Effect of viscous dissipation on mhd flow of a free convection power law fluid with a pressure
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    Effect of viscous dissipation on mhd flow of a free convection power law fluid with a pressure Effect of viscous dissipation on mhd flow of a free convection power law fluid with a pressure Document Transcript

    • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME302EFFECT OF VISCOUS DISSIPATION ON MHD FLOW OF A FREECONVECTION POWER-LAW FLUID WITH A PRESSURE GRADIENT1M N Raja ShekarDepartment of Mathematics,JNTUH College of Engineering,Nachupally, Karimnagar,2Shaik Magbul HussainProfessor,Dept. of Mechanical engineering.Royal Institute of technology and Science.Chevella.India.ABSTRACTThe paper deals with the study of steady two-dimensional flow of a electrically conductingpower-law fluid past a flat plate in the presence of transverse magnetic field under the influence of apressure gradient by considering viscous dissipation effects is studied. The resulting governing partialdifferential equations are transformed into set of non linear ordinary differential equations usingappropriate transformation. The set of non linear ordinary differential equations are first linearized byusing Quasi-linearization technique and then solved numerically by using implicit finite differencescheme. The system of algebraic equations is solved by using Gauss-Seidal iterative method. Theenergy equation for a special case for which similarity solution exist is also considered. The specialinterest is the effects of the power-law index, magnetic parameter, viscous dissipation and generalizedprandtl number on the velocity and temperature profiles. Numerical results are tabulated for skinfriction co-efficient. Velocity and Temperature profiles are drawn for different controlling parameterswhich reveal the tendency of the solution.Key words: Non-Newtonian fluids, Magnetic field effects, Prandtl number, Quasi-linearization, finitedifference method and viscous dissipation.INTRODUCTIONA non-Newtonian fluid is a fluid in which the viscosity changes with applied strain rate. As aresult non-Newtonian fluids may not have well defined viscosity. In modern technology and inindustrial applications, non-Newtonian fluids play an important role. Many processes in moderntechnology use non-Newtonian fluids as working fluids in heat exchangers. Heat transfercharacteristics of these fluids have been studied widely during the past decades due to the growing useof these non-Newtonian substances in various manufacturing and processing industries. In the recentINTERNATIONAL JOURNAL OF ADVANCED RESEARCH INENGINEERING AND TECHNOLOGY (IJARET)ISSN 0976 - 6480 (Print)ISSN 0976 - 6499 (Online)Volume 4, Issue 3, April 2013, pp. 302-307© IAEME: www.iaeme.com/ijaret.aspJournal Impact Factor (2013): 5.8376 (Calculated by GISI)www.jifactor.comIJARET© I A E M E
    • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME303years, the non-Newtonian fluids find an increasing applications in industries such as the flow ofnuclear fuel slurries, liquid metal and alloys, plasma and mercury, lubrication with heavy oils andgreases, coating of papers, polymer extrusion, continuous stretching of plastic films and artificialfibres and many others. During the past three decades there have been extensive research works onvarious aspects of non-Newtonian power-law fluids over bodies of different shapes which aredocumented in books by Skelland [1], Irvine and Karni [2] and others have presented an excellentreview of non-Newtonian fluids. Many flow problems, both internal and external flows, have beeninvestigated. The important concept of boundary layer was applied to power-law fluids by Schowalter[3].Similarity solutions were obtained by Kapur and Srivastava [4], Lee and Ames [5], Acrivoset.al [6], Berkoveskii [7], Hansen and Na [8] and others. Rao [9] investigated momentum and heattransfer phenomena on a continuous moving surface in power-law fluid. Pop et.al [10] considered thesteady laminar forced convection boundary layer of power-law non-Newtonian fluids on a continuallymoving cylinder with the surface maintained at a uniform temperature or uniform heat flux. Ananalysis of steady laminar forced convection heat transfer from a moving or stationary slendercylinder to a quiescent or flowing non-Newtonian fluid is presented by Tian-Yih Wang [11]. Agarwalet.al [12] presented the laminar momentum and thermal boundary layers of power-law fluids over aslender cylinder.Thomson and Snuder [13-14] studied the effect of injection on the flow of power-lawfluid over a flat plate. Liu [15] presented a class of asymptotic solutions for the flow of power-lawfluids over a flat plate with suction.In recent years, the non-Newtonian fluids in the presence of a magnetic field find increasingapplications in many areas such as chemical engineering, electromagnetic propulsion, nuclear reactorsetc. Sarpakaya [16] has given many possible applications of non-newtonian fluids in various fields.The flow of non-Newtonian power-law fluids in the presence of a magnetic field over two-dimensional bodies was investigated by Sarpakaya[16], Sapunkov [17], Vujanovic et.al [18] andDjukic [19], [20]. Anderson et.al [21] have studied the MHD flow of a non-Newtonian power-lawfluid over a stretching sheet in an ambient fluid. The effect of MHD heat transfer to non-Newtonianpower-law fluids flowing over a wedge in the presence of magnetic field taking into considerationviscous dissipation is studied by Kishan and Amrutha [22].Recently the steady two-dimensional incompressible flow of a conducting power-law fluid past a flatplate in the presence of a transverse magnetic field and under the influence of a pressure gradient wasstudied by T. C. Chiam [23].In this paper we have investigated the effect of viscous dissipation on the steady two-dimensional flow of a conducting power-law fluid past a flat plate in the presence of transversemagnetic field under the influence of a pressure gradient. The similarity transformations were appliedto partial differential equations governing the flow and heat transfer under boundary layerapproximations to transform the non-Newtonian two-dimensional steady boundary layer equationsinto non-linear ordinary differential equations system. Numerical solutions of out coming non-linearordinary differential equations are found by using an implicit finite difference method.FORMULATION OF THE PROBLEM:As in Sapunkov[17], we consider a steady two-dimensional incompressible conductivepower-law fluid flow past a semi-infinite flat plate under the influence of a pressure gradient and inthe presence of a transverse magnetic field with magnetic field intensity H. The continuity andmomentum equations are,0=∂∂+∂∂yvxu----- (1),)( 221Huuyuyuykdxduuyuvxuun−+∂∂∂∂∂∂+=∂∂+∂∂∞−∞∞ρσµρ----- (2)
    • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME304where x and y are cartesian coordinates along and normal to the body surface respectively, and u, vare the corresponding velocity components, k is the fluid consistency index, n the flow behaviorindex, ρ the fluid density, µ is the magnetic permeability and σ is the electrical conductivity of thefluid. The external velocity distribution u∞ is given byu∞= cxm, ----- (3)Where c and m are constants. The magnetic Reynolds number is assumed small.Sapunkov[17] has shown that similarity solutions exist if H = H0 x(m-1)/2.METHOD OF SOLUTIONSolution for Momentum equation:We shall transform equation (2) into a ordinary differential equation amenable to a numericalsolution by introducing a similarity variable η and a stream function ψ as follows:,21ααη yx= And )()1(1ηαψ fxc nr += ----- (4)Where r = (2n-1)m+1,1121)1(1)12( +−++−=nnnnnmkcρα ,11)2(2+−−=nnmαWhere the dimensionless stream function f satisfies the continuity equation withyu∂∂=ψandxv∂∂−=ψ.Under the transformation (4) the differential equations (2) is reduce to0)1()1( 21=′−+′−+′′+′′′′′−fMfffffnβ ----- (5)Where β=m(1+n)/r represents the flow behavior andM = σµ2H02(1+n)/cρr is the magnetic field parameter.Subject to the boundary conditions1)(0)0(,0)0( =∞′=′= fandff . ----- (6)Equation (5) has been solved numerically. As this system of equation is highly non-linear. We haveapplied Quasi-Linearization technique to linearize this system. This method converts the non-lineartwo-point boundary value problem into an iterative scheme of solution. This method is discussed indetail by Bellman and Kalaba [24]. This technique has been used successfully by my authors for thesolution of the Falkner-Skan type equations. Applying this technique to equation (5) we obtain0)1())(21(][}{ 2111=′−+′+′′−+′′−′′+′′+′′′′′−′′′′′+′′′′′−−−fMFfFFFFffFFFFffFnnnβ----- (7)Where F is assumed to be a known function and the above equation can be rewritten as
    • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME3051254310 ][][][][][][−′′−=+′+′′+′′′nfiAiAfiAfiAfiAfiA ----- (8)Where10 ][−′′=nFiA , FiA =][1 , FiA ′′′=][2 ,,2][3 MFiA −′−= β FiA ′′=][4 , MFFFFFiA n−′−−′′+′′′′′= − 215 )(][][ ββ ,Using implicit finite difference formulae, the equations (8)is transformed toC0 [i] f [i+2] + C1[i] f [i+1] + C2[i] f [i] + C3 [i] f [i-1] = C4[i] -----(9)Where C0[i] = 2A0[i], C1[i] = -6A0[i]+2hA1[i]+h2A3[i],C2[i] = -6A0[i]-4hA1[i]+2h3A3[i], C3[i] = -2A0[i]+2hA1[i]-h2A3[i],And ( )12534 ][][2][−′′−=nFiAiAhiCHere ‘h’ represents the mesh size in η direction. The transformed equation (9) is solvedunder the boundary conditions (6) by Gauss-Seidel iteration method and computations were carriedout by using C programming. The numerical solutions of ƒ are considered as (n+1)thorder iterativesolutions and F are the nthorder iterative solutions. After each cycle of iteration the convergencecheck is performed, and the process is terminated when 610fF −<− .Solution of the energy equation:By considering the viscous dissipation effects into account, the energy equation takes thesimplified form as :,122 +∂∂+∂∂=∂∂+∂∂nyuyTKyTvxTu µ ----- (10)Using∞−−=TTTTwwθThis equation can be transformed into the form,122 +∞ ∂∂−+∂∂=∂∂+∂∂nw yuTTyKyvxuµθθθ----- (11)Where T∞ is the uniform temperature of the free stream and Tw is the temperature at the wall. Weassume that the wall is isothermal (i.e Tw = constant). Lee and Ames [5] have shown that equation(11) possesses similarity solutions only for β = 0.5, which is the flow past a right-angled wedge whenthere is no imposed magnetic field. In this case, energy equation reduces to the ordinary differentialequation:02323Pr1 11111=′′+′+′′++−+−nnnnnfEcfθθ -----(12)Where the Prandtl number and Eckert are defined as)!()1(3121Pr +−+= nnncnkK ρAnd)TT(knxcEcw322∞−ρµ=And f is the solution of equation (05).The boundary conditions for equation (12) are.0)(1)0( =∞= θθ and ----- (13)
    • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME306Now, equation (12) can be expressed in the simplified form as0][][][ 210 =+′+′′ iBiBiB θθ ----- (14)WherePr1][0 =iB , fiBnn+−=11123][ and111223][++−′′=nnnfEciBTo solve the equation (10), apply the implicit finite difference scheme to the transformed equation(14) for obtaining0][]1[][][][]1[][ 3210 =+−+++ iDiiDiiDiiD θθθ -----(15)Where D0[i] = 2B0[i] + hB1[i], D1[i] = -4B0[i],D2[i] = 2B0[i] – hB1[i] and D3[i] = 2h2B2[i]Here ‘h’ represents the mesh size in η direction. Equation (14) is solved under the boundaryconditions (13) by Gauss-Seidel iteration method and computations were carried out by using Cprogramming.RESULTS AND DISCUSSIONSThe parametric study is performed to explore the effect of magnetic field parameter M on thevelocity profiles for different values of the flow behavior parameter β for both the pseudo plastic anddilatant fluids. And the effect of prandtl number Pr and Eckert number Ec on the temperature profilesfor various values of magnetic field parameter M is studied.The values of the skin friction coefficient )0(f ′′ for various values of magnetic fieldparameter M and flow behavior parameter β are tabulated in tables 1 and 2 for pseudo plastic anddilatant fluids respectively. It noticed that the skin friction coefficient )0(f ′′ increases with theincrease in the magnetic field parameter M for fixed value of β. For a constant magnetic fieldparameter M the skin friction coefficient )0(f ′′ increases with the increase of β for both pseudoplastic (n=0.5) and dilatant fluids (n=1.5). It can also be noticed that the skin friction coefficient)0(f ′′ increases as the power-law index n increases for a constant β value when there is no magneticfield, while in the presence of a magnetic field it decreases with the increase in power-law index n fora constant β value.The effect of magnetic field parameter M on the velocity profiles of pseudo plastic fluids(n=0.5) for β=0 (flat plate flow), β=0.5 and β=1 (stagnation point flow) are shown in figure 1. It isevident from these figures that the velocity profiles f ′increases with the increases with the increase ofmagnetic field parameter M. For dilatant fluids (n=1.5) the effect of magnetic field parameter Maccelerates the velocity profiles f ′in all cases i.e (a) for flat plate flow (β=0.0), (b) for flow withβ=0.5 and (c) for stagnation point flow (β=1.0) which is shown in figure 2.The set of temperature profiles are presented in the figures 3 and 4. It can be seen from figure3 that for a given n and β, the magnetic field decreases the thickness of thermal boundary layer fordifferent values of Pr. The effect of magnetic field parameter M on the temperature profiles is more incase of pseudo plastic fluid (n=0.5) than that of dilatant fluid (n=1.5). It is also noticed that thetemperature profiles decreases with the increase of Prandtl number Pr.The effect of viscous dissipation on the temperature profiles is shown in the figure 4. The viscousdissipation effects is to accelerates the temperature profiles rapidly for flow with β=0.5 in both thecases of pseudo plastic (n=0.5) and dilatant fluids (n=1.5). The viscous dissipation effect is more incase of dilatant fluid when compared to pseudo plastic fluids.
    • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME307REFERENCES[1] Skelland, A.H.D., 1967, Non-Newtonian flow and heat transfer, John Wiley, New York.[2] Irvine Jr., T.F., and Karni, J., 1987, “Non-Newtonian flow and heat transfer,” in : S. Kakac, R.K.Shah, W. Aung (Eds.), Handbook of Single-Phase Convective Heat transfer, John Wiley, NewYork, chapter 20, 20.1 - 20.57.[3] Schowalter, W.R., 1960, “The application of boundary-layer theory to power-law pseudo plasticfluids: similar solutions,” A.I.Ch.E. J., 6, pp. 24-28.[4] Kapur, J.N., Srisvastava, R.C, 1963, “Similar solutions of the boundary layer equations for power-law fluids,” ZAMP, 14, pp.383-388.[5] Lee, S. Y., Ames, W.F., 1960, “Similarity solutions for non-Newtonian fluids,” A.I.Ch.E. J., 12,pp. 700-708.[6] Acrivos, A., Shah, M. J. and Petersen, E. E., 1960, “Momentum and heat transfer in laminarboundary layer flows of non-Newtonian fluids past external surfaces,” AIChE. J., 6, pp. 312-317.[7] Berkovskii, B. M., 1966, “A class of self-similar boundary layer problems for rheological power-law fluids,” Int. Chem. Eng., 6, pp.187-201.[8] Hansen, A. G. and Na, R. Y., 1968, “Similarity solutions of laminar incompressible boundarylayer equations of non-Newtonian fluids,” Trans. ASME. J., Basic Eng., 40, pp.71-74.[9] Rao, J. H., Jeng, D. R. and De Witt, K. J., 1997, “Momentum and heat transfer on a continuousmoving surface in a power-law fluid,” Int. J. Heat Mass Transfer, 40, pp. 1853-1861.[10] Pop, I., Kumari, M., Nath, G., 1990, “Non-Newtonian boundary layer on a moving cylinder,” Int JEngineering science, 28, pp. 303-312.[11] Tian-Yih Wang, 1996, “Convective heat transfer between a moving cylinder and flowing non-Newtonian fluids,” Int communication Heat mass transfer, 23, pp. 101-114.[12] Agarwal, M., Chhabra, R. P., Eswaran, V., 2002, “Laminar momentum and thermal boundarylayers of power-law fluids over a slender cylinder,” Chem Engng Sci, 57, pp. 1331-1341.[13] Thomson, E. R. and Snuder, W. T., 1968, “Drag reduction of a non-Newtonian fluid by fluidinjection at the wall,” J. Hydronaut., 2, pp. 177-180.[14] Thomson E. R. and Snuder, W. T., 1970, “Laminar boundary-layer flows of Newtonian fluidswith non-Newtonian fluid injections,” J. Hydronaut., 4, pp. 86-91.[15] Liu, C. Y., 1973, “ Asymptotic suction flow of power-law fluids,” J. Hydronaut., 7, pp. 135-136.[16] Sarpakaya, T., 1961, “Flow of non-Newtonian fluids in a magnetic field,” AIChE J., 7, pp. 324-328.[17] Sapunkov, Ya. G., 1967,“Self-similar solutions of non-Newtonian fluid boundary layer in MHD”Fluid dynamics 2, pp. 42-47.[18] Vujanovic, B., Strauss, A.M, Djukic, D., 1972, ”A variational solution of Rayleigh problem forpower-law non-Newtonian conductive fluid,” Ing. Arch., 41, pp. 381-386.[19] Djukic, D.S., 1973,“On the use of Crocco equation for the flow of power-law fluids in a transversemagnetic field AIChE J., 19, pp. 1159-1163 (1973).[20] Djukic, D. S., 1974, “Hiemenz magnetic flow of power-law fluids. J. Appl. Mech., 4, pp. 822-823.[21] Anderson, H.I., Bach, K. H., Dandapat, B.S., 1992, “Magneto hydrodynamic flow of a power-lawfluid over a stretching sheet,” Int. J. Non-Linear Mech., 27, pp. 929-936.[22] Kishan, N., Amrutha, P., 2009, “MHD heat transfer to non-Newtonian power-law fluids flowingover a wedge with viscous dissipation,” Int. J. of applied mechanics and engineering, 14(4), pp.965-987.[23] Dr P.Ravinder Reddy, Dr K.Srihari, Dr S. Raji Reddy, “Combined Heat and Mass Transfer InMhd Three-Dimensional Porous Flow With Periodic Permeability & Heat Absorption”International Journal Of Mechanical Engineering & Technology (IJMET) Volume 3, Issue 2,2012.PP: 573 – 593, ISSN PRINT: 0976 – 6340, ISSN ONLINE: 0976 – 6359[24] Dr. Sundarammal Kesavan , M. Vidhya ,Dr. A. Govindarajan, “Unsteady Mhd Free ConvectiveFlow In A Rotating Porous Medium With Mass Transfer” International Journal Of MechanicalEngineering & Technology (IJMET) Volume 3, Issue 3,2012. PP: 214 - 228, ISSN PRINT: 0976 –6340, ISSN ONLINE: 0976 – 6359