Ferrofluid Lubrication equation for non-isotropic porous squeeze film bearing with slip velocity. Rajesh C. Shah * M.M.Parsania
|
|
- Elfrieda Stevenson
- 5 years ago
- Views:
Transcription
1 Matematics Today Vol.28(June-Dec-212)43-49 ISSN 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 dr_rcsa@yaoo.com M.M.Parsania Department of Matematics, Scool of Engineering, R.K. University, Rajkot, Gujarat State, India 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
2 44 Matematics Today Vol.28(June-Dec-212) 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.
3 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
4 46 Matematics Today Vol.28(June-Dec-212) 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 ) u 2 s 1 z z=. (2.18) Using (2.17) and (2.18), equation (2.16)becomes
5 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
6 48 Matematics Today Vol.28(June-Dec-212) 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, [2] Sina, P. C., Gupta, J. L.,(1973) :Hydromagnetic squeeze films between porous rectangular plates,j LubrTecnol, F95, [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, [5] Bat, M. V.,(1978) :Hydrodynamic Lubrication of a Porous Composite Slider Bearing,Jap J App Pys, 17, [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, [7] Morgan, V. T., Cameron, A.,(1957) :Mecanism of Lubrication in Porous MetalBearings,ProcConf Lubrication and Wear, Inst Mec Engrs, London, paper 89, [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, [1] Verma, P. D.S.,(1986) : Magnetic fluid-based squeeze film,int J Eng Sci, 24, [11] Bat, M. V., Deeri, G. M.,(1991) :Porous composite slider bearing lubricated wit magnetic fluid,jpn J App Pys, 3, [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)
7 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, [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), [15] Sparrow, E. M., Beavers, G. S., Hwang, I. T., (1972) : Effect of velocity slip on porous walled squeeze films,trans ASME, F 94, [16] Prakas, J., Vij, S. K., (1976) :Effect of velocity slip on te squeeze film between rotating porous annular discs,wear, 38, [17] Bat, M. V., Deeri, G. M., (1994) :Optimum Film Tickness in a Porous Slider Bearing Considering Slip Velocity, Mat Today, 12, 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
Ferrofluid lubrication equation for porous bearings considering anisotropic permeability and slip velocity
Indianlournal of Engineering & Materials Sciences Vol. 10, August 2003, pp. 277-281 Ferrofluid lubrication equation for porous bearings considering anisotropic permeability and slip velocity Rajesh C Shah"
More informationSLIP VELOCITY ON THE FERROFLUID LUBRICATION OF THE POROUS EXPONENTIAL SLIDER BEARING
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 http://www.iaeme.com/ijaret/issues.asp?jtype=ijaret&vtype=9&itype=3
More informationEFFECT OF SLIP VELOCITY ON MAGNETIC FLUID LUBRICATION OF ROUGH POROUS RAYLEIGH STEP BEARING.
Journal of Mechanical Engineering and Sciences (JMES) ISSN (Print): 2289-4659; e-issn: 2231-8380; Volume 4, pp. 532-547, June 2013 Universiti Malaysia Pahang, Pekan, Pahang, Malaysia DOI: http://dx.doi.org/10.15282/jmes.4.2013.17.0050
More informationRajesh C. Shah 1,*, Dilip B. Patel 2
Appied Matematics (5): 76-83 DOI:.593/j.am.5.5 Matematica Modeing of newy Designed Ferrofuid Based Sider Bearing Incuding Effects of Porosity Anisotropic Permeabiity Sip Veocity at bot te Ends Squeeze
More informationAnalysis of Static and Dynamic Load on Hydrostatic Bearing with Variable Viscosity and Pressure
Indian Journal of Science and Tecnology Supplementary Article Analysis of Static and Dynamic Load on Hydrostatic Bearing wit Variable Viscosity and Pressure V. Srinivasan* Professor, Scool of Mecanical
More informationFerrofluid Based Squeeze Film Lubrication between Rough Stepped Plates with Couple Stress Effect
Journal of Applied Fluid Mechanics, Vol. 11, No., pp. 597-61, 018. Available online at www.jafmonline.net, ISSN 175-57, EISSN 175-645. DOI: 10.18869/acadpub.jafm.7.46.7854 Ferrofluid Based Squeeze Film
More informationPrediction of Coating Thickness
Prediction of Coating Tickness Jon D. Wind Surface Penomena CE 385M 4 May 1 Introduction Tis project involves te modeling of te coating of metal plates wit a viscous liquid by pulling te plate vertically
More informationHYDROMAGNETIC BOUNDARY LAYER MICROPOLAR FLUID FLOW OVER A STRETCHING SURFACE EMBEDDED IN A NON-DARCIAN POROUS MEDIUM WITH RADIATION
HYDROMAGNETIC BOUNDARY LAYER MICROPOLAR FLUID FLOW OVER A STRETCHING SURFACE EMBEDDED IN A NON-DARCIAN POROUS MEDIUM WITH RADIATION MOSTAFA A. A. MAHMOUD, MAHMOUD ABD-ELATY MAHMOUD, AND SHIMAA E. WAHEED
More informationFerrofluid Lubrication of a Rough Porous Secant-Shaped Slider Bearing with Slip Velocity
Journal of the Serbian Society for Computational Mechanics / Vol. / No., 7 / pp. 69-8 (DOI:.4874/jsscm.7...7) Ferrofluid Lubrication of a Rough Porous Secant-Shaped Slider Bearing with Slip Velocity J.
More informationFlow of a Rarefied Gas between Parallel and Almost Parallel Plates
Flow of a Rarefied Gas between Parallel and Almost Parallel Plates Carlo Cercignani, Maria Lampis and Silvia Lorenzani Dipartimento di Matematica, Politecnico di Milano, Milano, Italy 033 Abstract. Rarefied
More informationOn the absence of marginal pinching in thin free films
On te absence of marginal pincing in tin free films P. D. Howell and H. A. Stone 6 August 00 Abstract Tis paper concerns te drainage of a tin liquid lamella into a Plateau border. Many models for draining
More informationInfluence of magnetic fluid through a series of flow factors on the performance of a longitudinally rough finite slider bearing
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 1 (2016), pp. 783 796 Research India Publications http://www.ripublication.com/gjpam.htm Influence of magnetic fluid through
More informationStudy of Convective Heat Transfer through Micro Channels with Different Configurations
International Journal of Current Engineering and Tecnology E-ISSN 2277 4106, P-ISSN 2347 5161 2016 INPRESSCO, All Rigts Reserved Available at ttp://inpressco.com/category/ijcet Researc Article Study of
More informationTheoretical Analysis of Flow Characteristics and Bearing Load for Mass-produced External Gear Pump
TECHNICAL PAPE Teoretical Analysis of Flow Caracteristics and Bearing Load for Mass-produced External Gear Pump N. YOSHIDA Tis paper presents teoretical equations for calculating pump flow rate and bearing
More informationComment on Experimental observations of saltwater up-coning
1 Comment on Experimental observations of saltwater up-coning H. Zang 1,, D.A. Barry 2 and G.C. Hocking 3 1 Griffit Scool of Engineering, Griffit University, Gold Coast Campus, QLD 4222, Australia. Tel.:
More informationLIMITS AND DERIVATIVES CONDITIONS FOR THE EXISTENCE OF A LIMIT
LIMITS AND DERIVATIVES Te limit of a function is defined as te value of y tat te curve approaces, as x approaces a particular value. Te limit of f (x) as x approaces a is written as f (x) approaces, as
More informationEffects of Radiation on Unsteady Couette Flow between Two Vertical Parallel Plates with Ramped Wall Temperature
Volume 39 No. February 01 Effects of Radiation on Unsteady Couette Flow between Two Vertical Parallel Plates wit Ramped Wall Temperature S. Das Department of Matematics University of Gour Banga Malda 73
More informationResearch Article Dynamic Performance Characteristics of a Curved Slider Bearing Operating with Ferrofluids
Advances in Tribology Volume 22 Article ID 278723 6 pages doi:.55/22/278723 Research Article Dynamic Performance Characteristics of a Curved Slider Bearing Operating with Ferrofluids Udaya P. Singh andr.s.gupta
More informationInertia Effects in Rheodynamic Lubrication of an Externally Pressurized Converging Thrust Bearing using Bingham Fluids
Journal of Applied Fluid Mecanics, Vol. 1, No., pp. 587-594, 19. Available online at www.jafmonline.net, ISSN 175-645, EISSN 175-645. DOI: 1.18869/acadpub.jafm.75.54.8914 Inertia Effects in Reodynamic
More informationINTRODUCTION DEFINITION OF FLUID. U p F FLUID IS A SUBSTANCE THAT CAN NOT SUPPORT SHEAR FORCES OF ANY MAGNITUDE WITHOUT CONTINUOUS DEFORMATION
INTRODUCTION DEFINITION OF FLUID plate solid F at t = 0 t > 0 = F/A plate U p F fluid t 0 t 1 t 2 t 3 FLUID IS A SUBSTANCE THAT CAN NOT SUPPORT SHEAR FORCES OF ANY MAGNITUDE WITHOUT CONTINUOUS DEFORMATION
More informationMagnetic Fluid Based Squeeze Film behavior between curved circular Plates and Surface Roughness Effect
opyright 2009 Tech Science Press FDMP, vol.5, no.3, pp.245-259, 2009 Magnetic Fluid Based Squeeze Film behavior between curved circular Plates and Surface Roughness Effect Nikhilkumar D. Abhangi 1 and
More informationMath 31A Discussion Notes Week 4 October 20 and October 22, 2015
Mat 3A Discussion Notes Week 4 October 20 and October 22, 205 To prepare for te first midterm, we ll spend tis week working eamples resembling te various problems you ve seen so far tis term. In tese notes
More informationEffect of L/D Ratio on the Performance of a Four-Lobe Pressure Dam Bearing
Vol:, No:8, 007 Effect of L/D Ratio on te Performance of a Four-Lobe Pressure Dam Bearin G Busan, S S Rattan, and N P Meta International Science Index, Mecanical and Mecatronics Enineerin Vol:, No:8, 007
More informationEvaluation and Accurate Estimation from Petrophysical Parameters of a Reservoir
American Journal of Environmental Engineering and Science 2016; 3(2): 68-74 ttp://www.aascit.org/journal/ajees ISSN: 2381-1153 (Print); ISSN: 2381-1161 (Online) Evaluation and Accurate Estimation from
More information6. Non-uniform bending
. Non-uniform bending Introduction Definition A non-uniform bending is te case were te cross-section is not only bent but also seared. It is known from te statics tat in suc a case, te bending moment in
More informationLecture XVII. Abstract We introduce the concept of directional derivative of a scalar function and discuss its relation with the gradient operator.
Lecture XVII Abstract We introduce te concept of directional derivative of a scalar function and discuss its relation wit te gradient operator. Directional derivative and gradient Te directional derivative
More informationFERROFLUID FLOW DUE TO A ROTATING DISK IN THE PRESENCE OF A NON-UNIFORM MAGNETIC FIELD
Int. J. of Applied Mechanics and Engineering, 2016, vol.21, No.2, pp.273-283 DOI: 10.1515/ijame-2016-0017 FERROFLUID FLOW DUE TO A ROTATING DISK IN THE PRESENCE OF A NON-UNIFORM MAGNETIC FIELD A. BHANDARI
More informationOn the modified Reynolds equation for journal bearings in a case of non-newtonian Rabinowitsch fluid model
MATEC Web of Conferences 45, 7 (8) NCTAM 7 ttps://doi.org/.5/matecconf/8457 On te modified Renolds equation for journal bearings in a case of non-newtonian Rabinowitsc fluid model Juliana Javorova,*, and
More information3. Using your answers to the two previous questions, evaluate the Mratio
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 0219 2.002 MECHANICS AND MATERIALS II HOMEWORK NO. 4 Distributed: Friday, April 2, 2004 Due: Friday,
More informationFilm thickness Hydrodynamic pressure Liquid saturation pressure or dissolved gases saturation pressure. dy. Mass flow rates per unit length
NOTES DERITION OF THE CLSSICL REYNOLDS EQTION FOR THIN FIL FLOWS Te lecture presents te derivation of te Renolds equation of classical lubrication teor. Consider a liquid flowing troug a tin film region
More informationCFD Analysis and Optimization of Heat Transfer in Double Pipe Heat Exchanger with Helical-Tap Inserts at Annulus of Inner Pipe
IOR Journal Mecanical and Civil Engineering (IOR-JMCE) e-in: 2278-1684,p-IN: 2320-334X, Volume 13, Issue 3 Ver. VII (May- Jun. 2016), PP 17-22 www.iosrjournals.org CFD Analysis and Optimization Heat Transfer
More informationMathematics 5 Worksheet 11 Geometry, Tangency, and the Derivative
Matematics 5 Workseet 11 Geometry, Tangency, and te Derivative Problem 1. Find te equation of a line wit slope m tat intersects te point (3, 9). Solution. Te equation for a line passing troug a point (x
More informationSqueeze film effect at elastohydrodynamic lubrication of plain journal bearings
Science, Juliana Engineering Javorova, & Education, Anelia Mazdrakova 1, (1), 16, 11- Squeeze film effect at elastoydrodynamic lubrication of plain journal bearings Juliana Javorova *, Anelia Mazdrakova
More informationDEVELOPMENT OF REYNOLDS EQUATION
CHAPTER DEVELOPMENT OF REYNOLDS EQUATION.1 Introduction.1.1 Inertia and Turbulent Effects in Lubrication.1. Termal effects in Lubrication.1.3 Non-Newtonian Lubricants. Matematical Modelling of a Bearing
More informationHeat Transfer and MHD Boundary Layer Flow over a Rotating Disk
J. Basic. Appl. Sci. Res., 4()37-33, 4 4, TextRoad Publication ISSN 9-434 Journal of Basic and Applied Scientific Researc www.textroad.com Heat Transfer and MHD Boundary Layer Flow over a Rotating Disk
More informationAnalysis of Two-Layered Journal Bearing Lubricated with Ferrofluid
MATEC Web of Conferences 1, 41 (14) DOI: 1.151/ matecconf/ 141 41 C Owned by the authors, published by EDP Sciences, 14 Analysis of Two-Layered Journal Bearing Lubricated with Ferrofluid T. V. V. L. N.
More informationHall Effcts Eon Unsteady MHD Free Convection Flow Over A Stretching Sheet With Variable Viscosity And Viscous Dissipation
IOSR Journal of Matematics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume, Issue 4 Ver. I (Jul - Aug. 5), PP 59-67 www.iosrjournals.org Hall Effcts Eon Unsteady MHD Free Convection Flow Over A Stretcing
More informationChapter 1 Functions and Graphs. Section 1.5 = = = 4. Check Point Exercises The slope of the line y = 3x+ 1 is 3.
Capter Functions and Graps Section. Ceck Point Exercises. Te slope of te line y x+ is. y y m( x x y ( x ( y ( x+ point-slope y x+ 6 y x+ slope-intercept. a. Write te equation in slope-intercept form: x+
More informationLecture 21. Numerical differentiation. f ( x+h) f ( x) h h
Lecture Numerical differentiation Introduction We can analytically calculate te derivative of any elementary function, so tere migt seem to be no motivation for calculating derivatives numerically. However
More informationA Modified Distributed Lagrange Multiplier/Fictitious Domain Method for Particulate Flows with Collisions
A Modified Distributed Lagrange Multiplier/Fictitious Domain Metod for Particulate Flows wit Collisions P. Sing Department of Mecanical Engineering New Jersey Institute of Tecnology University Heigts Newark,
More informationSeepage Analysis through Earth Dam Based on Finite Difference Method
J. Basic. Appl. Sci. Res., (11)111-1, 1 1, TetRoad Publication ISSN -44 Journal of Basic and Applied Scientific Researc www.tetroad.com Seepage Analysis troug Eart Dam Based on Finite Difference Metod
More informationFinding and Using Derivative The shortcuts
Calculus 1 Lia Vas Finding and Using Derivative Te sortcuts We ave seen tat te formula f f(x+) f(x) (x) = lim 0 is manageable for relatively simple functions like a linear or quadratic. For more complex
More informationA NEW INTERPRETATION OF PHOTON. Kunwar Jagdish Narain
1 A NW INTRPRTATION OF PHOTON a) b) Kunwar Jagdis Narain (Retired Professor of Pysics) Te resent interretation of oton is as: A oton = a quantum of radiation energy + energy, were te quantum of radiation
More informationQuantum Numbers and Rules
OpenStax-CNX module: m42614 1 Quantum Numbers and Rules OpenStax College Tis work is produced by OpenStax-CNX and licensed under te Creative Commons Attribution License 3.0 Abstract Dene quantum number.
More informationAvailable online at aeme.com/ijmet/issues.asp?jtype=ijmet&vtype= =9&IType=12 ISSN Print: and ISSN
International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue, December 8, pp. 98 9, Article ID: IJMET_9 9 Available online at http://www.ia aeme.com/ijmet/issues.asp?jtype=ijmet&vtype=
More informationThe structure of the atoms
Te structure of te atoms Atomos = indivisible University of Pécs, Medical Scool, Dept. Biopysics All tat exists are atoms and empty space; everyting else is merely tougt to exist. Democritus, 415 B.C.
More informationMATH CALCULUS I 2.1: Derivatives and Rates of Change
MATH 12002 - CALCULUS I 2.1: Derivatives and Rates of Cange Professor Donald L. Wite Department of Matematical Sciences Kent State University D.L. Wite (Kent State University) 1 / 1 Introduction Our main
More informationMath 34A Practice Final Solutions Fall 2007
Mat 34A Practice Final Solutions Fall 007 Problem Find te derivatives of te following functions:. f(x) = 3x + e 3x. f(x) = x + x 3. f(x) = (x + a) 4. Is te function 3t 4t t 3 increasing or decreasing wen
More informationDepartment of Statistics & Operations Research, Aligarh Muslim University, Aligarh, India
Open Journal of Optimization, 04, 3, 68-78 Publised Online December 04 in SciRes. ttp://www.scirp.org/ournal/oop ttp://dx.doi.org/0.436/oop.04.34007 Compromise Allocation for Combined Ratio Estimates of
More informationVelocity distribution in non-uniform/unsteady flows and the validity of log law
University of Wollongong Researc Online Faculty of Engineering and Information Sciences - Papers: Part A Faculty of Engineering and Information Sciences 3 Velocity distribution in non-uniform/unsteady
More informationExam in Fluid Mechanics SG2214
Exam in Fluid Mecanics G2214 Final exam for te course G2214 23/10 2008 Examiner: Anders Dalkild Te point value of eac question is given in parentesis and you need more tan 20 points to pass te course including
More informationA = h w (1) Error Analysis Physics 141
Introduction In all brances of pysical science and engineering one deals constantly wit numbers wic results more or less directly from experimental observations. Experimental observations always ave inaccuracies.
More informationModel development for the beveling of quartz crystal blanks
9t International Congress on Modelling and Simulation, Pert, Australia, 6 December 0 ttp://mssanz.org.au/modsim0 Model development for te beveling of quartz crystal blanks C. Dong a a Department of Mecanical
More informationFabric Evolution and Its Effect on Strain Localization in Sand
Fabric Evolution and Its Effect on Strain Localization in Sand Ziwei Gao and Jidong Zao Abstract Fabric anisotropy affects importantly te overall beaviour of sand including its strengt and deformation
More informationMANY scientific and engineering problems can be
A Domain Decomposition Metod using Elliptical Arc Artificial Boundary for Exterior Problems Yajun Cen, and Qikui Du Abstract In tis paper, a Diriclet-Neumann alternating metod using elliptical arc artificial
More informationLarge eddy simulation of turbulent flow downstream of a backward-facing step
Available online at www.sciencedirect.com Procedia Engineering 31 (01) 16 International Conference on Advances in Computational Modeling and Simulation Large eddy simulation of turbulent flow downstream
More informationf a h f a h h lim lim
Te Derivative Te derivative of a function f at a (denoted f a) is f a if tis it exists. An alternative way of defining f a is f a x a fa fa fx fa x a Note tat te tangent line to te grap of f at te point
More informationContinuity and Differentiability of the Trigonometric Functions
[Te basis for te following work will be te definition of te trigonometric functions as ratios of te sides of a triangle inscribed in a circle; in particular, te sine of an angle will be defined to be te
More informationUniversity Mathematics 2
University Matematics 2 1 Differentiability In tis section, we discuss te differentiability of functions. Definition 1.1 Differentiable function). Let f) be a function. We say tat f is differentiable at
More informationDistribution of reynolds shear stress in steady and unsteady flows
University of Wollongong Researc Online Faculty of Engineering and Information Sciences - Papers: Part A Faculty of Engineering and Information Sciences 13 Distribution of reynolds sear stress in steady
More informationA finite element approximation for the quasi-static Maxwell Landau Lifshitz Gilbert equations
ANZIAM J. 54 (CTAC2012) pp.c681 C698, 2013 C681 A finite element approximation for te quasi-static Maxwell Landau Lifsitz Gilbert equations Kim-Ngan Le 1 T. Tran 2 (Received 31 October 2012; revised 29
More informationNumerical Analysis MTH603. dy dt = = (0) , y n+1. We obtain yn. Therefore. and. Copyright Virtual University of Pakistan 1
Numerical Analysis MTH60 PREDICTOR CORRECTOR METHOD Te metods presented so far are called single-step metods, were we ave seen tat te computation of y at t n+ tat is y n+ requires te knowledge of y n only.
More informationFriction Coefficient s Numerical Determination for Hot Flat Steel Rolling at Low Strain Rate
American ournal of Applied Matematics 05; 3(5: -8 Publised online September 6, 05 (ttp://www.sciencepublisinggroup.com/j/ajam doi: 0.648/j.ajam.050305.3 ISS: 330-0043 (Print; ISS: 330-006X (Online riction
More informationAvailable online at ScienceDirect. Procedia Engineering 125 (2015 )
Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 15 (015 ) 1065 1069 Te 5t International Conference of Euro Asia Civil Engineering Forum (EACEF-5) Identification of aerodynamic
More informationVolume 29, Issue 3. Existence of competitive equilibrium in economies with multi-member households
Volume 29, Issue 3 Existence of competitive equilibrium in economies wit multi-member ouseolds Noriisa Sato Graduate Scool of Economics, Waseda University Abstract Tis paper focuses on te existence of
More informationVARIANCE ESTIMATION FOR COMBINED RATIO ESTIMATOR
Sankyā : Te Indian Journal of Statistics 1995, Volume 57, Series B, Pt. 1, pp. 85-92 VARIANCE ESTIMATION FOR COMBINED RATIO ESTIMATOR By SANJAY KUMAR SAXENA Central Soil and Water Conservation Researc
More informationFULL SHEAR DEFORMATION FOR ANALYSIS OF THICK PLATE
FULL SHAR DFORMATION FOR ANALYSIS OF THICK PLAT 1 IBARUGBULM, OWUS M., 2 NJOKU, KLCHI O., 3 ZIFULA, UCHCHI G. 1,2,3 Civil ngineering Department, Federal University of Tecnology, Owerri, Nigeria -mail:
More informationNumerical Analysis of Laminar flow of Viscous Fluid Between Two Porous Bounding walls
Numerical Analysis of Laminar flow of Viscous Fluid Between Two Porous Bounding walls Ramesh Yadav Department of Mathematics Babu Banarasi Das National Institute of Technology & Management Lucknow Uttar
More informationSome Results on the Growth Analysis of Entire Functions Using their Maximum Terms and Relative L -orders
Journal of Matematical Extension Vol. 10, No. 2, (2016), 59-73 ISSN: 1735-8299 URL: ttp://www.ijmex.com Some Results on te Growt Analysis of Entire Functions Using teir Maximum Terms and Relative L -orders
More informationIntroduction to Derivatives
Introduction to Derivatives 5-Minute Review: Instantaneous Rates and Tangent Slope Recall te analogy tat we developed earlier First we saw tat te secant slope of te line troug te two points (a, f (a))
More informationDevelopment and experimental verification of a numerical model for optimizing the cleaning performance of the doctor blade press roll tribosystem
Development and experimental verification of a numerical model for optimizing te cleaning performance of te doctor blade press roll tribosystem M. Rodríguez Ripoll, B. Sceicl,, D. Bianci, B. Jakab, F.
More informationSymmetry Labeling of Molecular Energies
Capter 7. Symmetry Labeling of Molecular Energies Notes: Most of te material presented in tis capter is taken from Bunker and Jensen 1998, Cap. 6, and Bunker and Jensen 2005, Cap. 7. 7.1 Hamiltonian Symmetry
More informationSimulation and verification of a plate heat exchanger with a built-in tap water accumulator
Simulation and verification of a plate eat excanger wit a built-in tap water accumulator Anders Eriksson Abstract In order to test and verify a compact brazed eat excanger (CBE wit a built-in accumulation
More informationCLOSED CONVEX SHELLS Meunargia T. Mixed forms of stress-strain relations are given in the form. λ + 2µ θ + 1
Seminar of I. Vekua Institute of Applied Matematics REPORTS, Vol. 43, 207 CLOSED CONVEX SHELLS Meunargia T. Abstract. If Ω is a closed convex sell, ten S : x 3 = 0 is an ovaloid. It is proved tat in tis
More informationCALCULATION OF COLLAPSE PRESSURE IN SHALE GAS FORMATION AND THE INFLUENCE OF FORMATION ANISOTROPY
CALCULATION OF COLLAPSE PRESSURE IN SHALE GAS FORMATION AND THE INFLUENCE OF FORMATION ANISOTROPY L.Hu, J.Deng, F.Deng, H.Lin, C.Yan, Y.Li, H.Liu, W.Cao (Cina University of Petroleum) Sale gas formations
More information232 Calculus and Structures
3 Calculus and Structures CHAPTER 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS FOR EVALUATING BEAMS Calculus and Structures 33 Copyrigt Capter 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS 17.1 THE
More informationFRICTIONAL EFFECT AND LOAD CARRYING CAPACITY IN POROUS INCLINED MULTI STEPPED COMPOSITE BEARINGS WITH COUPLE STRESS FLUIDS
VO. 1, NO. 17, SEPTEMBER 17 ISSN 1819-668 6-17 Asian Research Publishing Network (ARPN). All rights reserved. FRICTIONA EFFECT AND OAD CARRYING CAPACITY IN POROUS INCINED MUTI STEPPED COMPOSITE BEARINGS
More informationNumerical Solution to Parabolic PDE Using Implicit Finite Difference Approach
Numerical Solution to arabolic DE Using Implicit Finite Difference Approac Jon Amoa-Mensa, Francis Oene Boateng, Kwame Bonsu Department of Matematics and Statistics, Sunyani Tecnical University, Sunyani,
More informationDesalination by vacuum membrane distillation: sensitivity analysis
Separation and Purification Tecnology 33 (2003) 75/87 www.elsevier.com/locate/seppur Desalination by vacuum membrane distillation: sensitivity analysis Fawzi Banat *, Fami Abu Al-Rub, Kalid Bani-Melem
More informationEffect of the Dependent Paths in Linear Hull
1 Effect of te Dependent Pats in Linear Hull Zenli Dai, Meiqin Wang, Yue Sun Scool of Matematics, Sandong University, Jinan, 250100, Cina Key Laboratory of Cryptologic Tecnology and Information Security,
More informationThe Verlet Algorithm for Molecular Dynamics Simulations
Cemistry 380.37 Fall 2015 Dr. Jean M. Standard November 9, 2015 Te Verlet Algoritm for Molecular Dynamics Simulations Equations of motion For a many-body system consisting of N particles, Newton's classical
More informationJournal of Engineering Science and Technology Review 7 (4) (2014) 40-45
Jestr Journal of Engineering Science and Tecnology Review 7 (4) (14) -45 JOURNAL OF Engineering Science and Tecnology Review www.jestr.org Mecanics Evolution Caracteristics Analysis of in Fully-mecanized
More informationDECOMPOSITION OF RECURRENT CURVATURE TENSOR FIELDS IN A KAEHLERIAN MANIFOLD OF FIRST ORDER. Manoj Singh Bisht 1 and U.S.Negi 2
DECOMPOSITION OF RECURRENT CURVATURE TENSOR FIELDS IN A KAEHLERIAN MANIFOLD OF FIRST ORDER Manoj Sing Bist 1 and U.S.Negi 2 1, 2 Department of Matematics, H.N.B. Garwal (A Central) University, SRT Campus
More informationLecture 10: Carnot theorem
ecture 0: Carnot teorem Feb 7, 005 Equivalence of Kelvin and Clausius formulations ast time we learned tat te Second aw can be formulated in two ways. e Kelvin formulation: No process is possible wose
More informationTHE MODEL OF DRYING SESSILE DROP OF COLLOIDAL SOLUTION
THE MODEL OF DRYING SESSILE DROP OF COLLOIDAL SOLUTION I.V.Vodolazskaya, Yu.Yu.Tarasevic Astrakan State University, a Tatiscev St., Astrakan, 41456, Russia e-mail: tarasevic@aspu.ru e ave proposed and
More informationDELFT UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering, Mathematics and Computer Science
DELFT UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering, Matematics and Computer Science. ANSWERS OF THE TEST NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS (WI3097 TU) Tuesday January 9 008, 9:00-:00
More informationTHE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225
THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Mat 225 As we ave seen, te definition of derivative for a Mat 111 function g : R R and for acurveγ : R E n are te same, except for interpretation:
More informationAN ANALYSIS OF AMPLITUDE AND PERIOD OF ALTERNATING ICE LOADS ON CONICAL STRUCTURES
Ice in te Environment: Proceedings of te 1t IAHR International Symposium on Ice Dunedin, New Zealand, nd t December International Association of Hydraulic Engineering and Researc AN ANALYSIS OF AMPLITUDE
More informationProblem Solving. Problem Solving Process
Problem Solving One of te primary tasks for engineers is often solving problems. It is wat tey are, or sould be, good at. Solving engineering problems requires more tan just learning new terms, ideas and
More informationSection 3.1: Derivatives of Polynomials and Exponential Functions
Section 3.1: Derivatives of Polynomials and Exponential Functions In previous sections we developed te concept of te derivative and derivative function. Te only issue wit our definition owever is tat it
More informationWeak Solutions for Nonlinear Parabolic Equations with Variable Exponents
Communications in Matematics 25 217) 55 7 Copyrigt c 217 Te University of Ostrava 55 Weak Solutions for Nonlinear Parabolic Equations wit Variable Exponents Lingeswaran Sangerganes, Arumugam Gurusamy,
More informationNon-linear Dynamics of Inlet Film Thickness during Unsteady Rolling Process
5 CHINESE JOUNAL OF MECHANICAL ENGINEEING Vol 9aNo a16 DOI: 191/CJME161181 available online at wwwspringerlinkcom; wwwcjmenetcom Non-linear Dynamics of Inlet Film Tickness during Unsteady olling Process
More informationDifferentiation. Area of study Unit 2 Calculus
Differentiation 8VCE VCEco Area of stud Unit Calculus coverage In tis ca 8A 8B 8C 8D 8E 8F capter Introduction to limits Limits of discontinuous, rational and brid functions Differentiation using first
More informationUNIVERSITY OF SASKATCHEWAN Department of Physics and Engineering Physics
UNIVERSITY O SASKATCHEWAN Department of Pysics and Engineering Pysics Pysics 117.3 MIDTERM EXAM Regular Sitting NAME: (Last) Please Print (Given) Time: 90 minutes STUDENT NO.: LECTURE SECTION (please ceck):
More informationRevolving Ferrofluid Flow in Porous Medium with Rotating Disk
Vol:7, No:, 0 Revolving Ferrofluid Flow in Porous Medium with Rotating Disk Paras Ram, Vikas Kumar International Science Index, Aerospace and Mechanical Engineering Vol:7, No:, 0 waset.org/publication/999669
More informationMath Spring 2013 Solutions to Assignment # 3 Completion Date: Wednesday May 15, (1/z) 2 (1/z 1) 2 = lim
Mat 311 - Spring 013 Solutions to Assignment # 3 Completion Date: Wednesday May 15, 013 Question 1. [p 56, #10 (a)] 4z Use te teorem of Sec. 17 to sow tat z (z 1) = 4. We ave z 4z (z 1) = z 0 4 (1/z) (1/z
More informationVariance Estimation in Stratified Random Sampling in the Presence of Two Auxiliary Random Variables
International Journal of Science and Researc (IJSR) ISSN (Online): 39-7064 Impact Factor (0): 3.358 Variance Estimation in Stratified Random Sampling in te Presence of Two Auxiliary Random Variables Esubalew
More informationThe effects of shear stress on the lubrication performances of oil film of large-scale mill bearing
Universit of Wollongong Researc Online Facult of Engineering - Papers (Arcive) Facult of Engineering and Information Sciences 9 Te effects of sear stress on te lubrication performances of oil film of large-scale
More informationCarnot Factor of a Vapour Power Cycle with Regenerative Extraction
Journal of Modern Pysics, 2017, 8, 1795-1808 ttp://www.scirp.org/journal/jmp ISSN Online: 2153-120X ISSN Print: 2153-1196 arnot Factor of a Vapour Power ycle wit Regenerative Extraction Duparquet Alain
More informationPolynomial Interpolation
Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximatinga function fx, wose values at a set of distinct points x, x, x,, x n are known, by a polynomial P x suc
More information