EFFECTS OF SURFACE ROUGHNESS WITH MHD ON THE MICRO POLAR FLUID FLOW BETWEEN ROUGH ELLIPTIC PLATES

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1 International Journal of Mechanical Engineering and Technology (IJMET) Volume 9 Issue November 8 pp Article ID: IJMET_9 58 Available online at ISSN Print: and ISSN Online: IAEME Publication Scopus Indexed EFFECTS OF SURFACE ROUGHNESS WITH MHD ON THE MICRO POLAR FLUID FLOW BETWEEN ROUGH ELLIPTIC PLATES Brinda Halambi and Hanumagowda B. N. Department of Mathematics School of Applied Sciences REVA University Bangalore Karnataka India ABSTRACT This paper exhibits a theoretical analysis of effects of surface roughness on the squeeze film features of micro polar between two elliptic plates with hydro magnetic aspect under the application of an external magnetic field. The stochastic model of rough surfaces introduced by Christensen formed basis of modified Reynolds equation for hydrodynamic lubrication. The Reynolds equation is solved for lubrication characteristics such as squeeze film pressure squeeze film load carrying capacity and squeeze film time. It is shown that roughness structure and magnetic field may considerably impart a much appreciated load carrying capacity film pressure and establish a lengthened time of approach compared to classical case. Further the squeeze film lubrication features are found to increase for larger values of the Hartmann parameter fluid gap parameter and the coupling parameter. Keyword: Elliptic plates micro polar surface roughness squeeze film and magneto hydrodynamic. Cite this Article Brinda Halambi and Hanumagowda B. N Effects of Surface Roughness with Mhd on the Micro Polar Fluid Flow Between Rough Elliptic Plates International Journal of Mechanical Engineering and Technology 9() 8 pp INTRODUCTION Traditionally surface roughness has a straight influence on the electrical optical and mechanical solid thin film devices. Any solid material surface roughness will have important effect on the macroscopic contact angle evaluation on its flat surface which plays essential role in different process as moistening spreading and wetting. Various methods have been used to investigate outcomes of rough surfaces such as saw tooth curve model this is used on deformable surfaces by Davies [] Fourier approximation method is used on lubricant film by Burton [] Stochastic approach has been used on slider bearing by Tzeng and Saibel [3] and stochastic theory has been used for both longitudinal and transverse roughness by Christensen [4]. Many researchers utilized stochastic model of Christensen [4] to study surface roughness on hydrodynamic lubrication of solid bearings such as [5-8] studied slider and journal bearing Naduvinamaniet.al [9] studied a editor@iaeme.com

2 Brinda Halambi and Hanumagowda B. N sphere and flat plate Rajani et.al [] used on conical bearings. From their research it is analyzed that surface roughness provides an improvement for squeeze film pressure squeeze film load capacity and imparts lengthened squeeze film time of approach. Magneto hydrodynamics (MHD) studies functioning of an electrically conducting fluid acted on a magnetic field. The electrically conductive lubricant prevents viscosity variation of lubricant with temperature. So many are tempted of working with electrically conductive lubricant on the basis of hydro magnetic flow theory of Cowling []. Various kinds of studies have been conducted to analyze effects of hydro magnetic squeeze film for different bearings such as Hamza [] has investigated MHD effects between two rotating parallel disks Lin [3] has studied MHD effects on rectangular plates Naduvinamaniet.al [4] has applied for circular stepped plates Brinda and Hanumagowda [5] has analyzed MHD effects on elliptic plates all their investigations shown an increase in squeeze film pressure squeeze film load carrying capacity and delayed squeeze film time of approach for electrically conducting fluids. Micro polar fluid is a non- Newtonian fluid with microstructure. To analyze flow behavior of micro polar fluid as lubricant a micro continuum theory has been generated by Eringen [6]. In the theory of micro polar fluid influences of the couple stresses the body couples of microstructures and micro rotations are taken in to the account. Several kinds of squeeze film bearings have been studied using Eringen [6] micro continuum theory of micro polar fluids such as the porous spherical bearing by Zaheeruddin and Isa [7] the porous journal bearing by Naduvinamani and Santhosh [8] the finite width journal bearing by Tsai-Wang Hung [9] the slider bearing by Siddangouda et.al [] and elliptical plates by Roopa et.al []. From their studies it is proved that non-newtonian effects of micro polar fluids produce an improved film pressure load carrying capacity and lengthen time of approach. In the existing study surface roughness and hydro magnetic effects on micro polar squeeze film between elliptic plates under the application of external magnetic field are examined. Applying stochastic model of Christensen [4] together with the hydro magnetic flow theory of Cowling [] and micro-continuum theory of Eringen [6] non-newtonian hydro magnetic Reynolds equation is derived. Effects of surfaces roughness and hydro magnetic aspect on non Newtonian micro polar fluid through the variation of Hartmann number coupling number and fluid gap number are discussed.. MATHEMATICAL FORMULATION The physical configuration of the squeeze film geometry of the problem between rough elliptic plates of major and minor axes a and b respectively is displayed in figure.. The upper smooth elliptic plate proceed towards the fixed lower rough elliptic plate with fixed velocity v=dh/dt. Micro polar fluid which is an isothermal and incompressible in nature is taken as the lubricant and a constant magnetic field B is applied in the y direction editor@iaeme.com

3 Effects of Surface Roughness with Mhd on the Micro Polar Fluid Flow Between Rough Elliptic Plates Figure. Physical model of rough elliptic plates with micropolar fluids and magnetic field The thin film of thickness H is composed up of two parts H= H+ hs(x ξ z).where H=h(t) here h denotes smooth part of the film geometry hs is a randomly varying quantity of zero mean and is a segment due to the surface asperities measured from nominal smooth level and ξ is the index parameter which determines exact roughness pattern. The basic governing equations of micro polar fluid in Cartesian-coordinate system with thin film lubrication theory are u v w + + = x y z () χ u v p µ + + χ σ B u = y y x () χ w µ + χ σ B u = y v y p z (3) v γ u v y χ y χ = (4) γ v w v y χ + y χ = (5) Here variables (u v w) are lubricant velocity elements in the film region in the (x y z) directions respectively. v and v are micro rotational velocity components p represents film pressure σ denotes electrical conductivity of fluid denotes spin viscosity denotes viscosity co-efficient of micro polar fluid and μ denotes Newtonian viscosity. Relevant boundary conditions: At the upper plate surface u = w = v = dh / dt at y = h 6(a) editor@iaeme.com

4 Brinda Halambi and Hanumagowda B. N v = v = At the lower plate surface u = w = v = at y = -h 6(b) 6(c) v = v = 6(d) Equations () and (4) are solved to obtain the differential equation governing the velocity component u and similarly equations (3) and (5) are solved for differential equation governing the velocity component w. u y u y 4 α + βu = g 4 ( x) w y w y 4 α + β w = g 4 ( x) (7) (8) 4µχ + γσ B 4χσ B α = 4χ p 4χ p β = g( x) = g( x) = γ ( µ + χ) γ ( µ + χ) γ ( µ + χ) x and γ ( µ + χ) z The expressions for u and w are derived using boundary conditions (6a) to (6d) ( ) f SinhAh ( ) fsinhbh CoshAh CoshAy CoshBh CoshBy p u = fsinhbhcoshah fsinhahcoshbh σ B x (9) ( ) f SinhAh( ) fsinhbh CoshAh CoshAy CoshBh CoshBy p w = fsinhbhcoshah fsinhahcoshbh σ B z () A = α + α 4β B = f α α 4β ( ) f + B σ B µ χ = χ B ( ) + A σ B µ χ = χ A The following hydro magnetic Reynolds equation is derived by substituting u and w in equation () and integrating across film thickness with respect to y using boundary conditions (6a) to (6d) which describes pressure distribution in film region as p p dh G + = x z dt Where BfSinhBh( AhCoshAh SinhAh) AfSinhAh ( BhCoshBh SinhBh) G = σ B AB( fsinhbhcoshah fsinhahcoshbh ) () Incorporate roughness features by considering the expected values of equation () we get ( ) ( ) ( ) ( ) E p E G + E( G) E p = de H x z dt () editor@iaeme.com

5 Effects of Surface Roughness with Mhd on the Micro Polar Fluid Flow Between Rough Elliptic Plates Where the expectancy operator E ( ) is defined by E( ) = ( ) f ( hs ) dhs (3) Where hs is stochastic film thickness variable and f is its probability density function. By Christensen [4] we can accept that ( s ) c h c < hs < c f ( hs ) = 3c elsewhere (4) Where σ = 3 c is the standard deviation. There are two kinds of one dimensional roughness pattern they are longitudinal and transverse roughness structures. The one dimensional longitudinal roughness is assumed to have the form of long narrow ridges and furrows running in the x direction where as transverse roughness is of the form of long narrow ridges and furrows running in a direction perpendicular to the direction of sliding. The present analysis is carried out only for one dimensional longitudinal roughness and one dimensional transverse roughness can be obtained by just rotating the co-ordinate axes. In this case the film thickness takes the form H=H+hs(ξ z) then equation () becomes From (3) we have E( p) E( p) de( H) E( G) + = x z dt E G (5) E (H ) = H (6) E( p) E( p) dh / dt + R = x z E( G) (7) Where R = E( G) E G 3 ( ) 7 s 3c c c ( c hs ) E G = G( c h ) dhs E = 7 G 3c G c c 3 dh s Introducing the dimensionless variables and parameters x z H H + h H h a x = z = H = = = + = H + h a = a b H H H H b s s s 3 hs H E( p) H s µ c C = h = H = andp = H H H ab dh dt Corresponding boundary conditions for pressure are ( / ) E(p(x z )) = at x z + = (8) a b 59 editor@iaeme.com

6 Brinda Halambi and Hanumagowda B. N Where a and b are major and minor axis in x and z directions respectively. Solving equation (7) using (8) we can obtain non dimensional pressure as 3 E( p) H a 6 P = = ( x z ) µ ( dh dt) ab Ra + E{ G ( H N L M )} (9) Where ( ) 4( ξ ξ ) = M A B ξ3 G H N L M { } {.5 } ( ).5 ( ) ( ) ξ = B f Sinh.5B H A H Cosh.5A H Sinh.5A H ( ) ( ) ( ) ξ = A f Sinh.5A H B H Cosh.5B H Sinh.5B H ξ ( ) ( ) ( ) ( ) 3 = f Sinh.5B H Cosh.5A H f Sinh.5A H Cosh.5B H ( ) M N A = fh = f N A f ( ) M N B = fh = N B α A = AH = α + α 4β ( ) N + M N L = α H = L β B = BH = 4 N M = β H = L M = B ( ) / H σ / µ N α α 4β χ = µ + χ / ( ) / γ / 4µ L = H c 35 3 ( ) = ( )( ) 7 s 3c c ( ) E G H N L M G H N L M c h dh c 3 c hs 7 c 35 ( ) E ( G ( H N L M )) = dhs 3 c G ( H N L M ) And R = E( G ( H N L M )) E(/ G ( H N L M )) Here N denotes non Newtonian coupling number H denotes initial film thickness L denotes fluid gap interacting number and M denotes Hartmann number C represents dimensionless roughness parameter and P is the dimensionless film pressure. Load carrying capacity can be get by integrating film pressure equation (9) s a b a x a = E( W ) E( p) dz dx a b a x a () 59 editor@iaeme.com

7 Effects of Surface Roughness with Mhd on the Micro Polar Fluid Flow Between Rough Elliptic Plates W After the integration the dimensionless load capacity can be acquired as 3 E( W) H a 3 = = () µ ( dh dt) a b Ra + E{ G ( H N L M )} The dimensionless approaching time can be obtained as T t t E( W ) H dt a 3 = = dh µa b Ra + E H { G ( H N L M )} () 3. RESULTS AND DISCUSSION The study of significant effects of surface roughness on the micro polar squeeze film features between two elliptic plates with hydro magnetic aspect under the application of an external magnetic field are investigated. The squeeze film features of bearing such as fluid film pressure load carrying capacity squeeze time are procured as functions of dimensionless roughness C Hartmann number M aspect ratio a coupling number N film thickness H and fluid gap number L. In the limiting case when surface is smooth (C )an excellent agreement has been found between the present study results and study of Brinda and Hanumagowda [5]the numerical values of present analysis and Brinda Halambi and Hanumagowda [5] is listed for various values of L under both magnetic and nonmagnetic case in table.. Also in table. a comparative study of Roopa et.al [] with existing result is listed for various values of N under both smooth (C ) and nonmagnetic (M ) case and an excellent accuracy has been observed. Calculations of the work are taken for following numerical values. C = to.5 M = to 8 a = and N = to.4 L = to.3 and H =.5 to editor@iaeme.com

8 Brinda Halambi and Hanumagowda B. N 3.. Squeeze Film Pressure Figure. depict dimensionless pressure P variation with x for distinct values of C and M. It is observed that an increase in surface roughness and magnetic field increases the film pressure. This is because of applied magnetic field and surface asperities which minimizes the velocity of lubricant and sidewise leakage of the fluid respectively.figure.3 represents non-dimensional pressure P variation with x for distinct values of a and N. It is noticed that larger values of aspect ratio a decreases film pressure and larger values of coupling number N increases the pressure of fluid film. Figure.4 represents dimensionless pressure P variation with x for distinct values of H and L. It is found that impact of surface roughness and magnetic field provides a larger fluid film pressure for smaller values of film height H editor@iaeme.com

9 Effects of Surface Roughness with Mhd on the Micro Polar Fluid Flow Between Rough Elliptic Plates 3.. Load Carrying Capacity Figure.5 signifies the dimensionless load carrying capacity W variation with aspect ratio log ( a ) for different values of C and L. It is noticed that load carrying capacity in film region increases as C and L increases. The dimensionless load capacity W variation with aspect ratio log ( a ) for distinct values of H and C is shown in figure.6. The effect of surface roughness provides an increase in load carrying capacity for smaller values of H. Figure.7 expresses dimensionless load capacity W with log ( a ) for distinct values of M and C. It is noticed that there exits a critical value of the aspect ratio a c of a at which effects of roughness vanishes. When a > ac the load carrying capacity increases and when a < a c the load carrying capacity decreases. Also it is observed that increase in Hartmann number increases load carrying capacity editor@iaeme.com

10 Brinda Halambi and Hanumagowda B. N 3.3. Squeeze Film Time Figure.8 gives the dimensionless time T variation with film height H for distinct values of C and M. It is noted that existence of roughness and applied magnetic field increases the squeezing time compared to non-conducting lubricant case. Figure.9 represents dimensionless time T variation with film height H for distinct values of a and N. It is shown that increase in aspect ratio a reduces time of approach. Further higher values of coupling parameter gives a much delayed time of approach.figure. shows non-dimensional time T variation with film height Hfor distinct values of C and L. It is noticed that squeeze time of approach is delayed with increasing values of fluid gap parameter L and also as C increases the time of approach delays compared to smooth case editor@iaeme.com

11 Effects of Surface Roughness with Mhd on the Micro Polar Fluid Flow Between Rough Elliptic Plates 4. CONCLUSIONS On the basis of stochastic model of Christensen together with the hydro magnetic flow theory of Cowling and micro-continuum theory of Eringen the impact of surface roughness and hydro magnetic aspect on micro polar squeeze film between two elliptic plates under the application of an external magnetic field is examined and as a result below conclusions are drawn. Surface roughness has substantial influence on fluid squeeze film pressure squeeze load carrying capacity and squeeze time as compared to smooth case. The impact of micro polar fluid is to enhance fluid squeeze film pressure squeeze load carrying capacity and squeeze time of approach as compared to corresponding Newtonian case editor@iaeme.com

12 Brinda Halambi and Hanumagowda B. N The squeeze film pressure squeeze film load carrying capacity and squeeze time of approach are more for larger values of Hartmann number fluid gap number and coupling number. An excellent agreement has been found between the present study and study of Brinda and Hanumagowda [5] for smooth case (C ) and also with that of Roopa et.al [] under both smooth (C ) and nonmagnetic (M ) case respectively. Table Comparison between Brinda Halambi and Hanumagowda [5] and present analysis for various values of L when N =.3 H =.5 and a=. Table. Comparison between Roopa et.al [] and existing study for various values of a and N with L =.3 and H = editor@iaeme.com

13 Effects of Surface Roughness with Mhd on the Micro Polar Fluid Flow Between Rough Elliptic Plates REFERENCES: [] Davies M. G. The Generation of Pressure between Rough Fluid Lubricated Moving Deformable Surfaces. Lubrication Engineering pp.46. [] Burton R. A. Effects of Two Dimensional Sinusoidal Roughness on the Load Support Characteristics of a Lubricant Film. J. Basic engineering Trans ASME 85() 963 pp.46. [3] Tzeng S. T. and Saibel E. Surface Roughness Effect on Slider Bearing Lubrication. ASLE Trans 967 pp.334. [4] Chirstensen H. Stochastic Models for Hydrodynamic Lubrication of Rough Surfaces. Proc. Inst. Mech. Eng. JJ Eng. Tribol. 84 (55) pp.3-. [5] Chirstensen H. and Tonder K. The Hydrodynamic Lubrication of Rough Journal Bearing Surfaces of Finite Width. Journal of Lubrication Technology Trans ASME Series F pp.34. [6] Chirstensen H. and Tonder K. The Hydrodynamic Lubrication of Rough Journal Bearing. Journal of Lubrication Technology Trans ASME Series F pp.66. [7] Tonder K. and Chirstensen H. Waviness and Roughness in Hydrodynamic Lubrication. Proc Inst Mech Engrs Tribology Groups pp. 87. [8] Chirstensen H. A Theory of Mixed Lubrication. Proc. Instn. Mech. Engrs. Tribology Group pp.4. [9] Naduvinamani N.B. Hiremath P.S. and Gurubasvaraj G. Effect of Surface Roughness on the Couple Stress Squeeze Film between a Sphere and Flat Plate. Tribology Internation 38 5 pp [] Rajani C.B. Hanumagowda B.N. and Vijayalaxmi S. Shigehalli. Effects of Surface Roughness on Conical Squeeze Film Bearing with Micropolar Fluid. IOP Conf. Series Journal of Physics 8. [] Cowling T.G. Magneto Hydrodynamics. Inter Science Publishers New York. [] Hamza A. The Magneto Hydrodynamic Squeeze Film. J Tribol Technical Briefs 988 pp [3] Lin J.R. Magneto-Hydrodynamic Squeeze Film Characteristics for finite Rectangular Plates. Ind. Lubr. Tribol 55() 3 pp [4] Naduvinamani N.B. Hanumagowda B.N. and Siddangouda A. Effects of Surface Roughness on Magnto Hydrodynamic Couple-Stress Squeeze Film Lubrication between Circular Stepped Plates. Lubrication Science 4 pp [5] Brinda Halambi and Hanumagowda B.N. Effect of Hydromagnetic Squeeze Film Lubrication of Micropolar Fluid between two Elliptical Plates. International Journal of Mechanical Engineering and Technology 9 8 pp [6] Eringen A.C. Theory of Micropolar Fluids. Journal of Mathematics and Mechanics pp.-6. [7] Zaheeruddin K.H. and Isa M. Characteristics of Micropolar Lubricant in a Squeeze Film Porous Spherical Bearing pp.-. [8] Naduvinamani N.B. and Santhosh S. Micropolar Squeeze Film Lubrication of Finite Porous Journal Bearing. Tribology International 44 pp [9] Tsai-Wang Huang and Chen-I Wehg and Chao-Kuang Chen. Analysis of Finite Width Journal Bearings with Micropolar Fluids. Wear 4 99 pp [] Siddangouda A. Biradar T.V and Naduvinamani N.B. Combined Effect of Micro Polarity and Surface Roughness on the Hydrodynamic Lubrication of Slider Bearing. J Braz. Soc. Mech. Sci. Eng. 3. [] Roopa Rajashekar Anagod. Hanumagowda B.N. and Santhosh Kumar J. Effect of Micropolar Fluids on the Sqeeze Film Elliptical Plates. IOP Conf. Series Journal of Physics 8 pp editor@iaeme.com