Available online European Journal of Advances in Engineering and Technology, 2018, 5(3):

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1 Available olie Europea Joural of Advaces i Egieerig ad Techology, 018, 5(3): Research Article ISSN: X Iclied Magetic Field ad Viscous Dissipatio Effects o Taget Hyperbolic Naofluid Flow with Zero Normal Flu of Naoparticles at the Stretchig Surface N Saidulu 1, T Gagaiah ad A Vekata Lakshmi 3 1 Departmet of Mathematics, Osmaia Uiversity, Hyderabad, Tealagaa, Idia Departmet of Mathematics, Govt. Degree College, Macherial, Telagaa, Idia 3 Departmet of Mathematics, UCT, Osmaia Uiversity, Hyderabad, Telaaga, Idia ampellysaidulu8@gmail.com ABSTRACT This article presets the effect of iclied magetic field o the MHD boudary layer flow of taget hyperbolic fluid with aoparticles past a stretchig surface with viscous dissipatio, chemical reactio ad covective boudary coditio. Coditio of zero ormal flu of aoparticles o the wall is used for the cocetratio boudary coditio, which is the curret topic that have yet to be studied etesively. The partial differetial systems are reduced to ordiary differetial systems by usig appropriate similarity trasformatios. The reduced systems are solved umerically by Ruge-Kutta fourth order method with shootig techique. The velocity, temperature ad aoparticle volume fractio profiles are discussed for differet physical parameters. As well as the Ski frictio, Nusselt ad Sherwood umbers are ehibited ad aalyzed. It is foud that the viscous dissipatio ehaces the effective thermal diffusivity ad the temperature rises. It is also observed that the iclied magetic force decreases the velocity field, showig a icreasig behavior of temperature ad aoparticle volume fractio profiles. Key words: MHD, Taget hyperbolic Naofluid, Zero ormal flu, Iclied magetic field, Viscous dissipatio. INTRODUCTION The mometum ad heat trasfer of the boudary layer flow over a stretchig surface have bee applied i umerous chemical egieerig processes, such as polymer etrusio processes ad metallurgical processes, which ivolve coolig of a molte liquid. Sakiadis [1] iitiated studyig the boudary layer flow over a stretched surface movig with a costat velocity ad formulated boudary layer equatios for two dimesioal ad aisymmetric flows. Crae [] ivestigated the flow caused by a stretchig sheet. O the other had, Gupta [3] stressed that realistically, stretchig surface is ot ecessarily cotiuous. Magyari ad Keller [4] aalyzed the steady boudary layers o a epoetially stretchig cotiuous surface with a epoetial temperature distributio. Elbashbeshy [5] ivestigated the heat trasfer over a epoetially stretchig cotiuous surface with suctio. Fathizadeh et al [6] proposed a powerful modificatio of the homotopy perturbatio method for MHD flow over a stretchig sheet. The most importat o-newtoia liquid model is taget hyperbolic liquid model ad which has certai advatages over other o-newtoia formulatios. Pop ad Igham [7] preseted the taget hyperbolic fluid model ad it is etesively used i differet laboratory eperimets. After that, Nadeem et al [8] studied the peristaltic trasport of a hyperbolic taget fluid withi a asymmetric chael. The taget hyperbolic fluid model is used by Friedma et al [9] for large-scale mageto-rheological fluid damper coils. I aother study, peristaltic flow of taget hyperbolic fluid i a curved chael is studied by Nadeem et al [10] ad they eplored the behavior of various parameters o pressure rise agaist flow rate ad plotted stream lies to uderstad the patter of the flow. Akbar et al [11] ivestigated the steady MHD flow of taget hyperbolic fluid over a stretchig sheet. They foud that velocity profile decreases by icreasig power law ide ad Weisseberg umber but demostrates opposite results for ski frictio. A aofluid is a liquid cotaiig aometer-sized solid particles, called aoparticles, which basically icreasig thermal coductivity of the base fluids accordig to a ivestigatio of Choi [1]. Pak ad Cho [13] ascribed the icreased heat trasfer coefficiets oticed i aofluids to the dispersio of suspeded particles. Xua ad Li [14] 14

2 Saidulu et al Euro. J. Adv. Egg. Tech., 018, 5(3): proposed that the heat trasfer ehacemet was the result of the icrease i turbulece iduced by aoparticle motio. A large amout of literature is available, which deals with studyig aofluids ad their applicatios [15 17]. Akram ad Nadeem [18] ivestigated the impact of peristaltic trasport of a taget hyperbolic fluid i the presece of aoparticles uder the ifluece of iclied magetic field. They have foud that icrease i the Browia motio ad thermophoresis leads to ehace temperature profile. Saidulu ad Lakshmi [19] described the boudary layer flow of a o-newtoia Casso fluid accompaied by heat ad mass trasfer towards a epoetially stretchig sheet with viscous dissipatio ad chemical reactio. Bala [0] ivestigated the theoretical study of the steady two-dimesioal MHD covective boudary layer flow of a Casso fluid over a epoetially iclied permeable stretchig surface i the presece of thermal radiatio ad chemical reactio. Prabhakar et al [1] aalyzed the effect of iclied Loretz forces o hyperbolic taget aofluid flow with zero ormal flu of aoparticles at the stretchig sheet. Recetly, may researchers discussed the taget hyperbolic fluid flows over stretchig surfaces [ 8]. The aim of the preset study is to ivestigate the umerical solutio of the taget hyperbolic aofluid over a sheet with viscous dissipatio, chemical reactio ad covective boudary coditio. Usig similarity trasformatios, the partial differetial equatios are reduced to a set of oliear ordiary differetial equatios, which are solved by Ruge-Kutta fourth order method with shootig techique. Graphical results are plotted ad discussed for various parameters o the velocity, temperature, ad cocetratio profiles. MATHEMATICAL FORMULATION Cosider the steady two-dimesioal boudary layer flow of a icompressible viscous ad electrically coductig taget hyperbolic aofluid flow over a stretchig surface which coicides with the plae. The fluid flow is cofied to. The -ais is take alog the stretchig sheet i the directio of motio while the -ais is perpedicular to the sheet. Two equal ad opposite forces are applied alog the -ais so that the wall is stretched keepig the origi fied. The flow takes place i the upper half plae. Alog with this we cosidered a iclied magetic field, viscous dissipatio ad chemical reactio to the flow. The costitutive equatio of taget hyperbolic fluid is: 0 tah, (1) i the above epressio is a etra stress tesor, is a ifiite shear rate viscosity, is the zero shear rate y 0 is the power law ide i.e. flow-behavior ide ad de- viscosity, is the time-depedet material costat, fied as: T y 0 y ij ji () i j 1 where tr gradv gradv. We cosider Eq. 1, for the case whe 0 because it is ot possible to discuss the problem for the ifiite shear rate viscosity ad sice we are cosiderig taget hyperbolic fluid that describig shear thiig effects so 1. The Eq. 1 takes the followig form: (3) The goverig equatios for the taget hyperbolic fluid model after applyig the boudary layer approimatios ca be defied as follows u v 0, (4) y u u u u u B u u v 1 si ( ), y y y y 3 T T k T u u C T DT T u v (1 ) D, B y cp y cp y cp y y y T y C C C D T u v DB k y T T 1(C C ). y y 0 y (5) (6) (7) 143

3 Saidulu et al Euro. J. Adv. Egg. Tech., 018, 5(3): The correspodig boudary coditios are: Where u ad v T C DT T u U, v V, k hf Tf T, DB 0 at y 0 ( zero ormal flu), y y T y are the velocities i the u 0, T T, C C as y. (9) - ad y directios, respectively, the fluid desity (assumed costat), is the coefficiet of fluid viscosity, is variable magetic field, is a icliatio agle, is costat temperature of the fluid i the viscid free stream, k is the thermal coductivity, T c p (8) is the kiematic viscosity, is is the electrical coductivity, B ( ) is the fluid temperature, T is the specific heat at costat pressure, ( c) ( c) is the ratio betwee the effective heat capacity of the aoparticle material to the heat capacity of the base fluid, is the desity of the particles, c f is the volumetric epasio coefficiet, C p f p is the aoparticle volume fractio, T f D B is the Browia diffusio coefficiet, ad D T is the thermophoretic diffusio coefficiet ad k 1 is reactio rate. U a is the stretchig velocity, is the covective fluid temperature below the movig sheet, the covective heat trasfer coefficiet, V 0 is the velocity of suctio ad V 0 is the velocity of blowig. Method of Solutio Itroducig the similarity variables as Where is the similarity variable, ( ) a y, u af ( ), v a f ( ), T T C C ( ), ( ). T T C f ( ) f is the dimesioless stream fuctio, ( ) h f (10) is the dimesioless temperature, is the dimesioless cocetratio ad primes deote differetiatio with respect to. The trasformed ordiary differetial equatios are: 1 f ff ( f ) Wef f Msi ( ) f 0, (11) 1 3 f ( 1 )Ec(f ) WeEc(f ) Nb Nt( ) 0, Pr (1) Nt LePrf LePr K I 0, Nb (13) ad the boudary coditios take the followig form: f (0) S, f (0) 1, (0) Bi (0) 1, Nb (0) Nt (0) 0, (14) f ( ) 0, ( ) 0, ( ) 0 as. (15) B a where the prime deotes differetiatio with respect to, M is the magetic parameter, We U is a V c p the Weisseberg umber, S 0 or ( 0 ) is the suctio (or blowig) parameter, Pr is the Pradtl a k umber, Le is Lewis umber, D B DC B Nb is the Browia motio parameter, k is the thermal diffusivity, c p h f Bi k a is the Biot umber, DT( Tf T ) Nt is the thermophoresis parameter, T U k1 Ec is a Eckert umber ad KI is the chemical reactio parameter. The importat physical cp( Tf T ) a quatities of this problem are the ski frictio coefficiet C f ad the local Nusselt umber Nu, which represet the wall shear stress ad the heat trasfer rate respectively. 144

4 Saidulu et al Euro. J. Adv. Egg. Tech., 018, 5(3): The ski frictio coefficiet ad the local Nusselt umber Here Re U C f Nu is give by 1 C f Re (1 ) ( ) ( ), f We f is give by is a local Reyold umber. Nu Re 1 0 (16) (0), (17) NUMERICAL PROCEDURE The system of coupled o-liear ordiary differetial equatios (11) - (13) alog with the boudary coditios (14) ad (15), which are solved umerically by Ruge-Kutta fourth order method with shootig techique. The step size take as is used to obtai the umerical solutio, ad the boudary coditio is approimated by ma The solutios are obtaied with a absolute error tolerace of 10 6 i all cases. I order to get a clear isight of physical problem, umerical results are displayed with the help of graphical illustratios. Also, to calculate the accuracy of the preset umerical results, compariso with those obtaied by Fathizadeh et al [6] are show i Table 1. Table-1 Values of ski frictio coefficiet for several values of magetic parameter M i the absece of Ec KI Bi S Nb Nt Le 0 M Numerically HPM [6] Preset study RESULTS AND DISCUSSION This sectio is focused o the physical isight of differet parameters o the velocity ( ) f ( ), temperature aoparticle volume fractio profiles. Figures 1a ad 1b show the effect of the power law ide o the velocity, temperature ad aoparticle volume fractio profiles. Here, the velocity ad the associated boudary layer thickess show reducig but the reverse behavior is obtaied for temperature ad aoparticle volume fractio profiles with larger values of the power law ide. The effect of the magetic parameter M o the velocity, temperature ad aoparticle volume fractio profiles are show i Figs. a ad b. It is see that the velocity is a decreasig fuctio of the magetic field parameter M. It holds because with the icrease i M, the Loretz force icreases which produces the retardig effect o the fluid velocity. From Fig. b the effect of magetic field is to ehace the temperature ad aoparticle volume fractio profiles. Clearly, larger magetic parameter yields larger Loretz force which causes strog resistace i the fluid motio. Hece, more heat is produced which ehaces the temperature ad aoparticle volume fractio profiles. ( ) ad Fig. 1 Effect of o velocity ad temperature ad aoparticle volume fractio profiles 145

5 Saidulu et al Euro. J. Adv. Egg. Tech., 018, 5(3): Fig. Effect of M o velocity ad temperature ad aoparticle volume fractio profiles Fig. 3 Effect of S o velocity ad temperature ad aoparticle volume fractio profiles Fig. 4 Effect of We o velocity ad temperature ad aoparticle volume fractio profiles The effects of the suctio/blowig parameter S o the velocity, temperature ad aoparticle volume fractio profiles have bee aalyzed ad the results are preseted i Figs. 3a ad 3b. These figures show that the suctio/blowig has a profoud effect o the boudary layer thickess i which the suctio reduces the thermal boudary layer thickess whereas blowig thickes it. However, the et effect for the suctio parameter is to slow dow the flow velocity, temperature ad aoparticle volume fractio but the reverse is true for the blowig parameter. So, we ca coclude that the suctio ca be effectively used for the fast coolig of the sheet. 146

6 Saidulu et al Euro. J. Adv. Egg. Tech., 018, 5(3): Figures 4a ad 4b idicate the effect of the Weisseberg umber We o the velocity, temperature ad aoparticle volume fractio profiles. It is observed that the velocity profile decreases by the icreasig We. I fact, it is a ratio betwee the shear rate time ad the relaatio time. Hece, for larger Weisseberg umbers We, the fluid becomes thicker, ad cosequetly, the velocity ad the boudary layer thickess decrease. Hece velocity profile shows the decreasig behavior while temperature ad aoparticle volume fractio profiles are icreasig with icreasig values of Weisseberg umber We. Figures 5a ad 5b give the isight for the ifluece of the agle of icliatio o the velocity, temperature ad aoparticle volume fractio profiles. It is oted that with the icrease i, the velocity profile icreases but the reverse behavior is obtaied for temperature ad aoparticle volume fractio profiles. This pheomeo ofte occurs due to stregthes i the applied magetic field. Due to ehaced magetic field a opposite force is produced to the flow, called Loretz force which resists the fluid flow; cosequetly, the velocity profile decreases. It is also see that for the agle the magetic field has ot effect o the velocity profile, 0 while maimum resistace is offered for the fluid particles whe / Figure 6a shows the impact of covective parameter called Biot umber o the temperature ad aoparticle volume fractio profiles. Physically Biot umber is the ratio of covectio at the surface to coductio withi the surface of a body. It holds that both temperature ad aoparticle volume fractio profiles are icreasig with a icreasig values of Biot umber Bi. Figure 6b idicates the effect of the viscous dissipatio parameter Ec o the temperature ad aoparticle volume fractio profiles. It is see that the temperature ad aoparticle volume fractio profiles are icreasig fuctios of the viscous dissipatio parameter Ec.. Bi Fig. 5 Effect of o velocity ad temperature ad aoparticle volume fractio profiles Fig. 6 Effects of Bi ad Ec o temperature ad aoparticle volume fractio profiles 147

7 Saidulu et al Euro. J. Adv. Egg. Tech., 018, 5(3): Fig. 7 Effects of Nb ad Nt o temperature ad aoparticle volume fractio profiles Fig. 8 Effects of Le ad K I o temperature ad aoparticle volume fractio profiles Nb The behavior of the Browia motio parameter o the temperature ad aoparticle volume fractio profiles are draw i Fig. 7a. It is see that the temperature profile is icreasig but reverse behavior is obtaied for aoparticle volume fractio profiles with icreasig values of the Browia motio parameter Nb. This is because that the radom motio of the particles ehaces by icreasig the Browia motio parameter ad, as a result, the temperature profile icreases. Figure 7b gives the isight for the ifluece of the thermophoresis parameter o the temperature ad aoparticle volume fractio profiles. The icrease i the thermophoresis parameter Nt leads to the ehacemet of both the temperature ad aoparticle volume fractio profiles. The differece betwee the wall ad referece temperatures icreases for larger thermophoresis parameter Nt, ad the aoparticles move from hot regio to cold regio. Hece, the temperature profile icreases. Figure 8a depicts the variatio of temperature ad aoparticle volume fractio profiles with coordiate for various values of Lewis umber Le. It is clear from the figure aoparticle volume fractio profile reduces with a icrease i Lewis umber, but temperature profile icreases. A icrease i the values of Lewis umber Le correspods to a weak Browia diffusio coefficiet which results i short peetratio depth for aoparticle volume fractio profile. As a result a rise i Le the aoparticle volume fractio decreases. It is also oticeable that the aoparticle volume fractio profile is affected more eve for small value of Lewis umber Le. Figure 8b shows the ifluece of the chemical reactio parameter o the temperature ad aoparticle volume fractio profiles withi the boudary layer. From this graph, it is observed that the aoparticle volume fractio reduces but temperature profile icreases with a icrease i the chemical reactio parameter. I order to determie the impact of viscous forces at the surface, ski frictio is aalyzed i Fig. 9a ad 9b with respect to the variatio of suctio/blowig parameter S, power law ide, magetic parameter M, Weisseberg umber We ad icliatio agle. It is observed that ski frictio depicts the decreasig behavior for both blowig ad suctio regio. From these figures local ski frictio is decreasig with various values of power law ide Nb Nt 148

8 Saidulu et al Euro. J. Adv. Egg. Tech., 018, 5(3): , magetic parameter M ad Weisseberg umber We ad icliatio agle. I Figs. 10a ad 10b, variatio is obtaied for local Nusselt umber with suctio/blowig parameter, viscous dissipatio parameter, Biot umber, Pradtl umber ad thermophoresis parameter. It is see that the local Nusselt umber is decreasig with icreasig values of suctio/blowig parameter, viscous dissipatio parameter ad thermophoresis parameter, but reverse behavior is obtaied for Biot umber ad Pradtl umber. Fig. 9 Variatio of ski frictio with various values of S, M, We ad Fig. 10 Variatio of ski frictio with various values of S, Ec Bi, Pr ad Nt CONCLUSIONS I this paper, we studied the impact of iclied magetic field, viscous dissipatio ad chemical reactio o MHD boudary layer flow of taget hyperbolic aofluid over stretchig sheet with suctio/blowig ad covective boudary coditio. The mai fidigs of this study are as follows: Velocity profile decreases with icreasig magetic parameter M but temperature ad aoparticle volume fractio profiles are icreases i this case. Iclied agle reduces the velocity profile, but it icreases the temperature ad aoparticle volume fractio profiles. The surface temperature of a sheet icreases with viscous dissipatio parameter Ec. This pheomeo is ascribed to a higher effective thermal diffusivity. As the thermophoresis parameter Nt ehaces, both temperature ad aoparticle volume fractio profiles icreases. The effect of Browia motio Nb is to icrease the temperature ad decrease the aoparticle volume fractio profiles. The ski frictio decreases with the icreasig values of S,, M, We ad. The local Nusselt umber decreases with the icreasig values of Ec ad thermophoresis parameter Nt. 149

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