Effects of Thermal Radiation and Viscous Dissipation on Magnetohydrodynamic Stagnation Point Flow and Heat Transfer of Nanofluid Towards a Stretching Sheet

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Effects of Thermal Radiation and Viscous Dissipation on Magnetohydrodynamic Stagnation Point Flow and Heat Transfer of Nanofluid Towards a Stretching Sheet
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  Delivered by Publishing Technology to: Guest UserIP: 49.206.99.255 On: Sat, 16 Nov 2013 19:05:29Copyright: American Scientific Publishers A RT I     C L  E  Copyright © 2013 by American Scientific Publishers All rights reserved.Printed in the United States of America Journal of Nanofluids  Vol. 2, pp. 283–291, 2013 (www.aspbs.com/jon) Effects of Thermal Radiation and ViscousDissipation on MagnetohydrodynamicStagnation Point Flow and Heat Transfer of Nanofluid Towards a Stretching Sheet  Yohannes Yirga ∗ and Bandari Shankar  Department of Mathematics, Osmania University, Hyderabad 500007, Andhra Pradesh, India In this paper, the effects of thermal radiation and viscous dissipation on magnetohydrodynamic (MHD) stag-nation point flow and heat transfer of nanofluids towards a stretching sheet are investigated numerically. Theeffects of Brownian motion and thermophoresis on the flow and heat transfer were considered. The systemof partial differential equations governing the flow was reduced to a system of non linear ordinary differen-tial equations by using similarity transformations. The transformed equations were numerically solved by anexplicit finite difference scheme known as the Keller box method. The velocity, temperature, and concentrationprofiles were obtained and utilized to compute the skin-friction coefficient, the local Nusselt number, and localSherwood number for different values of the governing parameters. The study reveals that the skin friction andheat transfer rate at the surface increases with the magnetic parameter when the free stream velocity exceedsthe stretching velocity, i.e.,   >  1, and decrease when   <  1. It is also found that the local Sherwood number increases with velocity ratio parameter, Brownian motion parameter, and Lewis number. In addition to this, thethermal boundary layer become thicker by increasing the value of the Eckert number and becomes thinner byincreasing the thermal radiation parameter. A comparison of the numerical results of the present study withpreviously published data revealed an excellent agreement. KEYWORDS:  Thermal Radiation, Viscous Dissipation, Magnetohydrodynamic, Stagnation Point, Nanofluids. 1. INTRODUCTION The study of the stagnation point flow and heat transferof a viscous incompressible fluid has been the concern of many researchers because of its wide applications in man-ufacturing and natural process which include cooling of electronic devices by fans, cooling of nuclear reactors dur-ing emergency shutdown, cooling of an infinite metallicplate in a cooling bath, textile and paper industries, solarcentral receivers exposed to wind currents, MHD genera-tors, plasma studies and blood flow problems.The classical two dimensional stagnation point flow ona flat plate was first studied by Hiemenz in 1911. Sincethen many researchers have extended the idea to differ-ent aspect of the stagnation point flow problems. In 1936,Homann extended the Hiemenz flow to the axisymmetricthree-dimensional case. Ariel 1 studied the influence of anexternal magnetic field on Hiemenz flow. The effect of  ∗ Author to whom correspondence should be addressed.Email: yohannesalbin@yahoo.comReceived: 7 June 2013Accepted: 23 June 2013 uniform suction on Homann problem where the flat plateis oscillating in its own plane is considered by Weidmanand Mahalingam. 2 The two dimensional MHD steady stag-nation point flow towards a stretching surface was ana-lyzed by Mahaparta and Gupta, 3 Ishak  4 and Ishak et al. 5 Their results showed that the heat transfer rate at the sur-face increases with the magnetic parameter when the freestream velocity exceeds the stretching velocity and theopposite is observed when the free stream velocity is lessthan the stretching velocity.Fluid heating and cooling are important in manyindustries such as power, manufacturing, transportation,and electronics. Effective cooling techniques are greatlyneeded for cooling any sort of high-energy device. Com-mon heat transfer fluids such as water, ethylene glycol,and engine oil have limited/poor heat transfer capabilitiesdue to their low heat transfer properties. In contrast, met-als have thermal conductivities up to three times higherthan these fluids, so it is natural that it would be desiredto combine the two substances to produce a heat transfermedium that behaves like a fluid, but has the thermal con-ductivity of a metal. A lot of experimental and theoretical J. Nanofluids 2013, Vol. 2, No. 4   2169-432X/2013/2/283/009 doi:10.1166/jon.2013.1070  283  Delivered by Publishing Technology to: Guest UserIP: 49.206.99.255 On: Sat, 16 Nov 2013 19:05:29Copyright: American Scientific Publishers Effects of Thermal Radiation and Viscous Dissipation on MHD Stagnation Point Flow and Heat Transfer of Nanofluid  Yirga and Shankar        A      R      T      I      C      L      E researches have been made to improve the thermal con-ductivity of these fluids.In 1993, during an investigation of new coolants andcooling technologies at Argonne national laboratory inU.S, Choi invented a new type of fluid called Nanofluid. 6 Nanofluids are fluids that contain small volumetric quan-tities of nanometer-sized particles, called nanoparticles.The nanoparticles used in nanofluids are typically madeof metals, oxides, carbides, or carbon nanotubes. Commonbase fluids include water, ethylene glycol and oil. Nanoflu-ids commonly contain up to a 5% volume fraction of nanoparticles to see effective heat transfer enhancements.Nanofluids are studied because of their heat transfer prop-erties: they enhance the thermal conductivity and convec-tive properties over the properties of the base fluid. Typicalthermal conductivity enhancements are in the range of 15–40% over the base fluid and heat transfer coeffi-cient enhancements have been found up to 40%. 7 Increas-ing in thermal conductivity of this magnitude cannot besolely attributed to the higher thermal conductivity of theadded nanoparticles, and there must be other mechanismsattributed to the increase in performance.After the pioneer investigation of Choi, thriving experi-mental and theoretical researches were undertaken to dis-cover and understand the mechanisms of heat transfer innanofluids. The knowledge of the physical mechanismsof heat transfer in nanofluids is of vital importance asit will enable the exploitation of their full heat transferpotential. Masuda et al. 8 observed the characteristic fea-ture of nanofluid is thermal conductivity enhancement.This observation suggests the possibility of using nanoflu-ids in advanced nuclear systems. 9 A comprehensive sur-vey of convective transport in nanofluids was made byBuongiorno, 10 who says that a satisfactory explanation forthe abnormal increase of the thermal conductivity and vis-cosity is yet to be found. He focused on further heat trans-fer enhancement observed in convective situations. Prabhatet al. 11 studied the convective heat transfer enhancementin nanofluids. They showed that significant deviationsbetween the data and the predictions of the heat transfercorrelations can occur in the laminar flow regime, partic-ularly in the entrance region; the enhancement becomesmore pronounced at higher Reynolds number and higherparticle concentration. Kuznetsov and Nield 12 have exam-ined the influence of nanoparticles on natural convec-tion boundary-layer flow past a vertical plate using amodel in which Brownian motion and thermophoresis areaccounted for. The authors have assumed the simplest pos-sible boundary conditions, namely those in which boththe temperature and the nanoparticle fraction are constantalong the wall. Nield and Kuznetsov 13  14 have studied theCheng and Minkowycz 15 problem of natural convectionpast a vertical plate in a porous medium saturated by ananofluid. The model used for the nanofluid incorporatesthe effects of Brownian motion and thermophoresis forthe porous medium. Nadeem and Lee 16 used the homo-topy analysis method to study the boundary layer flow of nanofluid over an exponentially stretching sheet. Kumariand Gorla, 17 studied the mixed convective boundary layerflow over a vertical plate embedded in a porous mediumsaturated with a nanofluid. In addition to this, Gorla andKumari, 18 studied the mixed convection flow of a Non-Newtonian fanofluid over a non-linearly stretching sheet.Recently, Gorla and Llapa, 19 studied the heat transfer ina nano thin-film region of an evaporating meniscus; Gorlaand Kumari, 20 also studied the combined convection on avertical cylinder in a non-Newtonian nanofluid. The mag-netic field effects on free convection flow of a nanofluidpast a vertical semi-infinite flat plate has been discussedby Hamad et al. 21 and Mahajan et al. 22 studied the convec-tion in magnetic nanofluids. Haddad et al. 23 experimentallyinvestigated natural convection in nanofluds by consideringthe role of thermophoresis and Brownian motion in heattransfer enhancement. They indicated that neglecting therole of Brownian motion and thermophoresis deterioratethe heat transfer and this deterioration elevated by increas-ing the volume fraction of nanoparticles. Anbuchezhian 24 studied the thermoporesis and Brownian motion effects onboundary layer flow of nanofluid in the presence of ther-mal stratification due to solar energy. In addition to thesestudies, several researchers have investigated the flow andheat transfer of nanofluids. 25–35 An adequate understanding of radiative heat transferin flow processes is very important in engineering andindustries, especially in the design of reliable equipments,nuclear plants, gas turbines and various propulsion devicesfor aircraft, missiles, satellites and space vehicles. Thermalradiation effects are extremely important in the context of flow processes involving high temperature. The effects of thermal radiation on the boundary layer flow have alsobeen considerably researched. 36–40 However, all the papers in the literature are restrictedto the study of boundary layer flow of a nanofluid pasta differently stretching sheet. The stagnation point flowof nanofluids has been given less attention. Moreover,partly or completely the effects of thermal radiation, vis-cous dissipation, magnetic field, Bronian motion and ther-mophoresis are neglected in many of the studies conductedon stagnation point flow and heat transfer of nanoflu-ids. A few recent studies focused on a stagnation pointflow. Recently, Mustafa et al. 41 used the homotopy analysismethod to study the Stagnation-point flow of a nanofluidtowards a stretching sheet; Bachok et al. 42 studied theStagnation-point flow over a stretching/shrinking sheet ina nanofluid. They analyzed the effects of the solid vol-ume fraction and the type of the nanoparticles on the fluidflow and heat transfer characteristics of a nanofluid over astretching/shrinking sheet. Very recently, Wubshet et al. 43 studied the MHD stagnation point flow and heat transferdue to nanofluid towards a stretching sheet. 284  J. Nanofluids, 2, 283–291, 2013   Delivered by Publishing Technology to: Guest UserIP: 49.206.99.255 On: Sat, 16 Nov 2013 19:05:29Copyright: American Scientific Publishers Yirga and Shankar   Effects of Thermal Radiation and Viscous Dissipation on MHD Stagnation Point Flow and Heat Transfer of Nanofluid A RT I     C L  E  The aim of the present paper is to study the effects of thermal radiation and viscous dissipation on MHD stag-nation point flow and heat transfer of a nanofluid towardsa stretching sheet by taking the brownian motion andthermophoresis in to account. The effects of the gov-erning parameters such as magnetic parameter, velocityratio parameter, Prandtl number, thermal radiation param-eter, Echart number, Lewis number, Brownian motionparameter, and thermophoresis parameter on the flow andheat transfer characteristics are investigated. The combinedeffect of all the above mentioned parameters has not beenreported so far in the literature, which makes the presentpaper unique.The governing highly nonlinear partial differential equa-tions of momentum, energy and nano particle volumefraction has been simplified by using suitable similar-ity transformations and then solved numerically with thehelp of a powerful, easy to use method called the Kellerbox method. This method has already been successfullyapplied to several non linear problems corresponding toparabolic partial differential equations. As discussed inRef. [44] the exact discrete calculus associated with theKeller-box scheme is shown to be fundamentally differentfrom all other mimic numerical methods. The box-schemeof Keller, 44 is basically a mixed finite volume method,which consists in taking the average of a conservation lawand of the associated constitutive law at the level of thesame mesh cell. The paper is outlined as follows.In Section 2 the problem is formulated and similaritytransformation has been used to reduce the formulatedsystem of partial differential equation in to a non lin-ear ordinary differential equations. In Section 3 a numeri-cal solution using Keller box method has been discussed.In Section 4 the numerical results are discussed in detailgraphically and in table form. Finally, in Section 5, con-cluding remarks are given. 2. MATHEMATICAL FORMULATION Consider a steady two-dimensional stagnation point flowof a nanofluid towards a stretching sheet kept at a constanttemperature  T  w  and concentration   w . The ambient tem-perature and concentration far away from the sheet respec-tively, are  T    and    . The flow is subjected to a constanttransverse magnetic field of strength  B 0  which is assumedto be applied in the positive  y -direction, normal to the sur-face. The induced magnetic field is assumed to be smallcompared to the applied magnetic field and is neglected.It is further assumed that the base fluid and the suspendednanoparticles are in thermal equilibrium and no slip occursbetween them. We choose the coordinate system such that x -axis is along the stretching sheet and  y -axis is normalto the sheet. The physical flow model and coordinate sys-tem is shown in Figure 1. Under the above approximationand using the usual boundary layer approximations, the Fig. 1.  Physical flow model and coordinate systems. governing equation of the conservation of mass, momen-tum, energy and nanoparticles fraction for nanofluids in thepresence of magnetic field, radiation and viscous dissipa-tion towards a stretching sheet can be written in Cartesiancoordinates  x  and  y  as. 40  43  45 dudx + dvdx = 0 (1) uux + vuy = U   U   y  + v nf   2 uy 2 + B 20 x nf  U   − u  (2) uT x  + vT y =  nf   2 T y 2  + u nf  c p  nf   uy  2 + c p  s c p  f  ×  D B yT y  + D T  T    T y  2  − 1 c p  nf  q  r  y  (3) ux + vy  = D B  2 y 2  + D T  T     2 T y 2   (4)Where  u  and  v  are the velocity components in the  x and  y  directions respectively,  T   is the temperature of the nanofluid,    is the nanoparticle volume fraction, q  r   is the radiative heat flux;  U   B 0 c p  s c p  f  D B  and  D T   are the free stream velocity, electrical conduc-tivity, magnetic field, heat capacity of the nanoparticle,heat capacity of the base fluid, the Brownian diffu-sion and thermophoretic diffusion coefficient respectively,and  v nf   nf   nf   nf  K  nf   and  c p  nf   are the kinematicviscosity, thermal diffusivity, density, viscosity, thermalconductivity, and heat capacitance of the nanofluid, respec-tively, which are given as: 24  45 v f   =  f   f   v nf   =  nf   nf    nf   = k nf  c p  nf   nf   =  1 −  f  +  s   nf   =  f   1 −  2  5 c p  nf   =  1 − c p  f  + c p  s k nf  k f  = k s + 2 k f  − 2 k f  − k s k s + 2 k f  + K  f  − k s  (5) J. Nanofluids, 2, 283–291, 2013   285  Delivered by Publishing Technology to: Guest UserIP: 49.206.99.255 On: Sat, 16 Nov 2013 19:05:29Copyright: American Scientific Publishers Effects of Thermal Radiation and Viscous Dissipation on MHD Stagnation Point Flow and Heat Transfer of Nanofluid  Yirga and Shankar        A      R      T      I      C      L      E In which  v f   f   f  k f   are the kinematic viscosity, vis-cosity, density and thermal conductivity of the base fluidrespectively, and   s k s c p  s  are the density, thermalconductivity and heat capacitance of the nanoparticlerespectively.Using the Rosseland approximation for radiation, theradiative heat flux is simplified as: q  r   =− 4   ∗ 3 k ∗ T  4 y  (6)Where   ∗  and k ∗  are the Stefan-Boltzmann constant and themean absorption coefficient, respectively. The temperaturedifferences within the flow are assumed to be sufficientlysmall so that  T  4 may be expressed as a linear function of temperature  T   using a truncated Taylor series about the freestream temperature  T    and neglecting higher-order terms,weget: T  4  4 T  3  T   − 3 T  4   (7)If we take  N  r  = k nf  k ∗ / 4   ∗ T  3    as a radiation parameter,then making use of Eqs. (3), (6) and (7), becomes: uT x  + vT y  =  nf  k 0  2 T y 2  +  nf  c p  nf   uy  2 + c p  s c p  f   D B yT y  + D T  T    T y  2   (8)Where  K  0 =  3 N  r  / 3 N  r  + 4  . It is worth mentioning herethat as  N  r   →  (i.e.,  k 0 → 1), we get the classical solu-tion to the energy Eq. (3) without the influence of thermalradiation.The associated boundary conditions are u = U  w = ax v = 0  T  = T  w   =  w   at  y = 0 u = U   = bx v = 0  T  → T    →     as  y → (9)In which  x  and  y  represent coordinate axes along thecontinuous surface in the direction of motion and nor-mal to it, respectively.  U  w  T  w   and   w  are the refer-ence velocity of the sheet, temperatures of the sheet andnanoparticles volume fraction of the sheet, respectively,and  U   T     and     are the free stream velocity, ambi-ent temperature and nanoparticle volume fraction far awayfrom the sheet.Introducing the following similarity transformations  = y    av f   u = axf    v =−√  av f  f = T   − T   T  w − T    g =  −    w −   (10)The equation of continuity (1) is automatically satisfiedand Eqs. (2), (4) and (8) reduce to the following ordinaryequations f   + ff   − f   2 + M − f    +  2 = 0 (11)  1 + 43 N  r     + Pr f  + Ecf   2 + Nb  g  + Nt  2  = 0 (12) g  + Lefg  + NtNb  = 0 (13)With boundary conditions f 0  = 0  f    0  = 1   0  = 1  g 0  = 1 f      =     = 0  g   = 0   as   → (14)Where prime denote differentiation with respect to thesimilarity variable   .  f     and  g  represent the dimen-sion less velocity, temperature and particle concentration,respectively.  M Pr N  r  EcNbNt  and  Le  denote themagnetic parameter, velocity ratio parameter, Prandtl num-ber, radiation parameter, Eckert number, Brownian motionparameter, thermophoresis parameter, and Lewis number,respectively. These governing parameters are given as M  = B 20  nf    = U   U  w ba  Pr = v f   f   N  r  = k ∗ k nf  4   ∗ T  3  Ec = U  2 w c p T  w − T    Nb = D B c p  s c p  f   w −   v nf  Nt = D T  T   c p  s c p  f  T  w − T   v nf   Le = v f  D B (15)The quantities of practical interest in this study are thelocal skin friction coefficient  C  f  , the local Nusselt num-ber  Nu x  and the local Sherwood number  Sh x , which aregiven as 40  43 √  Re x C  f  =  nf   f  f    0  = 1  1 −  2  5 f    0 Nu x √  Re x =−  k nf  k f  + 43 N  r      0  Sh x √  Re x =− g   0  (16)Where Re x = U  w /v f  x  is the local Reynolds number. 3. NUMERICAL SOLUTION The non linear boundary value problem represented byEqs. (11)–(13) and (14) is solved numerically using theKeller box method. In solving the system of non lin-ear ordinary differential Eqs. (11)–(13) together with theboundary condition (14) using the Keller box method thechoice of an initial guess is very important. The successof the scheme depends greatly on how much good thisguess is to give the most accurate solution. Thus, an initialguess of  f  0  = 1 − e −  + 12   2     0  = e −  g 0  = e −  (17) 286  J. Nanofluids, 2, 283–291, 2013   Delivered by Publishing Technology to: Guest UserIP: 49.206.99.255 On: Sat, 16 Nov 2013 19:05:29Copyright: American Scientific Publishers Yirga and Shankar   Effects of Thermal Radiation and Viscous Dissipation on MHD Stagnation Point Flow and Heat Transfer of Nanofluid A RT I     C L  E  These choices have been made based on the convergencecriteria together with the boundary conditions in consid-eration. As in Cebeci and Pradshaw, 46 the values of thewall shear stress, in our case  f    0   is commonly used asa convergence criteria. This is because in the boundarylayer flow calculations the greatest error appears in thewall shear stress parameter. In the present study this con-vergence criteria is used. In this study a uniform grid of size   = 0  1 is chosen to satisfy the convergence criteriaof 10 − 5 , which gives about a four decimal places accuracyfor most of the prescribed quantities. 4. RESULTS AND DISCUSSION The transformed non linear Eqs. (11)–(13) subjected tothe boundary condition (14) was solved numerically usingKeller box method, which is described in Cebeci andBradshaw. 46 The velocity, temperature, and concentrationprofiles were obtained and utilized to compute the skin-friction coefficient, the local Nusselt number, and localSherwood number in Eq. (18). The results of the veloc-ity profiles, temperature profile and concentration profilefordifferent values of the governing parameters viz., mag-netic parameter  M  , velocity ratio parameter   , Prandtlnumber  Pr  , radiation parameter  N  r  , viscous dissipationparameter Eckert number  Ec , Brownian motion parameter  Nb , thermophoresis parameter  Nt  , and Lewis number  Le are presented in graphs, while the values of the skin fric-tion coefficient, Nusselt number and local Sherwood num-ber for some values of the parameters are presented ingraphs and tables.To validate the accuracy of our results a comparisonhas been made with previously reported works by Mustafaet al. 41 and Wubshet et al. 43 The comparisons are found tobe in an excellent agreement as shown in Tables I and II.Table I depicts the values of the local skin friction coef-ficients at the surface  − f    0   for various values of thevelocity ratio parameter   , when  M   = 0. Table II con-tains the values of the local Nusselt number −    0   whichshows the heat transfer rate at the surface, for some valuesof the Prandtl number  Pr   and velocity ratio parameter   ,when  M   = 0,  N  r   → ,  Ec  = 0,  Nt  = 0,  Nb  → 0 and Le = 2  0. Table I.  Comparison of values of local friction coefficient  f    0   withMustafa et al. 41 and Wubshet et al. 43 for some values of    , when  M  = 0, N  r   → ,  Ec = 0,  Nt = 0,  Nb → 0, and  Le = 2  0.   Mustafa 35 Wubshet et al. 37 Present study0.01  − 0  99823  − 0  9980  − 0  99870.1  − 0  96954  − 0  9694  − 0  96970.2  − 0  91813  − 0  9181  − 0  91840.5  − 0  66735  − 0  6673  − 0  66762.0 2  01767 2  0175 2  02013.0 4  72964 4  7292 4  7393 Table II.  Comparison of values of local friction coefficient −    0   withMustafa et al. 41 and Wubshet et al. 43 for some values of     and  Pr  , when M   = 0,  N  r   → ,  Ec = 0,  Nt = 0,  Nb → 0, and  Le = 2  0. Pr   Mustafa 35 Wubshet et al. 37 Present study1.0 0  1 0  6022 0  602156 0  60200  2 0  6245 0  624467 0  62440  5 0  6924 0  692460 0  69261.5 0  1 0  7768 0  776802 0  77680  2 0  7971 0  797122 0  79720  5 0  8648 0  864771 0  8652 Figures 2 and 3 show the effects of the magnetic param-eter  M   and velocity ratio parameter    on the flow fieldvelocity  f    . Figure 2, shows the effect of the magneticparameter  M   on the flow field velocity  f     for three dif-ferent values of the velocity ratio parameter   , i.e.,   = 0  = 1   and   = 3. When   = 0   the velocity field andthe boundary layer thickness decrease with an increasein  M  . When   = 1, the variation of   M   has no influenceon the flow velocity. On the other hand when   = 3, theflow velocity increases and the boundary layer thicknessdecreases with an increase in  M  .Figure 3, shows the effect of the velocity ratio parame-ter    on the flow field velocity  f    . It can be observedthat when the stretching velocity of the sheet exceeds thefree stream velocity (i.e.,  < 1), the flow velocity and theboundary layer thickness increases with an increase in   .Moreover, when   >  1   the flow velocity increases andthe boundary layer thickness decreases with an increase in$ \ lambda$. Furthermore, from Figures 2 and 3 it can beobserved that, when   >  1, the flow has boundary layerstructure but when  < 1, the flow has an inverted bound-ary layer structure.Figures 4–7 demonstrate the effects of various param-eters on the nanofluid temperature profile   . Figure 4,shows the effects of the Prandtl number  Pr   on temperatureof the nanofluid. The temperature of the nanofluid at a 00.511.522.533.5400.511.522.53 η    f        ′    (      η    ) M = 0, 1, 5, 10 λ   = 3.0 λ   = 1.0 λ   = 0.0M = 0, 1, 5, 10M = 0, 1, 5, 10 Fig. 2.  Velocity profile  f     for different values of   M  , when  Pr  = 1  0, Nr   = 3,  Ec = 1  0,  Nt = 0 = Nb = 0  5, and  Le = 2  0,   = 0  0  1  0  3  0. J. Nanofluids, 2, 283–291, 2013   287
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