- Email: [email protected]
On the squeeze film characteristics between a sphere and a flat porous plate using ferrofluid
Accepted Manuscript
On the squeeze film characteristics between a sphere and a flat porous plate using ferrofluid Rajesh C. Shah , Ramesh C. Kataria PII: DOI: Reference:
S0307-904X(15)00576-4 10.1016/j.apm.2015.09.042 APM 10741
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
4 April 2015 20 July 2015 23 September 2015
Please cite this article as: Rajesh C. Shah , Ramesh C. Kataria , On the squeeze film characteristics between a sphere and a flat porous plate using ferrofluid, Applied Mathematical Modelling (2015), doi: 10.1016/j.apm.2015.09.042
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights
ACCEPTED MANUSCRIPT
Ferrofluid lubrication with oblique variable magnetic field analysed.
Variable magnetic field because uniform field does not enhance performances
Porous plate considered because of added advantageous property of selflubrication
The Reynolds's type equation is developed using ferrohydrodynamics theory
Result indicates the better performance of the system when FF is used as
AC
CE
PT
ED
M
AN US
CR IP T
lubricant
ACCEPTED MANUSCRIPT
On the squeeze film characteristics between a sphere and a flat porous plate using ferrofluid RAJESH C. SHAH
RAMESH C. KATARIA
CR IP T
Department of Applied Mathematics, Faculty of Technology and Engineering, The M. S. University of Baroda, Vadodara – 390 001, Gujarat State, India e-mail [email protected]
AN US
Department of Mathematics, Som-Lalit Institute of Computer Applications, Ahmedabad – 380 009, Gujarat State, India e-mail [email protected]
Abstract Ferrofluid (FF) lubricated bearing design system formed by upper spherical surface and a flat porous plate considering variable magnetic field, which is oblique to the lower plate, have been analysed. The porous plate is considered because of having its advantageous property
M
of self-lubrication. The analytical model with squeeze behaviour of the phenomenon is developed, considering the validity of the Darcy’s law in the porous region, using equation of continuity, and equations from ferrohydrodynamics theory. The expressions for squeeze film
ED
characteristics are obtained and calculated numerically. The results indicate the better
PT
performance of the system, when FF is used as lubricant.
AC
CE
Keywords Ferrofluid, Squeeze film, Porous plate, Magnetic field
1
ACCEPTED MANUSCRIPT
1. Introduction When two lubricated surfaces approach each other with a normal velocity (known as squeeze velocity), then squeeze film phenomenon arise [1]. Study of squeeze film behaviour are observed in many fields of real life such as in machine tools, gears, rolling elements, hydraulic systems, engines, clutch plates, etc. Also, it is observed in the study of human knee joints and other skeletal joints as bio-lubrication [2]. Squeeze film with the attachment of porous layer
CR IP T
(region or plate or matrix or surface) are widely used in industry because of its advantageous property of self- lubrication and no need of exterior lubricant supply.
In recent years, many theoretical and experimental inventions are made on the bearing design systems as well as on the lubricating substances in order to increase the efficiency of the
AN US
bearing performances. One of the major revolutions in the direction of lubricating substances is an invention of ferrofluids. Ferrofluids (FFs) or Magnetic fluids (MFs) [3] are stable colloidal suspensions containing fine ferromagnetic particles dispersing in a liquid, called carrier liquid, in which a surfactant is added to generate a coating layer preventing the flocculation of the particles. When an external magnetic field H is applied, FFs experience magnetic body forces
M
(M )H which depends upon the magnetization vector M of ferromagnetic particles. Owing to these features, FFs are useful in many applications like in sensors, sealing devices, filtering
ED
apparatus, elastic damper, bearings, etc. [4-7]. Many researchers have also tried to find its application as lubricant in squeeze film bearing design systems. Verma [8] studied effects of MF on squeeze film bearing design system under an externally applied magnetic field oblique to the
PT
lower surface with the lower porous surface composed of three thin layers with different porosities. Explicit solutions for velocity, pressure, load carrying capacity and response time are
CE
obtained. It is found that upper plate takes longer time to come down in this case as compared to conventional lubricant based squeeze film. Kumar et. al. [9] studied squeeze film for spherical
AC
and conical bearings using FF as lubricant with the effects of rotation of particles and constant magnetic field in transverse direction, and numerically studied various bearing characteristics. Bhat and Deheri [10] discussed about curved porous circular discs squeeze film with the effect of MF and shown that pressure, load capacity and response time increases with the increase of magnetization. Prajapati [11] discussed various designed squeeze films with MF effect and shown the superiority performance of the MF lubricant than conventional lubricant. Shah et. al. [12] studied squeeze film between porous annular curved plates with the effects of rotational inertia as well as MF and found that the increase in pressure and load capacity depended only on 2
ACCEPTED MANUSCRIPT
the magnetization whereas response time dependent on magnetization, fluid inertia and speed of rotation of the plates. In [13], Shah and Bhat studied squeeze film in a long journal bearing using FF as lubricant and found that load capacity and response time increased with the increasing values of the eccentricity ratio. In [14], Patel and Deheri discussed about MF based squeeze film between porous conical plates and found that negative effect induced by the porosity can be neutralized by the positive effect caused by the magnetization parameter. Andharia and Deheri in
CR IP T
[15] studied MF based squeeze film for truncated conical plates with the effect of longitudinal roughness and found that load capacity can be increased with magnetization as well as negatively skewed roughness. The pressure and response time also found to increase with magnetization. Shah and Patel [16] discussed impact of various porous structures on curved porous circular plates squeeze film using FF as lubricant and found that globular sphere model have more impact
AN US
on increase of load capacity as compare to capillary fissures model. Lin et. al. in [17] studied squeeze film characteristics for conical plates with the effect of fluid inertia and FF, and shown the better performance of the system as compared to non-inertia non-magnetic case. In [18], Lin et. al. studied squeeze film characteristics of parallel circular discs with the effects of FF and non-Newtonian couple stresses using transverse magnetic field. With these effects, it was shown
M
that higher load capacity and lengthens approaching time obtained.
The purpose of the present paper is to study newly designed squeeze film bearing made
ED
up of a sphere and a flat plate using water based FF as lubricant, which is controlled by oblique and variable magnetic field, with the effects of porosity, slip velocity and squeeze velocity. With
PT
these effects, the paper studied about the impact of squeeze film height, permeability and width of the porous layer. Expressions for pressure, load carrying capacity and response time are
CE
obtained from Reynolds’s equation. The dimensionless pressure distribution, load carrying capacity and response time are calculated and presented graphically.
AC
2. Development of the Mathematical Model Figure 1 shows the schematic representation of the bearing design system under study.
The upper surface is a rigid sphere of radius a, and lower surface is a flat porous plate which is
formed when a porous layer of thickness H0 is attached to the impermeable flat surface. The porous layer is considered because of added advantage of self-lubricating property of the system. The region between the sphere and flat porous plate is known as film region, which is filled with FF and controlled by variable magnetic field of strength H, which is oblique to the lower plate. 3
ACCEPTED MANUSCRIPT
As far as dynamical part is concern, the upper surface (sphere) approaches to lower flat porous plate with a uniform velocity, known as squeeze velocity hm , and is defined as dh hm m , dt
where hm is minimum film thickness and t is time. The shape of the film thickness h is defined as
r2 ; r a, 2a
AN US
h hm
CR IP T
(1)
where r is the radial co-ordinate.
(2)
The basic flow equations for the motion of FFs (lubricant flow) in the film region based
M
on ferrohydrodynamics theory by R. E. Rosensweig with continuity equation are given by (refer [3, 19])
q q q p 2q 0 (M )H , t
AC
CE
PT
ED
(3) q 0 ,
(4)
H 0 , (5) MH,
(6) (H M ) 0 , 4
ACCEPTED MANUSCRIPT
(7) where
, p, , q, 0 M, H, are density of fluid, film pressure, fluid viscosity, fluid ,
velocity, free space permeability, the magnetization vector, magnetic field vector, and magnetic susceptibility respectively.
q (r, r, z ) (u, rv, w),
CR IP T
Also,
(8)
where (r , , z ) are cylindrical co-ordinates and dot (.) represents derivative w.r.t. t.
form [20]
AN US
As mentioned above, the variable magnetic field of strength H considered here is of the
K r2 ( a r) H , a
(9)
M
2
ED
where K is chosen to suit the dimensions of both sides. By combining equations (3) to (8) under usual assumptions of lubrication, neglecting
PT
inertia terms, and that the derivatives of fluid velocities across the film predominate, the equation
2u 1 d 1 2 p 0 H . 2 dr z 2 (10)
AC
CE
governing the lubricant flow in the film region in cylindrical co-ordinates yields
As porous layer is attached to the lower impermeable surface, solving equation (10)
using slip boundary condition at the lower porous surface (that is, at z 0 ) as ( refer [21, 22] ) u
5
1 u 1 ; r r , s z s 5
ACCEPTED MANUSCRIPT
and condition at the upper spherical surface as u 0 when z h,
yields velocity profile in the film region as
h z (1 sh) ( z h) 2 (1 sh)
d 1 2 p 0 H , dr 2
CR IP T
u
(11)
where s is a slip parameter, and r , r are the permeability and porosity of the porous region in the radial direction respectively.
AN US
The continuity equation for the film region in cylindrical co-ordinates is given by
1 w 0, (ru ) z r r
(12)
M
which on integrating over the film thickness h; that is, over the interval (0, h) and making use of
ED
equation (11) with the insertion of squeeze velocity effect wh w
z h
hm , yields
PT
1 rh3 (4 sh) d 1 2 p 0 H hm w0 , where w0 w r r 12 (1 sh) dr 2
z 0
.
CE
(13)
Assuming the validity of the Darcy’s law in the porous region, the radial and axial
AC
velocity components of the fluid in the porous region are given by u
r
1 2 r P 2 0 H , (14)
6
ACCEPTED MANUSCRIPT
w
z
1 2 z P 2 0 H , (15)
where z is the permeability of the fluid in the porous region in axial direction, and P is the pressure there.
(16)
AN US
1 w 0. (r u ) z r r
CR IP T
The continuity equation for the FF flow in the porous region is given by
Substituting equations (14) and (15) in equation (16), and integrating over the thickness of the porous matrix H0; that is, over the interval (-H0,0) yields
M
1 1 1 r H 0 r P 0 H 2 z P 0 H 2 r r r 2 z 2 z 0
(17)
ED
Using Morgan-Cameron approximation [23] that the pressure P in the porous region can be replaced by the average pressure p with respect to the bearing wall thickness as
CE
PT
1 1 2 2 P 0 H p 0 H , r 2 2 r
AC
equation (17) becomes
r H 0 1 1 2 r p 0 H 2 . P 0 H 2 2 rz r r z z 0 (18)
Assuming that the normal (axial) component of velocity across the film–porous interface at the lower plate are equal, therefore
7
ACCEPTED MANUSCRIPT
w
z 0
w
z 0
. (19)
Using equations (13), (15), (18) and (19), one yields
CR IP T
1 h3 (4 sh) 1 2 12 H r r 0 p 0 H 12 hm , (1 sh) r 2 r r
(20)
which is known as Reynolds’s type equation of the considered phenomenon. Introducing dimensionless quantities
ah h H a r , hm m , h 2 hm , H 0 0 , r r3 , r r H 0 , a a a hm hm
K 0 hm3 h3 p , p m2 . a hm hm
M
s s hm , *
AN US
R
(21)
ED
Equation (20) becomes
PT
d d 1 2 GR p * R (1 R ) 12 R, dR dR 2
CE
AC
where
8
(22)
G 12 r
(h )3 (4 s h ) . (1 s h ) (23)
ACCEPTED MANUSCRIPT
3. Solution Solving equation (22) under the boundary conditions for the pressure field as
p 0 when R 1 and
dp 0 when R 0 , dR
CR IP T
(24) yields
1
R
6R dR . G
(25)
AN US
1 p * R 2 (1 R ) 2
The load carrying capacity W can be obtained by using the definition
a
W 2 pr dr .
M
0
ED
(26)
Using dimensionless quantities defined in equation (21), equation (26) implies
PT
dimensionless load carrying capacity as
W hm3 R3 * 3 0 G dR , 40 2 a 4 hm 1
(27)
AC
CE
W
where G is defined in equation (23). Again, introducing dimensionless quantities for calculating response time t as hi
hm , hi
1*
K 0 a 4 h2 W t a , r* r 3 , r* r* H 0 , t i 4 , s* shi , a W hi (28)
9
ACCEPTED MANUSCRIPT
where hi is the initial film thickness. Using (28), the dimensionless response time t to reach a film thickness hm starting with an initial film thickness hi is given by 3R 3 dR G*
h3W dt i4 0 , dhi a hm 1 1* 2 40
which implies
1
t
0
3R 3 dR G*
1 1* 2 40
(29)
dhi ,
(30)
M
hi
AN US
1
CR IP T
1
ED
where
(hi )3 h (4 s* hi h ) G* * and G * 12 . r (hi )3 (1 s* hi h )
PT
G
(31)
CE
4. Results and Discussion
AC
The values of the dimensionless pressure distribution ( p ), load carrying capacity ( W )
and response time ( t ) have been calculated for the representative values of different parameters given in section 4.1 using Simpson’s 1/3-rule with step size 0.1. The FF used here is water based. The variable magnetic field considered is oblique to the lower plate and its strength is of O(102) in order to get maximum magnetic field strength at r 2a/3. The calculation of magnetic field strength is shown in section 4.1. The other order of magnetic field strength with different K is shown in figure 2. 10
ACCEPTED MANUSCRIPT
When FF is used as lubricant, then the variation in p is due to the first term of the equation (25). The same conclusion happens for W in equation (27). Also, it is clear from equation (29) that, the variation in t is due to the term 1* 40 , when FF is used as lubricant. Discussion on squeeze film pressure
CR IP T
Figure 3 shows the variation in dimensionless pressure distribution p as a function of dimensionless radial co-ordinate R for different values of dimensionless minimum film thickness hm . It is observed that
p increases as hm increases, but the increase rate is more
when 0 R 0.5 . When R > 0.5, then p remains almost same for all hm . That is, in general, as R increases, p decreases. Hence, p is maximum nearer to origin of radial axis. The variation in
AN US
p as a function of dimensionless radial permeability parameter of the porous region r is shown in figure 4. It is observed that when 2.92 106 r 2.92 103 , p is constant, and then it starts decreasing when r 2.92 103 . Figure 5 shows the variation in p as a function of dimensionless thickness of the porous matrix H 0 . It is observed that for
0 H 0 0.01 , p
M
remains constant. When H 0 0.01 , p starts decreasing. It should be noted here that H 0 = 0
ED
indicates solid case; that is, the bearing design system without porous surface at bottom. It is a general observation (refer [22, 24]) that, the insertion of porous surface decreases load carrying capacity and ultimately pressure. Moreover, porous surface is inserted because of
PT
advantageous property of self-lubrication. Thus, it is interesting to note from figure 5 that when 0 H 0 0.01 , p remains constant. It means that, p attains the same value as that of solid
CE
case ( H 0 = 0) even if 0 H 0 0.01 . This is the added advantage of obtaining p as that of solid case as well as maintaining self-lubricating property of bearing design system. From figures 4
AC
and 5, p attains its maximum value, when 2.92 106 r 2.92 103 and 0 H 0 0.01 . Discussion on load carrying capacity
Figure 6 shows the variation in dimensionless load carrying capacity W as a function of dimensionless thickness of the porous matrix H 0 for different values of dimensionless minimum film thickness hm . It is observed that W increases as hm increases. When 0.005 hm 0.009 11
ACCEPTED MANUSCRIPT
and 0.00001 H 0 0.01 , W attains almost constant value. For 0.001 hm 0.004 , W starts decreasing after H 0 = 0.0001 whereas before that it takes constant value. After hm 0.01 , W decreases for all hm . The variation in W as a function of dimensionless radial permeability parameter of the porous region r for hm 0.007 is shown in figure 7. It is observed that r has no effect on W , when 2.92 106 r 2.92 103 . After r 2.92 103 , W decreases. Thus,
CR IP T
from figures 6 and 7, it is observed that better load carrying capacity can be obtained for thin layer of FF, where 0.005 hm 0.009 , 0.00001 H 0 0.01 and 2.92 106 r 2.92 103 . Figure 8 shows that W attains the same value when H 0 = 0 (solid case) and H 0 = 0.0001. Thus, again as discussed in squeeze film pressure, added advantage of obtaining W as that of solid case
AN US
with self-lubricating property of bearing design system is observed. Discussion on squeeze film time
Solving equation (30) numerically using double integration procedure considering Simpson’s 1/3-rule with step size 0.1 for inner integral and step size given by following
hm 1 A step size A (Say), then hi 10
ED
hi
M
calculation for outer integral.
Figure 9 shows the variation in dimensionless response time t as a function of
PT
dimensionless thickness of the porous matrix H 0 for different values of dimensionless minimum
CE
film thickness hm . It is observed that, t decreases steeply when upper surface squeezing the FF film in the film region from height hi 0.1 to minimum film thickness hm 0.001 . When
AC
hm 0.02 , then decrease rate of t becomes low. For all other values of hm , t almost takes constant value. It is to be noted here from figure 9 that t is more when hm takes values from
0.009 to 0.001. The variation in t as a function of dimensionless radial permeability parameter of porous region r* for hm 0.007 is shown in figure 10 where t starts decreases after
r* 103 .
12
ACCEPTED MANUSCRIPT
4.1 Representative values and calculation of K The following representative values are taken
in computations. hi 0.0001(m), 0.05, 0.012 ( N s / m 2 ), r 0.25, 0 4 107 ( N / A2 ),
CR IP T
K 109 /1.48 ( A2 / m 4 ) , W= 25.0 N, hm = - 0.04 (m / s )
r 1011 (m2 ) (fixed for figures 3,5,6,8,9), H 0 = 0.0001 (m) (fixed for figures 3,4,7,10), a = 0.001 (m) (fixed for figure 3), a = 0.01(m) (fixed for figures 4-10),
AN US
r = 0.0001(m) (fixed for figures 4-10), hm = 0.00007(m) (fixed for figures 4,5). From equation (9),
H2
K r2 ( a r) , a
M
Max. H 2 1.48 105 K for a 0.01
where O indicates order.
PT
5. Conclusion
ED
For K 109 /1.48 , H= O(102) or O(H) = 2
On the basis of the ferrohydrodynamics theory, a FF lubricated squeeze film bearing
CE
design system formed by a sphere and a flat porous plate considering variable magnetic field, which is oblique to the lower plate, is theoretically analysed. It is noted here that porous plate is
AC
considered because of its advantageous property of self-lubrication and no need of exterior lubricant supply. The analytical model, known as Reynold’s equation, is derived using equation of continuity in film as well as porous region and equations from ferrohydrodynamics theory. The above model also considers the validity of Darcy’s law in the porous region. The following conclusions can be made from results and discussion. (1) 13
p is maximum nearer to R 0.1 .
ACCEPTED MANUSCRIPT
(2)
p is maximum and constant, when 2.92 106 r 2.92 103 .
(3)
p is maximum and constant, when 0 H 0 0.01 .
(4)
Better load carrying capacity can be obtained, when 0.005 hm 0.009 , 0.00001 H 0 0.01 and 2.92 106 r 2.92 103 . t is maximum for hm 0.001 , H 0 0.00001 and r* 106 103.
(6)
t has almost constant behaviour for 0.003 hm 0.009 .
CR IP T
(5)
It should be noted here that, according to [22], when porous layer is inserted then the
AN US
pressure of the porous medium provides a path for the fluid to come out easily from the bearing to the environment, which varies with permeability. Thus, the presence of the porous material decreases the resistance to flow in r – direction and as a consequence the load carrying capacity is reduced. The same behaviour also agrees with the conclusion of the Prakash and Tiwari [24] theoretically and experimentally by Wu [1]. In our case, the loss in W due to effect of porosity is
M
almost zero because of using FF as lubricant (which is controlled by oblique and variable magnetic field) for smaller values of H 0 and r . Moreover, because of porosity effect,
ED
self-lubrication property is an added advantage. The present case reduces to the case of conventional lubricant when * 0 and 1* 0 .
PT
It is to be noted here that variable magnetic field is used because uniform magnetic field
CE
does not enhance bearing performances as can be seen from equation (10).
AC
Acknowledgement The authors are grateful to the reviewers for their valuable comments.
14
ACCEPTED MANUSCRIPT
5. References [1]
Wu H. A review of porous squeeze films. Wear 1978; 47: 371-385.
[2]
Naduvinamani NB, Katti SG. Micropolar fluid poro-elastic squeeze film lubrication between a cylinder and a rough flat plate – A special reference to synovial joint lubrication.
CR IP T
Tribology Online 2014; 9(1): 21-30. [3]
Rosensweig RE. Ferrohydrodynamics. New York: Cambridge University Press; 1985.
[4]
Goldowsky M. New methods for sealing, filtering, and lubricating with magnetic fluids. IEEE Transactions on Magnetics, Mag. 1980; 16: 382-386.
[5]
Popa NC, Potencz I, Brostean L, Vekas L. Some applications of inductive Transducers with
[6]
AN US
magnetic fluids. Sensors and Actuators A 1997; 59: 197-200.
Mehta RV, Upadhyay RV. Science and technology of ferrofluids. Current Science 1999; 76(3): 305-312.
[7]
Liu J. Analysis of a porous elastic sheet damper with a magnetic fluid. Journal of Tribology 2009; 131: 0218011-15.
Verma PDS. Magnetic fluid-based squeeze film. International Journal of Engineering Science 1986; 24(3): 395-401.
Kumar D, Sinha P, Chandra P. Ferrofluid squeeze film for spherical and conical bearings.
ED
[9]
M
[8]
International Journal of Engineering Science 1992; 30(5): 645-656.
PT
[10] Bhat MV, Deheri GM. Magnetic-fluid-based squeeze film in curved porous circular discs.
Journal of Magnetism and Magnetic Materials 1993; 127: 159-162.
CE
[11] Prajapati BL. Magnetic-fluid-based porous squeeze films. Journal of Magnetism and
Magnetic Materials 1995; 149: 97-100.
AC
[12] Shah RC, Tripathi SR, Bhat MV. Magnetic fluid based squeeze film between porous
annular curved plates with the effect of rotational inertia. Pramana-Journal of Physics 2002; 58(3): 545-550.
[13] Shah RC, Bhat MV. Ferrofluid squeeze film in a long journal bearing. Tribology
International 2004; 37: 441- 446. [14] Patel RM, Deheri GM. Magnetic fluid based squeeze film between porous conical plates.
Industrial Lubrication and Tribology 2007; 59(3): 143-147. 15
ACCEPTED MANUSCRIPT
[15] Andharia PI, Deheri GM. Effect of longitudinal roughness on magnetic fluid based squeeze
film between truncated conical plates. FDMP 2011; 7(1): 111-124. [16] Shah RC, Patel DB. Squeeze film based on ferrofluid in curved porous circular plates with
various porous structure. Applied Mathematics 2012; 2(4): 121-123. [17] Lin JR, Lin MC, Hung TC, Wang PY. Effects of fluid inertia forces on the squeeze film
CR IP T
characteristics of conical plates-ferromagnetic fluid model. Lubrication Science 2013; 25: 429-439.
[18] Lin JR, Lu RF, Lin MC, Wang PY. Squeeze film characteristics of parallel circular disks
lubricated by ferrofluids with non-Newtonian couple stresses. Tribology International 2013; 61: 56-61.
Mathematical analysis of newly designed ferrofluid lubricated
AN US
[19] Shah RC, Patel DB.
double porous layered axially undefined journal bearing with anisotropic permeability, slip velocity and squeeze velocity. International Journal of Fluid Mechanics Research 2013; 40(5): 446-454.
[20] Shah RC, Bhat MV. Squeeze film based on magnetic fluid in curved porous rotating
M
circular plates. Journal of Magnetism and Magnetic Materials 2000; 208: 115-119. [21] Shah RC, Bhat MV. Anisotropic permeable porous facing and slip velocity on squeeze film
ED
in an axially undefined journal bearing with ferrofluid lubricant. Journal of Magnetism and Magnetic Materials 2004; 279: 224-230.
PT
[22] Sparrow EM, Beavers GS, Hwang IT. Effect of velocity slip on porous-walled squeeze
films. Journal of Lubrication Technology 1972; 94: 260-265.
CE
[23] Shah RC, Bhat MV. Effect of slip velocity in a porous secant-shaped slider bearing with a
AC
ferrofluid lubricant. FIZIKA A-ZAGREB 2003; 12(1): 1-8. [24] Prakash J, Tiwari K. Lubrication of a porous bearing with surface corrugations. Journal of
Lubrication Technology 1982; 104: 127-134.
16
ACCEPTED MANUSCRIPT
a
: radius of the sphere (m)
h
: film thickness defined in equation (2) (m)
hm
: minimum film thickness (m)
hm
:
dhm / dt , squeeze velocity (m s -1)
CR IP T
Nomenclature
: initial film thickness (m)
hm
: dimensionless minimum film thickness defined in equation (21)
hi
: dimensionless initial film thickness defined in equation (28)
h
: dimensionless quantity defined in equation (21)
H
: strength of variable magnetic field (Am -1)
H
: magnetic field vector
H0
: thickness of the porous layer (m)
H0
: dimensionless thickness of the porous layer defined in equation (21)
K
: defined in equation (9) (A2m -4)
M
: magnetization vector
p
: film pressure (N m -2)
p
: dimensionless pressure distribution defined in equation (25)
CE
PT
ED
M
AN US
hi
: pressure in the porous region (N m -2)
q
: defined in equation (8)
r
: radial co-ordinate (m)
R
: dimensionless radial co-ordinate defined in equation (21)
s
: slip parameter (m-1)
AC
P
17
ACCEPTED MANUSCRIPT
: dimensionless slip parameter for p and W defined in equation (21)
s*
: dimensionless slip parameter for t defined in equation (28)
t
: time(s)
t
: dimensionless response time defined in equation (30)
W
: load carrying capacity defined in equation (26) (N)
W
: dimensionless load carrying capacity defined in equation (27)
CR IP T
s
Greek symbols : fluid viscosity (N s m-2)
r
: porosity of the porous region in the radial direction
0
: free space permeability (N A-2)
: magnetic susceptibility
*
: dimensionless magnetization parameter for p and W defined in equation in (21)
1*
: dimensionless magnetization parameter for t defined in equation in (28)
: fluid density (N s2 m -4)
PT
ED
M
AN US
r
CE
r , z : permeabilities of the fluid in the radial and axial directions of the porous region (m2) : dimensionless radial permeability parameter of the porous region for p and W defined in
AC
equation (21)
r*
: dimensionless radial permeability parameter of the porous region for t defined in equation (28)
r
: dimensionless quantity defined in equation in (21)
r*
: dimensionless quantity defined in equation in (28)
18
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
Figure1. Squeeze film geometry between a sphere and a flat porous plate
19
with oblique and variable magnetic field
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
Figure 2. Order of magnetic field strength with different values of K
Figure 3. Variation in dimensionless pressure distribution p for different values of R and hm
20
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
Figure 4. Variation in dimensionless pressure distribution p for different values of r
Figure 5.Variation in dimensionless pressure distribution p for different values of H 0
21
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
Figure 6. Variation in dimensionless load carrying capacity W for different values of H 0 and hm
Figure 7. Variation in dimensionless load carrying capacity W for different values of r
22
CR IP T
ACCEPTED MANUSCRIPT
AN US
Figure 8. Variation in dimensionless load carrying capacity W for different values of
AC
CE
PT
ED
M
hm and H 0
Figure 9. Variation in dimensionless response time t for different values of H 0 and hm when squeezing takes place from height hi 0.1.
23
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 10. Variation in dimensionless response time t for different values of r* when
AC
CE
PT
ED
M
squeezing takes place from height hi 0.1.
24
On the squeeze film characteristics between a sphere and a flat porous plate using ferrofluid Rajesh C. Shah , Ramesh C. Kataria PII: DOI: Reference:
S0307-904X(15)00576-4 10.1016/j.apm.2015.09.042 APM 10741
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
4 April 2015 20 July 2015 23 September 2015
Please cite this article as: Rajesh C. Shah , Ramesh C. Kataria , On the squeeze film characteristics between a sphere and a flat porous plate using ferrofluid, Applied Mathematical Modelling (2015), doi: 10.1016/j.apm.2015.09.042
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights
ACCEPTED MANUSCRIPT
Ferrofluid lubrication with oblique variable magnetic field analysed.
Variable magnetic field because uniform field does not enhance performances
Porous plate considered because of added advantageous property of selflubrication
The Reynolds's type equation is developed using ferrohydrodynamics theory
Result indicates the better performance of the system when FF is used as
AC
CE
PT
ED
M
AN US
CR IP T
lubricant
ACCEPTED MANUSCRIPT
On the squeeze film characteristics between a sphere and a flat porous plate using ferrofluid RAJESH C. SHAH
RAMESH C. KATARIA
CR IP T
Department of Applied Mathematics, Faculty of Technology and Engineering, The M. S. University of Baroda, Vadodara – 390 001, Gujarat State, India e-mail [email protected]
AN US
Department of Mathematics, Som-Lalit Institute of Computer Applications, Ahmedabad – 380 009, Gujarat State, India e-mail [email protected]
Abstract Ferrofluid (FF) lubricated bearing design system formed by upper spherical surface and a flat porous plate considering variable magnetic field, which is oblique to the lower plate, have been analysed. The porous plate is considered because of having its advantageous property
M
of self-lubrication. The analytical model with squeeze behaviour of the phenomenon is developed, considering the validity of the Darcy’s law in the porous region, using equation of continuity, and equations from ferrohydrodynamics theory. The expressions for squeeze film
ED
characteristics are obtained and calculated numerically. The results indicate the better
PT
performance of the system, when FF is used as lubricant.
AC
CE
Keywords Ferrofluid, Squeeze film, Porous plate, Magnetic field
1
ACCEPTED MANUSCRIPT
1. Introduction When two lubricated surfaces approach each other with a normal velocity (known as squeeze velocity), then squeeze film phenomenon arise [1]. Study of squeeze film behaviour are observed in many fields of real life such as in machine tools, gears, rolling elements, hydraulic systems, engines, clutch plates, etc. Also, it is observed in the study of human knee joints and other skeletal joints as bio-lubrication [2]. Squeeze film with the attachment of porous layer
CR IP T
(region or plate or matrix or surface) are widely used in industry because of its advantageous property of self- lubrication and no need of exterior lubricant supply.
In recent years, many theoretical and experimental inventions are made on the bearing design systems as well as on the lubricating substances in order to increase the efficiency of the
AN US
bearing performances. One of the major revolutions in the direction of lubricating substances is an invention of ferrofluids. Ferrofluids (FFs) or Magnetic fluids (MFs) [3] are stable colloidal suspensions containing fine ferromagnetic particles dispersing in a liquid, called carrier liquid, in which a surfactant is added to generate a coating layer preventing the flocculation of the particles. When an external magnetic field H is applied, FFs experience magnetic body forces
M
(M )H which depends upon the magnetization vector M of ferromagnetic particles. Owing to these features, FFs are useful in many applications like in sensors, sealing devices, filtering
ED
apparatus, elastic damper, bearings, etc. [4-7]. Many researchers have also tried to find its application as lubricant in squeeze film bearing design systems. Verma [8] studied effects of MF on squeeze film bearing design system under an externally applied magnetic field oblique to the
PT
lower surface with the lower porous surface composed of three thin layers with different porosities. Explicit solutions for velocity, pressure, load carrying capacity and response time are
CE
obtained. It is found that upper plate takes longer time to come down in this case as compared to conventional lubricant based squeeze film. Kumar et. al. [9] studied squeeze film for spherical
AC
and conical bearings using FF as lubricant with the effects of rotation of particles and constant magnetic field in transverse direction, and numerically studied various bearing characteristics. Bhat and Deheri [10] discussed about curved porous circular discs squeeze film with the effect of MF and shown that pressure, load capacity and response time increases with the increase of magnetization. Prajapati [11] discussed various designed squeeze films with MF effect and shown the superiority performance of the MF lubricant than conventional lubricant. Shah et. al. [12] studied squeeze film between porous annular curved plates with the effects of rotational inertia as well as MF and found that the increase in pressure and load capacity depended only on 2
ACCEPTED MANUSCRIPT
the magnetization whereas response time dependent on magnetization, fluid inertia and speed of rotation of the plates. In [13], Shah and Bhat studied squeeze film in a long journal bearing using FF as lubricant and found that load capacity and response time increased with the increasing values of the eccentricity ratio. In [14], Patel and Deheri discussed about MF based squeeze film between porous conical plates and found that negative effect induced by the porosity can be neutralized by the positive effect caused by the magnetization parameter. Andharia and Deheri in
CR IP T
[15] studied MF based squeeze film for truncated conical plates with the effect of longitudinal roughness and found that load capacity can be increased with magnetization as well as negatively skewed roughness. The pressure and response time also found to increase with magnetization. Shah and Patel [16] discussed impact of various porous structures on curved porous circular plates squeeze film using FF as lubricant and found that globular sphere model have more impact
AN US
on increase of load capacity as compare to capillary fissures model. Lin et. al. in [17] studied squeeze film characteristics for conical plates with the effect of fluid inertia and FF, and shown the better performance of the system as compared to non-inertia non-magnetic case. In [18], Lin et. al. studied squeeze film characteristics of parallel circular discs with the effects of FF and non-Newtonian couple stresses using transverse magnetic field. With these effects, it was shown
M
that higher load capacity and lengthens approaching time obtained.
The purpose of the present paper is to study newly designed squeeze film bearing made
ED
up of a sphere and a flat plate using water based FF as lubricant, which is controlled by oblique and variable magnetic field, with the effects of porosity, slip velocity and squeeze velocity. With
PT
these effects, the paper studied about the impact of squeeze film height, permeability and width of the porous layer. Expressions for pressure, load carrying capacity and response time are
CE
obtained from Reynolds’s equation. The dimensionless pressure distribution, load carrying capacity and response time are calculated and presented graphically.
AC
2. Development of the Mathematical Model Figure 1 shows the schematic representation of the bearing design system under study.
The upper surface is a rigid sphere of radius a, and lower surface is a flat porous plate which is
formed when a porous layer of thickness H0 is attached to the impermeable flat surface. The porous layer is considered because of added advantage of self-lubricating property of the system. The region between the sphere and flat porous plate is known as film region, which is filled with FF and controlled by variable magnetic field of strength H, which is oblique to the lower plate. 3
ACCEPTED MANUSCRIPT
As far as dynamical part is concern, the upper surface (sphere) approaches to lower flat porous plate with a uniform velocity, known as squeeze velocity hm , and is defined as dh hm m , dt
where hm is minimum film thickness and t is time. The shape of the film thickness h is defined as
r2 ; r a, 2a
AN US
h hm
CR IP T
(1)
where r is the radial co-ordinate.
(2)
The basic flow equations for the motion of FFs (lubricant flow) in the film region based
M
on ferrohydrodynamics theory by R. E. Rosensweig with continuity equation are given by (refer [3, 19])
q q q p 2q 0 (M )H , t
AC
CE
PT
ED
(3) q 0 ,
(4)
H 0 , (5) MH,
(6) (H M ) 0 , 4
ACCEPTED MANUSCRIPT
(7) where
, p, , q, 0 M, H, are density of fluid, film pressure, fluid viscosity, fluid ,
velocity, free space permeability, the magnetization vector, magnetic field vector, and magnetic susceptibility respectively.
q (r, r, z ) (u, rv, w),
CR IP T
Also,
(8)
where (r , , z ) are cylindrical co-ordinates and dot (.) represents derivative w.r.t. t.
form [20]
AN US
As mentioned above, the variable magnetic field of strength H considered here is of the
K r2 ( a r) H , a
(9)
M
2
ED
where K is chosen to suit the dimensions of both sides. By combining equations (3) to (8) under usual assumptions of lubrication, neglecting
PT
inertia terms, and that the derivatives of fluid velocities across the film predominate, the equation
2u 1 d 1 2 p 0 H . 2 dr z 2 (10)
AC
CE
governing the lubricant flow in the film region in cylindrical co-ordinates yields
As porous layer is attached to the lower impermeable surface, solving equation (10)
using slip boundary condition at the lower porous surface (that is, at z 0 ) as ( refer [21, 22] ) u
5
1 u 1 ; r r , s z s 5
ACCEPTED MANUSCRIPT
and condition at the upper spherical surface as u 0 when z h,
yields velocity profile in the film region as
h z (1 sh) ( z h) 2 (1 sh)
d 1 2 p 0 H , dr 2
CR IP T
u
(11)
where s is a slip parameter, and r , r are the permeability and porosity of the porous region in the radial direction respectively.
AN US
The continuity equation for the film region in cylindrical co-ordinates is given by
1 w 0, (ru ) z r r
(12)
M
which on integrating over the film thickness h; that is, over the interval (0, h) and making use of
ED
equation (11) with the insertion of squeeze velocity effect wh w
z h
hm , yields
PT
1 rh3 (4 sh) d 1 2 p 0 H hm w0 , where w0 w r r 12 (1 sh) dr 2
z 0
.
CE
(13)
Assuming the validity of the Darcy’s law in the porous region, the radial and axial
AC
velocity components of the fluid in the porous region are given by u
r
1 2 r P 2 0 H , (14)
6
ACCEPTED MANUSCRIPT
w
z
1 2 z P 2 0 H , (15)
where z is the permeability of the fluid in the porous region in axial direction, and P is the pressure there.
(16)
AN US
1 w 0. (r u ) z r r
CR IP T
The continuity equation for the FF flow in the porous region is given by
Substituting equations (14) and (15) in equation (16), and integrating over the thickness of the porous matrix H0; that is, over the interval (-H0,0) yields
M
1 1 1 r H 0 r P 0 H 2 z P 0 H 2 r r r 2 z 2 z 0
(17)
ED
Using Morgan-Cameron approximation [23] that the pressure P in the porous region can be replaced by the average pressure p with respect to the bearing wall thickness as
CE
PT
1 1 2 2 P 0 H p 0 H , r 2 2 r
AC
equation (17) becomes
r H 0 1 1 2 r p 0 H 2 . P 0 H 2 2 rz r r z z 0 (18)
Assuming that the normal (axial) component of velocity across the film–porous interface at the lower plate are equal, therefore
7
ACCEPTED MANUSCRIPT
w
z 0
w
z 0
. (19)
Using equations (13), (15), (18) and (19), one yields
CR IP T
1 h3 (4 sh) 1 2 12 H r r 0 p 0 H 12 hm , (1 sh) r 2 r r
(20)
which is known as Reynolds’s type equation of the considered phenomenon. Introducing dimensionless quantities
ah h H a r , hm m , h 2 hm , H 0 0 , r r3 , r r H 0 , a a a hm hm
K 0 hm3 h3 p , p m2 . a hm hm
M
s s hm , *
AN US
R
(21)
ED
Equation (20) becomes
PT
d d 1 2 GR p * R (1 R ) 12 R, dR dR 2
CE
AC
where
8
(22)
G 12 r
(h )3 (4 s h ) . (1 s h ) (23)
ACCEPTED MANUSCRIPT
3. Solution Solving equation (22) under the boundary conditions for the pressure field as
p 0 when R 1 and
dp 0 when R 0 , dR
CR IP T
(24) yields
1
R
6R dR . G
(25)
AN US
1 p * R 2 (1 R ) 2
The load carrying capacity W can be obtained by using the definition
a
W 2 pr dr .
M
0
ED
(26)
Using dimensionless quantities defined in equation (21), equation (26) implies
PT
dimensionless load carrying capacity as
W hm3 R3 * 3 0 G dR , 40 2 a 4 hm 1
(27)
AC
CE
W
where G is defined in equation (23). Again, introducing dimensionless quantities for calculating response time t as hi
hm , hi
1*
K 0 a 4 h2 W t a , r* r 3 , r* r* H 0 , t i 4 , s* shi , a W hi (28)
9
ACCEPTED MANUSCRIPT
where hi is the initial film thickness. Using (28), the dimensionless response time t to reach a film thickness hm starting with an initial film thickness hi is given by 3R 3 dR G*
h3W dt i4 0 , dhi a hm 1 1* 2 40
which implies
1
t
0
3R 3 dR G*
1 1* 2 40
(29)
dhi ,
(30)
M
hi
AN US
1
CR IP T
1
ED
where
(hi )3 h (4 s* hi h ) G* * and G * 12 . r (hi )3 (1 s* hi h )
PT
G
(31)
CE
4. Results and Discussion
AC
The values of the dimensionless pressure distribution ( p ), load carrying capacity ( W )
and response time ( t ) have been calculated for the representative values of different parameters given in section 4.1 using Simpson’s 1/3-rule with step size 0.1. The FF used here is water based. The variable magnetic field considered is oblique to the lower plate and its strength is of O(102) in order to get maximum magnetic field strength at r 2a/3. The calculation of magnetic field strength is shown in section 4.1. The other order of magnetic field strength with different K is shown in figure 2. 10
ACCEPTED MANUSCRIPT
When FF is used as lubricant, then the variation in p is due to the first term of the equation (25). The same conclusion happens for W in equation (27). Also, it is clear from equation (29) that, the variation in t is due to the term 1* 40 , when FF is used as lubricant. Discussion on squeeze film pressure
CR IP T
Figure 3 shows the variation in dimensionless pressure distribution p as a function of dimensionless radial co-ordinate R for different values of dimensionless minimum film thickness hm . It is observed that
p increases as hm increases, but the increase rate is more
when 0 R 0.5 . When R > 0.5, then p remains almost same for all hm . That is, in general, as R increases, p decreases. Hence, p is maximum nearer to origin of radial axis. The variation in
AN US
p as a function of dimensionless radial permeability parameter of the porous region r is shown in figure 4. It is observed that when 2.92 106 r 2.92 103 , p is constant, and then it starts decreasing when r 2.92 103 . Figure 5 shows the variation in p as a function of dimensionless thickness of the porous matrix H 0 . It is observed that for
0 H 0 0.01 , p
M
remains constant. When H 0 0.01 , p starts decreasing. It should be noted here that H 0 = 0
ED
indicates solid case; that is, the bearing design system without porous surface at bottom. It is a general observation (refer [22, 24]) that, the insertion of porous surface decreases load carrying capacity and ultimately pressure. Moreover, porous surface is inserted because of
PT
advantageous property of self-lubrication. Thus, it is interesting to note from figure 5 that when 0 H 0 0.01 , p remains constant. It means that, p attains the same value as that of solid
CE
case ( H 0 = 0) even if 0 H 0 0.01 . This is the added advantage of obtaining p as that of solid case as well as maintaining self-lubricating property of bearing design system. From figures 4
AC
and 5, p attains its maximum value, when 2.92 106 r 2.92 103 and 0 H 0 0.01 . Discussion on load carrying capacity
Figure 6 shows the variation in dimensionless load carrying capacity W as a function of dimensionless thickness of the porous matrix H 0 for different values of dimensionless minimum film thickness hm . It is observed that W increases as hm increases. When 0.005 hm 0.009 11
ACCEPTED MANUSCRIPT
and 0.00001 H 0 0.01 , W attains almost constant value. For 0.001 hm 0.004 , W starts decreasing after H 0 = 0.0001 whereas before that it takes constant value. After hm 0.01 , W decreases for all hm . The variation in W as a function of dimensionless radial permeability parameter of the porous region r for hm 0.007 is shown in figure 7. It is observed that r has no effect on W , when 2.92 106 r 2.92 103 . After r 2.92 103 , W decreases. Thus,
CR IP T
from figures 6 and 7, it is observed that better load carrying capacity can be obtained for thin layer of FF, where 0.005 hm 0.009 , 0.00001 H 0 0.01 and 2.92 106 r 2.92 103 . Figure 8 shows that W attains the same value when H 0 = 0 (solid case) and H 0 = 0.0001. Thus, again as discussed in squeeze film pressure, added advantage of obtaining W as that of solid case
AN US
with self-lubricating property of bearing design system is observed. Discussion on squeeze film time
Solving equation (30) numerically using double integration procedure considering Simpson’s 1/3-rule with step size 0.1 for inner integral and step size given by following
hm 1 A step size A (Say), then hi 10
ED
hi
M
calculation for outer integral.
Figure 9 shows the variation in dimensionless response time t as a function of
PT
dimensionless thickness of the porous matrix H 0 for different values of dimensionless minimum
CE
film thickness hm . It is observed that, t decreases steeply when upper surface squeezing the FF film in the film region from height hi 0.1 to minimum film thickness hm 0.001 . When
AC
hm 0.02 , then decrease rate of t becomes low. For all other values of hm , t almost takes constant value. It is to be noted here from figure 9 that t is more when hm takes values from
0.009 to 0.001. The variation in t as a function of dimensionless radial permeability parameter of porous region r* for hm 0.007 is shown in figure 10 where t starts decreases after
r* 103 .
12
ACCEPTED MANUSCRIPT
4.1 Representative values and calculation of K The following representative values are taken
in computations. hi 0.0001(m), 0.05, 0.012 ( N s / m 2 ), r 0.25, 0 4 107 ( N / A2 ),
CR IP T
K 109 /1.48 ( A2 / m 4 ) , W= 25.0 N, hm = - 0.04 (m / s )
r 1011 (m2 ) (fixed for figures 3,5,6,8,9), H 0 = 0.0001 (m) (fixed for figures 3,4,7,10), a = 0.001 (m) (fixed for figure 3), a = 0.01(m) (fixed for figures 4-10),
AN US
r = 0.0001(m) (fixed for figures 4-10), hm = 0.00007(m) (fixed for figures 4,5). From equation (9),
H2
K r2 ( a r) , a
M
Max. H 2 1.48 105 K for a 0.01
where O indicates order.
PT
5. Conclusion
ED
For K 109 /1.48 , H= O(102) or O(H) = 2
On the basis of the ferrohydrodynamics theory, a FF lubricated squeeze film bearing
CE
design system formed by a sphere and a flat porous plate considering variable magnetic field, which is oblique to the lower plate, is theoretically analysed. It is noted here that porous plate is
AC
considered because of its advantageous property of self-lubrication and no need of exterior lubricant supply. The analytical model, known as Reynold’s equation, is derived using equation of continuity in film as well as porous region and equations from ferrohydrodynamics theory. The above model also considers the validity of Darcy’s law in the porous region. The following conclusions can be made from results and discussion. (1) 13
p is maximum nearer to R 0.1 .
ACCEPTED MANUSCRIPT
(2)
p is maximum and constant, when 2.92 106 r 2.92 103 .
(3)
p is maximum and constant, when 0 H 0 0.01 .
(4)
Better load carrying capacity can be obtained, when 0.005 hm 0.009 , 0.00001 H 0 0.01 and 2.92 106 r 2.92 103 . t is maximum for hm 0.001 , H 0 0.00001 and r* 106 103.
(6)
t has almost constant behaviour for 0.003 hm 0.009 .
CR IP T
(5)
It should be noted here that, according to [22], when porous layer is inserted then the
AN US
pressure of the porous medium provides a path for the fluid to come out easily from the bearing to the environment, which varies with permeability. Thus, the presence of the porous material decreases the resistance to flow in r – direction and as a consequence the load carrying capacity is reduced. The same behaviour also agrees with the conclusion of the Prakash and Tiwari [24] theoretically and experimentally by Wu [1]. In our case, the loss in W due to effect of porosity is
M
almost zero because of using FF as lubricant (which is controlled by oblique and variable magnetic field) for smaller values of H 0 and r . Moreover, because of porosity effect,
ED
self-lubrication property is an added advantage. The present case reduces to the case of conventional lubricant when * 0 and 1* 0 .
PT
It is to be noted here that variable magnetic field is used because uniform magnetic field
CE
does not enhance bearing performances as can be seen from equation (10).
AC
Acknowledgement The authors are grateful to the reviewers for their valuable comments.
14
ACCEPTED MANUSCRIPT
5. References [1]
Wu H. A review of porous squeeze films. Wear 1978; 47: 371-385.
[2]
Naduvinamani NB, Katti SG. Micropolar fluid poro-elastic squeeze film lubrication between a cylinder and a rough flat plate – A special reference to synovial joint lubrication.
CR IP T
Tribology Online 2014; 9(1): 21-30. [3]
Rosensweig RE. Ferrohydrodynamics. New York: Cambridge University Press; 1985.
[4]
Goldowsky M. New methods for sealing, filtering, and lubricating with magnetic fluids. IEEE Transactions on Magnetics, Mag. 1980; 16: 382-386.
[5]
Popa NC, Potencz I, Brostean L, Vekas L. Some applications of inductive Transducers with
[6]
AN US
magnetic fluids. Sensors and Actuators A 1997; 59: 197-200.
Mehta RV, Upadhyay RV. Science and technology of ferrofluids. Current Science 1999; 76(3): 305-312.
[7]
Liu J. Analysis of a porous elastic sheet damper with a magnetic fluid. Journal of Tribology 2009; 131: 0218011-15.
Verma PDS. Magnetic fluid-based squeeze film. International Journal of Engineering Science 1986; 24(3): 395-401.
Kumar D, Sinha P, Chandra P. Ferrofluid squeeze film for spherical and conical bearings.
ED
[9]
M
[8]
International Journal of Engineering Science 1992; 30(5): 645-656.
PT
[10] Bhat MV, Deheri GM. Magnetic-fluid-based squeeze film in curved porous circular discs.
Journal of Magnetism and Magnetic Materials 1993; 127: 159-162.
CE
[11] Prajapati BL. Magnetic-fluid-based porous squeeze films. Journal of Magnetism and
Magnetic Materials 1995; 149: 97-100.
AC
[12] Shah RC, Tripathi SR, Bhat MV. Magnetic fluid based squeeze film between porous
annular curved plates with the effect of rotational inertia. Pramana-Journal of Physics 2002; 58(3): 545-550.
[13] Shah RC, Bhat MV. Ferrofluid squeeze film in a long journal bearing. Tribology
International 2004; 37: 441- 446. [14] Patel RM, Deheri GM. Magnetic fluid based squeeze film between porous conical plates.
Industrial Lubrication and Tribology 2007; 59(3): 143-147. 15
ACCEPTED MANUSCRIPT
[15] Andharia PI, Deheri GM. Effect of longitudinal roughness on magnetic fluid based squeeze
film between truncated conical plates. FDMP 2011; 7(1): 111-124. [16] Shah RC, Patel DB. Squeeze film based on ferrofluid in curved porous circular plates with
various porous structure. Applied Mathematics 2012; 2(4): 121-123. [17] Lin JR, Lin MC, Hung TC, Wang PY. Effects of fluid inertia forces on the squeeze film
CR IP T
characteristics of conical plates-ferromagnetic fluid model. Lubrication Science 2013; 25: 429-439.
[18] Lin JR, Lu RF, Lin MC, Wang PY. Squeeze film characteristics of parallel circular disks
lubricated by ferrofluids with non-Newtonian couple stresses. Tribology International 2013; 61: 56-61.
Mathematical analysis of newly designed ferrofluid lubricated
AN US
[19] Shah RC, Patel DB.
double porous layered axially undefined journal bearing with anisotropic permeability, slip velocity and squeeze velocity. International Journal of Fluid Mechanics Research 2013; 40(5): 446-454.
[20] Shah RC, Bhat MV. Squeeze film based on magnetic fluid in curved porous rotating
M
circular plates. Journal of Magnetism and Magnetic Materials 2000; 208: 115-119. [21] Shah RC, Bhat MV. Anisotropic permeable porous facing and slip velocity on squeeze film
ED
in an axially undefined journal bearing with ferrofluid lubricant. Journal of Magnetism and Magnetic Materials 2004; 279: 224-230.
PT
[22] Sparrow EM, Beavers GS, Hwang IT. Effect of velocity slip on porous-walled squeeze
films. Journal of Lubrication Technology 1972; 94: 260-265.
CE
[23] Shah RC, Bhat MV. Effect of slip velocity in a porous secant-shaped slider bearing with a
AC
ferrofluid lubricant. FIZIKA A-ZAGREB 2003; 12(1): 1-8. [24] Prakash J, Tiwari K. Lubrication of a porous bearing with surface corrugations. Journal of
Lubrication Technology 1982; 104: 127-134.
16
ACCEPTED MANUSCRIPT
a
: radius of the sphere (m)
h
: film thickness defined in equation (2) (m)
hm
: minimum film thickness (m)
hm
:
dhm / dt , squeeze velocity (m s -1)
CR IP T
Nomenclature
: initial film thickness (m)
hm
: dimensionless minimum film thickness defined in equation (21)
hi
: dimensionless initial film thickness defined in equation (28)
h
: dimensionless quantity defined in equation (21)
H
: strength of variable magnetic field (Am -1)
H
: magnetic field vector
H0
: thickness of the porous layer (m)
H0
: dimensionless thickness of the porous layer defined in equation (21)
K
: defined in equation (9) (A2m -4)
M
: magnetization vector
p
: film pressure (N m -2)
p
: dimensionless pressure distribution defined in equation (25)
CE
PT
ED
M
AN US
hi
: pressure in the porous region (N m -2)
q
: defined in equation (8)
r
: radial co-ordinate (m)
R
: dimensionless radial co-ordinate defined in equation (21)
s
: slip parameter (m-1)
AC
P
17
ACCEPTED MANUSCRIPT
: dimensionless slip parameter for p and W defined in equation (21)
s*
: dimensionless slip parameter for t defined in equation (28)
t
: time(s)
t
: dimensionless response time defined in equation (30)
W
: load carrying capacity defined in equation (26) (N)
W
: dimensionless load carrying capacity defined in equation (27)
CR IP T
s
Greek symbols : fluid viscosity (N s m-2)
r
: porosity of the porous region in the radial direction
0
: free space permeability (N A-2)
: magnetic susceptibility
*
: dimensionless magnetization parameter for p and W defined in equation in (21)
1*
: dimensionless magnetization parameter for t defined in equation in (28)
: fluid density (N s2 m -4)
PT
ED
M
AN US
r
CE
r , z : permeabilities of the fluid in the radial and axial directions of the porous region (m2) : dimensionless radial permeability parameter of the porous region for p and W defined in
AC
equation (21)
r*
: dimensionless radial permeability parameter of the porous region for t defined in equation (28)
r
: dimensionless quantity defined in equation in (21)
r*
: dimensionless quantity defined in equation in (28)
18
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
Figure1. Squeeze film geometry between a sphere and a flat porous plate
19
with oblique and variable magnetic field
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
Figure 2. Order of magnetic field strength with different values of K
Figure 3. Variation in dimensionless pressure distribution p for different values of R and hm
20
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
Figure 4. Variation in dimensionless pressure distribution p for different values of r
Figure 5.Variation in dimensionless pressure distribution p for different values of H 0
21
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
Figure 6. Variation in dimensionless load carrying capacity W for different values of H 0 and hm
Figure 7. Variation in dimensionless load carrying capacity W for different values of r
22
CR IP T
ACCEPTED MANUSCRIPT
AN US
Figure 8. Variation in dimensionless load carrying capacity W for different values of
AC
CE
PT
ED
M
hm and H 0
Figure 9. Variation in dimensionless response time t for different values of H 0 and hm when squeezing takes place from height hi 0.1.
23
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 10. Variation in dimensionless response time t for different values of r* when
AC
CE
PT
ED
M
squeezing takes place from height hi 0.1.
24