SLIP VELOCITY ON THE FERROFLUID LUBRICATION OF THE POROUS EXPONENTIAL SLIDER BEARING

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1 International Journal of Advanced Research in Engineering and Technology (IJARET) Volume 9, Issue 3, May - June 18, pp. 1 3, Article ID: IJARET_9_3_8 Available online at ISSN Print: and ISSN Online: IAEME Publication SLIP VELOCITY ON THE FERROFLUID LUBRICATION OF THE POROUS EXPONENTIAL SLIDER BEARING Sheetal A. Patel Department of Mathematics, Silver Oak College of Engineering and Technology, Ahmedabad, Gujarat, India A. R. Patel Department of Mathematics, Vishwakarma Government Engineering College, Ahmedabad, Gujarat, India G. M. Deheri Department of Mathematics, Sardar Patel University, Vallabh Vidhyanagar, Anand, Gujarat, India ABSTRACT An endeavor has been made to analyze the combined impact of roughness and slip velocity on the Ferrofluid lubrication of porous exponential slider bearing. The surface roughness is described in perspective of the stochastic model of the Christensen and Tonder. The magnetic fluid flow is administered by the model of Neuringer-Rosensweig. Beavers-Joseph's slips model represents the impact of slip velocity. The concern stochastically average Reynolds type equation is solved to get the pressure distribution in the bearing system then the load carrying capacity is figured numerically. The outcomes appeared in graphical structures confirm that the magnetization goes to the limited extent to counter the impact of slip velocity and roughness. Key words: Exponential slider bearing, Ferrofluid, Roughness, Porosity, Load carrying capacity. Cite this Article: Sheetal A. Patel, A. R. Patel and G. M. Deheri, Slip Velocity on The Ferrofluid Lubrication of The Porous Exponential Slider Bearing. International Journal of Advanced Research in Engineering and Technology, 9(3), 18, pp INTRODUCTION The slider bearings are mainly designed to help the transverse load in the engineering. The slip velocity and impact of surface roughness on the execution of bearing have pulled in numerous specialists as of late. The vital work has been finished by considering the impact of magnetic fluid also. The stochastic model displayed by Tzeng and Saibel [1] was additionally created by Christensen and Tonder [-4] to think about the impact of transverse and also longitudinal 1 editor@iaeme.com

2 Slip Velocity on The Ferrofluid Lubrication of The Porous Exponential Slider Bearing surface roughness on the execution of bearing system. It has been utilized by numerous researchers. Berthe and Godet [5] studied a more general form of Reynolds equation on roughness. Agrawal [6] considered theoretically the magneto-hydrodynamic leading composite slider bearing in the existance of a transverse magnetic field and it was built up that the load carrying limit increases as the thickness of the boundaries and the conductivity increases. Puri and Patel [7] considered the execution of a porous composite slider bearing considering the slip velocity at the interface of the fluid film and the porous matrix. Cameron [8] proposed the closest shape of an exponential form of the slider. Shah and Bhat [9] investigated a porous exponential slider bearing with porous and a Ferrofluid considering slip velocity and demonstrated that comparing inclined plane porous slider bearing, an exponential porous slider bearing has more load capacity, friction and the coefficient of friction. The decrease in load capacity of the bearing as a result of the slip velocity and the material parameter can be made great by increasing the magnetization of the fluid. Jaw-Ren Lin and Chi-Ren Hung [1] investigated the dynamic attributes of wide slider bearing for an exponential film profile and found the higher estimation of stiffness coefficient, load carrying limit and damping coefficient. These are predicted for the bearing with large values of the inlet-outlet film ratio. Ochonski [11] investigated the possibility of another outline of magnetic fluid based sliding bearing and to utilize them in current bearing innovation, in audiovisual equipment and new computer. Deheri and Patel [1] examined the impact of magnetic fluid on a rough, porous composite slider bearing. They demonstrated that the negative impact of porosity and standard deviation can be limited by the positive impact of magnetization by the suitably chosen length of curved and length of flat pads. Beavers and Joseph [13] calculated that in a naturally permeable material an assumption might not hold at the nominal boundary. Patel et al. [14] showed that magnetization may not go a long way for reducing the adverse impact of roughness, regardless of whether the slip parameter is reduced using ferrofluid lubrication of a rough, porous convex pad slider bearing considering slip velocity by Jenkins model. However, the situation improves when negatively skewed roughness occurs.. Rajesh shah and Nayan patel [15] studied the impact of various and arbitrary porous structure on the performance of the step bearing lubricated with magnetic fluid, the calculation shows that globular speare model have better performance for load carrying capacity than capillary fissures model to extend this work in the present paper, magnetic liquid has been taken as lubricate to consider an exponential slider bearing on for the impact of roughness and slip velocity. The articulations for pressure and load carrying capacity have been determined and the impact of different dimensionless parameters has been considered using Simpson's one-third rule.. BASIC EQUATIONS AND ANALYSIS Figure 1 Exponential Slider Bearing editor@iaeme.com

3 Sheetal A. Patel, A. R. Patel and G. M. Deheri In Figure 1, the configuration of the exponential slider bearing is given. The stochastically model of Christensen and Tonder [-4] have been taken in to account for the evaluation of roughness. Accordingly, the film thickness h is considered as: h = h + h s Where the mean film thickness is h and the probability density function is h. (1) 3 35 h 1 s, b h 3 s b f ( hs ) = b b, elsew here () Where b is the maximum deviation from the mean film thickness and determined by the relationships 3 α= E( hs), σ = E ( hs α), ε = E ( hs α) 3 Where E denotes the expected value defined by b E (R) = Rf ( h ) dh (4) b s s The basic flow equations of magnetic fluid in view of Rosensweig s model [16] are given as follows. q ρ + ( q ) q = p + η q+ µ ( M ) H (5) t H =, q =, ( 6) ( ) M = µ H, H M =, (7) Also. q = ui + vj + wk, (8) u, v, w are components of fluid film velocity in x, y, z directions respectively. The magnetic field oblique to the bottom surface is expressed as H = Kx ( B x ) ( 9) Where $(% & ' () ) is chosen to suitable the dimensions of both sides. With the aid Eqs. (5), (6), (7) and (8), the lubricant flow in decided by ( ) u 1 1 = p µ µ H z η x Associated slip boundary condition is (1) 1 u 1 k u = ; =, z = (11) β z β ξ 3 editor@iaeme.com

4 Slip Velocity on The Ferrofluid Lubrication of The Porous Exponential Slider Bearing The integral form of continuity equation for film region h udz + w z = h w z = = x Yields (1) h β 3 h ( + h) h 1 1 p H p H µ µ µ µ + V w (13) x 6 x x ( h 1) x z = = η η β + Where, dh w z = h = V = = & h (14) dt Eq. (14) represents the impact of squeeze velocity in the downward z direction. By the generalized Darcy s law in porous region the velocity components are govern by. k 1 u = p µ µ H ( x direction ) η x (15) k 1 w = p µ µ H ( z direction ) (16) η z Substituting Eqs. (15) and (16) in the continuity equation for porous region u w + = x z (1 7 ) 1 1 p µ µ H p µ µ H + = x z Take Integration in region ( 4,) 1 1 p µ µ H = H p µ µ H z * z = o x By 8 = 4 is a solid surface so that (18) 1 p µ µ H = (19) z * z = H Consider the normal components of velocity across the film porous interface is continuous, therefore w = w z = z = From Eqs. (13) and (16) () h β 3 h ( + h) h 1 1 k 1 p µ µ H p µ µ H + V = p µ µ H x 6η x η( h β + 1) x η z 4 editor@iaeme.com

5 Sheetal A. Patel, A. R. Patel and G. M. Deheri Using Eq. (18) we get h ( h β + ) * 1 h 1kH p µ µ H = 1 ηv (1) x ( h β + 1) x Now, we put H kx ( B x ) ( β ) = therefore Eq. (1) becomes 3 * h h 4 1 kh ( hβ 1) p µ µ kx( B x) = 1ηV x ( hβ + 1) x Introducing non- dimensional quantities, we get, 3 ( h β + 4) h + 1ψ ( h β + 1) ( h β + 1) 1 * p µ X(1 X ) = 1 X X Where 3 3 h * p * µ µ kh h1 σ α ε ε= 3 ηvb η h x h kh X =, h=, p=, µ =, β= hβ, ψ=, a=, σ=, α=, () B h V h h h h Take, So, ( ) G h = 3 ( h β + 4 ) h + 1 ψ ( h β + 1 ) ( h β + 1) 1 * ( ) µ G h p X (1 X ) = 1 (3) X X Here, And we take, h 4 + h g( h) β = h 1+ β 3 ( h 3 h 3 ( α ) ) σ h 3σ α 3 1 g( h ) = + α α + + ε + ψ (4) 1 3 Now, taking X ln a h = ae, X 1, with the boundary conditions at X =, P = w here h = a X = 1, P = w here h = 1 (5) Integrating Eq. (3) and using appropriate boundary condition we get, 1 X * 1 X p = µ X (1 X ) + 1 d X ( 6 ) G ( h ) 5 editor@iaeme.com

6 Slip Velocity on The Ferrofluid Lubrication of The Porous Exponential Slider Bearing W 1 = p dx * µ 1 1 = ( X ) G ( h ) d X ( 7 ) 3. RESULT AND DISCUSSION Figure Variation of L.C.C.( load carrying capacity) w.r.t. µ* Figure 3 Variation of L.C.C. w.r.t. µ* Vs. A 6 editor@iaeme.com

7 Sheetal A. Patel, A. R. Patel and G. M. Deheri Figure 4. Variation of L.C.C. w.r.t. µ* Vs. B Figure 5. Variation of L.C.C. w.r.t. µ* Vs. ψ Figure 6. Variation of L.C.C. w.r.t. µ* Vs. 1/D 7 editor@iaeme.com

8 Slip Velocity on The Ferrofluid Lubrication of The Porous Exponential Slider Bearing Figure 7. Variation of L.C.C. Vs. A Figure 8. Variation of L.C.C. Vs. B Figure 9. Variation of L.C.C. Vs. ψ 8 editor@iaeme.com

9 Sheetal A. Patel, A. R. Patel and G. M. Deheri Figure 1. Variation of L.C.C. Vs. 1/D Figure 11. Variation of L.C.C. w.r.t. B Vs. A Figure 1. Variation of L.C.C. w.r.t. B Vs. ψ 9 editor@iaeme.com

10 Slip Velocity on The Ferrofluid Lubrication of The Porous Exponential Slider Bearing Figure 13. Variation of L.C.C. w.r.t. A Vs. ψ Figure 14. Variation of L.C.C. w.r.t. A Vs. 1/D Figure 15. Variation of L.C.C. w.r.t. B Vs. 1/D 3 editor@iaeme.com

11 Sheetal A. Patel, A. R. Patel and G. M. Deheri Figure 16. Variation of L.C.C. w.r.t. ψ Vs. 1/D It is seen from Eqs. (6) and (7) that the dimensionless pressure increases by E & μ G(1 G) while the dimension less load carrying capacity moves up by μ 1in comparison with the usual fluid based bearing system. An increase in the magnetization parameter possesses increase in the load carrying capacity as it can be seen from Fig The fact that the standard deviation causes load reduction is given in Fig Fig. 11 and Fig. 13 suggest that increasing variance decrease the load carrying capacity while decreasing variance increases load carrying capacity. Fig. 1 increasing value of skewness decreases the load carrying capacity, better load can be observed for negative value of skewness. The significant impact of slip velocity on the performance characteristic is presented in Fig But, it is easily seen that an increase in slip velocity can causes considerable load reduction however, this impact is less when negative variance occurs. These figures underline that for a better performance the slip velocity must be added at a reduced level. 4. CONCLUSIONS This investigation establishes that the execution of the bearing system remains a little better in the case of negatively skewed roughness. In particular, the slip is added at a reduced level. The adverse impact of roughness augments when moderate to higher values of porosity are involved. If there is no flow in such type of bearing system sustains certain amount of load, which does not happen in the case of conventional lubricant based bearing system. NOMENCLATURE B Bearing length H Magnitude of the magnetic field H * Thickness of porous layer 4I Magnetic field vector h 1,h Maximum and minimum value of h h The mean film thickness JI Magnetization vector k Permeability of porous matrix P Lubricant film Pressure K Dimensionless pressure L Magnetic fluid velocity W Load carrying capacity MI Dimensionless load carrying capacity α A D B B N μ O μ ξ ρ Ψ Variance Non dimensional variance Slip parameter Skewness Skewness in dimensionless form Fluid viscosity Free space permeability Magnetic susceptibility Slip coefficient Magnetic fluid density Standard deviation Dimensionless standard deviation Porosity 31 editor@iaeme.com

12 Slip Velocity on The Ferrofluid Lubrication of The Porous Exponential Slider Bearing REFERENCES [1] Tzeng, S. T. and Saibel, E. Surface Roughness Effect on Slider Bearing Lubrication, Trans. ASLT, 1(3), 1967, pp [] Christensen, H. and Tonder, K. C. Tribology of rough surfaces: stochastic models of hydrodynamic lubrication, SINTEF, Report no. 1/69, 1969a. [3] Christensen, H. and Tonder, K. C. Tribology of rough surfaces: parametric study and comparison of lubrication models, SINTEF, Report no. /69, 1969b. [4] Christensen, H. and Tonder, K. C. The hydrodynamic lubrication of rough bearing surfaces of finite width, Proceedings of the Conference. ASME-ASLE Lubrication, Cincinnati, Ohio, USA, 197, pp [5] Berthe, D. and Godet, M. A More General Form of Reynolds Equation Application to Rough Surfaces, Wear, 7(3), 1974, pp [6] Agrawal, V. K. The Impact of Conductivity on the Load Capacity of Hydromagnetic Composite Slider Bearing, Japanese journal of Applied Physics, 9(11), 197. [7] Puri, V. K. and Patel, C. M. Analysis of a porous composite slider bearing with slip velocity, 78(3), 198, pp [8] Cameron, A. Basic Lubrication Theory, Wiley Eastern Ltd., 1987, pp. 6. [9] Shah, R. C. and Bhat, M. V. Analysis of a porous exponential slider bearing lubricated with a Ferrofluid considering slip velocity, J. BRAZ. Soc. Mech. Sci. & Eng. 5(3), July/Sept-3. [1] Lin, Jaw-Ren and Hung, Chi-Ren Analysis of dynamic characteristics for wide slider bearings with an exponential film profile, Journal of Marine science and Technology, 1(3), 4, pp [11] Ochonski, W. Sliding bearings lubricated with magnetic fluids, Industrial Lubrication and Tribology, 59(6), 7, pp [1] Patel, N. D., Deheri, G. M. and Patel, H. C. Magnetic Fluid Lubrication of a Rough, Porous Composite Slider Bearing, International Journal of Surface Engineering and Interdisciplinary Materials Science, 1(), 13, pp [13] Beavers, G. S. and Joseph, D. D. Boundary conditions at a naturally porous wall, Journal of Fluid Mechanics, 3(1), 1967, pp [14] Patel, P. A., Deheri, G. M. and Patel, A. R. Jenkins model based Ferrofluid lubrication of a Rough, Porous Convex pad slider bearing with slip velocity, Bio Info Mechanical Engineering, 4(1), 16, pp [15] Shah, Rajesh C. and Patel, Nayan I. Impact of various and arbitrary porous structure in the study of squeeze step bearing lubricated with magnetic fluid considering variable magnetic field, Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology, 9(5), 15, pp [16] Rosensweig, R. E. Ferro hydrodynamics. New York: Cambridge University Press, editor@iaeme.com