2015 American Journal of Engineering Research (AJER)

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1 American Journal o Engineering Research (AJER) 2015 American Journal o Engineering Research (AJER) e-issn: p-issn : Volume-4, Issue-7, pp ajer.org Research Paper Open Access The boundary layer lo o nanoluids over an isothermal stretching sheet inluenced by magnetic ield. Preeti Agarala 1, R. Khare 2 1 Department o Mathematics & Statistics, SHIATS, Allahabad, INDIA 2 Department o Mathematics & Statistics, SHIATS, Allahabad, INDIA ABSTRACT : An analysis is carried out to study the eect o the magnetic ield on the boundary layer lo o the nanoluids over an isothermal stretching surace. To types o nanoluids namely Ag ater and Cu ater are considered. Similarity transormation is used to convert the governing nonlinear equations into coupled higher order nonlinear ordinary dierential equations. Fourth-order Runge Kutta method ith shooting technique is employed or the numerical solution o the obtained equations. Numerical results are obtained or distribution o velocity, temperature and concentration, or both cases.the numerical results indicate that an increase in the nanoparticle volume raction ill decrease the velocity boundary layer thickness. Meanhile, the presence o nanoparticles results in an increase in the magnitude o the skin riction along the surace. Such eects are ound to be more evident in the Ag ater solution than in the Cu ater solution. The obtained numerical results have been presented graphically and discussed in details. KEYWORDS MHD Flo, Magnetic ield, Nanoluids, Runge-Kutta Method. I. INTRODUCTION A large number o theoretical investigations dealing ith magnetohydrodynamic (MHD) los o nanoluids have been perormed during the last decades due to their sitly increasing applications in many ields o technology and engineering, such as MHD poer generation,. Recently, the application o magnetohydrodynamics in the polymer industry and metallurgy has attracted the attention o many researchers. Heat and mass transer processes by means o external orce eects is one o the most important problems in modern applied physics. The study o the magnetic ield eect on heat and mass transer is o substantial signiicance in various industries. Sattar and Alam [1] studied the thermal diusion as ell as transportation eects on MHD ree convection and mass transer lo past an accelerated vertical porous plate. Choi [2] ho coined the term nanoluid proposed that the thermal conductivity o the base luid can be increased by adding lo concentration o nanoparticles o materials having higher thermal conductivity than the base luid.putra et,al [3]studied about the temperature dependence o thermal conductivity enhancement or nanoluids. Raptis and Perdikis [4] studied the viscous lo over a non-linearly stretching sheet in the presence o a chemical reaction and magnetic ield. Singh et al. [5] studied the eects o thermal radiation and magnetic ield on unsteady stretching permeable sheet in presence o ree stream velocity. Khan and Pop [6] examined the boundary-layer lo o a nanoluid past a stretching sheet. Momentum and energy equations or a nanoluid over a linearly impermeable stretching surace in the absence o slip and magnetic ield ere studied by Noghrehabadi et al [7] they also studied the theoretical investigations o SiO 2 -ater nanoluid heat transer enhancement over an isothermal stretching sheet. Makinde and Aziz [8] analyzed the boundary layer lo o nano luids over a stretching sheet ith convective heat transer boundary condition. Rana and Bhargava [9] numerically studied the lo and heat transer o a nanoluid over a nonlinearly stretching sheet. Yacob et al.[10] analyzed enhancement o to types o nanoluids, namely, Cu-ater and Ag-ater nanoluids over an stretching sheet ith convective boundary condition. Khare and Rai [11] examined the MHD lo o non-netonian luid through a rectangular channel. Khare and Srivastava [12] studied the eect o Hall current on MHD lo o a dusty viscoelastic liquid through porous medium past an Ininite Plane. Several researches investigated the MHD boundary layer lo [13,14,15] in various situations. [ T y p e t h e c o m p a n y a d d r e s s ] Page 33

2 American Journal o Engineering Research (AJER) 2013 In this paper, an analysis is carried out numerically to study the eect o the magnetic ield on the boundary layer lo o the nanoluids over an isothermal stretching surace. To types o nanoluids namely Ag ater and Cu ater are considered.. The eects o the governing parameters on the velocity and temperature have been discussed and presented in tables and graphs. II. FORMULATION OF MATHEMATICAL MODEL FLOW Consider an incompressible steady to-dimensional boundary layer lo past an isothermal stretching sheet in a ater-based nanoluid hich can contains dierent volume raction o Cu (copper) and Ag (silver) nanoparticles and uniorm magnetic ield is applied on it. It is assumed that the induced lo o nanoluid is laminar, and the base luid (i.e. ater) and the nanoparticles are in thermal equilibrium and no slip occurs beteen them. The thermophysical properties o the luid and nanoparticles are given in Table 1. It is assumed that the sheet surace has constant temperature o T, and the temperature o ambient luid is T. The luid outside the boundary layer is quiescent, and the stretching sheet velocity is U(x) = cx here c is a constant. Under the usual boundary layer approximations, the continuity, momentum and energy equations or the nanoluid, in the Cartesian coordinates can be represent as. The continuity equation; u x + v y = 0 (1) The momentum equation u u u + v = μ n 2 u x y ρ n 2 y The energy equation (2) u T T + v = α 2 T x y n (3) y 2 The initial and boundary conditions are; at y 0 ; v 0, u U( x), T T at y ; u 0, v 0, T T (4) here, u and v are the velocity components in the x and y directions respectively. T is temperature o the nanoluid, T temperature o the ambient nanoluid, B 0 the uniorm magnetic ield strength, σ electrical conductivity o base luid, ρ n eective density o the nanoluid, α n thermal diusivity o the nanoluid, μ n dynamic viscosity o the nanoluid are respectively given as: ρ n = 1 φ ρ + φρ s (5) α n = μ n = K n (ρc p ) n μ (1 φ) 2.5 (7) The eective density o the nanoluid is given by (ρc p ) n = 1 φ (ρc p ) + φ(ρc p ) s (8) (6) K n K = K s+2k 2 K K s K s +2K + K K s (9) [ T y p e t h e c o m p a n y a d d r e s s ] Page 34

3 American Journal o Engineering Research (AJER) 2013 Here θ, μ, ρ, K are the Kinematic viscosity, dynamic viscosity, density and thermal conductivity o the base luid respectively ρ s, K s, ρc P s are the density, thermal conductivity and heat capacitance o the nanoparticles respectively; φ is the solid volume raction o nanoparticles and K n is thermal conductivity o the nanoluid. Subscripts and s are used or base luid and nanoparticles respectively. In this case, ater as base luid ith nonoparticles o copper and silver are studied. Table1: Thermophysical Properties For Pure Water And Various Types O Nanoparticles. III. SOLUTION OF THE PROBLEM To simpliy the mathematical analysis o our study e introduce the olloing similarity transormations x, y x v c 1/ 2 ' u cx v v c 1/ 2 yc v 1/ 2 T T T T / ( 10) Where ψ x, y is the stream unction ith u = ψ y, v = ψ x θ η = dimensionless temperature. η = dimensionless velocity here primes denote dierentiation ith respect to the similarity variable η. By applying the introduced similarity transorms (10) on the governing equations (1-3), the equations are reduces as ollos, n s (1 ) ( 2 ' ) M 0 (11) [ T y p e t h e c o m p a n y a d d r e s s ] Page 35

4 American Journal o Engineering Research (AJER) 2013 s k c n p Pr 1 k cp ' 0 12 Subject to the ollo ing boundary conditions: 0, y 0, 0, 1, 1, y, 0, 0 (13) Pr Pr = Prandtl number. k The physical quantities o interest in this problem are the local skin riction coeicient C and the Nusselt number Nu x, hich are deined as C 2 u Nu k q ( T T ) (14) Where, is the surace shear stress and q is the surace heat lux,hich are given by u T n q kn 15 y y y0 y0 Using the similarity variable (10), e obtain 1/ 2 1 Re x C (0) 1/ 2 kn Rex Nu (0) k (16) Re u x / v x is the local Reynolds number. IV. RESULTS AND DISCUSSIONS In the present paper an analysis is carried out numerically to study the eect o the magnetic ield on the boundary layer lo o the nanoluids over an isothermal stretching surace. To types o nanoluids namely Ag ater and Cu ater are considered. It is ound that the behavior o the luid lo changes ith the change o nanoparticle type.as the governing boundary layer equations (11) and (12) are non linear, it is not possible to get the closed orm solutions. As a result, the equations ith the boundary conditions (13) are solved numerically using Runge - Kutta orth order method ith a systematic guessing o 0 and θ 0 by the shooting technique until the boundary conditions at ininity are satisied. The step size η = is used hile obtaining the numerical solution. The numerical computations are carried out or velocity, temperature at dierent values o magnetic parameter M and are presented in igures 1-6. [ T y p e t h e c o m p a n y a d d r e s s ] Page 36

5 American Journal o Engineering Research (AJER) 2013 Table2: Thermophysical Properties o Ag- ater Nanoluid. Table3: Thermophysical Properties o Cu- ater Nanoluid. Fig1. It shos the dimensionless velocity '(η) or various values o magnetic parameter M. As the value o magnetic parameter M increases,the retarding orce increases because o interaction o electric and magnetic ields and consequently the velocity decreases. Fig2. Exhibits the eect o shear stress distribution or various values o M. It is observed that the magnitude o the all o shear stress given by (1/(1-Ф) 2.5 ) "(0) decreases hen the value o magnetic parameter M increases. Fig 3: Dipicts the dimensionless velocity proiles or selected values o volume raction Ф. it shos that increase o nanoparticle volume raction have not signiicant eect on the velocity proile. Fig 4: Shos eect o Prandtl number Pr on volume raction or Cuater and Ag-ater nanoluids. It is obvious that as prandtl number Pr decreases volume raction increases in both the cases and the decrease o Pr is slightly more or Ag- ater nanoluid than Cu- ater nanoluid. Fig 5: Reveals the eect o volume raction on density o the Cu-ater and Ag-ater nanoluids.it is observed that hen volume raction increases, density o all nanoluids increases and the increase is more or Ag ater nanoluid than other nanoluid. Fig. 6. It exhibits the temperature proile or selected values o volume raction Ф. It is very much evident rom the ig that temperature o nanoluids increases as the value o volume raction increases. [ T y p e t h e c o m p a n y a d d r e s s ] Page 37

6 American Journal o Engineering Research (AJER) 2013 Figure1.Eect o magnetic parameter M on dimensionless velocity proiles '(η) Figure2: Shear stress distribution or various values o M Figure3: Dimensionless velocity proiles or selected values o volume raction Ф [ T y p e t h e c o m p a n y a d d r e s s ] Page 38

7 American Journal o Engineering Research (AJER) 2013 Figure 4: Eect o Prandtl number Pr on volume raction or Cu-ater and Ag-ater nanoluids Figure 5: Eect o volume raction on density o the Cu-ater and Ag-ater nanoluids Figure 6: Temperature proile or selected values o volume raction Ф [ T y p e t h e c o m p a n y a d d r e s s ] Page 39

8 American Journal o Engineering Research (AJER) 2013 V. CONCLUSION In this paper the eect o the magnetic ield on the boundary layer lo o the nanoluids over an isothermal stretching surace as presented. Velocity and temperature distribution in the lo and thermal boundary layers studied. Numerical results prove that nanoluids under the inluence o magnetic ield lead to drop o dimensionless velocity and magnitude o the all o shear stress at the surace. Hoever, it as also observed that hen volume raction increases, density and temperature o all nanoluids increases. Such eects ere ound to be more evident in the Ag ater solution than in the Cu ater solution REFERENCES [1] Sattar. M.A.,Alam.M.M, Thermal Diusion as ell as transportation eects on MHD ree convection and mass transer lo past an accelerated vertical porous plate, Indian Journal o pure applied Mathematics, Vol25(6), pp ,1994. [2] Choi,S.U.S. Enhanching thermal conductivity o luids ith nanoparticales, in:the proceedings o the 1995 ASME International Mechanical Engineering Congress and Exposition,San Fransisco,USA, 66,pp [3] Putra N,Thiesen P, and Roetzel W.,Temperature dependence o thermal conductivity enhancement or nanoluids, Journal o Heat Transer.125(2003),pp [4] Raptis, A. and Perdikis, C. Viscous lo over a non-linearly stretching sheet in the presence o a chemical reaction and magnetic ield. International Journal o Non-Linear Mechanics, Vol41, pp [5] Singh, P., Jangid, A., Tomer, N.S. and Sinha, D.: Eects o thermal radiation and magnetic ield on unsteady stretching permeable sheet in presence o ree stream velocity. International Journal o Inormation and Mathematical Sciences. 6(3), pp , (2010). [6] Khan, W.A.and Pop,I. Boundary-layer lo o a nanoluid past a stretching sheet, International Journal o Heat and Mass Transer., Vol 53,pp ,2010. [7] Noghrehabadi, A, Ghalambaz,M and Ghalambaz,M, Theoretical Investigations o Sio2-ater Nanoluid Heat Transer Enhancement over an Isothermal Stretching Sheet, International journal O Multidisciplinary Science and Engineering. vol. 9, May [8] Makinde, O.D. and Aziz, A., "Boundary layer lo o a nanoluid past a stretching sheet ith a convective boundary condition," International Journal o Thermal Sciences, vol. 50, pp , [9] Rana.P and Bhargava.R, "Flo and heat transer o a nanoluid over a nonlinearly stretching sheet: A numerical study," Communications in Nonlinear Science and Numerical Simulation, [10] Yacob, N. A. Ishak,A Pop. I and Vajravelu.K, "Boundary layer lo past a stretching/shrinking surace beneath an external uniorm shear lo ith a convective surace boundary condition in a nanoluid," Nanoscale research letters, vol. 6, article no. 314, [11] Khare R. and Rai A. MHD lo o non-netonian luid through a rectangular channel. Journal o International Academy o Physical Science. vol. 17(1) pp [12] Khare, R. and Srivastava, S. Eect o Hall Current on MHD lo o a dusty viscoelastic liquid through porous medium past an Ininite Plane. Research Journal o Mathematical and Statistical Sciences, vol 2 (10) pp [13] Nandy.S.K,Sumanta Sidui.S,Mahapatra.T.R: Unsteady MHD boundary layer lo and heat transer o nanoluid over a permeable shrinking sheet in the presence o thermal radiation. Alxendria eng.journal, 53, ,2014. [14] Hunegna.D, Naikoti.K, Unsteady MHD Flo o Heat and Mass Transer o Nanoluids over Stretching Sheet ith a Non- Uniorm Heat/Source/Sink Considering Viscous Dissipation and Chemical Reaction, International Journal o Engineering Research in Arica, Vol.14,pp [15] Na T.Y.and pop I., Unsteady lo past a stretching sheet,mechanics Research Communications, vol.23 pp [ T y p e t h e c o m p a n y a d d r e s s ] Page 40