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Derivation of ferrofluid lubrication equation of cylindrical squeeze films with convective fluid inertia forces and application to circular disks
Tribology International 49 (2012) 110–115
Contents lists available at SciVerse ScienceDirect
Tribology International journal homepage: www.elsevier.com/locate/triboint
Short Communication
Derivation of ferrofluid lubrication equation of cylindrical squeeze films with convective fluid inertia forces and application to circular disks Jaw-Ren Lin n Department of Mechanical Engineering, Nanya Institute of Technology, P.O. Box 324-22-59, Jhongli 320, Taiwan
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 July 2011 Received in revised form 7 November 2011 Accepted 8 November 2011 Available online 15 November 2011
Based upon the ferrohydrodynamic flow model, a lubrication equation in cylindrical coordinates for ferrofluid squeeze films including the effects of convective inertia forces and the presence of transverse magnetic fields has been derived for engineering application. As an application, the problem of parallel circular disks is illustrated. It is found that the ferrofluid circular squeeze film considering fluid inertia effects provides a higher load capacity and a longer elapsed time as compared to the non-inertia nonferrofluid case. These improved characteristics are further emphasized for larger values of the density parameter, the Langevin parameter and the volume concentration of particles. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Ferrofluids Squeeze films Convective inertia Lubrication equation
1. Introduction Ferrofluids or magnetic fluids [1] are stable colloidal suspensions containing fine ferromagnetic particles dispersing in liquid carriers, in which a surfactant is added to generate a coating layer preventing the flocculation of the particles. Ferrofluids can experience magnetic body forces depending on the magnetization of ferromagnetic particles when an external magnetic field is applied. Owing to these features, ferrofluids have been applied in many areas of industrial engineering and applied science, such as the medicine treatment for drug targeting by Rosensweig [1], the flow sensors by Popa et al. [2], the sealing devices by Zou et al. [3], the loudspeaker system by Odenbach [4], and the filtering apparatus by Goldowsky [5]. In the lubrication area, many investigators have also applied the ferrofluids as lubricants to study the squeeze film behavior between approaching surfaces. Some papers are focused on lubricating flows in different geometries such as rectangular plates by Verma [6], the circular disks by Prajapati [7], the annular plates by Bhat and Deheri [8] and the curved circular disks by Bhat and Deheri [9]. Observing these contributions [6–9], it has been shown that squeeze film plates with ferrofluids as lubricants provide an increased load capacity and a longer response time as compared to the case with a conventional lubricant. Since ferrofluids are stable colloidal suspensions containing fine ferromagnetic particles, it takes more time for squeeze film surfaces to squeeze the lubricant out of the bearings as compared to the situation of a
n
Tel.: þ886 3 4361070x6312; fax: þ886 3 4384670. E-mail address: [email protected]
0301-679X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2011.11.006
non-ferrofluid lubricant. Therefore, a larger value of the film pressure and load capacity is obtained for ferrofluid squeeze films. However, these contributions neglect the effects of rotation of the ferromagnetic particles in their studies. According to the analysis by Shliomis [10,11], the relaxation ways of the magnetization of a ferrofluid include the rotation of ferromagnetic particles described by the Brownian relaxation time parameter and the rotation of the magnetic moment of suspended particles described by the rotational magnetic moment relaxation time parameter. Therefore, the rotation of the particles themselves should be considered in ferrohydrodynamic equations. Based upon the ferrohydrodynamic flow model of Shliomis [10,11], excellent studies concerning the rotation of the ferromagnetic particles in squeeze films have been presented for the long journal bearings by Shah and Bhat [12] and the curved annular plates by Shah and Bhat [13]. According to their results, the load capacity and the response time are shown to increase with increasing values of the volume concentration of the solid phase and the strength of applied magnetic fields. However, the effects of fluid inertia forces are not considered in these studies [12,13]. Since the influences of inertia forces are important in the squeeze film pressure, the fluid inertia effects have been investigated on some squeeze film bearings such as the circular plates with pseudo-plastic fluids by Hashimoto and Wada [14] and the parallel plates and thrust bearings with micropolar fluids by Mahanti and Ramanaiah [15]. Therefore, a further analysis for ferrofluid squeeze films including the effects of convective inertia forces is motivated in the present study. On the basis of the ferrohydrodynamic flow model of Shliomis [10,11], a lubrication equation in cylindrical coordinates for ferrofluid squeeze films including the effects of convective inertia
J.-R. Lin / Tribology International 49 (2012) 110–115
Nomenclature a fm h h0 h* H !0 H kB m ME ! M n p pa p* r,z r* R t
the radius of the film periphery modified pressure gradient function film thickness initial film thickness non-dimensional film thickness, h* ¼h/h0 transverse magnetic field applied magnetic field vector Boltzmann constant magnetic moment of a particle equilibrium magnetization, M E ¼ nmðcoth x1=xÞ magnetization vector number of particles per unit volume film pressure reference pressure non-dimensional film pressure, p* ¼p/pa radial and vertical coordinates non-dimensional radial coordinate, r* ¼ r/a 4 density parameter, R ¼ rh0 pa =6Z0 2 a2 elapsed time
forces and the presence of transverse magnetic fields will be derived in this study for engineering application. As an example, the squeeze film problem between parallel circular disks is illustrated. Thereafter, the effects of the fluid inertia forces, the rotational viscosity and the strength of magnetic fields on the load capacity and the elapsed time are presented and discussed by comparing with the corresponding non-inertia non-ferrofluid situation.
2. Formulation Fig. 1 describes the squeeze film geometry. The ferrofluid film is sandwiched between two curved surfaces in the presence of ! external magnetic field H ¼ ð0,0,H0 Þ. The film is axially symmetric, and a is the radius of the film periphery. The upper surface approaches the lower one with a squeezing velocity vs ¼ qh/qt. Assume that the radii of curvature are much larger than the film thickness, therefore the thin-film theory of lubrication is applicable in the present study. According to the ferrohydrodynamic flow model of Shliomis [10,11], the Maxwell equations, the
2
non-dimensional elapsed time, t n ¼ h0 pa t=6Z0 a2 temperature velocity components in the r and z directions fluid velocity vector squeezing velocity, vs ¼ qh/qt non-dimensional squeezing velocity, 3 V n ¼ 6Z0 a2 vs =pa h0 load capacity non-dimensional load capacity, W n ¼ 8W=pa2 pa
t* T u,w ! u vs V* W W*
Greek symbols
Z0 Z m0
viscosity of the main liquid viscosity of the suspension permeability of free space volume concentration of particles fluid density Langevin parameter, x ¼ m0 mH0 =kB T rotational viscosity parameter, t ¼ ð3f=2Þðxtanh xÞ= ðx þ tanh xÞ
f
r x
t
continuity equation, and the momentum equations including the effects of convective fluid inertia forces can be expressed in vector form. !
r H ¼0
ð1Þ
! !
rUð H þ M Þ ¼ 0
ð2Þ
r U! u ¼0
ð3Þ
h i ! ! 1 ! ! ! 2! r ð! u UrÞ u ¼ rp þ Zr u þ m0 ðM Ur Þ H þ m0 r ðM H Þ 2
ð4Þ
where ! ! H t ! ! þM E ðO H Þ M ¼ ME H0 H0
ð5Þ
ME ¼ nmðcoth x1=xÞ
ð6Þ
!
!
O ¼ ð1=2Þr u
t¼ x¼
Fig. 1. Squeeze film geometry between cylindrical curved plates lubricated with a ferrofluid.
111
6Zf nkB Tð1 þ x coth xÞ
m0 mH0 kB T
ð7Þ ð8Þ
ð9Þ
! In these equations, M is the magnetization vector, r is the fluid density, Z is the viscosity of the suspension, m0 is the permeability of free space, ME is the equilibrium magnetization, n is the number of particles per unit volume, m is the magnetic moment of a particle, f is the volume concentration of particles, kB is the Boltzmann constant, T is the temperature, and x is the Langevin parameter. For the axially symmetric flow, the momentum equations governing the radial velocity component u in the r-direction can be simplified as @u @u dp @2 u @2 u ¼ þ Z 2 þ Zt 2 r u þw ð10Þ @r @z dr @z @z
112
J.-R. Lin / Tribology International 49 (2012) 110–115
where t is the rotational viscosity parameter, 3 xtanh x t¼ f 2 x þ tanh x
ð11Þ
When the inertia terms are neglected, Eq. (10) reduces to the non-inertia case for the curved annular plates derived by Shah and Bhat [13]. Now, it needs to obtain the radial velocity component u of the squeeze film. Since the fluid film is thin, the convective inertia forces can be treated as a constant over the film thickness. Similar to the procedure by Hashimoto and Wada [14] and Mahanti and Ramanaiah [15], the convective inertia terms in the momentum equation can be approximated by the mean value averaged across the film thickness. ! Z z ¼ þ h2 Z z ¼ þ h2 r @u @u dp @2 u dz þ dz ¼ þ Zð1 þ tÞ 2 ð12Þ u w @r @z dr h @z z ¼ h1 z ¼ h1 The boundary conditions for the velocity components are: u ¼0, w¼0 at z ¼ h1; and u ¼0, w¼ vs at z¼ þh2. Applying the continuity Eq. (3) and performing the above integrations, one can rewrite the momentum integral equation. ! Z Z @2 u dp r @ z ¼ þ h2 2 1 z ¼ þ h2 2 þ Zð1þ tÞ 2 ¼ u dzþ u dz ð13Þ dr r z ¼ z ¼ h1 h @r z ¼ h1 @z In order to obtain the expression for the pressure gradient dp/ dr, a modified pressure gradient function fm is introduced. ! Z Z dp r @ z ¼ þ h2 2 1 z ¼ þ h2 2 þ fm ¼ u dzþ u dz ð14Þ dr r z ¼ h1 h @r z ¼ h1 Then the momentum integral Eq. (13) is expressed in terms of the modified pressure gradient. @2 u 1 f ¼ Zð1þ tÞ m @z2
ð15Þ
Since the modified pressure gradient is independent of z, the radial velocity component u can be obtained by integrating the equation with respect to z. u¼
1 f ðz2 hzÞ 2Zð1 þ tÞ m
ð16Þ
Thereafter, one can obtain the following integrals. Z
3
z ¼ þ h2
u dz ¼ z ¼ h1
Z
z ¼ þ h2
h f 12Zð1 þ tÞ m 5
u2 dz ¼
z ¼ h1
h
120Z2 ð1 þ tÞ2
ð17Þ
2
fm
presence of transverse magnetic fields. The convective inertia 2 term 9rv2s r=ð10h Þ is the same as that obtained by Hashimoto and 2 Wada [14]. When neglecting the inertia term: 9rv2s r=ð10h Þ, the pressure gradient Eq. (21) reduces to lim
non-inertia
ð22Þ
This equation agrees with the non-inertia ferrofluid lubrication equation for the long journal bearings and for the curved annular plates by Shah and Bhat [12,13], respectively.
3. Squeeze film characteristics of circular disks To illustrate the application of the derived lubrication equation, the squeeze film problem for parallel circular disks with an incompressible ferrofluid under a uniform magnetic field H0 in the z-direction is shown in Fig. 2. The boundary conditions for the film pressure are: dp/dr¼ 0 at r¼ 0; and p ¼pa at r ¼a, where pa denotes the reference pressure. Integrating the lubrication equation and using the pressure conditions, one can obtain the film pressure. 1 6Zð1 þ tÞ 9r 2 p ¼ pa v þ ðv Þ ð23Þ ðr 2 a2 Þ s s 3 2 2 h 10h From Shliomis [10], the viscosity of the suspension can be described by the Einstein formula. 5 Z ¼ Z0 1 þ f ð24Þ 2 where Z0 denotes the viscosity of the main liquid. Similar to those by Mahanti and Ramanaiah [15], introduce the non-dimensional parameter and variables as follows: rn ¼
r , a
n
h ¼
h , h0
pn ¼
p , pa
Vn ¼
6Z0 a2 vs 3 pa h0
,
R¼
rh40 pa 6Z20 a2
ð25Þ
where R denotes the density parameter. Then the non-dimensional film pressure is given by 1 5 ð1 þ tÞ n 3R 1þ f pn ¼ 1 þ V þ V n2 ð1r n2 Þ ð26Þ n 3 n 2 2 2 h 20h The load-carrying capacity can be obtained by integrating the film pressure over the film region. Z þa ðppa Þ2pr dr ð27Þ W¼ r¼0
ð18Þ
On the other hand, the equality of the axial and radial flow rates gives Z z ¼ þ h2 1 u dz ¼ rvs ð19Þ 2 z ¼ h1
dp 6Zð1 þ tÞ ¼ vs r 3 dr h
Using the results of the film pressure and performing the integration, the non-dimensional load capacity can be obtained. Wn ¼
8W
pa2 pa
¼ ðW nN þ W nM ÞV n þW I RV n2
ð28Þ
Combining Eqs. (17) and (19) leads to the expression for the modified pressure gradient. fm ¼
6Zð1 þ tÞ 3
h
vs r
ð20Þ
Substituting Eqs. (20) and (18) into the definition of the modified pressure gradient (14), then the lubrication equation for the pressure gradient is derived after arrange the equation. dp 6Zð1 þ tÞ 9r 2 ¼ vs r v r 3 2 s dr 10h h
ð21Þ
The derived lubrication Eq. (21) can be applied to the study of cylindrical curved squeeze films lubricated with ferrofluids including the effects of convective fluid inertia forces and the
Fig. 2. Squeeze film configuration between parallel circular disks lubricated with a ferrofluid.
J.-R. Lin / Tribology International 49 (2012) 110–115
4. Results and discussion
where W nN ¼ W nM ¼ W nI ¼
2
ð29Þ
n3
h
2t þ5f þ 5ft
Based upon the above analysis, the squeeze film performances of parallel circular disks are characterized by three parameters: the density parameter, R; the volume concentration of particles, f; and the Langevin parameter x. In addition, one can obtain the corresponding cases from the specific values of the parameters.
ð30Þ
n3
h 3
ð31Þ
n2
10h
(1) R¼0: the non-inertia case; (2) f ¼0: the non-ferrofluid case; and (3) x ¼0; the case without magnetic fields.
For the squeeze film problem, it is also necessary to realize the time that will elapse from the initial state for the ferrofluid film to be reduced to a specific height. From Eq. (28), the non-dimensional squeezing velocity is given by Vn ¼
ðW nN þW nM Þ þ ½ðW nN þ W nM Þ2 þ4RW n W nI 1=2 2RW nI
Some representative values used for ferrofluids can be observed in many articles such as Rosensweig [1], Odenbach [4], Shliomis [10,11] and Shah and Bhat [12]. To justify the range of parameters used, the data are illustrated as follows:
ð32Þ
m0 ¼ 4p 107 kg m s2 UA2 , kB ¼ 1:38 1023 kg m2 s2 K1 , T ¼ 25 1C, r ¼ 895 kg m3 , Z0 ¼ 0:002 kg m1 s1 ,
Now, introduce the non-dimensional elapsed time. 2
tn ¼
113
h0 pa t 6Z0 a2
ð33Þ
m ¼ 2 1019 A m2 , h0 ¼ 0:15 mm,
Using this definition, the non-dimension squeezing expressed as n
V n ¼ dh =dt n
a ¼ 10:0 mm,
ð34Þ
Using W* ¼1 as by Mahanti and Ramanaiah [15], the elapsed time can be numerically obtained.
9
R=0, h*=0.5
8
vz ¼ 40 cm s1
R=8, h*=0.5
8
= 10, = 0.05 = 5, = 0.05 = 5, = 0.02
7
pa ¼ 1:01325 105 N m2 ,
According the definition of parameters, one can obtain x ffi6.1, Rffi 19.1, V ffi1.4; to present the results of this study, the values and the range of parameters are taken as: the density parameter: R¼0–20; the Langevin parameter: x ¼0–10; the volume concentration of particles: f ¼0–0.05; the non-dimensional squeezing velocity: V*¼ 1. Fig. 3 presents the variation of the non-dimensional film pressure p* with the non-dimensional radial coordinate r* for different values of x and f at the film height h*¼ 0.5. Under the non-inertia situation (R¼0) without magnetic fields (x ¼0) in
From Eqs. (32) and (34) together with the initial condition is: h ðt n ¼ 0Þ ¼ 1, one can obtain the elapsed time required for the film thickness to be reduced from the initial value to a final value hn. Z hn ¼ 1 W nI n dh ð35Þ t n ¼ 2R n n 2 n ½ðW N þ W M Þ þ 4RUW n W nI 1=2 ðW nN þ W nM Þ h n
9
H0 ¼ 1 105 A m1 ,
7
= 0, = 0.02
6
5
5
p*
p*
= 0, = 0 6
4
4
3
3
2
2
1
1
= 10, = 0.05 = 5, = 0.05 = 5, = 0.02 = 0, = 0.02 = 0, = 0
0
0.25
0.5 r*
0.75
1
0
0.25
0.5 r*
0.75
1
Fig. 3. Variation of the film pressure p* with the radial coordinate r* for different x and f at the film height h* ¼0.5. (a) Non-inertia situation and (b) with inertia.
J.-R. Lin / Tribology International 49 (2012) 110–115
55
50 45 40
= 10, = 5, = 5, = 0, = 0,
= 0.05 = 0.05 = 0.02 = 0.02 =0
R=8
W*
30 25 20 R=0
10 5 0 1
0.8
0.6
R=0
0.9
= 10, = 0.05 = 5, = 0.05 = 5, = 0.02 = 0, = 0.02 = 0, = 0
0.8 0.7 0.6 0.5 0.4 1
0
2
3
4
5
6
7
t* 1
= 10, = 0.05 = 5, = 0.05 = 5, = 0.02 = 0, = 0.02 = 0, = 0
R=8
0.9 0.8 0.7 0.6 0.5 0.4 0
1
2
3
4
5
6
7
t* Fig. 5. Variation of the response time t* with the film height h* for different values of x and f. (a) Non-inertia situation and (b) with inertia.
density parameter. Since the convective inertia terms in the momentum equation have been approximated by the mean value averaged across the film thickness, a linear dependency of W* on R is expected. Comparing with the situation without magnetic fields, it is also observed that under the application of a magnetic field (x ¼10) the effects of R on the load capacity are more pronounced for the ferrofluid-lubricated squeeze film disks with a higher value of the volume concentration of particles (f ¼ 0.05).
5. Conclusions
35
15
1
h*
Fig. 3(a), the effects of ferrofluids with volume concentration f ¼0.02 yield a higher pressure as compared to the conventional case of non-ferrofluid lubricant (f ¼0). When a magnetic field is applied (x ¼ 5, f ¼ 0.02), further higher values of the pressure are obtained. In addition, larger increments of the pressure are obtained by increasing values of the Langevin parameter and the volume concentration (x ¼5, f ¼0.05; x ¼10, f ¼0.05). When the influence of temporal acceleration of fluids is considered (R¼8) in Fig. 3(b), the effects of convective inertia forces are observed to signify an increase in the pressure as compared to the non-inertia situation. Fig. 4 shows the variation of the nondimensional load capacity W* with the non-dimensional film height h* for different values of x and f. Since the squeeze film with ferrofluids under applied magnetic fields results in a higher pressure as compared to the conventional non-ferrofluid case, the integrated load capacity is similarly influenced. Comparing with the situation without inertia forces, the effects of convective inertia forces are observed to increase the ferrofluid squeeze film load capacity; and larger increments are obtained with decreasing values of h* and increasing values of x and f. Fig. 5 describes the variation of the non-dimensional elapsed time t* with the nondimensional film height h* for different values of x and f. Under the non-inertia situation (R¼ 0) without magnetic fields (x ¼0), the influence of ferrofluids with volume concentration f ¼0.02 provide a longer elapsed time as compared to the non-ferrofluid case (f ¼ 0). With the application of a magnetic field (x ¼ 5, f ¼0.02), larger increments of the elapsed time are predicted. Increasing values of x and f (x ¼5, f ¼0.05; x ¼10, f ¼0.05) provide further increments of the elapsed time for ferrofluid squeeze film disks. On the other hand, when the convective inertia forces are included (R¼ 8), the effects of temporal acceleration of fluids are observed to bring forth longer response times as compared to the case of the non-inertia ferrofluid circular squeeze films. The effect of variation of R on the load capacity W* for different values of f at the film height h* ¼0.5 is displayed in Fig. 6. The load capacity is observed to grow linearly with the
h*
114
0.4
h* Fig. 4. Variation of the load capacity W* with the film height h* for different values of x and f.
Based upon the above formulation and the results discussed, conclusions can be drawn as follows. A lubrication equation of cylindrical curved squeeze films lubricated with ferrofluids including the effects of convective inertia forces and the presence of transverse magnetic fields has been derived on the basis of the ferrohydrodynamic flow model of 2 Shliomis [10,11]. The convective inertia term 9rv2s r=ð10h Þ is the same as that obtained by Hashimoto and Wada [14]. When neglecting this inertia term, the derived pressure gradient equation agrees with the non-inertia ferrofluid lubrication equation for the long journal bearings by Shah and Bhat [12] and for the curved annular plates by Shah and Bhat [13]. As an example, an analytical solution for the squeeze film characteristics between parallel circular disks has been obtained. Comparing with the non-inertia non-ferrofluid case, the use of ferrofluids as lubricants including the effects of fluid inertia provides a higher load-carrying capacity as well as a longer elapsed time; the improved characteristics are more pronounced for the squeeze film disks operating under larger values of the density
J.-R. Lin / Tribology International 49 (2012) 110–115
45
h*=0.5, =0
h*=0.5, =0
40
40
35
35
W*
W*
45
115
30
25
20
30
25
= 0.05 = 0.04 = 0.03 = 0.02
20
= 0.05 = 0.04 = 0.03 = 0.02
= 0.01
= 0.01
=0
=0
15
15 0
5
10
15
20
25
R
0
5
10
15
20
25
R
Fig. 6. Effect of variation of R on the load capacity W* for different values of f at the film height h* ¼0.5. (a) Without magnetic fields and (b) with magnetic fields.
parameter, the Langevin parameter and the volume concentration of particles. References [1] Rosensweig RE. Ferrohydrodynamics. New York: Cambridge University Press; 1985. [2] Popa NC, Potencz I, Brostean L, Vekas L. Some applications of inductive transducers with magnetic fluids. Sensors and Actuators A 1997;59:197–200. [3] Zou J, Li X, Lu Y, Hu J. Numerical analysis on the action of centrifuge force in magnetic fluid rotating shaft seals. Journal of Magnetism and Magnetic Materials 2002;252:321–3. [4] Odenbach S. Magnetoviscous effects in ferrofluids. Berlin Heidelberg: Springer-Verlag; 2002. [Chapter 2]. [5] Goldowsky M. New methods for sealing, filtering, and lubricating with magnetic fluids. IEEE Transactions on Magnetics 1980;Mag.-16:382–6. [6] Verma PDS. Magnetic fluid-based squeeze film. International Journal of Engineering Sciences 1986;24:395–401.
[7] Prajapati BL. Magnetic fluid-based porous squeeze films. Journal of Magnetism and Magnetic Materials 1995;149:97–100. [8] Bhat MV, Deheri GM. Squeeze film behaviour in porous annular disks lubricated with magnetic fluids. Wear 1991;151:123–8. [9] Bhat MV, Deheri GM. Magnetic fluid-based squeeze film in curved porous circular discs. Journal of Magnetism and Magnetic Materials 1993;127: 156–62. [10] Shliomis MI. Effective viscosity of magnetic suspensions. Soviet Physics Jetp 1972;34:1291–4. [11] Shliomis MI. Magnetic fluids. Soviet Physics Usp 1974;17:153–69. [12] Shah RC, Bhat MV. Ferrofluid squeeze film in a long journal bearing. Tribology International 2004;37:441–6. [13] Shah RC, Bhat MV. Ferrofluid squeeze film between curved annular plates including rotation of magnetic particles. Journal of Engineering Mathematics 2005;51:317–24. [14] Hashimoto H, Wada H. The effects of fluid inertia forces in parallel circular squeeze film bearings lubricated with pseudo-plastic fluids. Journal of Tribology 1986;108:282–7. [15] Mahanti AC, Ramanaiah G. Inertia effect of micropolar fluid in squeeze bearings and thrust bearings. Wear 1976;39:227–38.
Contents lists available at SciVerse ScienceDirect
Tribology International journal homepage: www.elsevier.com/locate/triboint
Short Communication
Derivation of ferrofluid lubrication equation of cylindrical squeeze films with convective fluid inertia forces and application to circular disks Jaw-Ren Lin n Department of Mechanical Engineering, Nanya Institute of Technology, P.O. Box 324-22-59, Jhongli 320, Taiwan
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 July 2011 Received in revised form 7 November 2011 Accepted 8 November 2011 Available online 15 November 2011
Based upon the ferrohydrodynamic flow model, a lubrication equation in cylindrical coordinates for ferrofluid squeeze films including the effects of convective inertia forces and the presence of transverse magnetic fields has been derived for engineering application. As an application, the problem of parallel circular disks is illustrated. It is found that the ferrofluid circular squeeze film considering fluid inertia effects provides a higher load capacity and a longer elapsed time as compared to the non-inertia nonferrofluid case. These improved characteristics are further emphasized for larger values of the density parameter, the Langevin parameter and the volume concentration of particles. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Ferrofluids Squeeze films Convective inertia Lubrication equation
1. Introduction Ferrofluids or magnetic fluids [1] are stable colloidal suspensions containing fine ferromagnetic particles dispersing in liquid carriers, in which a surfactant is added to generate a coating layer preventing the flocculation of the particles. Ferrofluids can experience magnetic body forces depending on the magnetization of ferromagnetic particles when an external magnetic field is applied. Owing to these features, ferrofluids have been applied in many areas of industrial engineering and applied science, such as the medicine treatment for drug targeting by Rosensweig [1], the flow sensors by Popa et al. [2], the sealing devices by Zou et al. [3], the loudspeaker system by Odenbach [4], and the filtering apparatus by Goldowsky [5]. In the lubrication area, many investigators have also applied the ferrofluids as lubricants to study the squeeze film behavior between approaching surfaces. Some papers are focused on lubricating flows in different geometries such as rectangular plates by Verma [6], the circular disks by Prajapati [7], the annular plates by Bhat and Deheri [8] and the curved circular disks by Bhat and Deheri [9]. Observing these contributions [6–9], it has been shown that squeeze film plates with ferrofluids as lubricants provide an increased load capacity and a longer response time as compared to the case with a conventional lubricant. Since ferrofluids are stable colloidal suspensions containing fine ferromagnetic particles, it takes more time for squeeze film surfaces to squeeze the lubricant out of the bearings as compared to the situation of a
n
Tel.: þ886 3 4361070x6312; fax: þ886 3 4384670. E-mail address: [email protected]
0301-679X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2011.11.006
non-ferrofluid lubricant. Therefore, a larger value of the film pressure and load capacity is obtained for ferrofluid squeeze films. However, these contributions neglect the effects of rotation of the ferromagnetic particles in their studies. According to the analysis by Shliomis [10,11], the relaxation ways of the magnetization of a ferrofluid include the rotation of ferromagnetic particles described by the Brownian relaxation time parameter and the rotation of the magnetic moment of suspended particles described by the rotational magnetic moment relaxation time parameter. Therefore, the rotation of the particles themselves should be considered in ferrohydrodynamic equations. Based upon the ferrohydrodynamic flow model of Shliomis [10,11], excellent studies concerning the rotation of the ferromagnetic particles in squeeze films have been presented for the long journal bearings by Shah and Bhat [12] and the curved annular plates by Shah and Bhat [13]. According to their results, the load capacity and the response time are shown to increase with increasing values of the volume concentration of the solid phase and the strength of applied magnetic fields. However, the effects of fluid inertia forces are not considered in these studies [12,13]. Since the influences of inertia forces are important in the squeeze film pressure, the fluid inertia effects have been investigated on some squeeze film bearings such as the circular plates with pseudo-plastic fluids by Hashimoto and Wada [14] and the parallel plates and thrust bearings with micropolar fluids by Mahanti and Ramanaiah [15]. Therefore, a further analysis for ferrofluid squeeze films including the effects of convective inertia forces is motivated in the present study. On the basis of the ferrohydrodynamic flow model of Shliomis [10,11], a lubrication equation in cylindrical coordinates for ferrofluid squeeze films including the effects of convective inertia
J.-R. Lin / Tribology International 49 (2012) 110–115
Nomenclature a fm h h0 h* H !0 H kB m ME ! M n p pa p* r,z r* R t
the radius of the film periphery modified pressure gradient function film thickness initial film thickness non-dimensional film thickness, h* ¼h/h0 transverse magnetic field applied magnetic field vector Boltzmann constant magnetic moment of a particle equilibrium magnetization, M E ¼ nmðcoth x1=xÞ magnetization vector number of particles per unit volume film pressure reference pressure non-dimensional film pressure, p* ¼p/pa radial and vertical coordinates non-dimensional radial coordinate, r* ¼ r/a 4 density parameter, R ¼ rh0 pa =6Z0 2 a2 elapsed time
forces and the presence of transverse magnetic fields will be derived in this study for engineering application. As an example, the squeeze film problem between parallel circular disks is illustrated. Thereafter, the effects of the fluid inertia forces, the rotational viscosity and the strength of magnetic fields on the load capacity and the elapsed time are presented and discussed by comparing with the corresponding non-inertia non-ferrofluid situation.
2. Formulation Fig. 1 describes the squeeze film geometry. The ferrofluid film is sandwiched between two curved surfaces in the presence of ! external magnetic field H ¼ ð0,0,H0 Þ. The film is axially symmetric, and a is the radius of the film periphery. The upper surface approaches the lower one with a squeezing velocity vs ¼ qh/qt. Assume that the radii of curvature are much larger than the film thickness, therefore the thin-film theory of lubrication is applicable in the present study. According to the ferrohydrodynamic flow model of Shliomis [10,11], the Maxwell equations, the
2
non-dimensional elapsed time, t n ¼ h0 pa t=6Z0 a2 temperature velocity components in the r and z directions fluid velocity vector squeezing velocity, vs ¼ qh/qt non-dimensional squeezing velocity, 3 V n ¼ 6Z0 a2 vs =pa h0 load capacity non-dimensional load capacity, W n ¼ 8W=pa2 pa
t* T u,w ! u vs V* W W*
Greek symbols
Z0 Z m0
viscosity of the main liquid viscosity of the suspension permeability of free space volume concentration of particles fluid density Langevin parameter, x ¼ m0 mH0 =kB T rotational viscosity parameter, t ¼ ð3f=2Þðxtanh xÞ= ðx þ tanh xÞ
f
r x
t
continuity equation, and the momentum equations including the effects of convective fluid inertia forces can be expressed in vector form. !
r H ¼0
ð1Þ
! !
rUð H þ M Þ ¼ 0
ð2Þ
r U! u ¼0
ð3Þ
h i ! ! 1 ! ! ! 2! r ð! u UrÞ u ¼ rp þ Zr u þ m0 ðM Ur Þ H þ m0 r ðM H Þ 2
ð4Þ
where ! ! H t ! ! þM E ðO H Þ M ¼ ME H0 H0
ð5Þ
ME ¼ nmðcoth x1=xÞ
ð6Þ
!
!
O ¼ ð1=2Þr u
t¼ x¼
Fig. 1. Squeeze film geometry between cylindrical curved plates lubricated with a ferrofluid.
111
6Zf nkB Tð1 þ x coth xÞ
m0 mH0 kB T
ð7Þ ð8Þ
ð9Þ
! In these equations, M is the magnetization vector, r is the fluid density, Z is the viscosity of the suspension, m0 is the permeability of free space, ME is the equilibrium magnetization, n is the number of particles per unit volume, m is the magnetic moment of a particle, f is the volume concentration of particles, kB is the Boltzmann constant, T is the temperature, and x is the Langevin parameter. For the axially symmetric flow, the momentum equations governing the radial velocity component u in the r-direction can be simplified as @u @u dp @2 u @2 u ¼ þ Z 2 þ Zt 2 r u þw ð10Þ @r @z dr @z @z
112
J.-R. Lin / Tribology International 49 (2012) 110–115
where t is the rotational viscosity parameter, 3 xtanh x t¼ f 2 x þ tanh x
ð11Þ
When the inertia terms are neglected, Eq. (10) reduces to the non-inertia case for the curved annular plates derived by Shah and Bhat [13]. Now, it needs to obtain the radial velocity component u of the squeeze film. Since the fluid film is thin, the convective inertia forces can be treated as a constant over the film thickness. Similar to the procedure by Hashimoto and Wada [14] and Mahanti and Ramanaiah [15], the convective inertia terms in the momentum equation can be approximated by the mean value averaged across the film thickness. ! Z z ¼ þ h2 Z z ¼ þ h2 r @u @u dp @2 u dz þ dz ¼ þ Zð1 þ tÞ 2 ð12Þ u w @r @z dr h @z z ¼ h1 z ¼ h1 The boundary conditions for the velocity components are: u ¼0, w¼0 at z ¼ h1; and u ¼0, w¼ vs at z¼ þh2. Applying the continuity Eq. (3) and performing the above integrations, one can rewrite the momentum integral equation. ! Z Z @2 u dp r @ z ¼ þ h2 2 1 z ¼ þ h2 2 þ Zð1þ tÞ 2 ¼ u dzþ u dz ð13Þ dr r z ¼ z ¼ h1 h @r z ¼ h1 @z In order to obtain the expression for the pressure gradient dp/ dr, a modified pressure gradient function fm is introduced. ! Z Z dp r @ z ¼ þ h2 2 1 z ¼ þ h2 2 þ fm ¼ u dzþ u dz ð14Þ dr r z ¼ h1 h @r z ¼ h1 Then the momentum integral Eq. (13) is expressed in terms of the modified pressure gradient. @2 u 1 f ¼ Zð1þ tÞ m @z2
ð15Þ
Since the modified pressure gradient is independent of z, the radial velocity component u can be obtained by integrating the equation with respect to z. u¼
1 f ðz2 hzÞ 2Zð1 þ tÞ m
ð16Þ
Thereafter, one can obtain the following integrals. Z
3
z ¼ þ h2
u dz ¼ z ¼ h1
Z
z ¼ þ h2
h f 12Zð1 þ tÞ m 5
u2 dz ¼
z ¼ h1
h
120Z2 ð1 þ tÞ2
ð17Þ
2
fm
presence of transverse magnetic fields. The convective inertia 2 term 9rv2s r=ð10h Þ is the same as that obtained by Hashimoto and 2 Wada [14]. When neglecting the inertia term: 9rv2s r=ð10h Þ, the pressure gradient Eq. (21) reduces to lim
non-inertia
ð22Þ
This equation agrees with the non-inertia ferrofluid lubrication equation for the long journal bearings and for the curved annular plates by Shah and Bhat [12,13], respectively.
3. Squeeze film characteristics of circular disks To illustrate the application of the derived lubrication equation, the squeeze film problem for parallel circular disks with an incompressible ferrofluid under a uniform magnetic field H0 in the z-direction is shown in Fig. 2. The boundary conditions for the film pressure are: dp/dr¼ 0 at r¼ 0; and p ¼pa at r ¼a, where pa denotes the reference pressure. Integrating the lubrication equation and using the pressure conditions, one can obtain the film pressure. 1 6Zð1 þ tÞ 9r 2 p ¼ pa v þ ðv Þ ð23Þ ðr 2 a2 Þ s s 3 2 2 h 10h From Shliomis [10], the viscosity of the suspension can be described by the Einstein formula. 5 Z ¼ Z0 1 þ f ð24Þ 2 where Z0 denotes the viscosity of the main liquid. Similar to those by Mahanti and Ramanaiah [15], introduce the non-dimensional parameter and variables as follows: rn ¼
r , a
n
h ¼
h , h0
pn ¼
p , pa
Vn ¼
6Z0 a2 vs 3 pa h0
,
R¼
rh40 pa 6Z20 a2
ð25Þ
where R denotes the density parameter. Then the non-dimensional film pressure is given by 1 5 ð1 þ tÞ n 3R 1þ f pn ¼ 1 þ V þ V n2 ð1r n2 Þ ð26Þ n 3 n 2 2 2 h 20h The load-carrying capacity can be obtained by integrating the film pressure over the film region. Z þa ðppa Þ2pr dr ð27Þ W¼ r¼0
ð18Þ
On the other hand, the equality of the axial and radial flow rates gives Z z ¼ þ h2 1 u dz ¼ rvs ð19Þ 2 z ¼ h1
dp 6Zð1 þ tÞ ¼ vs r 3 dr h
Using the results of the film pressure and performing the integration, the non-dimensional load capacity can be obtained. Wn ¼
8W
pa2 pa
¼ ðW nN þ W nM ÞV n þW I RV n2
ð28Þ
Combining Eqs. (17) and (19) leads to the expression for the modified pressure gradient. fm ¼
6Zð1 þ tÞ 3
h
vs r
ð20Þ
Substituting Eqs. (20) and (18) into the definition of the modified pressure gradient (14), then the lubrication equation for the pressure gradient is derived after arrange the equation. dp 6Zð1 þ tÞ 9r 2 ¼ vs r v r 3 2 s dr 10h h
ð21Þ
The derived lubrication Eq. (21) can be applied to the study of cylindrical curved squeeze films lubricated with ferrofluids including the effects of convective fluid inertia forces and the
Fig. 2. Squeeze film configuration between parallel circular disks lubricated with a ferrofluid.
J.-R. Lin / Tribology International 49 (2012) 110–115
4. Results and discussion
where W nN ¼ W nM ¼ W nI ¼
2
ð29Þ
n3
h
2t þ5f þ 5ft
Based upon the above analysis, the squeeze film performances of parallel circular disks are characterized by three parameters: the density parameter, R; the volume concentration of particles, f; and the Langevin parameter x. In addition, one can obtain the corresponding cases from the specific values of the parameters.
ð30Þ
n3
h 3
ð31Þ
n2
10h
(1) R¼0: the non-inertia case; (2) f ¼0: the non-ferrofluid case; and (3) x ¼0; the case without magnetic fields.
For the squeeze film problem, it is also necessary to realize the time that will elapse from the initial state for the ferrofluid film to be reduced to a specific height. From Eq. (28), the non-dimensional squeezing velocity is given by Vn ¼
ðW nN þW nM Þ þ ½ðW nN þ W nM Þ2 þ4RW n W nI 1=2 2RW nI
Some representative values used for ferrofluids can be observed in many articles such as Rosensweig [1], Odenbach [4], Shliomis [10,11] and Shah and Bhat [12]. To justify the range of parameters used, the data are illustrated as follows:
ð32Þ
m0 ¼ 4p 107 kg m s2 UA2 , kB ¼ 1:38 1023 kg m2 s2 K1 , T ¼ 25 1C, r ¼ 895 kg m3 , Z0 ¼ 0:002 kg m1 s1 ,
Now, introduce the non-dimensional elapsed time. 2
tn ¼
113
h0 pa t 6Z0 a2
ð33Þ
m ¼ 2 1019 A m2 , h0 ¼ 0:15 mm,
Using this definition, the non-dimension squeezing expressed as n
V n ¼ dh =dt n
a ¼ 10:0 mm,
ð34Þ
Using W* ¼1 as by Mahanti and Ramanaiah [15], the elapsed time can be numerically obtained.
9
R=0, h*=0.5
8
vz ¼ 40 cm s1
R=8, h*=0.5
8
= 10, = 0.05 = 5, = 0.05 = 5, = 0.02
7
pa ¼ 1:01325 105 N m2 ,
According the definition of parameters, one can obtain x ffi6.1, Rffi 19.1, V ffi1.4; to present the results of this study, the values and the range of parameters are taken as: the density parameter: R¼0–20; the Langevin parameter: x ¼0–10; the volume concentration of particles: f ¼0–0.05; the non-dimensional squeezing velocity: V*¼ 1. Fig. 3 presents the variation of the non-dimensional film pressure p* with the non-dimensional radial coordinate r* for different values of x and f at the film height h*¼ 0.5. Under the non-inertia situation (R¼0) without magnetic fields (x ¼0) in
From Eqs. (32) and (34) together with the initial condition is: h ðt n ¼ 0Þ ¼ 1, one can obtain the elapsed time required for the film thickness to be reduced from the initial value to a final value hn. Z hn ¼ 1 W nI n dh ð35Þ t n ¼ 2R n n 2 n ½ðW N þ W M Þ þ 4RUW n W nI 1=2 ðW nN þ W nM Þ h n
9
H0 ¼ 1 105 A m1 ,
7
= 0, = 0.02
6
5
5
p*
p*
= 0, = 0 6
4
4
3
3
2
2
1
1
= 10, = 0.05 = 5, = 0.05 = 5, = 0.02 = 0, = 0.02 = 0, = 0
0
0.25
0.5 r*
0.75
1
0
0.25
0.5 r*
0.75
1
Fig. 3. Variation of the film pressure p* with the radial coordinate r* for different x and f at the film height h* ¼0.5. (a) Non-inertia situation and (b) with inertia.
J.-R. Lin / Tribology International 49 (2012) 110–115
55
50 45 40
= 10, = 5, = 5, = 0, = 0,
= 0.05 = 0.05 = 0.02 = 0.02 =0
R=8
W*
30 25 20 R=0
10 5 0 1
0.8
0.6
R=0
0.9
= 10, = 0.05 = 5, = 0.05 = 5, = 0.02 = 0, = 0.02 = 0, = 0
0.8 0.7 0.6 0.5 0.4 1
0
2
3
4
5
6
7
t* 1
= 10, = 0.05 = 5, = 0.05 = 5, = 0.02 = 0, = 0.02 = 0, = 0
R=8
0.9 0.8 0.7 0.6 0.5 0.4 0
1
2
3
4
5
6
7
t* Fig. 5. Variation of the response time t* with the film height h* for different values of x and f. (a) Non-inertia situation and (b) with inertia.
density parameter. Since the convective inertia terms in the momentum equation have been approximated by the mean value averaged across the film thickness, a linear dependency of W* on R is expected. Comparing with the situation without magnetic fields, it is also observed that under the application of a magnetic field (x ¼10) the effects of R on the load capacity are more pronounced for the ferrofluid-lubricated squeeze film disks with a higher value of the volume concentration of particles (f ¼ 0.05).
5. Conclusions
35
15
1
h*
Fig. 3(a), the effects of ferrofluids with volume concentration f ¼0.02 yield a higher pressure as compared to the conventional case of non-ferrofluid lubricant (f ¼0). When a magnetic field is applied (x ¼ 5, f ¼ 0.02), further higher values of the pressure are obtained. In addition, larger increments of the pressure are obtained by increasing values of the Langevin parameter and the volume concentration (x ¼5, f ¼0.05; x ¼10, f ¼0.05). When the influence of temporal acceleration of fluids is considered (R¼8) in Fig. 3(b), the effects of convective inertia forces are observed to signify an increase in the pressure as compared to the non-inertia situation. Fig. 4 shows the variation of the nondimensional load capacity W* with the non-dimensional film height h* for different values of x and f. Since the squeeze film with ferrofluids under applied magnetic fields results in a higher pressure as compared to the conventional non-ferrofluid case, the integrated load capacity is similarly influenced. Comparing with the situation without inertia forces, the effects of convective inertia forces are observed to increase the ferrofluid squeeze film load capacity; and larger increments are obtained with decreasing values of h* and increasing values of x and f. Fig. 5 describes the variation of the non-dimensional elapsed time t* with the nondimensional film height h* for different values of x and f. Under the non-inertia situation (R¼ 0) without magnetic fields (x ¼0), the influence of ferrofluids with volume concentration f ¼0.02 provide a longer elapsed time as compared to the non-ferrofluid case (f ¼ 0). With the application of a magnetic field (x ¼ 5, f ¼0.02), larger increments of the elapsed time are predicted. Increasing values of x and f (x ¼5, f ¼0.05; x ¼10, f ¼0.05) provide further increments of the elapsed time for ferrofluid squeeze film disks. On the other hand, when the convective inertia forces are included (R¼ 8), the effects of temporal acceleration of fluids are observed to bring forth longer response times as compared to the case of the non-inertia ferrofluid circular squeeze films. The effect of variation of R on the load capacity W* for different values of f at the film height h* ¼0.5 is displayed in Fig. 6. The load capacity is observed to grow linearly with the
h*
114
0.4
h* Fig. 4. Variation of the load capacity W* with the film height h* for different values of x and f.
Based upon the above formulation and the results discussed, conclusions can be drawn as follows. A lubrication equation of cylindrical curved squeeze films lubricated with ferrofluids including the effects of convective inertia forces and the presence of transverse magnetic fields has been derived on the basis of the ferrohydrodynamic flow model of 2 Shliomis [10,11]. The convective inertia term 9rv2s r=ð10h Þ is the same as that obtained by Hashimoto and Wada [14]. When neglecting this inertia term, the derived pressure gradient equation agrees with the non-inertia ferrofluid lubrication equation for the long journal bearings by Shah and Bhat [12] and for the curved annular plates by Shah and Bhat [13]. As an example, an analytical solution for the squeeze film characteristics between parallel circular disks has been obtained. Comparing with the non-inertia non-ferrofluid case, the use of ferrofluids as lubricants including the effects of fluid inertia provides a higher load-carrying capacity as well as a longer elapsed time; the improved characteristics are more pronounced for the squeeze film disks operating under larger values of the density
J.-R. Lin / Tribology International 49 (2012) 110–115
45
h*=0.5, =0
h*=0.5, =0
40
40
35
35
W*
W*
45
115
30
25
20
30
25
= 0.05 = 0.04 = 0.03 = 0.02
20
= 0.05 = 0.04 = 0.03 = 0.02
= 0.01
= 0.01
=0
=0
15
15 0
5
10
15
20
25
R
0
5
10
15
20
25
R
Fig. 6. Effect of variation of R on the load capacity W* for different values of f at the film height h* ¼0.5. (a) Without magnetic fields and (b) with magnetic fields.
parameter, the Langevin parameter and the volume concentration of particles. References [1] Rosensweig RE. Ferrohydrodynamics. New York: Cambridge University Press; 1985. [2] Popa NC, Potencz I, Brostean L, Vekas L. Some applications of inductive transducers with magnetic fluids. Sensors and Actuators A 1997;59:197–200. [3] Zou J, Li X, Lu Y, Hu J. Numerical analysis on the action of centrifuge force in magnetic fluid rotating shaft seals. Journal of Magnetism and Magnetic Materials 2002;252:321–3. [4] Odenbach S. Magnetoviscous effects in ferrofluids. Berlin Heidelberg: Springer-Verlag; 2002. [Chapter 2]. [5] Goldowsky M. New methods for sealing, filtering, and lubricating with magnetic fluids. IEEE Transactions on Magnetics 1980;Mag.-16:382–6. [6] Verma PDS. Magnetic fluid-based squeeze film. International Journal of Engineering Sciences 1986;24:395–401.
[7] Prajapati BL. Magnetic fluid-based porous squeeze films. Journal of Magnetism and Magnetic Materials 1995;149:97–100. [8] Bhat MV, Deheri GM. Squeeze film behaviour in porous annular disks lubricated with magnetic fluids. Wear 1991;151:123–8. [9] Bhat MV, Deheri GM. Magnetic fluid-based squeeze film in curved porous circular discs. Journal of Magnetism and Magnetic Materials 1993;127: 156–62. [10] Shliomis MI. Effective viscosity of magnetic suspensions. Soviet Physics Jetp 1972;34:1291–4. [11] Shliomis MI. Magnetic fluids. Soviet Physics Usp 1974;17:153–69. [12] Shah RC, Bhat MV. Ferrofluid squeeze film in a long journal bearing. Tribology International 2004;37:441–6. [13] Shah RC, Bhat MV. Ferrofluid squeeze film between curved annular plates including rotation of magnetic particles. Journal of Engineering Mathematics 2005;51:317–24. [14] Hashimoto H, Wada H. The effects of fluid inertia forces in parallel circular squeeze film bearings lubricated with pseudo-plastic fluids. Journal of Tribology 1986;108:282–7. [15] Mahanti AC, Ramanaiah G. Inertia effect of micropolar fluid in squeeze bearings and thrust bearings. Wear 1976;39:227–38.