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Effect of viscous dissipation on mhd flow and heat transfer of a non newtonian power-law fluid
Effect of viscous dissipation on mhd flow and heat transfer of a non newtonian power-law fluid
Effect of viscous dissipation on mhd flow and heat transfer of a non newtonian power-law fluid
Effect of viscous dissipation on mhd flow and heat transfer of a non newtonian power-law fluid
Effect of viscous dissipation on mhd flow and heat transfer of a non newtonian power-law fluid
Effect of viscous dissipation on mhd flow and heat transfer of a non newtonian power-law fluid
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Effect of viscous dissipation on mhd flow and heat transfer of a non newtonian power-law fluid

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  • 1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME296EFFECT OF VISCOUS DISSIPATION ON MHD FLOW AND HEATTRANSFER OF A NON-NEWTONIAN POWER-LAW FLUID PAST ASTRETCHING SHEET WITH SUCTION/INJECTIONM N Raja ShekarDepartment of Mathematics,JNTUH College of EngineeringNachupally,Shaik Magbul HussainProfessor, Dept. of Mechanical engineering.Royal Institute of technology and Science,Chevella.India.ABSTRACTThis paper deals with the MHD effects on convection heat transfer of an electricallyconducting, non-Newtonian power-law stretched sheet with surface heat flux by considering theviscous dissipation. The effects of suction/injection at the surface are considered. The resultinggoverning equations are transformed into non linear ordinary differential equations using appropriatetransformation. The set of non linear ordinary differential equations are first linearized by usingQuasi-linearization technique and then solved numerically by using implicit finite difference scheme.Then the system of algebraic equations is solved by using Gauss-Seidal iterative method. The solutionis found to be dependent on six governing parameters including the magnetic field parameter M, thepower-law fluid index n, the sheet velocity exponent p, the suction/blowing parameter ƒw, Eckertnumber Ec and the generalized Prandtl number Pr. Numerical results are tabulated for skin friction co-efficient and the local Nusselt number. Velocity and Temperature profiles drawn for differentcontrolling parameters reveal the tendency of the solution.Keywords: MHD, Power-law stretched sheet, suction/injection, Surface heat flux and viscousdissipation.INTRODUCTIONMost of the fluids such as molten plastics, artificial fibers, drilling of petroleum, blood andpolymer solutions are considered as non-Newtonian fluids. In modern technology and in industrialapplications, non-Newtonian fluids play an important role. Increasing emergence of non-Newtonianfluids such as molten plastic pulp, emulsions, raw materials in a great variety of industries likepetroleum and chemical processes has stimulated a considerable amount of interest in the study of theINTERNATIONAL JOURNAL OF ADVANCED RESEARCH INENGINEERING AND TECHNOLOGY (IJARET)ISSN 0976 - 6480 (Print)ISSN 0976 - 6499 (Online)Volume 4, Issue 3, April 2013, pp. 296-301© IAEME: www.iaeme.com/ijaret.aspJournal Impact Factor (2013): 5.8376 (Calculated by GISI)www.jifactor.comIJARET© I A E M E
  • 2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME297behavior of such fluids when in motion. Exact solutions of the equations of motion of non- Newtonianfluids are difficult. The difficulty arises not only due to the non-linearity but also due to increase inthe order of differential equations. Many researchers have attempted to find the exact solution of non-Newtonian fluids.The study of flow and heat transfer problems due to stretching boundary has many practicalapplications in technological processes, particularly in polymer processing systems involving drawingof fibers and films or thin sheets, etc. Sometimes the polymer sheet is stretched while it is extrudedfrom a die. Usually the sheet is pulled through the viscous liquid with desired characteristics. Themoving sheet may introduce a motion in the neighboring fluid or alternatively, the fluid may have anindependent forced convection motion which is parallel to that of the sheet. Sakiadis [1] was the firstto investigate the flow due to sheet issuing with constant speed from a slit into a fluid at rest.Schowalter [2] has introduced the concept of the boundary layer in the theory of non-Newtonianpower-law fluids. Acrivos, Shas and Petersen [3] have investigated the steady laminar flow on non-Newtonian fluids over a plate.MATHEMATICAL FORMULATIONLet us consider a steady two-dimensional flow of an incompressible, electrically conductingfluid obeying the power-law model past a permeable stretching sheet. The origin is located at the slitthrough which the sheet is drawn through the fluid medium, the x-axis is chosen along the sheet andy-axis is taken normal to it. This continuous sheet is assumed to move with a velocity according to apower-law form, i.e. = C xp, and be subject to a surface heat flux. Also, a magnetic field of strength Bis applied in the positive y-direction, which produces magnetic effect in the x-direction. The magneticReynolds number is assumed to be small so that the induced magnetic field is negligible incomparison to the applied electric field and the Hall Effect is neglected.Under the foregoing assumptions and invoking the usual boundary layer approximations, the problemis governed by the following equations:0=∂∂+∂∂yvxu(1)ρσρuByuyuyKyUvxuun 21−∂∂∂∂∂∂=∂∂+∂∂−(2)122 +∂∂+∂∂=∂∂+∂∂np yuCKyTyTvxTuρα (3)Where u and ν are the velocity components, T is the temperature, B is the magnetic field strength, K isthe consistency coefficient, n is the flow behavior index, ρ is the density, σ is the electricalconductivity and α is the thermal diffusivity. The appropriate boundary conditions are given byuw(x) = C.xp, ν = νw,kqyT w−=∂∂at y = 0, x > 0 (4)u → 0, T → T∞ as y → ∞ (5)where νw is the surface mass flux and qw is the surface heat flux. It should be noted that positive pindicates that the surface is accelerated while negative p implies that the surface is decelerated fromthe slit. Also note that positive νw is for fluid injection and negative for fluid suction at the sheetsurface.METHOD OF SOLUTIONWe shall further transform equations (2) & (3) into a set of partial differential equationsamenable to a numerical solution. For this purpose we introduce the variables
  • 3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME298yxKC nnpnn)1(]1)2([)1(12+−−+−=ρη (6)fxKC nnpnn)1(]1)12([)1(121++−+−−=ρψ (7)kxqTTwnx)1(1Re)( +∞−=φ (8)Where the dimensionless stream function f satisfies the continuity equation withyu∂∂=ψandxv∂∂−=ψ.Under the transformations (6), (7) and (8), the differential equations (2) and (3) reduce to( ) 0)(11)12( 21=′−′−′′++−+′′′′′−fMfpffnnpffn(9)011)2(11)12(Pr1 1=′′+′+−−+′++−+′′+nfEcfnnpfnnpφφφ (10)With the boundary conditions=∞=∞′−=′+−+==′0)(,0)(1)0(,1)12(1)0(,1)0(φφffnpnff w(11)Where M is the magnetic field parameter, fw is the suction/injection parameter, Pr is the generalizedPrandtl number for the power law fluid, Ec is the Eckert number and primes indicate thedifferentiation with respective to η. The parameters M, fw, Pr and Ec are defined aswuxBMρσ 2= , )1(1Re +−= nxwwwuvf ,)1(2RePr +−= nxwxuα, )1(12Re +−= nxpwwCxqkuEcWhereKxu nnwx−=2Reρis the local Reynolds number. Note here that the magnetic field strength Bshould be proportional to x to the power (p-1)/2 to eliminate the dependence of M on x, i.e.B(x)=B0x(p-1)/2where B0 is a constant. Quantities of main interest include the velocity components uand ν, the skin friction coefficient, viscous dissipation and the local Nusselt number. In terms of thenew variables, the velocity components can be expressed asfuu w′= ′+−−+++−−= +−fnnpfnnpuv nxw η11)2(11)12(Re )1(1
  • 4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME299The wall shear stress is given by)0()0(Re1)1(1201ffuyuyuKnnxwynw′′′′=∂∂∂∂=−+−=−ρτTo solve the system of transformed governing equations (9) and (10) with the boundary conditions(11), first equation (9) is linearized using the Quasi linearization technique17. Then equation (9) ischanged to0)(2{][11)12(}][][]{[ 2111=′−′−′′−′′−′′+′′++−+′′′′′−′′′′′+′′′ −−−fMFfFpFFFffFnnpFFFffFn nnn(12)Where F is assumed to be a known function and the above equation can be rewritten as1154320 ][ −′′−=+′−′′+′′′ nfAAfAfAfAfA (13)Where 10 ][ −′′= nFnA , FnA ′′′=1 , FnnpA11)12(2++−= ,,23 MFpA −′= FnnpA ′′++−=11)12(4 ,215 )(11)12(][ FpFFnnpFFnA n′−′′++−+′′′′′= −,Using implicit finite difference formulae, the equations (13) and (10) are transformed toB0 [i] f [i+2] + B1 [i] f [i+1] + B2 [i] f [i] + B3 [i] f [i-1] = B4[i] (14)D1[i] φ[i+1] + D2[i] φ[i] + D3[i] φ[i-1]+D4[i] = 0 (15)where B0[i] = 2A0[i], B1[i] = 2hA2[i] – 6A0[i] – h2A3[i],B2[i] = 6A0[i] – 4hA2[i] +2 h3A4[i] , B3[i] = 2hA2[i]+h2A3[i] –2 A0[i],B4[i] = 2h3{ A5[i] – A1[i]( F2[i] )n-1},D1[i] = hC1[i] + 2, D2[i] = 2h2C2[i] – 4, D3[i] = 2 – hC1[i],C1[i] = fnnpPr11)12(++−C2[i] = fnnp′+−−Pr11)2(Here ‘h’ represents the mesh size in η direction. The system of equations (14) & (15) are solvedunder the boundary conditions (11) 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 410−<− fF .
  • 5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME300SKIN FRICTIONThe skin friction coefficient is defined as)0()0(Re22/1)1(12ffuCnnxwwf′′′′==−+−ρτOr)0()0(2Re1)1(1ffCnnxf′′′′=−+(16)HEAT TRANSFERThe local heat transfer coefficient is)0(Re )1(1φ+∞=−=nxww kTTqhThe local Nusselt number is given by)0(Re )1(1φ+==nxxkhxNuOr)0(1Re )1(1φ=+− nxxNu (17)RESULTS AND DISCUSSIONSThe effects of various parameters on the skin friction coefficient )0(f ′′− and the Nusseltnumber )0(1 φ are displayed in the tables. The values for )0(f ′′− are tabulated for various values ofn and M in Table 1. It can be seen from the table that the value of )0(f ′′− increases as magneticfield parameter M increases and decreases as power law fluid index n increases. Table 2 shows theresults of heat transfer obtained for a Newtonian fluid in the absence of magnetic field i.e. n = 1 andM = 0. It is obvious from the table that the Nusselt number )0(1 φ increases with the increase ofPrandtl number Pr and velocity exponent p. Also the effect of viscous dissipation is to reduce thevalue of Nusselt number )0(1 φ . Table 3 lists the calculations for the flow and heat transfercharacteristics, including the sheet surface temperature )0(φ , the skin-friction co-efficient)1(1Re +nxfC and the Nusselt number )1(1Re +− nxxNu for various values of n, M and ƒw with Pr = 5and p = 0.5. It is apparent from this table that the sheet surface temperature increases with increasingthe magnetic field parameter M, but it decreases with increasing the suction/injection parameter ƒw .With all other parameters fixed the magnitude of skin-friction coefficient increases with increasing themagnetic field parameter due to the fact that the magnetic field retards the fluid motion and thusincreases this coefficient. To impose suction is to increase the skin-friction coefficient, but fluidinjection decreases it. Also, the local Nusselt number is decreased as a result of the applied magneticfield. The effect of suction ( ƒw >0 ) is found to increase the Nusselt number, whereas injection has theopposite effect. More detailed discussions about the influence of various governing parameters on thelocal Nusselt number are presented latter by making use of figs. 14-18.
  • 6. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME301REFERENCES1. B.C. Sakiadis, Boundary-layer behavior on continuous moving solid surfaces, AIChEJ 7(1961) 26-28.2. W.R. Schowalter, The application of boundary-layer theory to power-law pseudo plastic fluid:Similar solutions, AIChEJ 6(1960) 24-28.3. A. Acrivos, M.J. Shah, E.E. Petersen, Momentum and heat transfer in laminar boundary layerflows of non-Newtonian fluids past external surfaces, AIChEJ 6(1960) 312-317.4. H.I. Andersson, K.H Bech, B.S. Dandapat, Magneto hydrodynamic flow of a power-law fluid overa stretching sheet, Int J Non-Linear Mech 27(1992) 929-9365. R. Cortell, A note on magneto hydrodynamic flow of a power-law fluid over a stretching sheet,Appl Math Comput 168(2005) 557-566.6. M.A.A. Mahmoud, M.A.E. Mahmoud, Analytical solutions of hydro magnetic boundary-layer flowof a non-Newtonian power-law fluid past a continuously moving surface, Acta Mech 181(2006),83-89.7. B. Singh, C. Thakur, An exact solution of plane unsteady MHD non-Newtonian fluid flows, IndianJournal of pure and applied maths 33(7) (2002), 993-1001.8. T.C. Chiam, Hydro magnetic flow over a surface stretching with a power-law velocity, Int J EngSci 33(1995) 429-435.9. I. Pop, T.Y. Na, A note on MHD flow over a stretching permeable surface, Mech Res Comm25(1998) 263-269.10. N. Kishan, B.S. Reddy, Quasi Linear approach to MHD effects on boundary layer flow of power-law fluids past a semi infinite flat plate with thermal dispersion, International Journal of non-linear science, 11(2011) 301-311.11. M. Kumari, G. Nath, MHD boundary- layer flow of a non-Newtonian fluid over a continuouslymoving surface with a parallel free stream, Acta Mech 146(2001)139-150.12. M. Kumari, G. Nath, Transient rotating flow over a moving surface with a magnetic field, Int JHeat Mass Transfer 48(2005) 2878-2885.13. S.J. Liao, on the analytic solution of magneto hydrodynamic flows of non-Newtonian fluids over astretching sheet, J Fluid Mech 488(2003) 189-212.14. Kavitha T , Rajendran A , Durairajan A and Shanmugam A, “Heat Transfer Enhancement usingNano Fluids and Innovative Methods - An Overview”, International Journal of MechanicalEngineering & Technology (IJMET) Volume 3, Issue 2, 2012. PP. 769 - 782, ISSN Print: 0976 –6340, ISSN Online: 0976 – 6359.15. Dr P.Ravinder Reddy, Dr K.Srihari and 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

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