58 ANALYSIS OF A DOUBLE LAYERED POROUS SLIDER BEARING WITH HYPERBOLIC
STATOR LUBRICATED WITH FERROFLUID Darshana A Patel1, Ramesh C Kataria2*
1Department of Mathematics and Humanities, Vishwakarma Government Engineering College, Ahmedabad – 382424
2Department of Mathematics, Som-Lalit Institute of Computer Applications, Ahmedabad – 380 009
Abstract - Ferrofluid lubricated a doublelayered porous slider bearing with the hyperbolic stator, using slip-squeeze velocity and the magnetic field, is analytically analyzed. The fluid flow of the system is governed by the Neuringer –Rosensweig model. Associated Reynolds equation has been derived. Results are presented which relate the non-dimensional film pressure and load-carrying capacity. With the increase of magnetic field strength, the non- dimensional film pressure increases. For the different values of film thickness ratio, permeabilities, and magnetization parameter, the values of non-dimensional film pressure and load-carrying capacity are obtained and a comparison is made between the performance characteristics of double-layered porous slider bearing and conventional porous slider bearing.The study reveals that double layered porous slider bearing has superior performance.
Keywords Ferrofluid, Permeability, Porosity, Load carrying capacity.
1. INTRODUCTION
Ferrofluid [1] is a colloidal dispersion containing fine ferromagnetic particles, like ferric oxide in a non-conducting carrier liquid. To prevent clumping, each magnetic particle is thoroughly coated with anappropriate surfactant, in a carrier liquid. When an external magnetic field is applied, ferrofluid experiences magnetic body force as per the magnetization of ferromagnetic particles.
A typical ferrofluid contains approximately 85% carrier liquid, 10%
surfactant, and 5% magnetic solids. There are many advantages of using ferrofluid as a lubricant. Ferrofluids are retained at the desired location. No rubbing takes place between solid materials, no external lubrication is required and no side leakage is possible. Because of these features, ferrofluids are useful in many applications like sealing, sensors, filtering apparatus, elastic damper, lubrication, etc. [2 -5].
Amongst the hydrodynamic bearings, slider bearing is the simplest and commonly used one. The slider was found bent due to elastic, thermal, and uneven wear effects. Cameron [6]
proposed that an exponential shape of the slider was found to be in its truest shape.
The contacting surfaces in slider bearings are separated by fluid to reduce wear and
friction. A fluid layer between two approaching surfaces is known as squeeze film. The squeeze film lubrication plays a significant role in the applications of engineering such as rolling elements, engine tools, ball bearings, matching gears,and human knee joints,etc. Due to this inspiration, a lot of theoretical and experimental investigations considering different viewpoints were carried out on squeeze film.
Recently, researchers have also focused on film shape as it influencesbearing characteristics.
Abramovitz [7] investigated the impact of pad curvatures on the performance of thrust bearing. Bagci and Singh [8]
studied hydrodynamically lubricated slider bearing having different geometrical film shapes. They investigated the outcomes of film shape on the performance which includes a coefficient of friction and load carrying capacity.Gethin [9] suggested that the film shapes influence the bearing performances significantly. Das [10], Dobrica, and Fillon [11] found out that the outcomes of converging wedge on the bearing performance are significant. The performance of a curved slider bearing considering ferrofluid as a lubricant has been investigated by Singh and Gupta
59 [12].Shah and Parsania [13] analyzed
parallel plate stator slider bearing with other such bearings taking ferrofluid as a lubricant. Shah and Parikh [14]
mathematicallystudied differentdesigned slider bearings using ferrofluid as a lubricant under the presence ofthe transversely applied magnetic field.
Porous bearings were used in mechanics because of their self- lubricating nature and cheapness. Porous bearings are widely used in industry due to their self-contained fluid reservoir.
These bearings are generally used in breaks, clutches, etc. because of low friction characteristics. Bhat [15] studied the performance characteristics of an exponential slider bearing having a porous facing stator respectively. Agrawal [16] examined the impacts of magnetic fluid on an inclined plane porous slider bearing. Heshowed that magnetization of the magnetic particles increases load- carrying capacity with no considerable effect on the friction of the moving slider.
Shah and Bhat [17] analyzed slip velocity and material constant effects on a porous secant-shaped slider bearing with FF lubricant. The result showed that load- carrying capacity, friction on the slider, and the coefficient of friction decrease as slip parameter increases whereas load- carrying capacity decreases and friction as well as coefficient of friction increases as material parameter increases. Shah and Bhat [18] studied magnetic fluid- based slider bearing which has a convex pad slider and porous layer attached to the stator with slip velocity. The various bearing characteristics such as load- carrying capacity, friction on the slider, coefficient of friction, and position of the center of pressure were studied.
Mathematical modeling of ferrofluid-based porous-pivoted slider bearing considering slip velocity was analyzed by Ahmad and Singh [19]. Shah and Patel[20]
investigated newly designed ferrofluid lubricated slider bearing considering porosity, anisotropic permeability, and slip velocity at both ends. Shah and Kataria [21] examined the attributes of double porous layered slider bearing with convex pad stator using ferrofluid lubricant with the help of the equation
suggested by Neuringer-Rosensweig.Patel and Kataria [22] analysed and discussed the performances of double-porous layered slider bearing with parabolic pad stator using ferrofluid lubricant.
In this research paper, ferrofluid has been taken as a lubricant to study slider bearing having hyperbolic pad stator and double porous layered attached to the slider under the presence of magnetic field oblique to the lower surface. Here, Porous structures are important because of the self-lubricating property of the bearing design systems.
Flow in the porous matrix satisfies Darcy’s law. A modified Reynolds type equation is derived. Expressions for various bearing characteristics like non- dimensional film pressure and load- carrying capacity are obtained and calculated.The Introduction of the double porous layer increases non-dimensional film pressure and load-carrying capacity.
2. SYSTEM OF BASIC EQUATIONS For incompressible, Newtonian ferrofluids, the Navier-Stokes -a momentum equation for a magnetic fluid by Neuringer and Rosensweig [1] is given by
2 m
D f .
Dt p
u
u
(1)(1) Mass conservation
u 0,
(2)Where
D Dt t
u u
u u
is the material derivative,u = ( , , ) u v w
is the velocity vector of the fluid,p
is the pressure,
is the density,
is the viscosity of the lubricant. Heref
m
0( M ) H
is the magnetic bodyforce due to ferrofluid, which can be obtained from the following electromagnetic equationsMaxwell’s relation
= 0,
B
(3)Ampere’s law in the absence of current
H 0
, (4) Also, assuming that M and H are parallel i.e.
M H
, (5)60 where B is the induced field, M is the
magnetization vector, H is the magnetic field vector and
is magnetic susceptibility. B, H, and M are scalar quantities, thus B =B, H
H and M.
M
These vectors are related by
=
0( + ).
B H M
(6)Here
0 is the permeability of the free space and the value of
0 4 10
7. 3. ANALYSISThe geometry of the problem is shown in Fig. 1 where the film thickness hisa function of xin which the upper surface is in the hyperbolic shapewhich is fixed to form theminimum film thickness
h
1at the outlet and the maximum film thicknessh
2 at the inlet. The upper hyperbolic surface moves in they-direction with the velocityV dh dt
normal to the bearing surface is called the squeeze velocity.Fig. 1 Porous hyperbolic slider bearing
The lower element –slider of length L moves with the velocity Uin the x- direction. The fluid-film region is created by the shape and relative motion of the surfaces.Anincompressible, viscous ferrofluid lubricant is contained between two surfaces,which is called the film region.The lower surface-wall is backed with double porous layers with thicknesses H1 and H2 as shown in Fig.
1.The pressure in the porous regions satisfies the Laplace equation.
The magnetic field for magnetization of magnetic fluid considered here is oblique to the lower surface and is defined as [23]
H
2 K x(L x),
(7) whereK (A2m-4) is chosen to suit the dimensions of both sides.Some basic lubrication assumptions mentioned here are as follows:
The flow is laminar.
The fluid is incompressible and possesses constant properties.
Porous matrix is homogeneous and isotropic.
Flow in the film and porous region is axisymmetric.
Velocity gradient across the film is predominant.
Velocities are continuous at the film- porous interface.
All the inertia terms can be neglected as viscous force is greater than the inertia force.
Pressure is invariant across the fluid film.
Introduction of these assumptions and use of eqns. (1) to (6), leads to amodified equationfor the film region
3 3
2 2
0 0
2
2
h 1 1 0
0
(4 +sh ) h 1 (4 +sh ) h 1
H H
x (1+sh ) x 2 z (1+sh ) z 2
sh 1
6 U 12 12 H
x (1+sh ) y 2 y
p p
V p
(8) Where the boundary conditions [24] are1 u U , u s y
wheny 0
andu 0
when y = h61
1 1
1v ; w s y s
, wheny 0
andw 0
when y = h (9)also, pressure
p
1of the inner porous layer satisfies2 2 2
2 2 2
1 0 1 0 1 0
2 2 2
1 1 1
H H H 0
x p 2 y p 2 z p 2
(10) and pressure
p
2of the outer porous layer satisfies2 2 2
2 2 2
2 0 2 0 2 0
2 2 2
1 1 1
H H H 0
x p 2 y p 2 z p 2
(11)
(x, )
1(x, 0, z)
p z p
(12)1
(x, H , z)
1 2(x, H , z)
1p p
(13)1 1
2 2
1 1 0 2 2 0
= - H y = -H
1 1
H H
y p 2
yy p 2
(14)The conditions are given by eqns. (12) and (13) result from the continuity of pressure at the interface between the inner porous layer and the film region, and between the two porous layers, respectively. Eqn. (14) results from the fact that the flow normal to the boundary between the two porous layers must be equal on the two sides.
As porous slider bearing fitted with an impermeable housing
1 2
2
2 0
y = - (H + H )
1 H 0
y p 2
(15)Integrating eqn. (10) with respect to yover the porous layer of thickness
H
1, gives1 1
0 2 2
2 2 2 2
1 0 2 1 0 2 1 0 1 0
H
0 H
1 1 1 1
H ( H ) ( H ) y + H
y p 2
y x p 2
z p 2
d y p 2
y
(16) From the condition (14), eqn. (16) reduce to
1 1
0 2 2
2 2 2 2 2
1 0 2 1 0 2 1 0 2 0
H 1
0 y = - H
1 1 1 1
H ( H ) ( H ) y + H
y p 2
y x p 2
z p 2
d
y p 2
(17) Integrating eqn. (11) with respect to y over the porous layer of thickness
H
2, gives1
1 2
1
H 2 2
2 2 2
2 0 2 2 0 2 2 0
(H H ) H
1 1 1
H ( H ) ( H ) y
y p 2
y x p 2
z p 2
d
(18)
Since 2
1
0 2y p 2 H
is zero at y
H1
H2
. From eqns. (17) and (18)62
1
1
1 2
0 2 2
2 2 2
1 0 2 1 0 2 1 0
H 0
H 2 2
2 2
2
2 0 2 0
2 2
1 (H H )
1 1 1
H ( H ) ( H ) y
y 2 x 2 z 2
1 1
( H ) ( H ) y
x 2 z 2
y
p p p d
p p d
(19) If the wall thicknesses
H
1 andH
2 are assumed to be too smalland with the help of Morgan-Cameron approximation [21], eqn. (19) reduces2 2
1 1 2 2
2 2 2
1 0 2 0 2 0
0 1
H H
1 1 1
H ( H ) ( H )
y p 2 y
x p 2 z p 2
(20) From eqn. (20), eqn. (8) gives
3
2
1 1 2 2 0
3 2
2
1 1 2 2 0 h
(4 +sh ) h 1
12 H 12 H H
x (1+sh ) x 2
(4 +sh ) h 1 sh
12 H 12 H H 6 U 12
z (1+sh ) z 2 x (1+sh )
p
p V
(21) Neglecting the side-leakage effect, eqn (21) reduces to
3 2
2
1 1 2 2 0 h
(4 + sh ) h 1 sh
12 H 12 H H 6 U 12
x (1+ sh ) x p 2 x (1+ sh ) V
(22) Since there is no normal velocity i.e
V
h 0 ,
eqn (22) reduces to3 2
2
1 1 2 2 0
(4 + sh ) h 1 sh
12 H 12 H H 6 U
x (1+ sh ) x p 2 x (1+ sh )
(23) Setting
s
in eqn. (23), the present analysis reduced to the corresponding no slipcase in Neuringer-Rosensweig model for the ferrofluid flow, we get h3 12 H
1 1 12
2H
2 1 0 H
2 6 U h
H
26 U h
x x p 2 x
. (24)Assuming the ends of the bearing to be exposed to the atmosphere, the boundary conditions for pressure are
0 ) ( )
( h
1 p h
2
p
(25)Solving eqn. (24) with boundary conditions (25), the non-dimensional pressure distributionis given as
2
h
1UL p p
63
1
1 2 1 2 1 2 1 2 1 2 1 2
2 3
1
1 2
3 3 2 2
1 2 1
1 (1 )
2
6 ln 12 6 ln
(S S ) (S S ) (L L ) 2 3(T T ) (L L ) 2 3(T T )
h + 2h - 6
2 ln 1 2 S S ln 2 3 tan
ln h h h 3 h
(L L ) 2 3(T T X X
a a
a a a
a a
1 2
) 6 lna a
(26) Where
3
1 3
S ln 1 a
,3 3
2 3
(1 ln )
S ln 1 a
a
21 2 2
L ln a
a a
,2 2 1
2
2 2
L ln
1 ln (1 ln ) 1 ln
a a a
a a a
1 1
(2 ) T tan
3 a
,1 2
2 T tan 1 ln
3 a
a
1
2 ln 2
A = a
a
a
, 22 ln 1
A 2 3 a
a a
and 1 lnh a
X a
(27)
The load carried by the slider is given as
2 1 1 2
0
1 1
1 2 1 2
* 2
1 1
1 2 1 2 1 2 2 1
2 3
1 2
h UBL
1 1 2 3 2 (1 ln ) 2
S S ln 2 6ln L L tan tan
3 3
2 3 2 (1 ln ) 2
A L L A T T tan tan
12 (ln ) 3 3
L L
W W p dX
a a a
a a
a a a
a a a a
S S
a a a
1 2
12 3 T T 6 lna a
(28)
4. RESULTS AND DISCUSSION
Figures 2-5 present the pressure distribution and load capacity as the function of permeability parameter
, the porous thicknessH1, H2, and film thickness ratio a.The
values range from 0.1 to 1.0 and those of afrom 1.5 to 4.Consider
1 0 . 0001 ,
2 0 . 01
for doublelayered porous slider bearing and, 01 .
1
0
2 0 . 01
for conventionalporous slider bearing, and H1= 0.01(m) and H2 = 0.009(m), also K= 108/0.56 .
A comparative study of variation in non-dimensional film pressure
p
for a double layered porous slider bearing and conventional porous slider bearing (i.e. a porous slider bearing having a permeability
2 and a wall thickness of H1+ H2 ) is shown in Fig. 2. In comparison with the conventional porous slider bearing, the non-dimensional film pressure increases significantly with the
64 double layered porous slider bearing.
Furthermore, a maximum increase rate difference is observed when X = 0.5. At this X, W = 0.3071960for double layered porous slider bearing and W = 0.1945085 for conventional porous slider bearing.
Therefore, the increase rate of W is almost 57.93 % more for double-layered porous slider bearing as compared to conventional porous slider bearing.
In Fig.3thevariation in non- dimensional film pressure
p
is plotted against non-dimensional parameter X for different values of a. Consider 0.1
, it is observed thatp
increases as the filmthickness ratio a increases. Also, it can be seen from Fig. 3 and Table 1 that
for a smaller value of film thickness h1
(keeping h2 fixed)
p
increases. This tendency of increasing pressure with the insertion of double layered porous slider bearing is visible as per Srinivasan [25].Table 1 shows the calculated results of
p
and W when h1 takes a higher value 0.05 to lower one 0.0166 considering 1.0
. It is observed thatp
increases from 0.1660950 to 0.2833295;
that means about 70.58%, whereas W increases from 0.6426005 to 0.7368423;
that means about 14.66%.
h1 0.05 0.0166 % increase in
p
% increase in W a 2 6p
0.1660950 0.2833295 70.58%W 0.6426005 0.7368423 14.66%
Table 1 Effects on W when h1 takes a higher value 0.05 to lower one 0.0166 taking
1.0
. Fig. 4 shows effect of permeabilityparameter
on the non-dimensional film pressurep
.The non-dimensional film pressure falls rapidly as the permeability parameter
increasesTable 2 shows the comparative study of non-dimensional load W for a double layered porous slider bearing with that of the conventional porous slider bearing when X=0.5. When
* 0
increase rate of Wis almost 79.75% more
for double layered porous slider bearing as compared to conventional porous slider bearing. Also, when
* 0
, W is nearly 74.76% more for double layered porous slider bearing as compared to conventional porous slider bearing. The non-dimensional load-carrying capacityW increases with the double-layered porous slider bearing. Also, it is found that using ferrofluid as lubricant both pressure and load capacity increasesignificantly.W
*
0
* 0
Conventional porous slider bearing 0.4382292 0.4674597 Double layered porous slider bearing 0.7877129 0.8169434
% increase in W 79.75 74.76
Table 2 Effects onW for
* 0
and
* 0
when X=0.5.When magnetic fluid is used as a lubricant, W increases more as compared to conventional lubricant is exhibited in Table 3. Also due to magnetic fluid as a lubricant, W increases more up to 3.71% for double layered porous slider bearing and 6.67% for conventional porous slider bearing.
65
W
without using magnetic fluid
lubricant (conventional
lubricant)
*
0
using magnetic fluid lubricant
*
0
due to magnetic fluid as a lubricant %
increase in W Conventional porous slider
bearing
0.4382292 0.4674597 6.67
Double layered porous slider bearing
0.7877129 0.8169434 3.71
Table 3 Comparative effect of conventional lubricant and magnetic fluid lubricant on W Fig.5 shows the comparative study
of non-dimensional load W as a function of magnetization parameter
* for a double layered porous slider bearing and conventional porous slider bearing when a = 2. In comparison with the conventional porous slider bearing, the load capacity increases significantly with the double-layered porous slider bearing.Also, as the magnetization parameter
*increases, load W increases. This happens because the increase rate of Wis more with the generation of stronger spikes of the ferrofluid due to the higher strength of the magnetic field.
Fig.6 illustratesthe comparative study of variation in non-dimensional load capacity versus non-dimensional parameter X for double layered porous slider bearing and conventional porous slider bearing when a=1.5.It is observed that W significantly increased for double layered porous slider bearing as compared to conventional porous slider bearing.
This is because fluid flow through the porous matrix depends on the permeability and thickness of the porous matrix. It can be seen from the equation
1 1 2 2
3 1
H H
h
that the porous effect becomes more significant for a large value of
1H1
2H2 and a smaller value of h1.Fig. 2 Pressure
p
against non- dimensional parameter X for doubleand conventional porous layers
Fig. 3 Pressure
p
against non- dimensional parameter X for variousvalues of film thickness ratio a
Fig. 4 Pressure
p
against non- dimensional parameter X for variousvalues of permeability parameter
66 Fig. 5 Load Was a function of
magnetization parameter
* for a double-layered porous slider and conventional porous slider bearingsFig. 6 Load Wverses non-dimensional parameter X for a double-layered porous slider and conventional porous
slider bearings 5. CONCLUSIONS
The following conclusions can be made to increase the efficiency of double-layered porous slider bearing.
For both double-layered porous slider bearing and conventional porous slider bearing, W increases with the increase of non-dimensional magnetization parameter. This may be because the magnetization increases the viscosity of the lubricant, which results in increased pressure and consequently load carrying capacity.Also, double-layered porous slider bearing remains superior to conventional porous slider bearing
As the permeability parameter increases, both the non-dimensional film pressure p and non-dimensional load-carrying capacity Wdecreases for both double-layered porous slider bearing and conventional porous slider bearing.
As the film thickness ratioaincreases, non-dimensional pressurep increases for double- layered porous slider bearing.
DECLARATIONS
Funding
Not applicable
Conflicts of interest
The authors declare that they have no conflicts of interest
Authors' contributions
Both the authors contributed to the study conception, design, analysis and writing.
The authors read and approved the final manuscript.
Nomenclature
B Bearing breadth
h Fluid film thickness (m), h2
, 0 L
1 ln L
x a
x
h1 Minimum film thickness(m) h2 Maximum film thickness (m)
h
Non- dimensional film thickness,
1
h h a Film thickness ratio, 2
1
h h
V
Squeeze velocity (m/s),dh dt, H Strength of variable magnetic field (A / m)H Magnetic field vector
H1 Wall thickness of the inner porous layer (m)
H
2 Wall thickness of the outer porous layer (m)K Quantity defined in equation (7) (A2/m4)
L Bearing length (m) M Magnetization vector
p Pressure in the film region (N/m2)
p
Non-dimensional pressure (N/ m2),2
h
1UL p
p
1 Pressure of inner porous layerp
2 Pressure of outer porous layer67 s Slip parameter (1 / m),
1t Time (s)
u, v, wComponents of film fluid velocity in x, y and z –directions (m/s)
U Velocity of slider (m/s) W Load-carrying capacity (N) W Non-dimensional load-carrying capacity,
2 1 2
h U B L W
x, y, z Co-ordinates (m) X x/L
Greek symbols
Fluid viscosity (N s / m2)
0 Free space permeability (N / A2)
Slip constant ( 12 )
13
Fluid density (N s2 /m 4)
Magnetic susceptibility* Non-dimensional
magnetization parameter,
2 0 KL h1
U
1 Permeability of the inner porous matrix (m2)
2 Permeability of the outer porous matrix (m2)
Permeability parameter, 1 1 32 21
H H
h
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Appendix
The basic equations governing the ferrofluid flow in the film region are
2
2 2 0
1 1
y x 2 H
u p
(1)2
2 2 0
1 1
y z 2 H
w p
(2) Solving equation (1) and (2) under the boundary conditions
1 u U , u s y
wheny 0
andu 0
when y = h1 1
1v ; w s y s
, wheny 0
andw 0
when y = h We get,
20
y h hy y h
(h y) 1
U H
1 h 2 (1 h) x 2
s s
u p
s s
and (3)
20
y h hy 1
2 (1 h) z 2 H
s y h
w p
s
(4)Continuity equation for the film region is given by
x y z 0
u v w
(5)Substituting eqns. (3) and (4) into the continuity eqn. (5) and integrating it across the film thickness, we obtain
2 3 3
2 2
0 0 0 h
h U h (4 h) 1 h (4 h) 1
H H
x 2 1 h 12 1 h x 2 z 12 1 h 2
s s s
p p V V
s s s z
(6) Since the velocity component in the z- direction must be continuous at the plate-film interface, one has
1 2
0 0 1 0
y 0
1 H
y 2
V V p
(7) Substituting eqn. (7) into eqn. (6), we get
3 3
2 2
0 0
2
2
h 1 1 0
y 0
h (4 h) 1 h (4 h) 1
H H
x 1 h x 2 z 1 h z 2
h 1
12 6 U 12 H
x 1 h y 2
s s
p p
s s
V s p
s
