Ferrofluid lubrication of circular squeeze film bearings controlled by variable magnetic field with rotations of the discs, porosity and slip velocity

Based on the Shliomis ferrofluid flow model (SFFM) and continuity equation for the film as well as porous region, modified Reynolds equation for lubrication of circular squeeze film bearings is derived by considering the effects of oblique radially variable magnetic field (VMF), slip velocity at the film–porous interface and rotations of both the discs. The squeeze film bearings are made up of circular porous upper disc of different shapes (exponential, secant, mirror image of secant and parallel) and circular impermeable flat lower disc. The validity of Darcy's Law is assumed in the porous region. The SFFM is important because it includes the effects of rotations of the carrier liquid as well as magnetic particles. The VMF is used because of its advantage of generating maximum field at the required active contact area of the bearing design system. Also, the effect of porosity is included because of its advantageous property of self-lubrication. Using Reynolds equation, general form of pressure equation is derived and expression for dimensionless load-carrying capacity is obtained. Using this expression, results for different bearing design systems (due to different shapes of the upper disc) are computed and compared for variation of different parameters.


Introduction
of the carrier liquid as well as magnetic particles. With the invention of FF [1], its applications as lubricant on various bearing design systems have been found from different viewpoints [2,3]. Everywhere it was shown that the better performances of the bearing characteristics were obtained. The NR model depends on the assumption that the magnetization vector is parallel to the magnetic field vector. Prajapati [2] studied effect of MF on different porous squeeze film bearing designs like circular, annular, elliptic, conical, etc. It was concluded that the load-carrying capacity increases with the increase of magnetization parameter. Montazeri [3] numerically discussed FF lubricated hydrodynamic journal bearings. It was shown that compared to conventional lubricant, FF improves hydrodynamic characteristics and provides a higher load capacity with the reduction in friction coefficient.
In the case of different angular velocities of rotations of the carrier liquid as well as magnetic particles, frictional forces arise. These forces cause an increase in the effective viscosity of the FF and it has major impact on the pressure when FF is used as lubricant. Shliomis [4] considered rotations of the carrier liquid as well as magnetic particles in the FF flow model with magnetic body force. Many authors [5][6][7][8][9][10][11][12][13][14] studied this model from different viewpoints. Shukla & Kumar [5] analysed FF lubricated slider and squeeze film bearings using uniform transverse magnetic field by neglecting relaxation time of particle rotation. In their study they derived pressure equation under the assumptions that the FF is saturated (so that the saturation magnetization is independent of the applied magnetic field) and the magnetic moment relaxation time is negligible. However, Shah & Bhat [6] derived pressure equation without above assumptions of [5] in their study on FF squeeze film between curved annular plates. It was concluded that the load-carrying capacity and approaching time of squeeze films can be enhanced by increasing the volume concentration of solid phase in FF and the intensity of external magnetic fields. Shah [7] extended the above analysis [6] with the insertion of rotation effect of the upper plate, and studied different shapes (secant, exponential and flat) of the upper plate. The results showed that loadcarrying capacity and response time increase with the increase of volume fraction of the particles and rotation of the upper plate. Also, it was shown that load-carrying capacity and response time increase with the increase of curvature of the exponential plate, whereas they decrease with the increase of curvature of the secant shape. Singh & Gupta [8] studied FF lubricated curved slider bearing with the effect of transverse magnetic field, and showed the improvement in stiffness and damping capacities due to the effects of rotation and volume concentration of magnetic particles. Lin [9] derived Reynolds equation for MF lubricated slider bearings using transverse magnetic field and showed the improvement in load-carrying capacity, dynamic stiffness and damping characteristics. Patel & Deheri [10] discussed FF lubrication of squeeze film in rotating rough curved circular discs with assorted porous structures. It was concluded that even if suitable magnetization is in force, roughness aspect must be accorded priority while designing the bearing system. Shah & Parikh [11] analysed FF lubrication of different shapes of slider bearings and compared dimensionless load-carrying capacity for the effect of squeeze velocity. It was concluded that the load-carrying capacity of all bearings remains constant with the increase of Langevin's parameter, whereas it has an increasing tendency with the increase of volume concentration of the particles. Lin et al. [12] studied effects of circumferential and radial rough surfaces on a non-Newtonian MF lubricated squeeze film. It was concluded that circumferential roughness effect increases the mean load-carrying capacity and lengthens the mean approaching time when compared with smooth discs. However, the radial roughness pattern showed the reverse trend. Huang & Wang [13] presented comprehensive review on FFs lubrication with some experimental studies. Nargund & Asha [14] studied load-carrying capacity of hyperbolic slider bearings and showed the better performance of the system.
All above studies based on the Shliomis model are with transverse magnetic field. It is observed that the study with oblique radially variable magnetic field (VMF) considering the effects of porosity, slip velocity at the film-porous interface and rotations of both the discs, is ignored. The Shliomis model is important because it includes the effects of rotations of the carrier liquid as well as magnetic particles, and it behaves differently in the case of VMF. The VMF is important because of having its advantage of generating maximum field at the required active contact area in the bearing design system. Also, looking to industrial applications, the above three effects are also important. The effect of porosity is included because of its advantageous property of self-lubrication. Hence, there is a need of the present paper. Thus, the aim of the present paper is to study lubrication of circular squeeze film bearings using the Shliomis FF flow model with the effects of oblique radially VMF, porosity, slip velocity at the filmporous interface and rotations of both the discs. The squeeze film bearings are made up of circular porous upper disc of different shapes (exponential, secant, mirror image of secant and parallel) and circular impermeable flat lower disc. While deriving the modified Reynolds equation, the validity of Darcy's Law is assumed in the porous region (matrix or layer). The continuity equation is also used in the film as well as porous region.         Figure 1 shows the physical configuration of the problem under consideration, which consists of two circular discs (case of both flat discs is shown) each of radius a. The upper disc is made by attaching a porous matrix of uniform thickness H * to the solid impermeable disc. The upper disc may also be curved (which may be either exponential or secant or mirror image of secant shapes as shown in figure 2) while the lower disc is solid impermeable flat. The central film thickness is h 0 . The region between two discs is known as film region (lubrication region), which is filled with FF. The upper disc moves normally towards lower one with a uniform velocity, known as squeeze velocityḣ 0 = dh 0 /dt, where t is time.

Analysis
The upper and lower discs rotated with rotational (angular) velocities Ω u and Ω l , respectively. Owing to the different shapes of the upper disc, the film thickness h takes following different forms. Case 1. For exponentially curved upper disc where β is curvature and r is the radial coordinate. Case 2. For secant curved upper disc where γ is curvature. Case 3. For mirror image of secant curved upper disc   With reference to the shape of the upper disc, the bearing designs for the Cases 1-4 are referred to here as exponential squeeze film bearing, secant squeeze film bearing, mirror image of secant squeeze film bearing and parallel squeeze film bearing, respectively. Neglecting inertia terms, assuming steady flow and other usual assumptions of lubrication, the basic flow equations governed by the Shliomis model [4,6,7] using cylindrical frame of reference can be written as follows. where p is the film pressure, η is the viscosity of the suspension, q is the fluid velocity vector, µ 0 is the permeability of free space, M is the magnetization vector, H is the applied magnetic field vector, ρ is the fluid density, v is the tangential component of q, M 0 is the equilibrium magnetization, H is the magnitude of H,Ω = 1 2 ∇ × q, τ B is the Brownian relaxation time, τ s is the magnetic moment relaxation time and I is the sum of moments of inertia of the particles per unit volume.

Equation of magnetization
Also, where (r, θ, z) are cylindrical polar coordinates and dot (·) represents derivative with respect to t.
Assuming the predomination of the velocity gradient across the film, v as a linear function of the axial coordinate z, and the axially symmetric flow in the film as well as magnetic field, the r-component of equation (2.5) with the help of equation (2.6) andΩ = 1 2 ∇ × q implies where Ω r = Ω u − Ω l and u is the radial component of q. The inclination θ of H = H(r)(cos θ, 0, sin θ), θ = θ(r, z) to the radial direction is assumed to be small and can be obtained from condition (2.8).
In order to consider active contact area in the neighbourhood of r = 2a/3, the magnetic field strength of radially VMF should be chosen (referring to [15]) as where K is the quantity chosen to suit the dimensions of both sides of equation ( where n is the number of magnetic particles per unit volume, m is the magnetic moment of a particle, ξ (Langevin's parameter) is the dimensionless form of H, k B is the Boltzmann constant, T is the temperature and ϕ is the volume concentration of the particles. Solving equation (2.14) using slip boundary conditions [15,16] (2.16) where k is permeability of the porous matrix, η r is porosity of the porous matrix in r-direction and s is the slip constant.
Substituting equation (2.16) into the integral form of continuity equation (2.7) in cylindrical polar coordinates for the film region (2.18) where w is the axial velocity component of q and w 0 = 0 as the lower disc is impermeable.
Moreover, the relation between viscosity of the suspension η and viscosity of the carrier liquid η 0 is given by [4,6] Assuming the validity of Darcy's Law, the radial and axial components (considering the contributions from the magnetic pressure and rotation of the upper disc) of the fluid velocity in the porous matrix yield, respectively, asū (2.25)

Solution
Introducing dimensionless quantities and using equation (2.12), equation (2.25) becomes and Solving equation (3.2) using boundary conditions The load-carrying capacity W of the bearing can be expressed in dimensionless form as where

Results and discussion
The results for the dimensionless load-carrying capacityW given by equation ( in computations are as follows [6,16,17]. These values remain fixed unless and until the calculation ofW is made with respect to the variation of the particular parameter. Using subscripts e, s, is, p for the concerned quantities when the squeeze film bearing designs are of exponential, secant, mirror image of secant and parallel shapes, respectively, equations (2.1)-(2.4) for computation becomeh The sketch of the above different shapes (except for parallel; for parallel shape refer to figure 1) are shown in figure 2 forβ = −0.6,β =γ =ᾱ = 0.6.
The computed values ofW for different parameters are displayed graphically. Before discussing graphs, it should be noted here that counterclockwise (positive) or clockwise rotations of either of the discs can be decided by looking from the top of the bearing system or with respect to the vertical axis. Figures 3-6 show the variation inW as a function of dimensionless rotational parameter Ω f considering |Ω l | > Ω u (that is, when the lower disc is rotated faster than the upper disc either in counterclockwise or clockwise direction) for different geometry of the squeeze film bearings like exponential, secant, mirror image of secant and parallel, respectively. The additional fixed values taken in computations are as follows: (1) Ω u = 30π , which indicates the rotation of the upper disc in counterclockwise direction with a fixed value 30π . (2) Dimensionless curvature parametersβ =γ =ᾱ = 0.6.
It is observed, in general from figures 3-6, thatW decreases in the case when Ω f increases along the positive axis or decreases along the negative axis. That is,W decreases in the case when the speed of rotation of the lower disc increases more than 30π either in counterclockwise or clockwise direction. It is also observed thatW is more in the case of clockwise rotation of the lower disc and maximum nearer to Ω f = −1. Moreover, this behaviour ofW is same for all bearing designs. Table 2 shows the maximum value ofW, when the lower disc is rotated in different directions with speed 40π . It is observed that when the lower disc is rotated in clockwise direction, the increase rate ofW is more and approximately 0.90% for exponential squeeze film bearing, 1 Table 3. Values ofW when upper disc is rotated in different directions either with Ω u = −30π (clockwise) or Ω u = 30π (counterclockwise) considering Ω l = 30π andβ =γ =ᾱ = 0.6. secant squeeze film bearing and 1.47% for parallel squeeze film bearing. It should be noted here that the mirror image of secant squeeze film bearing design shape is introduced for the first time in the study because such type of shape exists in industry while manufacturing the disc. Figures 7-10 show the variation inW as a function of Ω f considering Ω l ≤ |Ω u | (that is, when the upper disc is rotated faster or equal speed than the lower disc either in counterclockwise or clockwise direction) for different geometry of the squeeze film bearings like exponential, secant, mirror image of secant and parallel, respectively. Here, the additional fixed values taken as Ω l = 30π andβ =γ =ᾱ = 0.6. It is observed, in general, thatW increases in the case when Ω f increases along the positive axis or decreases along the negative axis. That is,W increases in the case when the speed of rotation of the upper disc moves from higher values to 30π either in counterclockwise or clockwise direction. It is also observed thatW is more in the case of clockwise rotation of the upper disc and maximum at Ω f = −1. Moreover, this behaviour ofW is same for all bearing designs. Table 3 shows the maximum value of W, when the upper disc is rotated in different directions with speed 30π . It is observed that when the upper disc is rotated in clockwise direction, the increase rate ofW is more and approximately 0.45% for exponential squeeze film bearing, 1.92% for secant squeeze film bearing, 1.05% for mirror image of secant squeeze film bearing and 1.47% for parallel squeeze film bearing.
It should be noted here that for the same data values, counterclockwise rotation of the upper disc and clockwise rotation of the lower disc, or clockwise rotation of the upper disc and counterclockwise   rotation of the lower disc, gives the same results due to kinematics of the rotation. That is, when Ω u = −30π andβ =γ =ᾱ = 0.6 are fixed, and when the lower disc is rotated faster than the upper disc either in clockwise or counterclockwise direction, then the same results are obtained as shown in figures 3-6 since Ω f = Ω l /Ω u . The same is also true for figures 7-10.  Table 4. Effects onW when the rotation of the lower disc is zero (that is, Ω l = 0) irrespective of the rotation of the upper disc in different directions (that is, either counterclockwise or clockwise) consideringβ =γ =ᾱ = 0.6.   Table 5. Effects onW when Ω f takes negative values (that is, either Ω u is rotated in counterclockwise direction and Ω l is rotated in clockwise direction or Ω u is rotated in clockwise direction and Ω l is rotated in counterclockwise direction) considerinḡ β =γ =ᾱ = 0.6.  Since Ω f = Ω l /Ω u , when Ω f = 0 (that is, when there is no rotation of the lower disc irrespective of the rotation of the upper disc in different directions), it is observed from table 4 that for all bearing designs, W decreases as speed of rotations of the upper disc increases. Again, in this calculationβ =γ =ᾱ = 0.6 is fixed. Table 5 represents the results ofW when Ω f takes negative values (that is, either Ω u is rotated in counterclockwise direction and Ω l is rotated in clockwise direction or Ω u is rotated in clockwise direction and Ω l is rotated in counterclockwise direction). Table 6 represents the results ofW when Ω f takes positive values (that is, either both the discs are rotated in counterclockwise direction or clockwise direction). For both the tablesβ =γ =ᾱ = 0.6 is fixed. It is observed from both the tables thatW decreases as the speed of rotations of both the discs increases. Further,W is more in the case when the speed of rotations of both the discs is in different directions to each other.  Table 6. Effects onW when Ω f takes positive values (that is, either both the discs rotated in counterclockwise direction or in clockwise direction) consideringβ =γ =ᾱ = 0.6. Ω u 30π (or −30π ) 4 0 π (or −40π ) 5 0 π (or −50π ) 6 0 π (or −60π ) Ω l 30π (or −30π ) 4 0 π (or −40π ) 5 0 π (or −50π ) 6 0 π (or −60π ) While discussing figures 3-10 and tables 2-6, the following behaviours ofW are observed in general.
(1) MaximumW is obtained in the case of exponential squeeze film bearing while minimumW in the case of secant squeeze film bearing. Thus, for all bearing designsW can be obtained in the order W e >W is >W p >W s . This may be because of the following reason.
Referring to figure 2, in the case of exponential squeeze film bearing the curvature of the upper disc at the centre is in downward direction (that is, upper disc is concave with respect to lower disc), whereas in the case of secant shape the curvature is in upward direction (that is, upper disc is convex with respect to lower disc). Table 7 shows the behaviour of curvatures of all the upper discs by referring to figure 2. As the maximum magnetic field is taken at r = 2a/3, so because of such magnetic field and exponential shape of the upper disc, nearly closed bearing design system appears which leads to less possibility of leakage. Thus, maximum pressure generation is possible, which implies increase inW. The curvature of the upper disc of mirror image of secant squeeze film bearing is less in downward direction when compared with exponential shape, soW is less in this case. Thus, as curvature of the upper disc at the centre increases in downward direction (that is, as concavity of the upper disc increases with respect to lower disc),W increases significantly. This may be the reason of obtainingW in the orderW e >W is >W p >W s .
(2) MaximumW is obtained in the case when both the discs are rotated in different directions to each other. This may be because of the following reason.
When the discs are rotated, then there is an appearance of centrifugal force (which points outward) in the rotating fluid system and it increases linearly with the radial distance r. In the case of rotation of both the discs in different directions to each other, the effect of centrifugal force is reduced (when compared with rotations of both the discs in the same direction) and as a result the leakage possibility is reduced. This may be because of twisted nature of the generated spikes. Nearly similar type of behaviour is obtained when only one disc is rotated. Figure 11 shows different shapes of the exponential squeeze film bearing for −0.6 ≤β ≤ 0.6. Here, the shape of the upper disc changes from convex to concave with respect to lower disc. Figure 12 shows various shapes of secant and mirror image of secant squeeze film bearings for 0.2 ≤γ ≤ 0.6 and 0.2 ≤ᾱ ≤ 0.6, respectively. The upper disc of the secant squeeze film bearing takes convex shape, which shows less curvature to more curvature in upward direction asγ moves from 0.2 to 0.6. For the upper disc of mirror image of secant squeeze film bearing, concave shape is obtained which shows less curvature to more curvature in downward direction asᾱ moves from 0.2 to 0.6. Variation inW for different values of dimensionless curvature parameterγ and Ω f = −1 (that is, either Ω u = 30π and Ω l = −30π or Ω u = −30π and Ω l = 30π ) forh =h s = sec (γ R 2 ).  show the variation inW when Ω f = −1 (that is, either Ω u is rotated in counterclockwise direction and Ω l is rotated in clockwise direction or Ω u is rotated in clockwise direction and Ω l is rotated in counterclockwise direction with a fixed value of 30π ). Figure 13 shows the variation inW as a function of dimensionless curvature parameterβ for exponential squeeze film bearing. It is observed thatW increases asβ moves from − 0.6 to 0.6. That means concave shape with more curvature at the centre has more impact on the increase ofW. Figure 14 shows the variation inW as a function of dimensionless curvature parameterγ for secant squeeze film bearing. It is observed thatW decreases asγ moves from 0.2 to 0.6. That means convex shape with less curvature at the centre in upward direction has more impact on the increase ofW. Moreover, secant function is an even function, so symmetric behaviour ofW with respect to vertical axis is obtained. Figure 15 shows the variation inW as a function of dimensionless curvature parameterᾱ for mirror image of the secant squeeze film bearing. It is observed thatW increases asᾱ moves from 0.2 to 0.6. That means concave shape with more curvature at the centre has more impact on the increase ofW. Again, mirror image of secant is even function, so symmetric behaviour ofW is obtained.
Patel & Deheri [10] studied squeeze film bearing system formed by upper exponential disc using the Shliomis model with transverse magnetic field. It was shown thatW increases with the increasing values ofβ. Also, maximumW is obtained about 1.6 whenβ = 1.9. In the present study, the similar behaviour ofW is obtained with the advantage of having maximumW as 2.24 (refer to tables 2, 3 and 5) at smaller  value ofβ = 0.6. Shah & Bhat [6] also observed the similar type of increasing behaviour ofW with the increasing values ofβ. Figure 16 shows the comparative study of variation inW as a function of dimensionless porous thickness parameter ψ for all bearing designs. It is observed thatW increases in all cases when ψ → 0; that is,W increases when width of the porous matrix decreases. This may be because of following reason.
According to [16], when porous matrix is attached with any one of the discs, then there is a possibility that the pressure in the porous medium provides a path for the fluid to come out easily to the environment (leakage problem). Of course this varies with permeability of the porous matrix. Thus, the presence of the porous material decreases the resistance to flow in r-direction and as a consequence the load-carrying capacity is reduced. The same behaviour also agrees with the theoretical conclusion of the Prakash & Tiwari [18], and experimental results of Wu [19]. The similar type of behaviour is also observed in [10], where they have considered the Shliomis model based on [5] using transverse magnetic field. Figure 17 shows the comparative study of variation inW as a function of dimensionless squeeze velocity parameter V sq for all bearing designs. It is observed that squeeze velocity parameter has no effect onW.
Both figures 16 and 17 show that the secant and parallel squeeze film bearings show almost same behaviour. Moreover, it is observed thatW is maximum for exponential squeeze film bearing, and minimum for secant and parallel squeeze film bearings withW e >W is >W s ≈W p . Here, for both figures 16 and 17,β =ᾱ = 0.6 andγ = 0.2 are fixed.

Conclusion
On the basis of ferrohydrodynamic theory by Shliomis and equation of continuity for film as well as porous region, modified Reynolds equation for lubrication of circular squeeze film bearings is derived by considering the effects of oblique radially VMF, slip velocity at the film-porous interface and rotations of both the discs. The squeeze film bearings are made up of circular porous upper disc of different shapes (exponential, secant, mirror image of secant and parallel) and circular impermeable flat lower disc. The validity of Darcy's Law is assumed in the porous region. The FF flow by the Shliomis model is important because it includes the effects of rotations of the carrier liquid as well as magnetic particles. Moreover, the VMF is used because of its advantage of generating maximum field at the required active contact area of the bearing design system. Also, the effect of porosity is included because of its advantageous property of self-lubrication. Using Reynolds equation, pressure equation is derived and expression for dimensionless load-carrying capacity is obtained. Using this expression, results for different bearing design systems are computed and compared for variation of different parameters like rotation, curvature of the upper discs, thickness of the porous matrix and squeeze velocity. The pressure equation derived in the present case is more general in nature and different from all previous studies. Moreover, the present analysis considers the effect of sample magnetic field and it can be extended to other forms of fields similarly. Further, the mirror image of secant design shape is introduced for the first time because such type of shape exists in industry while manufacturing the disc.
The following conclusions can be drawn from the results and discussion: