Matematics Today Vol.28(June-Dec-212)43-49 ISSN 976-3228 Ferrofluid Lubrication equation for non-isotropic porous squeeze film bearing wit slip velocity Rajes C. Sa * Department of Applied Matematics, Faculty of Tecnology and Engineering, Te M. S. University of Baroda, Vadodara 39 1, Gujarat State, India e-mail :dr_rcsa@yaoo.com M.M.Parsania Department of Matematics, Scool of Engineering, R.K. University, Rajkot, Gujarat State, India e-mail : mmp781992@yaoo.co.in Abstract Te aim of tis paper is to derive general ferrofluid lubrication equation for nonisotropic permeable porous squeeze film bearing wit slip velocity from momentum and continuity equations.various special cases are also deduced. KeywordsFerrofluid, Lubrication, Squeeze film bearing, Slip velocity AMS 76D8 1. Introduction Many investigators [1-3] in teir analyses used porous bearing wit conventional lubricants. Some simplified teir analysis [4-6] by using Morgan-Cameron approximation [7] to avoid series solutions of bearing caracteristics. Ferrofluids or magnetic fluids [8] are stable colloidal suspensions containing fine ferromagnetic particles dispersing in liquid carriers, in wic a surfactant is added to generate a coating layer preventing te flocculation of te particles. Ferrofluids can experience magnetic body forces depend upon te magnetization of ferromagnetic particles wen an externally magnetic field is applied. Owing to tese features, ferrofluids ave been applied in many areas of industrial engineering and applied science, suc as te medicine treatment for drug targeting by Rosensweig [8], te flow sensors by Popaet. al.[9]. Wit te invention of ferrofluids, studies [1-12] were made wit ferrofluid-based porous bearings using Neuringer-Rosensweig model for te lubricant flow. Beavers and Josep [13], Beavers et. al. [14] and Sparrow et. al.[15] proved analytically and experimentally tat te assumption of no slip at te porous interface could not old wen te porous matrix was
44 Matematics Today Vol.28(June-Dec-212) 43-49 formed wit naturally permeable material like foam. Investigations [16-17] were made using te * Corresponding autor boundary conditions proposed by Beavers and Josep [13] and sparrow et. al.[15]. In tis paper our aim is to deriveferrofluidlubrication equation for non-isotropic porous squeeze film bearing wit slip velocity from momentum and continuity equations. 2. Matematical formulation Te configuration of te bearing sown in figure1 consists of two surfaces( or plates )wic is separated by film tickness ( metre ) wic is filled wit ferrofluid flowing as per Neuringer-Rosensweigs model [8]. Te lower plate is made up of naturally permeable material and te upper plate is solid. Te upper plate moves normally towards permeable flat lower plate wit uniform velocity = d dt. Figure 1.Configuration of te problem. Te following assumptions are made in te derivation (i) tere is no variation of pressure across te film, (ii) velocity gradients across te film predominate, (iii) te porous region is non-isotropic permeable, (iv) te flow in te porous region is governed by Darcy s law, (v) pressure and normal velocity components are continuous at te interfaces and (vi) tere is a slip velocity at te porous interface.
R. Sa&M.Parsania- Ferrofluid Lubrication equation for 45 Te basic equations governing te above penomenon wic obtained from Navier- Stroke sequation are 2 u z 2 = 1 2 v z 2 = 1 x (p 1 2 μ μ H 2 ), (2.1) y (p 1 2 μ μ H 2 ), (2.2) were(u,v,w) are film fluid velocities in x, y, z directions respectively, is te fluid viscosity, p is pressure of te fluid in te film region,μ is te permeability of free space, μ is te magnetic susceptibility and H is te strengt of te magnetic field wit its curl and divergence vanising. In te porous material aving non-isotropic permeability te flow is taken to be Darcian so tat q = 1 φ x + φ x y + φ y z (P 1 μ z 2 μ H 2 ) (2.3) =u + v + w, (2.4) were u = φ x v = φ y x (P 1 2 μ μ H 2 ), (2.5) y (P 1 2 μ μ H 2 ), (2.6) w = φ z z P 1 2 μ μ H 2, (2.7) wereq is te Darcian velocity,φ x, φ y, φ z are te permeabilities in x,y and z- directions respectively, P is te pressure in porous region. At te permeable boundary, from continuity consideration P x, y, z z= = p(x, y). (2.8) Terefore, equation (2.3) becomes q = q z= = 1 φ x x p 1 2 μ μ H 2 + φ y y p 1 2 μ μ H 2 +φ z z P 1 2 μ μ H 2 z= (2.9) = u + v + w. (2.1) In te film region te equation of continuity is expressed by
46 Matematics Today Vol.28(June-Dec-212) 43-49 x ( ρ u dz) + y ( ρ v dz) + w w + ρ dz =, (2.11) t wereρ is te mean density of te fluid wic is taken to be independent of z coordinate. Since w denotes te z componentof te film fluid velocity at z =, terefore w = w = φ z z P 1 2 μ μ H 2 z=. (2.12) Using slip boundary condition [15] at z =, u = 1 s 1 u z wen z = and u = wenz =, (2.13) v = 1 v wenz = and v = wen z =, (2.14) s 2 z were 1 s 1 = x φ x 5, 1 s 2 = y φ y 5 ; are slip parameters, x and y being te porosities in te x and y directions respectively. Integrating equation (2.1) wit respect to z, we obtain u z = 1 x (p 1 2 μ μ H 2 )z + A, (2.15) were A is a constant of integration. Again, integrating equation (2.15) wit respect to z, we obtain u = 1 (p 1 μ x 2 μ H 2 ) z 2 + Az + B, (2.16) 2 were B is a constant of integration. Using first boundary condition of equation (2.13), equation (2.16) yields 1 u s 1 z z= = B. (2.17) Again, using second boundary condition of equation (2.13), equation (2.16) yields A= 1 1 (p 1 μ x 2 μ H 2 ) 2 + 1 u 2 s 1 z z=. (2.18) Using (2.17) and (2.18), equation (2.16)becomes
R. Sa&M.Parsania- Ferrofluid Lubrication equation for 47 u = 1 ( z 2 2 z 2 ) x (p 1 2 μ μ H 2 )+ 1 s 1 1 z u z z=. (2.19) Similarly, using boundary conditions (2.14), equation (2.2)yields v = 1 ( z 2 2 z 2 ) y (p 1 2 μ μ H 2 ) + 1 s 2 1 z v z z=. (2.2) Substituting te values of u and v from equation (2.19) and (2.2) in equation (2.11), we obtain linear partial differential equation of second order in two variables as x 3 12 x (p 1 2 μ μ H 2 ) + u 2s 1 z z= + 3 (p 1 μ y 12 y 2 μ H 2 ) + v 2s 2 z z= + + φ z z P 1 2 μ μ H 2 z= + 1 ρ ρ dz =. (2.21) t Tis is te desired ferrofluid lubrication equation for non-isotropic porous squeeze film bearing wit slip velocity. Tis equation is of general nature and can be reduced to te following simpler forms. 3. Discussion of Special Cases (1) Setting s 1, s 2 in Equation (2.21), we obtain te no slip version corresponding to te present case as [ 3 x 12 (p 1 μ x 2 μ H 2 ) ] + [ 3 y 12 y (p 1 2 μ μ H 2 )] + + φ z z P 1 2 μ μ H 2 z= + 1 ρ ρ dz =. t (2) Setting H = in Equation (2.21), we obtain te equation for te conventional lubricant considering slip velocity and anisotropic permeability as 3 p x 12 x + u 2s 1 z z= + 3 p y 12 y + 2s 2 v z z= + + φ z P z z=+ 1 ρ ρ dz =. t 4. Conclusions
48 Matematics Today Vol.28(June-Dec-212) 43-49 Equation (2.21) is te most general form, using Neuringer-Rosensweig model, for squeeze film bearing for non-isotropic porosity and slip velocity. It can be used to study te performance of te squeeze film bearing of various sapes.also, it can be furter developed using oter equations for porous matrix. Acknowledgement Te autors are grateful to te Referee for teir valuable suggestions. References [1] Wu, H.,(1972): An analysis of te squeeze film betweenporous rectangular plates,j LubrTecnol, F94, 64-68. [2] Sina, P. C., Gupta, J. L.,(1973) :Hydromagnetic squeeze films between porous rectangular plates,j LubrTecnol, F95, 394-398. [3] Bat, M. V., Hingu, J. V.,(1978):A Study of te HydromagneticSqueeze Film between Two Layered PorousRectangular Plates,Wear, 5, 1-1. [4] Prakas, J., Vij, S. K.,(1973) :Hydrodynamic lubrication of a porous slider,j MecEngSci, 15,232-234. [5] Bat, M. V.,(1978) :Hydrodynamic Lubrication of a Porous Composite Slider Bearing,Jap J App Pys, 17, 479-481. [6] Gupta, J. L., Vora, K. H., Bat, M. V., (1982) :Te effect of rotational inertia on te squeeze film load between porous annular curved plates,wear, 79,235-249. [7] Morgan, V. T., Cameron, A.,(1957) :Mecanism of Lubrication in Porous MetalBearings,ProcConf Lubrication and Wear, Inst Mec Engrs, London, paper 89,151-157. [8] Rosensweig, R. E.,( 1985): Ferroydrodynamics, New York : Cambridge University Press. [9] Popa, N. C., Potencz, I., Brostean, L., Vekas, L., (1997) :Some applications of inductive transducers wit magnetic liquids,sensors and Actuators, 59, 197-2. [1] Verma, P. D.S.,(1986) : Magnetic fluid-based squeeze film,int J Eng Sci, 24,395-41. [11] Bat, M. V., Deeri, G. M.,(1991) :Porous composite slider bearing lubricated wit magnetic fluid,jpn J App Pys, 3,2513-2514. [12] Sa, R. C., Tripati, S. R., Bat, M. V.,(22): Magnetic lfuid based squeeze film between porous annular curved plates wit te effect of rotational inertia, Pramana-J Pys, 58(3) 545-55.
R. Sa&M.Parsania- Ferrofluid Lubrication equation for 49 [13] Beavers, G. S., Josep, D. D.,(1967) :Boundary conditions at a naturally permeable wall,j Fluid Mec, 3,197-27. [14] Beavers, G. S., Sparrow, E. M., Magnuson, R. A., (197) :Experiments on coupled parallel flows in a canneland a bounding porous medium, Journal of BasicEngineering, D 92 (4), 843-848. [15] Sparrow, E. M., Beavers, G. S., Hwang, I. T., (1972) : Effect of velocity slip on porous walled squeeze films,trans ASME, F 94, 26-265. [16] Prakas, J., Vij, S. K., (1976) :Effect of velocity slip on te squeeze film between rotating porous annular discs,wear, 38, 73-85. [17] Bat, M. V., Deeri, G. M., (1994) :Optimum Film Tickness in a Porous Slider Bearing Considering Slip Velocity, Mat Today, 12, 27-3. NOMENCLATURE Squeeze velocity ( m / s ) u, v, w Film fluid velocities ( m / s ) x, y, z Coordinates ( m ) p Pressure of te fluid in te film region ( kg / ms 2 ) Fluid viscosity (kg / ms ) Free space permeability ( kg m / s 2 A 2 ) Magnetic Susceptibility H Strengt of magnetic field ( A / m ) q Darcian velocity ( m / s ) x, y, z Permeabilities in x, y and z directions ( m 2 ) P Pressure of te fluid in te porous region ( kg / ms 2 ) ρ Mean density of te fluid ( kg / m 3 ) x, y Porosities in x and y directions