Dynamic Performance Characteristics of a Curved Slider Bearing Operating with Ferrofluids

Abstract

In the present theoretical investigation, the effect of ferrofluid on the dynamic characteristics of curved slider bearings is presented using Shliomis model which accounts for the rotation of magnetic particles, their magnetic moments, and the volume concentration in the fluid. The modified Reynolds equation for the dynamic state of the bearing is obtained. The results of dynamic stiffness and damping characteristics are presented. It is observed that the effect of rotation of magnetic particles improves the stiffness and damping capacities of the bearings.

1. Introduction

In the field of engineering and technology, slider bearings are often designed to bear the transverse loads. The study of performance characteristics of slider bearings with different shape and different lubricants has been done from time to time by the researchers. Gupta and Kavita [1] analysed the effect of frame rotation for porous slider bearing, Singh and Gupta [2] investigated the performance of a pivoted curved slider bearings for pseudoplastic and dilatant lubricants, Pascovici et al. [3] presented an experimental evidence of cavitation effects in a Rayleigh step slider, Venkateswarlu and Rodkiewicz [4] discussed the thrust bearing characteristics considering the terminal speed of the slider, Williams and Symmons [5] analysed the performance of hydrodynamic slider bearings for non-Newtonian lubricants, and Sharma and Pandey [6] presented an experimental comparison of the slider bearing performance for different shapes.

In last few decades, the researches shown that the performances of the bearings can be improved, and enhanced pressure and load carrying capacity can be obtained by use of the magnetic lubricants and magnetic fields [7–10]. The applications of magnetic lubricant are widely found in dampers, seals, sensors, loudspeakers, gauges, steppers, and coating systems [11]. Investigators have used the Jenkins model [12] for the lubricant flow. On the other hand, Shliomis [13, 14] proposed a ferrofluid flow model, in which the effects of rotation of magnetic particles, their magnetic moments, and the volume concentration are included. Ram and Verma [15] used the Shliomis model to investigate the performance of a porous inclined slider bearing and reported an increased pressure and load capacity. Shah and Bhat [16] used this model to study the ferrofluid-based squeeze film characteristics of curved annular plates and obtained similar results. Yamaguchi [17] presented a detailed analysis and simplification of the Shliomis model for different lubricating conditions. Recently, the Shliomis model was used by Lin [18] to study the squeeze film characteristics of circular plates under magnetic field. It was observed that the volume concentration and the intensity of magnetic field provide an increase in the load capacity and the time of approach.

All these researchers have analyzed the steady-state characteristics of the bearings lubricated with magnetic fluids. However, the study of the dynamic (damping and stiffness) characteristics of the bearings lubricated with magnetic fluids is yet to be investigated for its importance in the bearing design.

In the present theoretical analysis, an attempt has been made to investigate the effects of the particle rotation and volume concentration of magnetic particles on the dynamic characteristics of curved slider bearings lubricated with magnetic fluids in the presence of transversely uniform magnetic fields.

2. Constitutive Equations and Boundary Conditions

The physical configuration of a curved slider bearing lubricated with a magnetic fluid in the presence of a transverse magnetic field is described in Figure 1. It is assumed that the thin film lubrication theory is applicable for the present analysis.

Figure 1 

Schematic diagram of a curved slider bearing under applied magnetic field with a pivot at the centre of pressure (C.P.) and 𝐻𝑐 at 𝐵/2.

After the ferrohydrodynamic flow model by Shliomis [13, 14], the electromagnetic field equations are ∇×𝐻^→=0,∇⋅𝐻^→+𝑀^→=0, (1) where 𝐻^→ is the applied magnetic field and 𝑀^→ is the magnetization vector.

Constitutive equations for incompressible magnetic fluid with internal rotation of magnetic particles [13, 14, 17] are ∇⋅^→𝑣𝜌𝐷=0,_→𝑣𝐷𝑡=−∇𝑝+𝜂∇^2→𝑣+𝜇𝑀_𝑜^→⋅∇𝐻^→+12𝜏_𝑠∇×^→𝑆−𝐼Ω^→, (2) where ^→𝑣 is the fluid velocity vector, 𝜌 is the fluid density, t is the time, p is the pressure, 𝜂 is the viscosity of the suspension, 𝜇_𝑜 is the free space permeability, 𝜏_𝑠 is the spin relaxation time, and 𝐼 is the sum of moments of inertia of the particles per unit volume. The local angular velocity of the Ω^→, equilibrium magnetization 𝑀^→, and the internal angular moment ^→𝑆 are given as ^→1Ω=2∇×^→𝑣,_→𝑀=𝑀_𝑜^→𝐻𝐻_𝑜+𝜏_𝐵𝐼^→𝑆×𝑀^→,_→𝑆=𝐼Ω^→+𝜇_𝑜𝜏𝑀_𝑠^→×𝐻^→. (3) where 𝜏_𝐵 is the Brownian relaxation time of magnetic particles.

Assuming the slider to be infinitely wide, (2) for one dimensional flow under isothermal condition [15, 16] are 𝜕𝑢+𝜕𝑥𝜕𝑤𝜕𝜕𝑧=0,₂𝑢𝜕𝑧²=1𝜂1+𝜇_𝑜𝑀_𝑜𝐻_𝑜𝐺𝜏/4𝜂𝜕𝑝,𝜕𝑥 (4)(5) where 𝜏𝜏=^𝐵1+𝜇_𝑜𝜏_𝐵𝜏_𝑠𝑀_𝑜𝐻_𝑜./𝐼 (6) In the Langevin relationship, the magnetization depends upon the intensity of magnetic fields for a system of spherical particles [13, 14] and it is given as 𝑀_𝑜(𝛼)−1=𝑛𝑚Coth𝛼,𝜇𝛼=^𝑜𝑚𝐻_𝑜𝑘_𝐵𝑇,𝜏_𝑠=6𝜂𝜙𝐼,𝜏_𝐵=3𝜂𝜙𝑛𝑘_𝐵𝑇, (7) where 𝑛 is the number of magnetic particles per unit volume, m is the magnetic moment of a particle, 𝛼is the Langevin parameter, 𝑘_𝐵 is the Boltzmann constant, T is the temperature, and 𝜙is the volume fraction of the dispersed solid phase.

Substituting (6) and (7) in (5) and introducing the magnetoviscosity, the additional viscosity (𝜂_𝑚) due to rotation of magnetic particles [13, 14]: 𝜂_𝑚=32(𝛼)𝜂𝜙𝛼−tanh(𝛼),𝛼+tanh (8) (5) can be rewritten as 𝜕²𝑢𝜕𝑧²=1𝜂+𝜂_𝑚𝜕𝑝.𝜕𝑥 (9) Integrating (9) under the no-slip boundary conditions: 𝑢=𝑈at𝑧=0,𝑢=0at𝑧=ℎ, (10) the expression of 𝑢 is obtained as 1𝑢=𝜂+𝜂_𝑚𝜕𝑝𝑧𝜕𝑥²𝑧−𝑧ℎ+𝑈1−ℎ. (11) Substituting (11) in (4) and integrating under the boundary conditions: 𝑤=0at𝑧=0,𝑤=𝑉=−𝜕ℎ𝜕𝑡at𝑧=ℎ, (12) the modified Reynolds equation is obtained as 𝜕ℎ𝜕𝑥³𝜕𝑝𝜕𝑥=6𝜂+𝜂_𝑚𝑈𝜕ℎ𝜕𝑥+12𝜂+𝜂_𝑚𝜕ℎ.𝜕𝑡 (13) Here, the effective viscosity 𝜂[14] is given by 𝜂=𝜂_𝑜(𝛼)1+4𝛼+tanh(𝛼)𝜙,𝛼+tanh (14) where 𝜂_𝑜denotes the viscosity of the carrier fluid.

For the one-dimensional curved slider bearings (Figure 1), the film thickness is taken as suggested by Abramovitz [19] in the form: ℎ=𝐻_𝑐4𝑥𝐵−12²−1+ℎ₁−ℎ₂𝑥1−𝐵+ℎ₂, (15) which can be rewritten as ℎ(𝑥,𝑡)=ℎ_𝑠(𝑥)+ℎ_𝑚(𝑡), (16) where ℎ_𝑠determines the steady state film profile, given by ℎ_𝑠=ℎ₁−ℎ₂+4𝐻_𝑐²16𝐻_𝑐−ℎ₁−ℎ₂+4𝐻_𝑐𝐵𝑥+4𝐻_𝑐𝐵²𝑥², (17) and ℎ_𝑚(𝑡)is the time dependent minimum film thickness which, in steady state, can be given as ℎ_𝑚(0)=()1−1−𝑟2−(1−𝑟)²ℎ16𝛿−𝛿₂, (18) where, 𝛿=𝐻_𝑐/ℎ₂ and 𝑟=ℎ₁/ℎ₂.

Introducing the dimensionless quantities: 𝑥𝑥=𝐵,ℎℎ=ℎ₂,ℎ_𝑚=ℎ_𝑚ℎ₂,ℎ_𝑠=ℎ_𝑠ℎ₂,𝑝=𝑝ℎ²₂𝜂_𝑜,𝑈𝐵𝑡=𝑈𝑡𝐵,𝑉=𝐵𝑉=𝑑ℎ𝑈ℎ_𝑚𝑑𝑡,𝜂𝜂=𝜂_𝑜,𝜂_𝑚=𝜂_𝑚𝜂_𝑜, (19) the modified Reynolds equation can be expressed in dimensionless form: 𝜕𝜕𝑥ℎ³𝜕𝑝𝜕𝑥=−6𝜂+𝜂_𝑚{(1−4𝛿−𝑟)+8𝛿𝑥}+2𝑉,ℎ=ℎ_𝑆+ℎ_𝑚=(𝑟−1+4𝛿)−()16𝛿𝑟−1+4𝛿𝑥+4𝛿𝑥²+ℎ_𝑚(𝑡). (20)(21) Integrating (20) under the pressure boundary conditions 𝑝=0 at 𝑥=0,1, the film pressure is obtained as 𝑝=−6𝜂+𝜂_𝑚1−4𝛿−𝑟+2𝑉𝑓₁(𝑥)+4𝛿𝑓₂(𝑥)+𝐶𝑓₃(𝑥), (22) where 𝑓₁(𝑥)=^𝑥₀𝑥ℎ³𝑑𝑓𝑥,₂(𝑥)=^𝑥₀𝑥²ℎ³𝑑𝑓𝑥,₃(𝑥)=^𝑥₀1ℎ³𝑑𝑥,𝐶=6𝜂+𝜂_𝑚1−4𝛿−𝑟+2𝑉𝑓₁(1)+4𝛿𝑓₂(1)𝑓₃(1). (23) The film force is obtained by integrating the film pressure over the slider length as follows: 𝐹=𝐿^𝐵_𝑥=0𝑝𝑑𝑥. (24) It takes the dimensionless form: 𝐹=¹₀𝑝𝑑𝑥=−6𝜂+𝜂_𝑚1−4𝛿−𝑟+2𝑉¹₀𝑓₁(𝑥)𝑑𝑥+4𝛿¹₀𝑓₂(𝑥)𝑑𝑥+𝐶¹₀𝑓₃(𝑥)𝑑𝑥, (25) where 𝐹=(ℎ²₂/𝜂_𝑜𝑈𝐿𝐵²)𝐹.

The steady state load capacity (𝐹_𝑠) and centre of pressure (𝑥) can be obtained from the film force and film pressure, respectively, under the steady state (i.e., ℎ_𝑚=ℎ_𝑚(0)and 𝑉=0) as follows: 𝐹_𝑆=[𝐹]_{(𝑡=0,𝑉=0)},∫𝑥=^𝐵0𝑥𝑝𝑑𝑥∫^𝐵₀𝑝𝑑𝑥_{(𝑡=0,𝑉=0)}. (26)

The dynamic damping coefficient 𝐶_𝐷 and the dynamic stiffness coefficient 𝑆_𝐷 can be obtained as 𝐶_𝐷=𝜕𝐹𝜕𝑉_{(𝑡=0,𝑉=0)},𝑆_𝐷=𝜕𝐹𝜕ℎ_{𝑚(𝑡=0,𝑉=0)}, (27) which take the dimensionless form: 𝐶_𝐷=ℎ³₂𝜂_𝑜𝐿𝐵³𝐶_𝐷=−12𝜂+𝜂_𝑚¹₀𝑓₁(𝑥)𝑑𝑥−𝑓₁(1)𝑓₃(1)¹₀𝑓₃(𝑥)𝑑𝑥_𝑡=0,𝑆_𝐷=ℎ³₂𝜂_𝑜𝑈𝐿𝐵³𝑆_𝐷=−6𝜂+𝜂_𝑚(1−4𝛿−𝑟)¹₀𝜕𝜕ℎ_𝑚𝑓₁(𝑥)𝑑𝑥+4𝛿¹₀𝜕𝜕ℎ_𝑚𝑓₂(𝑥)𝑑𝑥_𝑡=0+𝐶¹₀𝜕𝜕ℎ_𝑚𝑓₃(𝑥)𝑑𝑥_𝑡=0+𝜕𝐶𝜕ℎ_𝑚¹₀𝑓₃(𝑥)𝑑𝑥_𝑡=0. (28)

3. Results and Discussion

To study the ferrohydrodynamic effects on the dynamic characteristics of curved slider bearings, the numerical results for the coefficients of dynamic damping (𝐶_𝐷) and stiffness (𝑆_𝐷)have been calculated. The effect of magnetic field is analyzed by use of parameters 𝜙and 𝛼which stand for the amount of volume concentration of particles and Langevin parameter, respectively. In the analysis, 𝛼=0and 𝜙=0refer to the cases of “no magnetic field” and common lubricants (nonmagnetic lubricant), respectively. In order to validate the present results and to enhance its practical applicability, the steady state load carrying capacity of simple inclined slider (𝛿=0)is compared with the results of Taylor and Dowson [20] in Figure 2.

Figure 2 

Variation of steady-state load capacity of plane inclined slider with respect to film thickness ratio 𝑟 for different values of 𝛼 and 𝜙.

The values of the parameters taken in the present analysis are as follows:

(i) α: 0–10, (ii) ϕ: 0–1.0, (iii) δ: 0–0.5, and (iv) r: 1.0–2.0,

Figure 2 shows the variation of dimensionless steady state load capacity with respect to the inlet outlet film thickness ratio 𝑟 for plane inclined slider (𝛿=0)compared with the results of Taylor and Dowson [20]. It is clear from the figure that the load carrying capacity obtained in the present analysis for ferrohydrodynamic parameters 𝛼=0,𝜙=0is in the good agreement with the results obtained by Taylor and Dowson [20]. Further, it is observed that the load capacity increases with the increase of Langevin parameter 𝛼as well as the volume concentration parameter 𝜙.

Figure 3 presents the variation of dimensionless dynamic damping coefficient 𝐶_𝐷 with respect to the volume concentration parameter 𝜙 for Langevin parameter 𝛼=0,2.5,5,7.5,10 and curvature parameter 𝛿=0,0.5. It is observed that the coefficient of dynamic damping (𝐶_𝐷) increases with the increase of 𝜙in both the cases: (i) when the magnetic field is absent (𝛼=0) and (ii) when the uniform transverse magnetic field is applied (𝛼>0). However, the presence of magnetic field increases the damping coefficient which further increases with the increase of intensity of magnetic field (𝛼), that is, the value of damping coefficient is higher for higher values of Langevin parameter 𝛼. Further, the nature of variation of damping coefficient for plain and curved sliders (𝛿=0and𝛿=0.5) is found similar.

Figure 3 

Variation of dynamic damping coefficient with respect to volume concentration parameter 𝜙 for film thickness ratio 𝑟=2 and different values of 𝛼.

Figure 4 presents the variation of dimensionless dynamic stiffness coefficient 𝑆_𝐷 with respect to the Langevin parameter 𝛼 for 𝜙=0,0.1,0.2,0.3,0.4 and curvature parameter 𝛿=0,0.5.It is observed that the coefficient of stiffness obtained with common lubricants (𝜙=0)is less than that of a ferrofluid based lubricant regardless of the intensity of the magnetic field (𝛼). Further, with the ferrofluid analysis, the stiffness increases with the increase in Langevin parameter (𝛼). It is also observed that the stiffness increases with 𝛼 at a higher rate upto 𝛼≈2 and for 𝛼>4, the effect of increasing the field intensity (i.e., increasing 𝛼) decays in a sequential manner. The results are still sustained for both the plain and curved sliders (𝛿=0and𝛿=0.5).

Figure 4 

Variation of dynamic stiffness coefficient with respect to Langevin parameter 𝛼 for film thickness ratio 𝑟=2 and different values of 𝜙.

4. Conclusions

In the present theoretical investigation, the dynamic characteristics of curved slider bearings are studied in the presence of transverse magnetic field. The ferrofluid model given by Shliomis is adopted to obtain modified Reynolds equation. Based on the results, so obtained, the following conclusions have been drawn.(1)Magnetic fluid under the transverse magnetic field provides an improvement in the dynamic stiffness and damping characteristics of slider bearings.(2)On comparing with the conventional lubricants, both the dynamic stiffness and the dynamic damping coefficients are higher for ferrofluids even if the magnetic field is absent.(3)Under the applied magnetic field, both the stiffness and the damping increase with the increase of volume concentration of magnetic particles.

Nomenclature

𝐵:	Length of the bearing
𝐶𝐷,𝐶𝐷:	Dynamic damping coefficients, 𝐶𝐷=(ℎ32/𝜂𝑜𝐿𝐵3)𝐶𝐷
𝐹,𝐹:	Film force, 𝐹=(ℎ22/𝜂𝑜𝑈𝐵2𝐿)𝐹
𝐹𝑠:	Dimensionless steady state load capacity
ℎ,ℎ:	Film thickness, ℎ=ℎ/ℎ2
ℎ𝑚,ℎ𝑚:	Time dependent minimum film thickness, ℎ𝑚=ℎ𝑚/ℎ2
ℎ𝑠,ℎ𝑠:	Steady state film thickness, ℎ𝑠=ℎ𝑠/ℎ2
𝐻→:	Applied magnetic field
𝐼:	Sum of moments of inertia of the particles per unit volume
𝑘𝐵:	Boltzmann constant
𝐿:	Width of the bearing
𝑚:	Magnetic moment of a particle
𝑀→:	Magnetization vector
𝑛:	Number of magnetic particles per unit volume
𝑝,𝑝:	Film pressure, 𝑝=𝑝ℎ22/(𝜂𝑜𝑈𝐵)
𝑟:	ℎ1/ℎ2,
→𝑆:	Internal angular moment
𝑆𝐷,𝑆𝐷:	Dynamic stiffness coefficients, 𝑆𝐷=(ℎ32/𝜂𝑜𝑈𝐿𝐵3)𝑆𝐷
𝑡,𝑡:	Time, 𝑡=𝑈𝑡/𝐵
𝑇:	Temperature
→𝑣:	Velocity vector
𝑉:	𝑑ℎ𝑚/𝑑𝑡
𝑥,𝑥:	Coordinate distance, 𝑥=𝑥/𝐵
𝑥:	Centre of pressure
𝛼:	Langevin parameter
𝛿:	𝐻𝑐/ℎ2
𝜂,𝜂:	Effective viscosity (Shliomis model), 𝜂=𝜂/𝜂𝑜
𝜂𝑚,𝜂𝑚:	Additional viscosity due magnetization, 𝜂𝑚=𝜂𝑚/𝜂𝑜
𝜂𝑜:	Viscosity of the carrier fluid
𝜇𝑜:	Free space permeability
Ω→:	Local angular velocity
𝜙:	Volume fraction of the dispersed solid phase
𝜌:	Fluid density
𝜏𝑠:	Spin relaxation time
𝜏𝐵:	Brownian relaxation time of magnetic particles.

Acknowledgment

The authors, hereby, thank Dr. V. K. Kapur (Former Professor and Chairman, K. N. Institute of Technology, Sultanpur, India) for providing useful materials and the valuable guideline to enhance the content of this paper.