MAGNETIC FIELD EFFECT ON MIXED CONVECTION FLOW IN A NANOFLUID UNDER CONVECTIVE BOUNDARY CONDITION

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1 NTRNATONA OURNA OF CANCA NNRN AND TCNOOY (T) nternational ournal o echanical ngineering and Technology (T), SSN (Print), SSN (Print) SSN (Online) Volume 6, ssue 4, April (015), pp A: ournal mpact Factor (015): (Calculated by S) T A ANTC FD FFCT ON XD CONVCTON FOW N A NANOFUD UNDR CONVCTV BOUNDARY CONDTON 1 Subhas Abel, Pamita axman Rao, 3 agadish V.Tawade* 1, Department o athematics, ulbarga University, ulbarga , arnataka, ndia 3 Department o mathematics, Bheemanna handre nstitute o Technology, Bhalki ABSTRACT An analysis is carried out to investigate the inluence o the prominent magnetic eect on mixed convection heat and mass transer in the boundary layer region o a semi-ininite vertical lat plate in a nanoluid under the convective boundary conditions. The transormed boundary layer, ordinary dierential equations are solved numerically using Runge-utta Fourth order method. A wide range o parameter values is chosen to bring out the eect o agnetic ield parameter on the mixed convection process with the convective boundary condition. The eect o mixed convection, agnetic ield and Biot parameters on the low, heat and mass transer coeicients is analyzed. The numerical results obtained or the velocity, temperature and volume raction proiles are presented graphically and discussed. eywords: ixed Convection, Nanoluid, agnetic ect, Convective Boundary Condition, Numerical Solution. 1. NTRODUCTON A Nanoluid is a luid containing nanometer sized particles, called nanoparticles. These luids are engineered colloidal suspensions o nanoparticles in a base luid. The nanoparticles used in nanoluids are typically made o metals, oxides, carbides, or carbon nanotubes. Common base luids include water, ethylene glycol and oil. Nanoluids have novel properties that make them potentially useul in many applications in heat transer, including microelectronics, uel cells pharmaceutical processes, and hybrid-powered engines, engine cooling/vehicle thermal management, domestic rerigerator, chiller, heat exchanger, in grinding, machining and in boiler lue gas temperature 7

2 reduction. They exhibit enhanced thermal conductivity and the convective heat transer coeicient compared to the base luid. nowledge o the rheological behavior o nanoluids is ound to be very critical deciding their suitability or convective heat transer applications. erkin [1] investigated the mixed convection boundary layer low on a semi-ininite vertical lat plate when the buoyancy orces aid, and oppose the development o the boundary layer. Ding et al. [] Nanoluid are also important or the production o nanostructured 73 materials, or the engineering o complex luids, as well as or cleaning oil rom suraces due to their excellent wetting and spreading behavior. Ater the poineering work by Sakiadis [3], a large amount o literature is available on boundary layer low o Newtonian and non-newtonian luids over linear and nonlinear stretching surace. A detailed review o mixed convective heat and mass transer can be ound in the book by Bejan [4]. Recently, Subhashini et al. [5] discussed the simultaneous eects o thermal and concentration diusion on a mixed convection boundary layer low over a permeable surace under convective surace boundary condition. The general topic o heat transer in nanoluids has been surveyed in a review article by Das and Choi [6] and a book by Das et al. [7]. Buongiorno [8] noted that the nanoparticle absolute velocity can be viewed as the sum o the base luid velocity and a relative velocity. e considered in turn seven slip mechanism: inertia, Brownian diusion, thermophoresis, diusiophoresis, agnus eect, luid drainage and gravity setting. e concluded that in the absence o turbulent eects, it is the Brownian diusion and the thermophoresis that will be important. Buongiorno proceeded to write down conservation equations based on these two eects. The problem o natural convection in a regular luid past a vertical plate is a classical problem irst studied theoretically by. Pohlhausen in contribution to an experimental study by Schmidt and Beckmann[9]. unortunately the boundary layer scaling used by early researchers and text book authors did not properly incorporate the papers by uiken [10, 11]. An extension to the case o heat and mass transer was made by hair and Bejan [1]. akinde, and Aziz [13] considered to study the eect o a convective boundary condition on boundary layer low, heat and mass transer and nanoparticle raction over a stretching surace in a nanoluid. uznetsov and Nield [14] studied, the natural convection boundary layer low, heat and mass transer o nanoluid past a vertical plate. The transormed non-linear ordinary dierential equations governing the low are solved numerically by the Runge-utta Fourth order method. These authors discussed about the convective heat transport in nanoluids. They studied natural convective low o nanoluids over a vertical plate and their similarity analysis is identical with our parameters governing the transport process, namely a ewis number e, a Buoyancy- ratio Number r, a Brownian motion number Nb and a thermophoresis number Nt, A.V. uznetsov, D. A. Nield [010]. Ali et al. [011] included the idea o induced magnetic ield to the problem o Ashra [011] and analyzed D stagnation-point low and heat transer towards stretching sheet with induced magnetic ield. The application o agnetic Field to convection process will play as a control actor in the convection by damping both the low and temperature oscillation material manuacturing ields. otivated by the above reerenced work, and the vast possible industrial applications, it is paramount interest to consider the eect o magnetic parameter on mixed convective low along a vertical plate in a nanoluid under the convective boundary condition. A similarity solution is presented. This solution is depends on Prandtl number Pr, a ewis number e, a Brownian motion number Nb, a nanoparticle buoyancy ratio Nr, thermophoresis number Nt, Biot number Bi, mixed convection number λ and a magnetic number.the dependency o the Skin riction coeicient, Nusselt number, nanoparticle Sherwood number and regular Sherwood number on these six parameters is numerically investigated. Consideration o the nanoluid and the convective boundary conditions enhanced the number o non-dimensional parameters considerably. The eect

3 o mixed convection, agnetic and Biot parameters on the physical quantities o the low, heat and mass transer coeicients are analyzed. To examine the convergence o the numerical code written to solve the present problem, we compare the present result or the clear luid mixed convection results with previously published works with convective boundary conditions and the comparison shows that the results are in very good agreement.. ATATCA FORUATON Choose the coordinate system such that the x-axis is along the vertical plate and y-axis normal to the plate. The physical model and coordinate system are shown in ig. 1. Consider the steady laminar two-dimensional mixed convection heat and mass transer along a lat vertical surace embedded in a nanoluid having T and φ as the temperature and nanoparticle volume raction respectively in the ambient medium. Also assume that a ree stream with uniorm velocity u goes past the lat plate. The plate is either heated or cooled rom let by convection rom a luid o temperature T with T > T corresponding to a heated surace (assisting low) and T > T corresponding to a cooled surace (opposing low) respectively. On the wall the nanoparticle volume raction is taken to be constant and is given by φ respectively. w By employing Oberbeck-Boussinesq approximation, making use o the standard boundary layer approximation and eliminating pressure, the, the governing equations or the nanoluid are given by Fig: 1 u v + = 0 x y (1) ρ u u u u v g (1 )[ ( T T )] x y µ + = + y ρ φ β T 74

4 σb0 u ( ρp ρ ) g( φ φ ) () ρ D T T T φ T T T u + v = α + τ D + x y m B y y y T y (3) φ φ φ D T T u + v = D + x y B y T y (4) where u and v are the velocity components along the x and y axes, respectively, T is the temperature, φ is the nanoparticle volume raction, g is the gravitational acceleration, α = k /( ρc) is the thermal diusivity o the luid, ν = µ / ρ m is the kinematic viscosity coeicient and τ = ( ρc) p /( ρc). Further, ρ is the density o the base luid and ρ, µ, k, β T and β are the density, viscosity, thermal conductivity, volumetric thermal expansion coeicient and c volumetric solutal expansion coeicient o the nanoluid, while is ρ p the density o the nanoparticles, ( ρc) is the heat capacity o the luid and ( ρ c) p is the eective heat capacity o the nanoparticle material. The coeicients that appear in qs. (3) and (4) are the Brownian diusion coeicient D, the thermophoretic diusion coeicient D B T. For, details o the derivation o equations (1) - (4), one can reer the papers by Buonggiorno [8] and Nield and uznetsov ([14, 15]). The boundary conditions are T u = 0, v = 0, k = h ( T T ), φ = φw y at y=0 (5a) u u, T T, φ φ as y here, h is the convective heat transer coeicient and the subscripts w and conditions at the surace and at the outer edge o the boundary layer respectively. n view o the continuity equation (1), we introduce the stream unction ψ by (5b) indicate the u ψ ψ =, v = y x (6) Substituting (6) in eqs. ()-(4) and then using the ollowing non-dimensional transormation = u / ν y, ψ = u ν x ( ), (7a) 75

5 T T φ φ θ ( ) =, γ ( ) = T T φ φ with the local Reynolds's number Re x dierential equations as w (7b) u x =, we get the transormed system o ordinary υ ''' '' ' ' + + λ[ θ Nry] = 0 (8) 1 '' ' ' ' ' Pr θ θ θ θ + + Nb y + Nt = 0 (9) '' ' Nt '' y + ey + 0 Nb θ = (10) where the primes indicate dierentiation with respect to. n usual notations, Prandtl number and buoyancy ratio. Further, υ e = is the ewis number. D B Nr = ν Pr = is the α ( ρ p ρ )( φw φ ) is the nanoluid ρ β ( T T )(1 φ ) T ( ρc) p DB ( φw φ ) Nb = is the Brownian motion parameter, ( ρc) ν ( ρc) p DT ( T T ) Nt = is the thermophoresis parameter, rx = g(1 φ ) βt ( T T ) is the ( ρc) υt r local rasho number and λ = x the mixed convection parameter. Finally = is the u x magnetic number Boundary conditions (5) in terms o, θ, γ become m θ θ γ ' ' = 0 : (0) = 0, (0) = 0, (0) = Bi[1 (0)], (0) = 1 (11a) ' : ( ) 1, ( ) 0, ( ) 0 (11b) θ γ where Bi= c k ν is the Biot number. This boundary conditions will be ree rom the local variable u x when we choose h 1/ = cx. The Biot number Bi is a ratio o the internal thermal resistance o the plate to the boundary layer thermal resistance o the hot luid at the bottom o the surace. t is important to note that this boundary value problem reduces to the classical problem o low and heat and mass transer due to the Blasius problem o low when Nb and Nt are zero. ost nanoluids examined to date, have large values or the ewis number e >1.For water nanoluids at room temperature with nanoparticles o 1-100nm diameters, the Brownian diusion coeicient D 76 B

6 m / s ranges rom to Furthermore, the ratio o the Brownian diusivity coeicient to the thermophoresis coeicient or particles with diameters o 1-100nm can be varied in the ranges o - or alumina, and rom to 0 or copper nanoparticles, ence, the variation o the non-dimensional parameters o nanoluids in the present study is considered in the mentioned range. 3. SN FRCTON, AT AND ASS TRANSFR COFFCNTS The primary objective o this study is to estimate the skin riction coeicientc, Nusselt number Nu and the nanoparticle Sherwood number NSh. These parameters characterize the surace drag, the wall heat and nanoparticles mass transer respectively. The shearing stress, local heat and local nanoparticle mass rom the vertical plate can be obtained rom u T τ µ, q k φ = w = y w y 0 y and q = D n B = y y = 0 y = 0 τ w The non -dimensional shear stress C = ρ u q x and the nanoparticle Sherwood number NSh = n x D ( φ φ ) B w C (Re ) 1/ = '' (0), x Nu (Re / ) 1/ = θ ' (0), x x N S h 1 / ' x (R e x / ) = γ (0 ), the Nusselt number, are given by (1) q x Nu = w x k( T T ) ect o the various parameters involved in the investigation on these coeicients is discussed in the ollowing section. 4. RSUT AND DSCUSSON The resulting transport qns are non-linear, coupled, ordinary dierential equations, which possess no closed-orm solution. Thereore, these are solved numerically subject to the boundary conditions given by qn. 1. The computational domain in the -direction was made up o 196 non-uniorm grid points. t is expected that most changes in the dependent variable occur in the region close to the plate where viscous eect dominate. owever, small changes in the dependent variables are expected ar away rom the plate surace. For these reasons, variable step sizes in the -direction are employed. The initial step size 1 and growth actor * employed 3 such that i+ 1 = i (where the subscript i indicates the grid location) were 10 and respectively. These values were ound (by perorming many numerical experimentations) to give accurate and grid-independent solutions. The solution convergence criterion employed in the present work was based on the dierence between the values o the dependent variables at the current and 77

7 5 the previous iteration. When this dierence reached10, the solution was assumed converged and the iteration process was terminated. To have a better understanding o the low characteristics, numerical results or the velocity, temperature, volume raction are calculated or dierent sets o values o the parameters, e, λ, Nb, Nt, Nr, Bi also, the eect o these parameters on skin raction, non-dimensional heat and nanoparticle mass coeicients is discussed. Fig. (a)-(c) show the non-dimensional velocity, temperature and volume raction proiles or various values o magnetic along with varying values o the ewis number e or a given Nt = 0., Nb = 0., Nr = 0., Pr = 1.0, λ = 0.4, Bi = 1. The magnetic number accounts or the additional mass diusion due to the temperature gradients. t is noticed rom Fig. that an increase in the magnetic number resulted in an increase in the velocity while decrease in temperature and nanopartical volume raction is noted within the boundary layer. The present analysis shows that the low ield is appreciably inluenced by magnetic parameter. t is clear rom these igures that an increase in ewis number e increased the momentum boundary layer thickness, while reduction in the thermal and nanopartical volume raction boundary layer thickness is noted. The non-dimensional velocity or dierent values o Biot number Bi with ixed values o the other parameters is plotted in Fig. 3(a). ncreased convective heating associated with an increase in is seen to thicken the momentum boundary layer. A reverse trend is seen in the case o nanoluid buoyancy ratio. iven that convective heating increases with Biot number simulates the isothermal surace, which is clearly seen rom Fig.3 (b), where, n act, a high Biot number indicates higher internal thermal resistance o the plate than the boundary layer thermal resistance. As a result, an increase in the Biot number leads to increase o luid temperature eiciently, these igures conirm this act also. As the parameter values and increases, the volume raction increased or the ixed values o the other parameters seen rom Fig.3(c). Fig. 4. presents the eect o the Brownian motion Nb and thermophoresis Nt parameters on the velocity, temperature and volume raction. t is observed that the momentum boundary layer thickness increases with the increase in values o Nb but it decreases with increasing values o Nt. The nanoparticle volume raction decreased with increase in Nb and it increased with increasing values o Nt. t is also noticed that the nanoparticle volume raction increased with an increase in Nb in the case o orced convection low. >0 indicates a cold surace while negative <0 corresponds a hot surace, in case o hot surace, thermophoresis tends to blow the nanoparticle volume raction away rom the surace since a hot surace repels the sub-micron sized particles rom it, thereby orming a relative particle-ree layer near the surace. Variation o non-dimensional velocity and temperature against the similarity variable, is shown respectively in Fig.5, or a ew set o values o λ and Pr with ixed values o other parameters. As the parameter λ and Pr increase, the velocity increased whereas temperature o nanoluid decreased. 78

8 '() = =1.1...=1. e=0.,0.3, Nt=0. Nb=0. Nr=0. Pr=1.0 am=0.4 Bi=1 C F Fig. (a): ect o and e on velocity '( ) 0 = =1.1...=1. 15 θ() e=0.,0.3,0.5 Nt=0. Nb=0. Nr=0. Pr=1.0 am=0.4 Bi=1 N Fig. (b): ect o and e on temperature θ ( ) 79

9 γ() Nt=0. Nb=0. Nr=0. Pr=1.0 am=0.4 Bi=1 = =1.1...=1. N O P e=0.,0.3, Fig. (c): ect o and e on volume raction γ ( ) Nr= Nr=0.3...Nr=0.5 '() Bi=0.1,0.3,0.5 Nt=0. Nb=0. Pr=1.0 =1.0 am=1.0 e=0. C F Fig. 3(a): ect o Bi and Nr on the velocity ' ( ) 80

10 θ() Bi=0.1,0.3,0.5 Nt=0. Nb=0. Pr=1.0 =1.0 am=1.0 e= Nr=0.1 Nr=0.3 Nr=0.5 N Fig. 3(b): ect o Bi and Nr on the temperature θ ( ) γ() Nr= Nr=0.3...Nr=0.5 Bi=0.1,0.3,0.5 Nt=0. Nb=0. Pr=1.0 =1.0 am=1.0 e=0. N O P Fig.3(c): ect o Bi and Nr on the volume raction γ ( ) 81

11 Nt=0.1,0.5,0.9 Nb= Nb= Nb= '() am=0.4, Nr=0., Pr=1.0, =1.0, e=0. Bi= C F Fig. 4(a): ect o Nb and Nt on the volume velocity '( ) θ() Nb= Nb=0.5...Nb=0.9 am=0.4, Pr=1.0, =1.0, e=0., Bi=0.1, Nr=0. N 0.1 Nt=0.1,0.5, Fig. 4(b): ect o Nb and Nt on the temperature θ ( ) 8

12 Nb= Nb=0.5...Nb=0.9 γ() Nt=0.1,0.5,0.9 am=0.4 Nr=0. Pr=1.0 =1.0 e=0. Bi=0.1 N O P Fig. 4(c): ect o Nb and Nt on the volume raction γ ( ) Pr= Pr=.0...Pr=3.0 '() lam=0.4,0.8,1. Nt=0. Nb=0. Nr=0. =1.0 e=0.4 Bi= C F Fig. 5(a): ect o λ and Pr on Velocity '( ) 83

13 5 θ() 0 15 am=0.4,0.8,1. Nt=0. Nb=0. Nr=0. =0. e=0.4 Bi=1 N Pr= Pr=.0...Pr= Fig. 5(b): ect o λ and Pr on Temperature θ ( ) 5. CONCUSON n this paper, the eect o agnetic parameter on mixed convection low along a vertical plate in a nanoluid is analyzed under the convective boundary conditions. Using the similarity variables, the governing equations are transormed into a set o non-dimensional parabolic equations. These equations are solved numerically using the Runge-utta Fourth order method. The numerical results are obtained or a wide range o values o the physical parameters. To ascertain the convergence o the numerical method adopted. The nanoparticle is considered in the analysis. The skin riction, heat and nanopartical mass coeicients are obtained or a physically realistic values o governing parameters. The results are analyzed thoroughly or dierent values o, Bi, and λ on the low, thermal and solutal ield. The major conclusion is that the magnetic eect enhanced the skin riction, heat and nanoparticle mass in the medium. NONCATUR Bi c D B D T Biot number constant Brownian diusion coeicient Thermophoretic diusion coeicient Dimensionless steam unction g ravitational acceleration r x ocal rasho number 84

14 h τ k e Nb Nr Nt Nu Pr q n x Convective heat transer coeicient Ratio between the eective heat capacity o the nano- particle material and heat capacity o the luid Thermal conductivity o the nanoluid ewis number Brownian motion parameter Nanopartical buoyancy ratio Thermophoresis parameter ocal Nusselt number Prandtl number Nanoparticle mass lux at the wall q w eat lux at the wall Re x ocal Reynolds number NSh x ocal nanoparticle Sherwood number T T T u agnetic number Temperature Temperature o the hot luid Ambient temperature Characteristic velocity u, v x, y coordinates along and normal to the plate α m γ λ θ φ φ w φ Velocity components in x and y direction Thermal diusivity Similarity variable Dimensionless volume raction ixed convection parameter Dimensionless temperature Nanoparticle volume raction Nanoparticle volume raction at the wall Nanoparticle volume raction at large values o y(ambient) µ Dynamic viscosity o the base luid υ inematic viscosity ρ Density o the luid Density o the base luid ρ ρ p Nanoparticle mass density ( ρ c) eat capacity o the luid ( ρ c) p ective heat capacity o the nanoparticle material τ w ψ Wall shear stress Stream unction 85

15 Subscripts w Wall condition Ambient condition c Concentration T Temperature RFRNCS 1... erkin, The eects o buoyancy orces on the boundary layer low over a semi-ininite vertical lat plate in a uniorm stream,. Fluid ech.35 (1996) Y. Ding,. Chen,. Wang, C.-Y. Yang, y. e, W. Yang, W. P. ee,. Zhang, R. uo, eat transer intensiication using nanoluids, ona 5 (007) B. C. Sakiadas, Boundary layer behavior on continuous solid suraces: Boundary layer equations or two dimensional and low, ACh.7 (1961) A. Bejan, Convection eat Transer, ohn Wiley, New York, S. V. Subhashini, Samuel Nancy,. Pop, Double-diusive convection rom a permeable vertical surace under convective boundary condition, nt. Commun. eat ass Transer 38 (011) S.. Das, S.U.S. Choi, A review o heat transer in nanoluids, Adv. eat Transer 41(009) S.. Das, S.U.S. Choi, W. Yu, T. Pradeep, Nanoluids: Science and Technology. Wiley, oboken, NY, Buongiorno, Convective transport in nanoluids, AS. eat Transer 18 (006) Schmidt, W. Beckmann, Das Temperature-und eschwindikeitseld voneiner warme abgebenden senkrechten platte bei naturlicher konvection,. Die Versuche und ihre rgibnisse, Forcsh, ngenieurwes 1 (1930) uiken, An asymptotic solution or large Prandtl number ree convection,. ngng. ath. (1968) uiken, Free convection at low Prandtl numbers,. Fluid ech 39 (1969) R. hair, A. Bejan, ass transer to natural convection boundary-layer low driven by heat transer, AS. eat Transer 107 (1985) O.D. ankinde, A. Aziz, Boundary layer low o a nanoluid past a stretching sheet with a convective boundary condition,nt.. Therm. Sci. 50(011) A.V. uznetsov, D.A. Nield, natural convective boundary-layer low o a nanoluid past a vertical plate, nt.. Therm. Sci. 49 (010) A.V. uznetsov, D.A. Nield, double-diusive natural convective boundary-layer low o a nanoluid past a vertical plate, nt.. Therm. Sci. 50 (011) Dr.N..Narve and Dr.N..Sane, xperimental nvestigation o aminar ixed Convection eat Transer n The ntrance Region o Rectangular Duct nternational ournal o echanical ngineering & Technology (T), Volume 4, ssue 1, 013, pp , SSN Print: , SSN Online: Dr. B.Tulasi akshmi Devi, Dr. B.Srinivasa Reddy,.V.P.N.Srikanth and Dr..Srinivas, ydromagnetic ixed Convection icro Polar Flow Driven by A Porous Stretching Sheet A Finite lement Study nternational ournal o echanical ngineering & Technology (T), Volume 5, ssue, 014, pp. 5-63, SSN Print: , SSN Online: