Acta Polytechnica


https://doi.org/10.14311/AP.2022.62.0488
Acta Polytechnica 62(4):488–497, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence

Published by the Czech Technical University in Prague

A COMPARATIVE STUDY OF FERROFLUID LUBRICATION ON
DOUBLE-LAYER POROUS SQUEEZE CURVED ANNULAR PLATES

WITH SLIP VELOCITY

Niru C. Patela, Jimit R. Patela, ∗, Gunamani M. Deherib

a Charotar University of Science and Technology (CHARUSAT), P. D. Patel Institute of Applied Sciences,
Department of Mathematical Sciences, CHARUSAT campus, Changa 388 421, Gujarat, India

b Sardar Patel University, Department of Mathematics, V. V. Nagar 388 120, Anand, Gujarat, India
∗ corresponding author: patel.jimitphdmarch2013@gmail.com

Abstract.
This article makes an effort to present a comparative study on the performance of a Shliomis

model-based ferrofluid (FF) lubrication of a porous squeeze film in curved annular plates taking slip
velocity into account. The modified Darcy’s law has been adopted to find the impact of the double-
layered porosity, while the slip velocity effect has been calculated according to Beavers and Joseph’s slip
conditions. The modified Reynolds equation for the double-layered bearing system is solved to compute
a dimensionless pressure profile and load-bearing capacity (LBC). The graphical results of the study
reveal that the LBC increases in the case of magnetization, volume concentration and upper plate’s
curvature parameter while it decreases with other parameters for both the film thickness profile. A
comparative study suggests that the exponential film thickness profile is more suitable to enhance LBC
for the annular plates lubricated by ferrofluid, including the presence of a slip. The study shows that
the slip model performed quite well and there is a potential for improving the performance efficiency.
Besides, multiple methods have been presented to enhance the performance of the above mentioned
bearing system by selecting various combinations of parameters governing the system.

Keywords: Shliomis model, curved annular plates, double-layered porous, slip velocity, exponential
and hyperbolic film profile, ferrofluid.

1. Introduction
Porous materials seem to be ubiquitous and play a
notable role in many aspects of day-to-day life. They
are extensively used in various areas, such as energy
management, vibration automotive, heat insulation,
processes of sound, turbine industries and fluid filtra-
tion.

Due to the phenomenon known as self-lubrication,
the porous bearing has a porous film filled with some
amount of lubricants so that it does not require more
lubrication throughout the life period of the bearing.
The lubricant comes out of the porous layer and is
deposited between the annular plates to inhibit fric-
tion and wear, as well as withstanding the original
load applied to the annular plates. Therefore, the
impact of lubrication due to the double porous layer
is better than that of the single porous layer. Due to
the remarkable mechanical properties and wide appli-
cations of the annular plates, many researchers have
been focused on analysing annular bearing systems,
such as Lin [1], Shah and Bhat [2], Bujurke et al. [3],
Deheri et al. [4], Fatima et al. [5] and Hanumangowda
et al. [6].

Also, numerous studies (Ting [7], Gupta et al. [8],
Bhat and Deheri [9], Shah et al. [10], Shimpi and De-
heri [11], Patel and Deheri [12], Rao and Agarwal [13],
Vasanth et al. [14] and Shah et al. [15]) have been

carried out to examine the impact of porosity on the
effectivity of annular plates.

A synthetic fluid, namely “ferrofluid”, is a mixture
of colloidal dispersions containing ferromagnetic par-
ticles in a liquid carrier. Besides being used in elastic
dampers to reduce noise, FFs are used in cooling and
heating cycles, long-term sealing of rotating shafts,
and reducing unwanted resonances in loudspeakers.
In the last four decades, several investigators (Kumar
et al. [16], Sinha et al. [17], Shah and Bhat [18], Patel
and Deheri [19], Shah and Shah [20] and Munshi et
al. [21]) have worked on a FF lubrication theory to
examine the behaviour of various bearing systems.

Alternative physical boundary conditions were
proposed in the advanced study of Beavers and
Joseph [22] that allowed a non-zero tangential velocity
(called slip velocity) at the surfaces and uncovered
that slip velocity had a broad effect on the bearing per-
formance. Several studies have been documented in
the literature about slip velocities for different condi-
tions in bearing systems (Chattopadhyay and Majum-
dar [23], Shah and Parsania [24], Shah and Patel [25],
Venkata et al. [26], Deheri and Patel [27], Patel and
Deheri [28], Shah et al. [29], and Mishra et al. [30]).

In their studies, Fragassa et al. [31], Janevski et
al. [32] and Geike [33] analysed the theory of static-
dynamic load and lubricated contacts, respectively.
These investigations confirm that the load profile re-

488

https://doi.org/10.14311/AP.2022.62.0488
https://creativecommons.org/licenses/by/4.0/
https://www.cvut.cz/en


vol. 62 no. 4/2022 A comparative study of ferrofluid lubrication . . .

mains crucial for the bearing design. Patel and De-
heri [34] examined the influence of variations of viscos-
ity of the ferrofluid on long bearings. It was noticed
that the viscosity variation does not help to increase
the LBC in the case of long bearings. Patel and
Deheri [35] presented a comparison of a porous struc-
ture on shliomis-model-based ferrofluid lubrication of
a squeeze film between rotating rough-curved circu-
lar plates. It has been ascertained that the Kozeny-
Carman model has an edge over the Irmay’s model in
improving the LBC. A study of thin film lubrication
at nanoscale appears in Patel and Deheri [36], where
a ferrofluid-based infinitely long rough porous slider
bearing has been considered. It has been found that
the magnetic fluid induced a higher load and showed
a further improvement when the thin film lubrication
at nanoscale took place.

Very few studies have been made regarding the fer-
rofluid lubrication in multi-layered porous plates in
the presence of slip velocity. And even lesser amount
of studies has been done concerning the comparative
studies on the performance of ferrofluid lubricated
porous squeeze film in the multi-layered bearing sys-
tem considering slip velocity. Thus, it was thought
proper to put forward a comparative study regard-
ing the performance of a ferrofluid-based squeeze film
in two-layered porous annular plates when the slip
velocity is taken into account. To what extent can
the ferrofluid lubrication counter the adverse effect of
porosity and slip velocity? This fundamental question
has been addressed while presenting the comparison.

2. Analysis
Figure 1 involves two annular disks (inner and outer
radius b and a, respectively (b < a)) with curved
(exponential and hyperbolic film) upper surface and
flat lower surface.

Figure 1. Diagram of the porous annular bearing

In view of Murti [37], Shah and Bhat [2] and Patel
and Deheri [38], the thickness profile h of the film is

assumed as

h(r) = h0e−βr
2
, b ≤ r ≤ a

for the exponential and

h(r) =
h0

1 + βr
, b ≤ r ≤ a (1)

for the hyperbolic profile.
As per the discussions of Shliomis [39] and Ku-

mar [40], and neglecting the assumptions of Shukla
and Kumar [41], the governing equations for the flow
of FF as suggested by Shliomis [39] are

−∇p+η∇2q+µ0(M ·∇)H +
1

2τs
∇×(S−IΩ) = 0 (2)

S = IΩ + µ0τs(M × H), (3)

M = M0
H

H
+
τB
I

(S × M), (4)

with the continuity equation (∇ · q = 0), equations of
Maxwell ∇ ×H = 0, ∇ · (H +M) = 0 and Ω = 12 ∇ ×q
(Bhat [42]).

Above mentioned equation(2) reduces to

−∇p+η∇2q+µ0(M ·∇)H+
1
2
µ0∇×(M ×H) = 0 (5)

and
M =

M0
H

[
H + τ(Ω × H)

]
,

where
τ =

τB

1 +
µ0τBτs
I

M0H

with the help of equations (3) and (4), as given in
Shliomis [39].

Equation(5) takes the following form, as discussed
in Bhat [42] and Patel and Patel [43] with u, H =
(0, 0,H0) and an axially symmetric flow,

∂2u

∂x2
=

1
η(1 + τ)

dp
dr

(6)

where
τ =

3
2
ϕ
ξ − tan hξ
ξ + tan hξ

.

Because of Beavers and Joseph’s [22] slip boundary
conditions

u(z = 0) = 0,

u(z = h) = −
1
s

∂u

∂z
,

ηa = η(1 + τ),

the solution of equation (6) can be transformed to

u = −
1
ηa

z2 − zsh(z − h)
2(1 + sh)

dp
dr
. (7)

489



N. C. Patel, J. R. Patel, G. M. Deheri Acta Polytechnica

With the help of the above expression (7), one can find
the continuity equation

(1
r

d
dr
∫ h

0 rudz +wh −w0 = 0
)
,

as

1
r

d
dr

(
h3r(2 + sh)

1 + sh
dp
dr

)
= 12ηa(wh − w0). (8)

Assuming the upper surface having a double layered
porous facing and the lower flat surface being solid in
the annular plate bearing.

In the present study, P1 and P2 of the porous
region satisfy the following equations, respectively
(Bhat [42]),

1
r

∂

∂r

(
∂P1
∂r

)
+
∂2P1
∂z2

= 0 and

1
r

∂

∂r

(
∂P2
∂r

)
+
∂2P2
∂z2

= 0. (9)

Using the Morgan-Cameron approximation, one gets(
∂P1
∂z

)
z=h1

=
H1
r

d
dr

(
r

dp
dr

)
,(

∂P2
∂z

)
z=h2

=
H2
r

d
dr

(
r

dp
dr

)
. (10)

Since the lower surface is solid and the upper sur-
face has a double layer porous facing, the velocity
component along z-direction is

w0 = 0

wh = ḣ0 −
[
k1
ηa

(
∂P1
∂z

)
z=h1

+
k2
ηa

(
∂P2
∂z

)
z=h2

]
.

(11)

Incorporating equation (9) and (10), equation (11)
turns into

w0 = 0

wh = ḣ0 −
[
k1
ηa

(
H1
r

d
dr

(
r

dp
dr

))

+
k2
ηa

(
H2
r

d
dr

(
r

dp
dr

))]
.

(12)

Using equation (12), and ηa = η0
(

1 + 52ϕ
)

(1 + τ),
equation (8) yields

1
r

d
dr

({
h3(2 + sh)

1 + sh
+ 12k1H1 + 12k2H2

}
r

dp
dr

)

= 12η0
(

1 +
5
2
ϕ

)
(1 + τ)ḣ0. (13)

Upon introduction of the non-dimensional measures:

R =
r

b
,

h =
h

h0
,

β = βb2 (exponential),
β = βb (hyperbolic),

P = −
h0

3p

η0b2ḣ0
,

s = sh0,

ψ1 =
k1H1

h0
3 ,

ψ2 =
k2H2

h0
3 , (14)

and using above equation (14), equation (13) trans-
forms to

1
R

d
dR

{[
h

3
(2 + sh)
1 + sh

+ 12(ψ1 + ψ2)
]
R

dP
dR

}

= −12
(

1 +
5
2
ϕ

)
(1 + τ). (15)

Considering boundary conditions of annular plates,

P(1) = P(k) = 0, (16)

one can find the dimensionless P as

P =
∫ R

1

(
−

6ER
G

)
dR + C1

∫ R
1

(
1
GR

)
dR, (17)

where

C1 =
∫ k

1
(6ER

G

)
dR∫ k

1
( 1

GR

)
dR

,

G =
h

3
(2 + sh)
1 + sh

+ 12(ψ1 + ψ2),

E =
(

1 +
5
2
ϕ

)
(1 + τ),

while the non-dimensional LBC of the annular plates
can be found as,

W = −
h0

3W

2πη0b4ḣ0
=
∫ k

1
RP dR

= −
1
2

(∫ k
1

−
6ER3

G
dR + C1

∫ k
1

R

G
dR

)
(18)

3. Result and discussion
The results for the double-layered porous medium
and slip velocity on exponential and hyperbolic film
profiles of the annular bearing are discussed in this sec-
tion. Equation (17) establishes the non-dimensional

490



vol. 62 no. 4/2022 A comparative study of ferrofluid lubrication . . .

pressure, while equation (18) represents a dimension-
less LBC. In addition, expression (18) is linear in
terms of the magnetization parameter, which indi-
cates an improvement of the LBC of annular plates
mathematically.

As far as LBC is concerned, a comparison of the
film profile is exhibited graphically in Figures 2–13
with double-porous facing in the occurrence of a slip.
The first figure demonstrates the exponential film
shape and the second figure depicts the impact of the
hyperbolic profile. The performance of appearance of
the Shliomis’ FF lubricated double porous medium
annular plates is based on the foundation of several
non-dimensional parameters like magnetization, upper
plate’s curvature, porosity, volume concentration and
slip velocity. It can be observed that the exponential
film fares better.

It is noticed that equation (18) suggests the LBC
of a single layer porous medium when ψ2 → 0. With
ψ1 → 0 and ψ2 → 0, this investigation transfers to
the non-porous FF based annular bearing system with
the slip velocity. Also, this study reduces to a study
of a traditional annular bearing by removing the effect
of magnetization in the absence of the slip.

For the range of the parameters, one can refer below:
τ: 0.1–0.5, ϕ: 0.01–0.05, β: 1.5–1.9, 1/s: 0.02–0.1, ψ1:
0.001–0.005 and ψ2: 0.001–0.005.

The dispensation of LBC about the τ, for numerous
values of ϕ, β, ψ1, ψ2 and 1/s shown in Figures 2–5,
recommends that the LBC rises strictly due to the
FF lubricant. A closer examination of the figures
emphasizes that the functionality of bearing systems
as well as the increase in load is connected with all
the parameters for both the film profiles. Exponential
film profile registers a higher load as compared to the
hyperbolic shape in Figures 2–5.

The behaviour of the volume concentration parame-
ter concerning various parameters of LBC is illustrated
in Figures 6–8, respectively. Due to the rise of the
volume concentration parameter, the effect of LBC de-
creases with porosity (in Figure 7) and slip velocity (in
Figure 8), while the effect of β (in Figure 6) increases
the LBC. Moreover, Figure 8 suggests a marginally
improved effect of the slip velocity in an exponen-
tial film bearing, which indicates an enhancement of
the overall annular bearing’s performance up to some
extent.

The profile of a non-dimensional LBC with respect
to the β is described in Figures 9–11. If we increase
the β, then the capacity of the load is growing sharply
in the case of the hyperbolic function, while a re-
verse behaviour is observed with porosity and slip
velocity. However, the LBC increases slightly for the
exponential profile and follows the same trends for the
parameters mentioned above. One can visualize an
identical scenario for the curvature of exponential and
hyperbolic functions, which is shown in Figures 9–11.
The effect of the porosity on the load distribution

of the bearing is shown in Figures 12 and 13. In Fig-

ure 12, the effect of slip velocity is negligible for the
exponential profile. However, Figure 13 suggests that
the trends of both the porosity parameters are almost
the same. Both layers help to improve the lifespan
of the system by creating a film layer between the
surfaces.

The graphical representation makes it clear that
the following takes place.
(1.) The nature of both porous facings is almost the

same with the same values (ψ1 = ψ2), however,
that does not apply when the porosity values differ.
(meaning ψ1 > ψ2 or ψ1 < ψ2 ).

(2.) Figure 13 displays the maximum load among all
the figures, which means the double porous layer
improves the LBC in annular plates bearing sys-
tems.

(3.) Higher values of curvature parameter have a negli-
gible effect on the LBC in the case of the exponential
profile, but it does reflect on the LBC in the case
of the hyperbolic film profile.

(4.) The effect of slip velocity is satisfactory for the
exponential surface profile as compared to the hy-
perbolic surface.

(5.) Finally, this study helps to improve the LBC by
considering the proper selection of all parameters
and film shapes while designing the bearing system
of annular plates.

4. Conclusion
The effect of the double-porous layered on MF lubri-
cated curved annular bearing is investigated theoreti-
cally with the theory of Shliomis’ flow model of FF,
modified Darcy’s law for double layer, and Beavers
and Joseph’s more realistic slip conditions. In view
of the bearing’s life period, it is evident that some of
the parameters appear to have an opposite effect on
the performance of the bearing system. Hence, this
investigation clarifies that while designing the bearing
system, the porosity in two layers, and the slip velocity
must be considered. Interestingly, numerous factors
(like porosity and slip velocity) disturb the system
adversely even though the bearing can support a load
without flow, this does not apply in the case of tradi-
tional lubricants. Even the upper plate’s curvature,
either exponential or hyperbolic, may significantly
impact the performance of this bearing system, con-
sidering the moderate values of volume concentration,
slip velocity, and porosity. Lastly, the exponential
film profile exhibits a higher load-bearing capacity, in
the case of the double-layered porosity with Shliomis’
magnetic fluid flow when slip is in place. A pertinent
question is to elevate this analysis by incorporating
the effect of surface roughness and deformation. An
immediate concern is to carry out this analysis for
some other types of bearing systems including the
circular ones with slip velocity.

491



N. C. Patel, J. R. Patel, G. M. Deheri Acta Polytechnica

Figure 2. Variation of LBC as regards of τ and ϕ.

Figure 3. Variation of LBC as regards of τ and β.

Figure 4. Variation of LBC as regards of τ and ψ1.

Figure 5. Variation of LBC as regards of τ and 1/s.

492



vol. 62 no. 4/2022 A comparative study of ferrofluid lubrication . . .

Figure 6. Variation of LBC as regards of ϕ and β.

Figure 7. Variation of LBC as regards of ϕ and ψ2.

Figure 8. Variation of LBC as regards of ϕ and 1/s.

Figure 9. Variation of LBC as regards of β and ψ1.

493



N. C. Patel, J. R. Patel, G. M. Deheri Acta Polytechnica

Figure 10. Variation of LBC as regards of β and ψ2.

Figure 11. Variation of LBC as regards of β and 1/s.

Figure 12. Variation of LBC as regards of ψ1 and 1/s.

Figure 13. Variation of LBC as regards of ψ1 and ψ2.

494



vol. 62 no. 4/2022 A comparative study of ferrofluid lubrication . . .

List of symbols
a Outer radius of annular plates
b Inner radius of annular plates
h Film thickness
r Radial coordinates
p Pressure of fluid
u x – component of q
w z – component of q
H Magnitude of H
I A sum of moments of inertia of the particles per unit

volume
s Slip velocity
q Fluid velocity in the film region
M Magnetization vector
H An identical magnetic field
S Internal angular momentum
h0 Central film thickness
w0, wh Values of w at z = 0, h respectively
h1 Thickness of lubricant in the inner layer
h2 Thickness of lubricant in the outer layer
k1 Permeability of inner layer of the porous region
k2 Permeability of outer layer of the porous region
M0 Equilibrium magnetization
H0 Constant magnetic field
H1 Thickness of the inner layer adjacent to lubricant

layer
H2 Thickness of the outer layer adjacent to solid wall
P1 Pressure of inside layer in the porous region
P2 Pressure of outside layer in the porous region
η Viscosity of suspension
β Curvature of the upper plate
ξ Langevin’s parameter
τ Magnetization parameter
ϕ Volume concentration
τB Brownian relaxation time parameter
τS Relaxation time parameter
µ0 Permeability of free space
η0 Carrier fluid viscosity
ψ1 Inner layer porous structure parameter
ψ2 Outer layer porous structure parameter

References
[1] J. Lin. Magneto-hydrodynamic squeeze film

characteristics between annular disks. Industrial
Lubrication and Tribology 53(2):66–71, 2001.
https://doi.org/10.1108/00368790110384028.

[2] R. Shah, M. Bhat. Ferrofluid squeeze film between
curved annular plates including rotation of magnetic
particles. Journal of Engineering Mathematics
51(4):317–324, 2005.
https://doi.org/10.1007/s10665-004-1770-9.

[3] N. Bujurke, N. Naduvinamani, D. Basti. Effect of
surface roughness on the squeeze film lubrication
between curved annular plates. Industrial Lubrication
and Tribology 59(4):178–185, 2007.
https://doi.org/10.1108/00368790710753572.

[4] G. Deheri, R. Patel, N. Abhangi. Magnetic fluid-based
squeeze film behavior between transversely rough
curved annular plates: A comparative study. Industrial
Lubrication and Tribology 63(4):254–270, 2011.
https://doi.org/10.1108/00368791111140477.

[5] S. Fatima, T. Biradar, B. Hanumagowda.
Magneto-hydrodynamics couple stress squeeze film
lubrication of rough annular plates. International
Journal of Current Research 9(9):58007–58014, 2017.

[6] G. Hanumagowda, B.and Savitramma, A. Salma,
Noorjahan. Combined effect of piezo-viscous
dependency and non-newtonian couple stresses in
annular plates squeeze-film characteristics. Journal of
Physics: Conference Series 1000(1):1–8, 2018.
https://doi.org/10.1088/1742-6596/1000/1/012083.

[7] L. Ting. Engagement behavior of lubricated porous
annular disks. part i: Squeeze film phase - surface
roughness and elastic deformation effects. Wear
34(2):159–172, 1975.
https://doi.org/10.1016/0043-1648(75)90062-9.

[8] J. Gupta, K. Vora, M. Bhat. The effect of rotational
inertia on the squeeze film load between porous annular
curved plates. Wear 79:235–240, 1982.
https://doi.org/10.1016/0043-1648(82)90171-5.

[9] M. Bhat, G. Deheri. Squeeze film behaviour in porous
annular discs lubricated with magnetic fluid. Wear
151(1):123–128, 1991.
https://doi.org/10.1016/0043-1648(91)90352-u.

[10] R. Shah, S. Tripathi, M. Bhat. Magnetic fluid based
squeeze film between porous annular curved plates with
the effect of rotational inertia. Pramana - Journal of
Physics 58(3):545–550, 2002.
https://doi.org/10.1007/s12043-002-0064-x.

[11] M. Shimpi, G. Deheri. A study on the performance
of a magnetic fluid based squeeze film in curved porous
rotating rough annular plates and deformation effect.
Tribology International 47:90–99, 2012.
https://doi.org/10.1016/j.triboint.2011.10.015.

[12] J. Patel, G. Deheri. Theoretical study of shliomis
model based magnetic squeeze film in rough curved
annular plates with assorted porous structures. FME
Transactions 42(1):56–66, 2014.
https://doi.org/10.5937/fmet1401056p.

[13] P. Rao, S. Agarwal. Couple stress fluid-based squeeze
film between porous annular curved plates with the effect
of rotational inertia. Iranian Journal of Science and
Technology, Transactions A: Science 41(4):1171–1175,
2017. https://doi.org/10.1007/s40995-017-0295-9.

[14] K. Vasanth, J. Hanumagowda, J. Santhosh Kumar.
Combined effect of piezoviscous dependency and
non-newtonian couple stress on squeeze-film porous
annular plate. Journal of Physics: Conference Series
1000(1):1–8, 2018.
https://doi.org/10.1088/1742-6596/1000/1/012080.

[15] R. Shah, D. Patel, D. Patel. Ferrofluid-based annular
squeeze film bearing with the effects of roughness and
micromodel patterns of porous structures. Tribology -
Materials, Surfaces & Interfaces 12(4):208–222, 2018.
https://doi.org/10.1080/17515831.2018.1542192.

495

https://doi.org/10.1108/00368790110384028
https://doi.org/10.1007/s10665-004-1770-9
https://doi.org/10.1108/00368790710753572
https://doi.org/10.1108/00368791111140477
https://doi.org/10.1088/1742-6596/1000/1/012083
https://doi.org/10.1016/0043-1648(75)90062-9
https://doi.org/10.1016/0043-1648(82)90171-5
https://doi.org/10.1016/0043-1648(91)90352-u
https://doi.org/10.1007/s12043-002-0064-x
https://doi.org/10.1016/j.triboint.2011.10.015
https://doi.org/10.5937/fmet1401056p
https://doi.org/10.1007/s40995-017-0295-9
https://doi.org/10.1088/1742-6596/1000/1/012080
https://doi.org/10.1080/17515831.2018.1542192


N. C. Patel, J. R. Patel, G. M. Deheri Acta Polytechnica

[16] P. Kumar, D.and Sinha, P. Chandra. Ferrofluid
squeeze film for spherical and conical bearings.
International Journal of Engineering Science
30(5):645–656, 1992.
https://doi.org/10.1016/0020-7225(92)90008-5.

[17] P. Sinha, P. Chandra, D. Kumar. Ferrofluid
lubrication of cylindrical rollers with cavitation. Acta
Mechanica 98:27–38, 1993.
https://doi.org/10.1007/bf01174291.

[18] R. Shah, M. Bhat. Ferrofluid squeeze film in a long
journal bearing. Tribology International 37:441–446,
2004.
https://doi.org/10.1016/j.triboint.2003.10.007.

[19] J. Patel, G. Deheri. Performance of a ferrofluid based
rough parallel plate slider bearing: A comparison of
three magnetic fluid flow models. Advances in Tribology
2016:1–9, 2016.
https://doi.org/10.1155/2016/8197160.

[20] R. Shah, R. Shah. Derivation of ferrofluid lubrication
equation for slider bearings with variable magnetic field
and rotations of the carrier liquid as well as magnetic
particles. Meccanica 53(4-5):857–869, 2017.
https://doi.org/10.1007/s11012-017-0788-9.

[21] M. Munshi, A. Patel, G. Deheri. Lubrication of
rough short bearing on shliomis model by ferrofluid
considering viscosity variation effect. International
Journal of Mathematical, Engineering and Management
Sciences 4(4):982–997, 2019.
https://doi.org/10.33889/ijmems.2019.4.4-078.

[22] G. Beavers, D. Joseph. Boundary conditions at a
natural permeable wall. Journal of Fluid Mechanics
30(1):197–207, 1967.
https://doi.org/10.1017/s0022112067001375.

[23] A. Chattopadhyay, B. Majumdar. Steady state
solution of finite hydrostatic porous oil journal bearings
with tangential velocity slip. Tribology International
17(6):317–323, 1984.
https://doi.org/10.1016/0301-679x(84)90095-1.

[24] R. Shah, M. Parsania. Ferrofluid lubrication equation
for non-isotropic porous squeeze film bearing with slip
velocity. Mathematics today 28(2):43–49, 2012.

[25] R. Shah, D. Patel. Squeeze film based on ferrofluid in
curved porous circular plates with various porous
structure. Applied Mathematics 2(4):121–123, 2012.
https://doi.org/10.5923/j.am.20120204.04.

[26] J. Venkata, R. Murthy, M. Kumar. Effect of slip
parameter on the flow of viscous fluid past an
impervious sphere. International Journal of Applied
Science and Engineering 12(3):203–223, 2014.
https://doi.org/10.6703/IJASE.2014.12(3).203.

[27] G. Deheri, S. Patel. Combined effect of slip velocity
and surface roughness on a magnetic squeeze film for a
sphere in a spherical seat. Indian Journal of Materials
Science 2015:1–9, 2015.
https://doi.org/10.1155/2015/159698.

[28] J. Patel, G. Deheri. Numerical modeling of jenkins
model based ferrofluid lubrication squeeze film
performance in rough curved annular plates under the
presence of slip velocity. Facta Universitatis, Series:
Mathematics and Informatics 31(1):11–31, 2016.

[29] N. Shah, R.and Patel, R. Kataria. Some porous
squeeze film-bearings using ferrofluid lubricant: A
review with contributions. Proceedings of the Institution
of Mechanical Engineers, Part J: Journal of
Engineering Tribology 230(9):1157–1171, 2016.
https://doi.org/10.1177/1350650116629096.

[30] M. Mishra, S.and Barik, G. Dash. An analysis of
hydrodynamic ferrofluid lubrication of an inclined rough
slider bearing. Tribology - Materials, Surfaces &
Interfaces 12(1):17–26, 2018.
https://doi.org/10.1080/17515831.2017.1418280.

[31] C. Fragassa, G. Minak, A. Pavlovic. Measuring
deformations in the telescopic boom under static and
dynamic load conditions. Facta Universitatis, Series:
Mechanical Engineering 18(2):315–328, 2020.
https://doi.org/10.22190/FUME181201001F.

[32] G. Janevski, P. Kozic, A. Pavlovic, S. Posavljak.
Moment lyapunov exponents and stochastic stability of
a thin-walled beam subjected to axial loads and end
moments. Facta Universitatis, Series: Mechanical
Engineering 19(2):209–228, 2021.
https://doi.org/10.22190/FUME191127014J.

[33] T. Geike. Bubble dynamics-based modeling of the
cavitation dynamics in lubricated contacts. Facta
Universitatis, Series: Mechanical Engineering
19(1):115–124, 2021.
https://doi.org/10.22190/FUME210112027G.

[34] J. Patel, G. Deheri. Influence of viscosity variation
on ferrofluid based long bearing. Reports in Mechanical
Engineering 3(1):37–45, 2022.
https://doi.org/10.31181/rme200103037j.

[35] J. Patel, G. Deheri. Effect of various porous
structures on the shliomis model based ferrofluid
lubrication of the film squeezed between rotating rough
curved circular plates. Facta Universitatis, Series:
Mechanical Engineering 12(3):305 – 323, 2014.

[36] J. Patel, G. Deheri. A study of thin film lubrication
at nanoscale for a ferrofluid based infinitely long rough
porous slider bearing. Facta Universitatis, Series:
Mechanical Engineering 14:89–99, 2016.
https://doi.org/10.22190/FUME1601089P.

[37] P. Murti. Squeeze films in curved porous circular
plates. Journal of Lubrication Technology 97(4):650–652,
1975. https://doi.org/10.1115/1.3452699.

[38] J. Patel, G. Deheri. Jenkins model based magnetic
squeeze film in curved rough circular plates considering
slip velocity: A comparison of shapes. FME
Transactions 43(2):144–153, 2015.
https://doi.org/10.5937/fmet1502144p.

[39] M. Shliomis. Effective viscosity of magnetic
suspensions. Soviet Physics – JETP 34(6):1291–1294,
1972.

[40] D. Kumar. Lubrication with a Magnetic fluid. Ph.D.
thesis, IIT Kanpur, 1991.

[41] J. Shukla, D. Kumar. A theory for ferromagnetic
lubrication. Journal of Magnetism and Magnetic
Materials 65(2-3):375–378, 1987.
https://doi.org/10.1016/0304-8853(87)90075-8.

[42] M. Bhat. Lubrication with a Magnetic fluid. Team
Spirit (India) Pvt. Ltd., India, 2003.

496

https://doi.org/10.1016/0020-7225(92)90008-5
https://doi.org/10.1007/bf01174291
https://doi.org/10.1016/j.triboint.2003.10.007
https://doi.org/10.1155/2016/8197160
https://doi.org/10.1007/s11012-017-0788-9
https://doi.org/10.33889/ijmems.2019.4.4-078
https://doi.org/10.1017/s0022112067001375
https://doi.org/10.1016/0301-679x(84)90095-1
https://doi.org/10.5923/j.am.20120204.04
https://doi.org/10.6703/IJASE.2014.12(3).203
https://doi.org/10.1155/2015/159698
https://doi.org/10.1177/1350650116629096
https://doi.org/10.1080/17515831.2017.1418280
https://doi.org/10.22190/FUME181201001F
https://doi.org/10.22190/FUME191127014J
https://doi.org/10.22190/FUME210112027G
https://doi.org/10.31181/rme200103037j
https://doi.org/10.22190/FUME1601089P
https://doi.org/10.1115/1.3452699
https://doi.org/10.5937/fmet1502144p
https://doi.org/10.1016/0304-8853(87)90075-8


vol. 62 no. 4/2022 A comparative study of ferrofluid lubrication . . .

[43] N. Patel, J. Patel. Magnetic fluid-based squeeze film
between curved porous annular plates considering the
rotation of magnetic particles and slip velocity. Journal

of Serbian Society for Computational Mechanics
14(2):69–82, 2020.
https://doi.org/10.24874/jsscm.2020.14.02.05.

497

https://doi.org/10.24874/jsscm.2020.14.02.05

	Acta Polytechnica 62(4):488–497, 2022
	1 Introduction
	2 Analysis
	3 Result and discussion
	4 Conclusion
	List of symbols
	References