1 Introduction

Nanofluid is a homogeneous mixture of liquid, i.e. the base fluids like water, oil or ethylene-glycol and particles of solids like copper (Cu), silver (Ag), iron (Fe), etc. with size less than 100nm. The Ferrofluid (FF) belongs to a special class of synthetic fluids which undergo significant variation in flow behavior due to an external magnetic field. Ferrofluids are prepared by suspending magnetic solid particles in non-magnetic and non-conducting liquids such as di-esters, kerosene, hydrocarbons, fluorocarbons, water, etc. A fluid with solid Ferro particles of nanosize suspended in it is known as Ferro-nanofluid. These solid particles are in Brownian motion and migrate from higher temperature to the lower temperature region due to Brownian diffusion and thermophoresis forces [1]. In engineering, Fluid flow and Heat transfer phenomenon is a topic of keen interest. It is well known that thermal conductivity of nanofluid is higher than the base fluid because of the high conductivity of suspended particles [2]. Numerica [3] and experimental [4] study on forced and mixed convection proved that the transfer of heat is more in nanofluid as compared with the ordinary fluid. Moreover, the heat transfer rate increases with increase in the volume fraction [5]. In a study, A. Karimipour concluded that 2% volume fraction of Cu or Ag led to an increase of 30% (approx.) in the Nusselt number [6].

In recent years, many researchers have focused their work in examining the lubrication behavior of various Newtonian and non-Newtonian fluids. According to the classical theory of hydrodynamic lubrication, the lubricant was implicitly assumed as a Newtonian viscous fluid. But in modern industrial technology, the fluids like pulps, emulsions, molten plastics, slurries, grease etc. exhibiting non-Newtonian behavior, are also widely used as a lubricant. Therefore, the non-Newtonian behavior of lubricant is also of considerable interest. In order to prevent the viscosity variation with heat, the particles of insoluble solids having different material properties can be used as suspensions in the lubricant to enhance the viscosity index. Such type of lubricants exhibits the non-Newtonian behavior [7]. In lubricants, a considerable variation in the viscosity can be observed due to the temperature generated by frictional heating inside the film. Consequently, a mathematical mapping between the viscosity and the temperature exists [8]. In rotating disk problems, it is a well-reported fact that the heat transfer rate increases with the increase in the magnetic rotation parameter and the Prandtl number while the trend is opposite with porous permeability parameter, heat generation/ absorption parameter and Eckert number [9, 10]. In free convective flow, the enhancement in the magnetic parameter leads to decrease in the velocity, i.e., the magnetic force behaves like a retarding force [11]. Dual nature of temperature profile has been noticed in an unsteady convective flow of magnetic fluid in the presence of stretching effects [12]. Due to long-term stability and high thermal conductivity, magnetic fluids have attracted the researchers working on problems pertaining to various kinds of geometries like cylinder, rotating disk, sliding planar motion table, helical pipes, etc. [13, 14, 15, 16, 17].

In bearings, magnetic fluids or Ferrofluid (FF) [18] are known to play an important role to enhance the load carrying capacity and heat transfer. Depending on the requirements, the researchers usually adopt three basic magnetic fluid flow models, i.e. Jenkins model [19, 20, 21, 22, 23], Shliomis model [24, 25] and Neuringer-Rosensweig model (N-R model) [26, 27]. For higher loads, Shliomis model is suitable whereas N-R and Jenkins models are preferred for lower and moderate loads. Shliomis model accounts for the rotation of magnetic particles, their volume fraction and magnetic moments. The N-R model modifies the pressure while Jenkins model modifies both pressure and velocity of the magnetic fluid through an additional term pertaining to the co-rotational derivative of magnetization, which is missing in N-R model. This additional term takes care of velocity effects in the fluid flow. In a bearing design, the boundary slip and the surface texture are two important aspects and have similar effects on its tribological performance [28]. According to Navier boundary condition, the slip velocity is proportional to the rate of surface shear [29]. The position and the size of surface texture or slip region may either enhance or impair the lifting force of a bearing. Therefore, only a properly designed slip surface or surface texture can improve the bearing performance in terms of the lifting force [30, 31].

In the present problem, the work by Ram et al. [21] has been extended to analyze the rate of heat transfer by introducing the thermal effects in the porous inclined slider bearing together with the slip conditions. The term pertaining to the co-rotational derivative of magnetization has also been taken into account because of its significant impact on the bearing characteristics, and it was ignored by Singh and Ahmad [20]. The expressions for mean temperature, pressure and the lifting force (load capacity) have been derived as a function of slip, magnetic, permeability, material and thermal parameters. Using Simpson’s 1/3^rd rule, the values of the mean temperature and the lifting force have been computed for random values of various non-dimensional parameters by dividing the range of integration into 100 equal parts. The behavior of mean temperature has been investigated across the length of the bearing to examine the rate of heat transfer. Under the influence of material and magnetic parameter, the effects of slip and permeability on the lifting force have also been examined.

2 Formulation of the Problem

The governing equations pertaining to the flow of Ferrofluid in vector notation due to Ram and Verma [19] in the domain D = {(x, y) :   0 ≤ x ≤ L and 0 ≤ y ≤ h} are:

and

where

Fig. 1

Porous inclined slider bearing with ferrofluid lubrication

The equation of continuity in the porous region is given by

The simplified form of Eqs. (1) - (6) are given as:

Using (11) and (12) into (7), we obtain

The relevant boundary condition for the velocity field in the lubrication region is

where, U is the uniform sliding velocity component along the x-axis and h is the dimensional film height.

The slip velocity at the porous matrix [29] is given as:

where k is the porosity of the matrix and α^* is the slip coefficient.

For pressure, the appropriate boundary conditions are

where, L is the width of the bearing.

Solving Eq. (8) with the boundary conditions (14) and (15), we obtain the velocity component as

where s=−α∗k12 is the slip parameter.

Using Eqs. (9), (13) and (16), we can get the Reynolds equation in the following form:

Integration of equation (18) gives:

3 Heat Transfer Problem

Assuming the flow of lubricant thermally active and surfaces thermally inactive [20], the energy equation is simplified to get

Let the viscosity μ varies exponentially with temperature raised by frictional heat generated by flow i.e.

where

In the fluid film region, the boundary conditions for temperature are

4 Solution of the Problem

Introducing the following non-dimensional quantities:

The dimensionless form of (17), (20), (21) and (22) are:

Using Eqs. (24), (25) and boundary conditions T̄ = 1 at ȳ = 0 and at ȳ = h̄, we obtain the non- dimensional temperature field as:

where ϕ=∂∂x¯(p¯−μ¯∗2x¯(1−x¯))(1−α¯2x¯(1−x¯)).

Putting the value of T̄ into equation (27) and on simplifying we obtain the non-dimensional mean temperature as

The dimensionless form of equation (19) is

or

where the film thickness h̄(x̄) of the inclined slider bearing is given by h̄(x̄) = a − (a − 1)x̄, and a = h₁/h₀,   0 ≤ x̄ ≤ 1.

In particular, we take the inlet-outlet ratio a = h₁/h₀ = 2 ⇒ h̄(x̄) = 2 − x̄ = 2.

Now by integrating Eq. (30) and applying the boundary condition p̄ = 0 at x̄ = 0 and x̄ = 1, we obtain the value of as:

The lifting force or the load carrying capacity is given by:

Using Eqs. (29) and (33), we obtain the non-dimensional load capacity of the bearing as:

5 Results and Discussion

On behalf of the computation and investigations in the present problem, the following results are carried out.

Fig. 2 shows the variation of mean temperature (T̄_m) w.r.t. the permeability parameter (β̄). It is noticed that the mean temperature increases exponentially for β̄ > 0. So, it is concluded that the porosity of the matrix causes an increase in the mean temperature of the slider bearing. Therefore, for controlling the mean temperature of the slider, the porosity of the matrix should be adjusted suitably.

Fig. 2

Mean temperature vs. Permeability parameter at 1/s̄ = 1,   ᾱ² = 0.8,   γ̄² = 1.2,   P_r.E = 1.2,   x̄ = 0.6

From Fig. 3 the variation in the mean temperature (T̄_m) w.r.t. the slip parameter (1/s̄), has been observed. The mean temperature decreases with an increase in the slip parameter; this is because the slip velocity reduces the friction between the fluid and the boundary and hence the heat production is reduced which causes a fall in the mean the temperature.

Fig. 3

Mean temperature vs. Slip parameter at β̄ = 1.3,   ᾱ² = 0.8,   γ̄² = 1.2,   P_r.E = 1.2,   x̄ = 0.6

In Fig. 4 and Fig. 5, the behavior of mean temperature (T̄_m) has been noted with thermal (P_r.E) and magnetic parameter (μ̄^*) respectively. It is seen that the mean temperature of the bearing increases linearly with thermal as well as magnetic parameter. So, the mean temperature is a linear function of P_r.E and μ̄^*.

Fig. 4

Mean temperature vs. Thermal parameter at 1/s̄ = 1,   ᾱ² = 0.8,   β̄ = 1.3,   γ̄² = 1.2,   x̄ = 0.6

Fig. 5

Mean temperature vs. Magnetic parameter at 1/s̄ = 1,   ᾱ² = 0.8,   β̄ = 1.3,   γ̄² = 1.2   P_r.x = 1.2,   x̄ = 0.6

The variations of mean temperature (T̄_m) across the fluid film region have been analyzed for different values of the thermal parameter (P_r.E) and the material parameter (ᾱ²) in Fig. 6 and Fig. 7 respectively. The area under the mean temperature curve is large for the inner half part (0 ≤ x̄ ≤ 0.5) and negligible for the outer half part (0.5 ≤ x̄ ≤ 1) of the slider. Therefore, the width of the thermal boundary layer is quite large or the heat dissipation is very slow in the inner half part as compared to the outer half part of the bearing. From Fig. 7, it is observed that the width of the thermal boundary layer is much affected by the material parameter (ᾱ²) in the inner half part of the slider. So, for a desirable rate of heat transfer, the value of the material parameter (ᾱ²) should be adjusted accordingly.

Fig. 6

Mean temperature vs. bearing length for various values of the thermal parameter at 1/s̄ = 0.5,   ᾱ² = 1.2,   β̄ = 1.3,   γ̄² = 1.2

Fig. 7

Mean temperature vs. bearing length for various values of the material parameter at 1/s̄ = 0.5,   β̄ = 1.3,   γ̄² = 1.2,   P_r.E = 0.8

Fig. 8 reveals the variation of mean temperature (T̄_m) with slip parameter (1/s̄) for different values of the material parameter (ᾱ²). It is seen that for a small value of slip, i.e. 1/s̄ ≤ 0.3, the material parameter doesn’t affect the mean temperature but for a large value of slip, i.e. 1/s̄ > 0.3, it has a notable effect on mean temperature. Therefore, for 1/s̄ > 0.3, the material parameter should be adjusted according to the boundary slip.

Fig. 8

Mean temperature vs. Slip parameter for different values of the material parameter at β̄ = 1.3,   γ̄² = 1.2,   P_r.E = 1.2,   x̄ = 0.6

The variations of the lifting force (W̄) versus permeability parameter (β̄) under the influence of material parameter (ᾱ²) have been observed in Fig. 9. It is noted that the influence of the permeability parameter is not remarkable for a smaller value of the material parameter but for larger values of the material parameter, the load capacity decreases with an increase in the permeability parameter.

Fig. 9

Lifting force vs. Permeability parameter for different values of the magnetic parameter at 1/s̄ = 1.2,   γ̄² = 1.2,   P_r.E = 1.2,   μ̄^* = 12

The behavior of the lifting force w.r.t. the thermal parameter (P_r.E) for different values of the magnetic parameter (μ̄^*) has been plotted in Fig. 10. It is observed that the lifting force decreases with an increase in the thermal parameter. It is because the enhancement in the thermal parameter causes an increase in the mean temperature. Due to this the heat dissipation becomes slow which results in a fall in the lifting force or the load carrying capacity. Sometimes the bearing may break due to overheating between the pads.

Fig. 10

Lifting force vs. Thermal parameter for different values of the magnetic parameter at 1/s̄ = 1,   ᾱ² = 1,   β̄ = 1.3,   γ̄² = 1.2

Fig. 11 reveals the variations in the lifting force (W̄) w.r.t. the inlet-outlet ratio (a) for different values of the magnetic parameter (μ̄^*). It is noted that the lifting force enhances with an increase in the inlet-outlet ratio. Therefore, the angle of inclination of the inclined pad has also an impact on the performance of the slider bearing. In Fig. 10 and 11 it is also noted that the magnetic parameter boosts the lifting force of the slider. It is because the magnetization increases the viscosity of the lubricant, which results in an increase in the pressure and consequently, the lifting force.

Fig. 11

Lifting force vs. Inlet-outlet ratio for different values of the magnetic parameter at 1/s̄ = 1,   ᾱ² = 0.25,   β̄ = 1.3,   γ̄² = 1.2,   P_r.E = 1.2

In Fig. 12, the variations in the lifting force with the slip parameter have been noted. It is observed that for smaller values of the material parameter, i.e. ᾱ² ≤ 0.6, the lifting force decreases with an increase in the slip parameter but the trend is reversed for larger values of the material parameter, i.e. ᾱ² > 0.6. Therefore, the performance of the bearing in terms of the lifting force w.r.t. the slip parameter depends on the material properties like density, magnetic susceptibility, the coefficient of viscosity, bearing width, slip region, surface texture, etc.

Fig. 12

Lifting force vs. slip parameter for different values of the magnetic parameter at β̄ = 1.3,   γ̄² = 1.2,   P_r.E = 1.2,   μ̄^* = 12

Validation of the Results: In Figures 2-5, the trends of the graphs are similar to Singh and Ahmad [20] and the deviations in the values of mean temperature and the lifting force is due to the additional term of the Co rotational derivative of magnetization. If we remove the effect of this term, the results obtained are similar to Singh and Ahmad [20]. In Figure 9 the trend of the graph is also similar to Singh and Ahmad [20] for small values of the material parameter.

For larger values of the material parameter in Fig. 12, the trend of the graph shows that the effect of boundary slip is same as obtained by Q. Lin [28].

Further, in Figure 11, it is noted that the load capacity or lifting force enhances with an increase in the inlet-outlet ratio. The same result is also obtained by Ram et al. [21], and hence the results in the present paper validate the existing results.

6 Conclusions

1. Jenkins Model for lubrication has been analyzed with slip and thermal effects using magnetic nanofluid as the lubricant. The present work recommends that the mean temperature has been accelerated by the permeability parameter and decelerated by slip velocity. Also, the mean temperature increases linearly with the thermal and the magnetic parameter.

2. The heat dissipation or the cooling is quite slow in the inner part of the slider as compared to its outer part. The material parameter also has a notable effect on the thermal boundary layer so for desirable heat transfer; its value should be adjusted accordingly.

3. For small values of the material parameter, the permeability parameter does not have much effect on the lifting force, but for a large value of the material parameter, the lifting force decreases with an increase in the permeability parameter. Therefore for large values of the material parameter, the permeability i.e., the porosity of the matrix should be adjusted suitably to get the maximum load capacity.

4. With an increase in the thermal parameter, the lifting force decreases due to overheating. Therefore, the value of the thermal parameter, i.e., Prandtl number and Eckert number should be adjusted in such a way that the loss due to heat can be minimized and tribological performance of the bearing can be improved.

5. For a large value of the material parameter, i.e.ᾱ² > 0.6, the lifting forcehas a dual nature with the slip parameter Firstly, the load decreases with increase in the slip parameter, and the trend is reversed after an optimum value of the slip parameter. Therefore, proper slip conditions which depend on the material properties of the bearing are required to get the better tribological performance of the bearing in terms of the lifting force.

Finally, it is suggested that while designing a slider bearing, special care has to be taken about the porosity of the matrix, material properties, slip boundary conditions, inlet-out ratio depending on the pad length, heat dissipation rate, etc. for its better tribological performance in terms of the lifting force.