Acta Polytechnica doi:10.14311/AP.2019.59.0144 Acta Polytechnica 59(2):144–152, 2019 © Czech Technical University in Prague, 2019 available online at http://ojs.cvut.cz/ojs/index.php/ap A STUDY OF FERROFLUID LUBRICATION BASED ROUGH SINE FILM SLIDER BEARING WITH ASSORTED POROUS STRUCTURE Mohmmadraiyan M. Munshia, ∗, Ashok R. Patelb, Gunamani B. Deheric a Gujarat Technological University, Alpha College of Engineering and Technology, Kalol, Gujarat, 382721, India b Gujarat Technological University, Vishwakarma Government Engineering College, Ahmedabad, Gujarat, 382424, India c Sardar Patel University, Department of Mathematics, Vallabh Vidhyanagar, Gujarat, 388120, India ∗ corresponding author: raiyan.munshi@gmail.com Abstract. This paper attempts to study a ferrofluid lubrication based rough sine film slider bearing with assorted porous structure using a numerical approach. The fluid flow of the system is regulated by the Neuringer-Rosensweig model. The impact of the transverse surface roughness of the system has been derived using the Christensen and Tonder model. The corresponding Reynolds’ equation has been used to calculate the pressure distribution which, in turn, has been the key to formulate the load carrying capacity equation. A graphical representation is made to demonstrate the calculated value of the load carrying capacity which is a dimensionless unit. The numbers thus derived have been used to prove that ferrofluid lubrication aids the load carrying capacity. The study suggests that the positive impact created by magnetization in the case of negatively skewed roughness helps to partially nullify the negative impact of the transverse roughness. Further investigation implies that when the Kozeny-Carman’s model is used, the overall performance is enhanced. The Kozeny-Carman’s model is a form of an empirical equation used to calculate permeability that is dependent on various parameters like pore shape, turtuosity, specific surface area and porosity. The success of the model can be accredited to its simplicity and efficiency to describe measured permeability values. The obtained equation was used to predict the permeability of fibre mat systems and of vesicular rocks. Keywords: Porous structure, roughness, ferrofluid, load carrying capacity. 1. Introduction The last decade has seen a considerable shift wherein many tribological researches have been dedicated to study surface roughness and the impact of hydrody- namic lubrication. This is because every solid surface carries some amount of surface roughness, the height of which is usually parallel to the mean separation between lubricated contacts. As many researchers have suggested, studying the surface roughness will help to improve the performance of a bearing system. Due to this reason, many researchers [1–3] studied the performance of various bearing systems using the stochastic concept of [4–6]. Amongst the biggest inventions in the field is the use of ferrofluid as a bearing system lubricant. A number of authors [7–11] have worked to explain the performance and applications of ferrofluid when used in different types of bearing systems. These studies have suggested that ferrofluid impacts the bearing performance positively. Many researchers have used different types of film geometries in order to study the effect of ferrofluid based squeeze film. Some of the researches conducted on this topic are listed, [12] studied exponential slider bearing, [13] worked on secant shaped slider bearing, [14–16] analysed inclined slider bearing, [17] stud- ied curved slider bearings, [18] examined parallel slider bearing, [19] evaluated on Rayleigh step bearing, [20] investigated parabolic slider bearing, [21] worked on hyperbolic slider bearing, [22] discussed sine film thrust bearing, [23] studied convex pad slider bear- ing and [24] examined infinitely long slider bearing. From all the articles above, it can clearly be seen that the characteristics of ferrofluid and its effect on load bearing capacity are positive. The lubrication theory of porous bearings was first studied by [25]. Porous structures are usually de- scribed using two common parameters, which are porosity and permeability. Porosity is a measure of existing voids within a dense material structure. Per- meability defines the ease with which fluids can flow through the material, in case of open cell porosity. Darcy’s law is generally used to determine the porosity. Porous metallic materials have a lot of applications including vibration and sound absorption, light mate- rials, heat transfer media, sandwich core for different panels, various membranes and during the last years as suitable biomaterial structures for the design of medical implants. Porous matrix decreases the load carrying capacity and increase the frictional force on the slider. The porous layer has a beneficial property of self-lubrication, making it an important area of 144 http://dx.doi.org/10.14311/AP.2019.59.0144 http://ojs.cvut.cz/ojs/index.php/ap vol. 59 no. 2/2019 A study of ferrofluid lubrication based rough sine film slider bearing. . . study. [26] worked on studying the comparisons of porous structures and their impact on the load carry- ing capacity of a magnetic fluid based rough and short bearing. The studies have found that while magneti- zation has a positive impact on the bearing system’s performance, transverse roughness impacts it nega- tively. However, in the case of Kozeny-Carman model, this negative impact is comparatively lower. In this model, the negative impact of porosity on the bearing performance can be neutralized with the negatively skewed roughness’ positive impact. [27] worked on in- vestigating the performance of a magnetic fluid based double layered rough porous slider bearing considering the combined porous structures. For a considerable range of combined porous structure, magnetization neutralizes the adverse effect of roughness. [28] stud- ied Shliomis model-based magnetic squeeze film in rotating rough curved circular plates: making a con- trast of two different porous structures. It was found out that by choosing a proper rotation ratio and ap- propriate curvature parameters, the negative impacts of transverse roughness on a bearing’s load carrying capacity can be nullified by the positive impact of magnetization with a negatively skewed roughness. [29] studied squeeze film based on ferrofluid in curved porous circular plates with various porous structures. The studies showed that, with concave plates and porous structure given by Kozeny-Carman, there was a considerable increase in the load bearing capacity. Different forms of modification of Darcy’s law have been studied in [30]. [31] investigated a bearing sys- tem based on a hyperbolic slider. They experiment with porous structure as well as roughness in accor- dance with the impact of sinusoidal magnetic field. Furthermore, the load bearing capacity is enhanced due to the influence of magnetization and the slip pa- rameter being within the limited boundary. Recently, [32] analysed inclined slider bearing. In this work, we can identify that they worked in detail with all aspects of surface roughness, porosity and magnetic field. Somehow, by surprise, the result was that the load bearing capacity differs and gives a very effective ability when the sinusoidal magnetic field is applied in the form which appears in the presented study. None of the above-mentioned researchers worked on the impact of sine films in a slider bearing. In order to explore this filed, this paper studies ferrofluid lubrication based rough sine film slider bearing with assorted porous structure. 2. Analysis Figure 1 represents the geometry and configuration of the given bearing system. U denotes the uniform velocity of the system in the direction x. The thickness h is considered as h = h + hs (1) where h is taken as [22]: h = h0 + (h1 −h0) [ 1 − sin (πx 2L )] using the works of [4–6]. Also, the study uses hs using the probability density function f(hs) = { 35 32c ( 1 − h 2 s c2 )3 ,−c ≤ hs ≤ c 0 ,elsewhere (2) c being the maximum deviation from the mean film thickness. α, σ and ε are considered by the relation- ships α = E(hs), σ2 = E(hs−α)2, ε = E(hs−α)3 (3) where E(•) denotes the expectancy operator given by E(•) = ∫ c −c (•)f(hs)dhs (4) [9] formulated explaining the steady flow of a magnetic fluid. It was: Equation of motion ρ(q ·∇)q = −∇p + η∇2q + µ0(M ·∇)H (5) Equation of magnetization M = µH (6) Equation of continuity ∇·q = 0 (7) Maxwell equations ∇×H = 0 (8) and ∇· (H + M) = 0 (9) where ρ, q, M,p and η are fluid density, fluid velocity, magnetization vector, film pressure and fluid viscosity respectively. Also, q = ui + vj + wk (10) where u, v, w are components of film fluid velocity in x, y and z - directions respectively. Further, the magnetic field’s magnitude is given by H2 = kx(L−x) (11) where k is a suitable constant and, assuming the external magnetic field to come up from a potential function, the inclination angle of the magnetic field φ = φ(x,z) satisfies the equation [7] cot φ ∂φ ∂x + ∂φ ∂z = 2x−L 2x(L−x) (12) The governing equation of motion of the fluid flow in the film region [33] is ∂2u ∂z2 = 1 η ∂ ∂x ( p− 1 2 µ0µH 2 ) (13) 145 M. M. Munshi, A. R. Patel, G. B. Deheri Acta Polytechnica Figure 1. Configuration of a sine film porous slider bearing including squeeze action [22]. By solving equation (13) following the no slip bound- ary conditions: u = 0 at z = h and u = U at z = 0 One can find u = 1 η ( z2 2 − h 2 z ) ∂p ∂x + U ( 1 − z h ) (14) Integrating equation (14) over the film region, yields∫ h 0 udz = −h3 12η dp dx + Uh 2 (15) Using equation (15) in continuity equation ∂ ∂x ∫ h 0 udz + wh −w0 = 0 (16) yields ∂ ∂x [ −h3 12η dp dx + Uh 2 ] + wh −w0 = 0 (17) where wh = −ḣ0 and w0 = 0 Equation (17) leads to: d dx [ h3 d dx ( p− 1 2 µ0µH 2 )] = 6ηU dh dx + 12ηḣ0 (18) which is the Reynolds’ equation [7, 21] modified ac- cording to the general hydrodynamic lubrication as- sumption. According to the stochastically average process of [4], Equation (18) becomes: d dx [ E(h3) d dx ( p− 1 2 µ0µH 2 )] = = 6ηU d dx [E(h)] + 12ηḣ0 Figure 2. Structure model of porous sheet given by Kozeny-Carman [34]. d dx [ g(h,α,σ,ε,ψ) d dx ( p− 1 2 µ0µH 2 )] = = 6ηU d dx [ g(h,α,σ,ε,ψ)1/3 ] + 12ηḣ0 (19) where g(h,α,σ,ε,ψ) = h3 + 3αh2 + 3(σ2 + α2)h+ + 3σ2α + α3 + ε + 12ψl1 (20) 2.1. A globular sphere model Globular particles (a mean particle size Dc) are used to fill a porous material which is given in Fig. 2. In fluid dynamics, the Kozeny-Carman equation [35] plays a major role in calculating the pressure drop when working with a fluid flowing in a packed bed of solids. Although, the equation only remains valid for a laminar flow. This equation makes use of few general experimental trends, which makes it an efficient quality control tool that can be used for both physical as well as digital experimental results. The equation is commonly displayed as permeability versus porosity, pore size and turtuosity. The pressure gradient is assumed to be linear here. Following the ideas of discussion [36] the use of Kozeny- 146 vol. 59 no. 2/2019 A study of ferrofluid lubrication based rough sine film slider bearing. . . Carman formula becomes: ψ = D2ce 3 72(1 −e)2 l l ′ where e is the porosity and l l ′ is the length ratio. The following dimensionless quantities are used, g ( h,α,σ,ε,ψ ) = g(h,α,σ,ε,ψ) h30 , X = x L , a = h1 h0 , α = α h0 , σ = σ h0 , ε = ε h30 , p = ph20 ηUL , µ∗ = µ0µh 2 0Lk ηU , β1 = − Uh0 2ḣ0L , h = h h0 = 1 + (a− 1) [ 1 − sin (π 2 X )] , l∗ = l l ′ , g ( h,α,σ,ε,ψ )1/3 = g (h,α,σ,ε,ψ)1/3 h0 , ψ = D2ce 3l 72(1 −e)2l′ , ψ = D2cl1 h30 (21) The associated boundary conditions are p = 0 at X = 0, 1 (22) With the aid of equation (22) the pressure distribu- tion in a non-dimensional form comes out to be p = 1 2 µ∗X(1−X) + 6 ∫ X 0 [ g ( h,α,σ,ε,ψ )1/3 − . . . g ( h,α,σ,ε,ψ ) · · ·−g ( 1,α,σ,ε,ψ )1/3 + β−11 (1 −X) g ( h,α,σ,ε,ψ ) ]dX (23) where g ( h,α,σ,ε,ψ ) = h 3 + 3αh 2 + 3 ( σ2 + α2 ) h+ + 3σ2α + α3 + ε + ψe3l∗ 6(1 −e)3 (24) The load bearing capacity in dimensionless form is obtained W = h20 ηUL2B W (25) W = µ∗ 12 + 6 ∫ 1 0 [ g ( h,α,σ,ε,ψ )1/3 − . . . g ( h,α,σ,ε,ψ ) · · ·−g ( 1,α,σ,ε,ψ )1/3 + β−11 (1 −X) g ( h,α,σ,ε,ψ ) ](1 −X)dX (26) where the load bearing capacity is calculated using W = ∫ L 0 pBdX (27) 3. Results and discussion The results calculated for the dimensionless load- carrying capacity W given by equation (26) are found using Simpson’s one-third rule with a step size of 0.2 for the Kozeny-Carman model. It proves that the load bearing capacity increases by: µ∗ 12 Equation (26) suggests that even in the absence of a flow, a bearing system can handle a given amount of load for the Kozeny-Carman model. By keeping the roughness zero, the study reduces to the impact of an assorted porous structure on the Neuringer- Rosensweig model based ferrofluid squeeze film for a slider bearing [7]. Considering the magnetization parameter as a zero, it reduces to the study of [37] in the absence of porosity. Equation (26) clearly suggests that the expression for W is linear with respect to the magnetization pa- rameter µ∗. Thus, when the Kozeny-Carman model is applicable, by increasing magnetization, the load bearing capacity can also be increased (Fig. 3). Fig- ures 3-10 display a graphical representation of the Kozeny-Carman model results. They suggest that: (1.) According to Fig. 4, it is evident that a stan- dard deviation has a relatively lower impact when compared to porosity. (2.) As the positive variance increases, the load carry- ing capacity decreases. A decrease in the negative variance leads to an increase in the load carrying capacity (Fig. 5). As suggested by Fig. 6, the im- pact of skewness on the load carrying capacity is similar to variance. (3.) Effect of ψ on W with respect to e and l∗ is seen to be adversely from Fig. 7. (4.) Figure 8 demonstrates the impact of porosity on the distribution of load carrying capacity. It suggests that porosity considerably reduces the load bearing capacity. In case of a measure of symmetry, this scenario is further exaggerated. (5.) Figure 9 displays the impact of the ratio l∗ on the load bearing capacity. It is evident that, with an increase in l∗, the load bearing capacity decreases. The rate of it is further increased with an increase in porosity parameter e. If we look on the results presented in Fig. 10a and correlate them with the results from Fig. 10, we can firmly conclude that the Kozeny-Carman model is highly activated in reference to the conventional poros- ity case. 4. Validation Undoubtedly, Tables 1-5 underline that an enhance- ment in the load bearing capacity by almost 5% is registered here. 147 M. M. Munshi, A. R. Patel, G. B. Deheri Acta Polytechnica (a). (b). (c). (d). Figure 3. Profile of load bearing capacity with re- gards to µ∗. (a). (b). (c). (d). Figure 4. Profile of load bearing capacity with re- gards to σ. 148 vol. 59 no. 2/2019 A study of ferrofluid lubrication based rough sine film slider bearing. . . (a). (b). Figure 5. Profile of load bearing capacity with re- gards to α. Figure 6. Profile of load bearing capacity with re- gards to ε. µ∗ Load bearing capacity calculated for (α = 0.01,σ = 0.01,ε = 0.05, ψ = 30, 1/β1 = 0.01, l∗ = 1.75, e = 0.15,ψ∗ = 0.02) Result for assorted porosity Result for conventional porosity 0.01 0.1651655 0.1567618 0.02 0.1659988 0.1575952 0.03 0.1668321 0.1584285 0.04 0.1676655 0.1592618 0.05 0.1684988 0.1600952 Table 1. Comparison of W calculated for µ∗. (a). (b). Figure 7. Profile of load bearing capacity with re- gards to ψ. Figure 8. Profile of load bearing capacity with re- gards to e. α Load bearing capacity calculated for (µ∗ = 0.02,σ = 0.01,ε = 0.05, ψ = 30, 1/β1 = 0.01, l∗ = 1.75, e = 0.15,ψ∗ = 0.02) Result for assorted porosity Result for conventional porosity -0.02 0.1716939 0.1623983 -0.01 0.1697629 0.1607764 0 0.1678648 0.1591755 0.01 0.1659988 0.1575952 0.02 0.1641643 0.1560354 Table 2. Comparison of W calculated for α. 149 M. M. Munshi, A. R. Patel, G. B. Deheri Acta Polytechnica (a). (b). Figure 9. Profile of load bearing capacity with re- gards to l∗. σ Load bearing capacity calculated for (µ∗ = 0.02,α = 0.01,ε = 0.05, ψ = 30, 1/β1 = 0.01, l∗ = 1.75, e = 0.15,ψ∗ = 0.02) Result for assorted porosity Result for conventional porosity 0.01 0.1659988 0.1575952 0.03 0.1658797 0.1574952 0.05 0.1656424 0.1572959 0.07 0.1652885 0.1569985 0.09 0.1648207 0.1566048 Table 3. Comparison of W calculated for σ. (a). (b). (c). Figure 10. Profile of load bearing capacity with regards to µ∗, σ and � for the comparison of e and ψ∗. ε Load bearing capacity calculated for (µ∗ = 0.02,α = 0.01,σ = 0.03, ψ = 30, 1/β1 = 0.01, l∗ = 1.75, e = 0.15,ψ∗ = 0.02) Result for assorted porosity Result for conventional porosity -0.02 0.1692074 0.1602794 -0.01 0.168718 0.1598716 0 0.1682335 0.1594672 0.01 0.1677536 0.1590663 0.02 0.1672784 0.1586686 Table 4. Comparison of W calculated for ε. 150 vol. 59 no. 2/2019 A study of ferrofluid lubrication based rough sine film slider bearing. . . e Load bearing capacity calculated for (µ∗ = 0.02,α = 0.01,σ = 0.01, ψ = 30,ε = −0.01, 1/β1 = 0.01, l∗ = 0.08,ψ∗ = 0.02) Result for assorted porosity Result for conventional porosity 0.1 0.1706303 0.1599767 0.15 0.1699394 0.1599767 0.2 0.1684007 0.1599767 0.25 0.1655293 0.1599767 0.3 0.1607807 0.1599767 Table 5. Comparison of W calculated for e. 5. Conclusions This paper has studied the effect of ferrofluid lubri- cation when used with a rough sine film slider bear- ing with an assorted porous structure on the load carrying capacity. A modified Reynolds’ equation used for the sine profile slider bearing lubrication has been derived with the ferrohydrodynamic theory by Neuringer-Rosensweig and equation of continuity for film as well as porous region. The Reynolds’ equation has also been used to determine the pressure equation and an expression for dimensionless load-carrying ca- pacity. From the numerical calculations, the following conclusions have been derived: (1.) By increasing the strength of the external mag- netic field, a bearing system’s pressure and its load bearing capacity can be increased consider- ably. Also, unlike conventional lubricants, this type of a system can carry a given amount of load even if there is no flow. Additionally, as suggested by equa- tion (13), when the Neuringer-Rosensweig ferrofluid flow model is applicable, a constant magnetic field does not increase the load bearing capacity. (2.) Comparing the present paper with [15] makes it evident that the system, in this case, enhances the load carrying capacity threefold at minimum. Also, when a sine film profile is used to design the slider bearing, it enhances the bearing capacity, as can be seen when compared with an inclined slider bearing. Lastly, the article determines that, when Kozeny- Carman’s model is appropriate, the surface roughness must be studied properly in order to design a more efficient bearing system. List of symbols a inlet-outlet ratio B breath of the bearing g function of different parameters h film thickness [mm] h mean film thickness [mm] hs deviation from mean level h1 maximum film thickness h0 minimum film thickness h non dimensional film thickness H external magnetic field ḣ0 squeeze velocity in z-direction H0 thickness of porous facing L length of the bearing p non-dimensional film pressure w0 values of w at z = 0 wh values of w at z = h W load capacity [N] W non-dimensional load capacity α variance [mm] α non-dimensional variance β1 squeeze parameter ε skewness [mm3] ε skewness in dimensionless form µ0 magnetic characteristic µ magnetic susceptibility of particles µ∗ dimensionless magnetization parameter σ standard deviation [mm] σ dimensionless standard deviation ψ permeability of porous region ψ∗ dimensionless conventional porosity Acknowledgements The authors would like to thank the reviewers for their comments and suggestions, which resulted in an improve- ment of the materials presented in the paper. 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PHI Private Limited, New-Delhi, India, 2009. 152 http://dx.doi.org/10.1063/1.1711103 http://dx.doi.org/10.1108/ILT-08-2016-0179 http://dx.doi.org/10.29252/jafm.11.03.27854 http://dx.doi.org/10.1108/00368790310470930 http://dx.doi.org/10.1155/2016/8197160 http://dx.doi.org/10.15282/jmes.4.2013.17.0050 http://dx.doi.org/10.4172/2168-9873.1000143 http://dx.doi.org/10.12783/dtetr/amita2016/3580 http://dx.doi.org/10.22190/FUME1601089P http://dx.doi.org/10.1504/IJCMSSE.2014.063760 http://dx.doi.org/10.5923/j.am.20120204.04 http://dx.doi.org/10.1080/17515831.2016.1235843 http://dx.doi.org/10.1080/17515831.2017.1418280 http://dx.doi.org/10.1016/0020-7225(86)90095-9 http://dx.doi.org/10.1115/1.3075870 Acta Polytechnica 59(2):144–152, 2019 1 Introduction 2 Analysis 2.1 A globular sphere model 3 Results and discussion 4 Validation 5 Conclusions List of symbols Acknowledgements References