load capacity and time-height relation Figure 4 shows the comparison of dimensionless load capacity if" vs permeability parameter @ at facing-radius ratio H = 0.01 predicted by using the BM, DM and SFM. It is found that the presence of the porous facing reduces the load-carrying capacity of the squeeze-film. Moreover, the decrement is enlarged as the value of ~o increases. This result
Viscous shear effects on the squeeze-film behavior l.O
-
381
-
0,9 0.8
_ _ ......
Solid DM SFM
0.7
.....
BM H=O.01
0.6
¢0=0.00001
vl 0.5 0.4 0.3 0,2 0.1
0.0
t~o
2~o
3~o
4~o
5~o
600
T
Fig. 3. Compared to that of a nonporous solution.
10---
9-
8-"
.... __
DM _
SFM
_ _
BM
6~
~=oo~
4321
01 o.oool
~ ~ ' ~ I~$~Jl
0.001
~
' I ,~,~,l
~
0.01
@o
T ~ I~1~11
2
T ~ ' ~
0,1
Fig, 4. Load capacity I~ vs Do f o r / ~ = 0.01 using the BM, D M and SFM.
can be explained from the fact that the presence of the porous facing provides a path for radial flow toward the environments. For the porous squeeze-film, only part of the lubricant will be squeezed out as two circular disks approach each other; the remaining part will simply flow out through the porous medium. With the porous facing in place, it becomes easier to flow out of the squeeze-film. In this way, the presence of the porous facing reduces the resistance to radial flow. As a consequence, it makes the film pressure evenly distributed and the load capacity is, then, reduced. By observing the results predicted from the three models, it is found that the viscous shear effects of the BM is to reduce the load capacity of the squeeze-film as compared to that of the DM; but the viscous shear effects provide an enhancement in value of the load capacity when comparing with that derived by the SFM. Since the introduction of the viscous shear effects by the velocity continuity and viscous shear continuity at the interface is to reduce the resistance encountered by the fluid flowing toward the environments, it then diminishes the load-carrying capacity of the squeeze-film
382
J a w - R e n Lin 1.0
-
-
0.9
.....
Solid
. . . .
DM 5FM
0.8 __
_
DM
0.7 0.6 h- 0.5 0.4 ,
0.3
¢0 =o.ol
0.2 0.1 0.0 I00
200
300
400
500
6()0
700
T Fig. 5. Time--height relation f o r / 1 = 0.01 using the BM, D M a n d SFM.
as compared to that of the DM. But for the artificial SFM the slip velocity may be over introduced. An increase in value of slip velocity reduces the radial flow resistence. As a result, the SFM predicts a lower load-carrying capacity. In this sense, the SFM seems to require a suitable value of slip velocity for the present problem to derive the velocity profile; but the BM better describes the flow phenomena of the squeeze-film behavior and, thus, provides an accurate prediction of the load-carrying capacity of porous circular disks. Figure 5 displays the comparison of time-height relation at H = 0.01 for different @o predicted by using the BM, SFM and DM. Comparing with that of the solid case, the presence of the porous facing reduces the time required to attain a given value of/Y, and the viscous shear effects of the BM are to bring about a still further reduction in value of the time. However, the viscous shear effects of the BM provide an increase in the response time as compared to that derived by the SFM. These results can also be explained from the same physical considerations that are already discussed in connection with the results of loadcarrying capacity. Since the BM results in a higher load capacity as compared to that of the SFM, a higher/~ is attained for the same time to be taken. Accordingly, the BM predicts quite different squeeze-film action to those derived by the SFM and DM. From above we can say that, comparing with that of the SFM, the viscous shear effects of the BM enhance the load capacity and lengthen the response time of the squeeze-film action; but these trends reverse as compared to that derived by the DM. 4.4. Designexample Let us consider a porous facing disk with thickness H for various permeabilities K. The data of the disk are: h0 = 0.005 cm H = 0.05 cm R=5cm K = 2.5 x 1 0 - t l ; 2.5 × 10-t°; 2.5 x 10-9; 2.5 x 10-8; 2.5 x 10 -7 cm 2. From these data we find that the facing-radius r a t i o / 4 H/R the permeability parameter ~o = =
=
0.01
KH/h3 =
0.0001; 0.001; 0.1; 1.
Viscous shear effects on the squeeze-film behavior
383
Table 1. Dimensionless load capacity ff vs ~o at /4 = 0.01 predicted by using the BM with c~ --- 1
0.0001 0.0002 0.0003 0.0006 0.0010 0.0018 0.0032 0.0056 0.0100 0.0178 0.0316 0.0562 0.1000 0.1778 0.3162 0.5623 1.000(3
4.6622 4.6430 4.6157 4.5763 4.5187 4.4337 4.3077 4.1218 3.8533 3.4807 2.9979 2.4299 1.8381 1.2979 0.8633 0.5489 0.3390
Table 2. Time height relation for the squeeze-film at/4 = 0.01 for various ~0 predicted by using the BM with ~ = 1 ~o=0.0001
1,00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00
0.0000 0.2517 0.5464 0.8942 1.3089 1.8086 2.4184 3.1729 4.1217 5.3376 6.9306 9.0732 12.0483 16.3436 22.8542 33.3473 51.6202 86.5530 159.5828 309.9997 522.6009
~o=0.001
0.0000 02437 0.5280 0.8624 1.2593 1.7351 2.3118 3.0194 3.8998 5.0120 6.4417 8.3146 10.8190 14.2395 19.0030 25,7229 35.1741 48.0357 64.3020 83.2144 110.1631
~o=001
0.0000 0.2060 0.4418 0.7131 1.0265 1.3903 18141 2.3098 2.8911 3,5735 43743 5.3111 6.3999 7,6530 9.(1752 1(/.6629 12.4(154 14.2974 16,3895 19.1162 290771
~o=0.1
0.0000 0.0953 0.1975 0.3071 0,4242 0.5492 0.6820 0.8229 0.9717 1.1283 1.2926 1.4645 1.6438 1.8308 2.0265 2.2336 2.4595 2.7267 3.1181 4.0999 12.2980
~o=1
0,0000 0,0171 0,0347 0.0526 0,0710 0.0898 0.1091 0.1288 0.1492 0.1702 0.1921 0.2151 0.2397 0.2668 0.2980 0.3368 0.3914 0.4847 0.7004 1.5050 9.5252
With the values of ho and K given, the height-permeability ratio 60 can be specified. The results of nondimensional load capacity if" and time-height relation are presented in Tables 1 and 2. It is hoped that this design example can demonstrate how the results could be used by a design engineer. 5. C O N C L U S I O N
On the basis of the BM, this paper predicts the influences of the viscous shear effects on the squeeze-film behavior between porous circular disks. The modified Reynolds equation is derived by using the Brinkman equations to account for the viscous shear effects. The film pressure is solved and, then, applied to predict the load-carrying capacity and the
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time-height relationship. As the value of permeability parameter approaches zero, the squeeze-film characteristics predicted by using the BM approach that of a nonporous solution. This close agreement supports the present prediction. According to the results calculated, the effect of the variation of c~ for different porous media on the squeeze-film behavior is slight and negligible; but, the influences of viscous shear stresses on the squeeze-film characteristics are significant and not negligible. It is also found that the BM predicts quite different squeeze action to those derived by using the SFM and DM. When comparing with that of the SFM, the viscous shear effects of the BM provide an increase in value of the load-carrying capacity and, then, lengthen the response time of the squeeze-film action; but these trends are reversed as compared to that derived by using the DM. REFERENCES 1. H. Wu, Squeeze-film behavior for porous annular disks. ASME J. Lubr. Techno. 92, 593-596 (1970). 2. P. R. K. Murti, Squeeze-film behavior in porous circular disks. ASME J. Lubr. Techno. 96, 206-209 (1974). 3. J. Prakash and S. K. Vij, Load capacity and time-height relations for porous squeeze-films. Wear 24, 309-322 (1973). 4. G. S. Beavers and D. D. Joseph, Boundary conditions at a naturally permeable wall, Part 1. J. Fluid Mech. 30, 191 207 (1967). 5. G. S. Beavers, E. M. Sparrow and R. A. Magnuson, Experiments on coupled parallel flows in a channel and a bounding porous medium. J. Basic En#ng, Trans~ ASME, Series S, 92, 843-848 (1970). 6. G. I. Taylor, A model for the boundary condition of a porous material, Part 1. J. Fluid Mech. 49, 319-326 (1971). 7. E. M. Sparrow, G. S. Beavers and 1. T. Hwang, Effect of velocity slip on porous-walled squeeze-films. A S M E J . Lubr. Techno. 260-265 (1972). 8. J. Prakash and S. K. Vij, Effect of velocity porous-walled squeeze-films. Wear 29, 363 372 (1974). 9. H. Wu, A review of porous squeeze-films. Wear 47, 371-385 (1978). 10. H. C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. AppL Sci. Res. A1, 27-34 (1947). 11. G. Neale and W. Nader, Practical significance of Brinkman's extension of Darcy's law: coupled parallel flows within a channel and a bounding porous medium. Canadian J. Chem. Engng 52, 475-478 (1974). 12. D. Poulikakos and M. Kazmierczak, Forced convection in a duct partially filled with a porous material. Trans. ASME, J. Heat Transfer 109, 653-662 (1986). 13. C. T. Hsu and P. Cheng, A singular perturbation solution for Couette flow over a semi-infinite porous bed. Trans. J. Fluids Enong 113, 113-142 (1991). 14. P. Vasseur and U Robillard, The Brinkman model for natural convection in a porous layer: effects of nonuniform thermal gradient. Int. J. Heat Mass Transfer 36, 4199-4206 (1993). 15. B. S. Bhatt and N. C. Sacheti, Flow past a porous spherical shell using the Brinkman model. J. Physics D: Appl. Phys 27, 37 41 (1994). 16. J. R. Lin and C. C. Hwang, Lubrication of short porous journal bearings--use of the Brinkman-extended Darcy model. Wear 161, 93-104 (1993). 17. J. R. Lin and C. C. Hwang, Static and dynamic characteristics of long porous journal bearings--use of the Brinkman-extended Darcy model. J. Phys. D: Appl. Phys. 27, 634-643 (1994). 18. J. R. Lin and C. C. Hwang, Hydrodynamic lubrication of finite porous journal bearings--use of the Brinkman-extended Darcy model. Int. J. Mech. Sci. 36, 631-644 (1994). 19. J. R. Lin and C. C. Hwang, Linear stability analysis of finite porous journal bearings--use of the Brinkmanextended Darcy model. Int. J. Mech. Sci. 36, 645-658 (1994). 20. J. R. Lin and C. C. Hwang, On the lubrication of short porous journal bearings--use of the Brinkman model. Trans. ASME, J. Tribology 117, 196-199 (1995).