Free Convective Heat Transfer in Radiative MHD Casson Fluid Flow over a Stretched Surface of Variable Thickness

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1 ISSN (Paper) ISSN (Online) Vol.54, 07 Free Convective Heat Transfer in Radiative MHD Casson Fluid Flow over a Stretced Surface of Variable Tickness M.Barat Kumar Lecturer in Science, Government Polytecnic, Arakere, Mandya-5745, Karnataka, India Abstract Tis study deals wit te free convective eat transfer in a D magnetoydrodynamic flow of Casson fluid over a non-uniform tickness stretcing seet in te presence of termal radiation and non-uniform eat source/sink effects. Te governing equations of te flow and eat transfer are transformed as te asset of nonlinear ODEs and solved numerically using bvp4c Matlab package. Te effect of pertinent parameters, namely, magnetic field parameter, Casson parameter, termal radiation parameter, non-uniform eat source/sink parameters on te flow and eat transfer is investigated wit te assistance of graps. Numerical results are computed for te friction factor and reduced Nusselt number. It is observed tat te termal radiation as tendency to enance te temperature field of Casson fluid. Keywords: MHD, Casson fluid, Termal radiation, Slendering seet, non-uniform eat source/sink. Nomenclature: u, v : Velocity components in x and y directions x : Direction along te surface y : Direction normal to te surface C p : Specific eat capacity at constant pressure f : Dimensionless velocity A : Coefficient related to stretcing seet B( x ) : Magnetic field parameter T : Temperature of te fluid Ra : Termal radiation parameter A, B : non-uniform eat source/sink parameters k T m T a b m Pr M C f σ k Nu x Re x : Termal conductivity : Mean fluid temperature : Temperature of te fluid in te free stream : Dimensional velocity slip parameter : Dimensional temperature jump parameter : Termal accommodation coefficient : Pysical parameter related to stretcing seet : Velocity power index parameter : Prandtl number : Magnetic interaction parameter : Dimensionless velocity slip parameter : Dimensionless temperature jump parameter : Skin friction coefficient : Stefan-Boltzmann constant : mean absorption coefficient : Local Nusselt number : Local Reynolds number Greek Symbols η : Similarity variable 5

2 ISSN (Paper) ISSN (Online) Vol.54, 07 σ θ ρ β υ λ : Electrical conductivity of te fluid : Dimensionless temperature : Density of te fluid : Casson fluid parameter : Kinematic viscosity : Wall tickness parameter. Introduction In recent days, te investigations on non-newtonian fluids are significantly improved owing to teir immense pragmatic usages in te arena of science and engineering. Viscosity of te non-newtonian fluids pivots on sear rate. Several stuffs in te pysical world suc as, paints, sampoos, blood, and toot paste are specimens of non- Newtonian fluids. Te constituting equations are igly nonlinear in nature in tese fluids. No single model exists in te literature to describe every asset of non-newtonian fluids. Te induced magnetic field effect on te peristaltic transport of Carreau Liquid was numerically investigated by Hayat et al. []. Termal radiation effect on free convective nanofluid flow over a vertical plate was teoretically investigated by Sandeep et al. []. Sulocana and Sandeep [3] studied te flow and termal transport in MHD dusty flow over a stretcing/srinking cylinder by considering te various temperatures. Cattaneo-Cristov eat flux model for flow of variable termal conductivity generalized Burgers fluid was studied by Waqas [4]. Inclined magnetic field effect on MHD nanofluid flow was teoretically studied by Sandeep [5]. Te teoretical investigation of magnetoydrodynamic nanofluid flow embedded wit te magnetite nanoparticles Sandeep et al. [6]. Jayacandra Babu and Sandeep [7] studied te upper convected Maxwell fluid flow over a melting surface wit double stratification. Analysis of boundary layer formed on an upper orizontal surface of a paraboloid of revolution witin a nanofluid flow in te presence of termoporesis and Brownian motion was studied by Koriko et al. [8]. Magnetoydrodynamic Oldroyd-B fluid flow wit Soret and Dufour effects was numerically explored by Sandeep and Gnaneswara Reddy [9]. Moan Krisna et al. [0] investigated te parabolic flow of MHD Carreau fluid wit buoyancy effects. Te frictional eating effect on ferrofluid flow over a slendering seet was reported by Ramana Reddy et al. []. MHD stagnation flow of Casson fluid over a stretced surface wit convective boundary conditions was numerically studied by Ibraim and Makinde []. Hayat et al. [3] studied te 3D boundary layer flow of Sisko nanofluid wit transverse magnetic field effect. Te mixed convection flow of Maxwell nanofluid in te presence of eat generation and obsorption was studied by Abbasi et al. [4]. 3D flow of radiative nanofluid wit magnetic field and internal eat source effects was studied by Abbasi et al. [5]. Hayat et al. [6] studied te cemical reaction effects on radiative MHD flow. Analytical solution for eat and mass transfer of te mixed ydrodynamic/termal slip flow over a stretcing seet was studied by Turkyilmazoglu [7]. Dual solutions for unsteady mixed convection flow of micropolar fluid over a stretcing and srinking seet was studied by Sandeep and Slocana [8]. Te effects of radiation on MHD convective flow over a permeable stretcing surface wit suction and eat generation was studied by Moan Krisna et al. [9]. Te researcers [0-3] studied te eat transfer in magnetic non-newtonian fluid flows over various flow geometries. Te effect os Termoporesis and Brownian moment on nanofluid flow over a microcannel was studied by Fani et al. [4]. Te boundary layer flow of a nanofluid past a stretcing seet wit a convective boundary conditions was examined by Makinde and Aziz [5]. Te MHD treedimensional boundary layer flow of Casson nanofluid past a linearly stretcing seet wit te convective boundary condition Nadeem et al. [6]. Very recently, te researcers [7-30] studied te eat transfer in MHD flows. Present study deals wit te free convective eat transfer in a D magnetoydrodynamic flow of Casson fluid over a non-uniform tickness stretcing seet in te presence of termal radiation and non-uniform eat source/sink effects. Te governing equations of te flow and eat transfer are transformed as te asset of nonlinear ODEs and solved numerically using bvp4c Matlab package. Te effect of pertinent parameters on te flow and eat transfer is investigated wit te assistance of graps. Numerical results are computed for te friction factor and reduced Nusselt number. 6

3 ISSN (Paper) ISSN (Online) Vol.54, 07. Formulation of te problem Fig. Pysical Model Consider a steady D flow of Casson fluid over a stretcing seet of variable tickness in suc a way tat te x-axis is along te seet and te y-axis is perpendicular to it. It is assumed tat y = A( x + b) m, m ( ) 0 u ( x) = x+ b U, v = 0, m. Te termal radiation and non-uniform eat source/sink effects are w w taken into account. A transverse magnetic field of strengt B 0 is imposed along te flow direction as depicted in Fig.. Wit te above assumptions, te governing equations can be expressed as u u + = 0, x y u u u σb ( x) u + v = v + u, x y β y ρ 3 T T k T 6σ T T u + v = + + q''', x y ρc y ρc 3ρc k y ρc p p p p Wit te conditions u T u = Uw( x) +, v = 0, T = Tw ( x) +, y y and u( ) = 0, T( ) = T, were ( ) m f γ ξ = ξ x + b, ξ =, f γ + Pr = ( ) 0 + ( ) w 0 ( Tw T ) ku w ( T T ) q''' = A f B xv + ( T T ) a = ξ + a ( x b) m B( x) B x b m, T T = T x+ b (6) In Eq. (3) We now introduce te similarity transformations as we now suggest te following similarity transformations: 0.5 m+ ( ) U0( x b), w m 0.5 m+ ( x + b) = y U0 ψ = f η ν + η m+ υ ψ ψ If stream function ψ be described as u = and v = y x m is te non-uniform eat source/sink parameter.,, () () (3) (5) ( T ( x) T ) = T T (7) θ w 7

4 ISSN (Paper) ISSN (Online) Vol.54, 07 ( ) m '( ) νu0 m m u = U0 x + b f η and v ( m ) ( x b) f '( η) η = + + f ( η) + m+ wit te elp of (7) and (8), te equations ()-(3) converted as m + f ''' + f '' f f ' M f ' = 0, β m+ 4 m + θ + θ θ + + θ = 3 m+ Ra '' Pr f ' Pr f ' A f ' B 0, and te corresponding conditions are m f (0) = λ [ + f ''(0)], f '(0) = [ + f ''(0)], m + θ(0) '(0), [ θ ] = + f ' = 0, θ = 0, φ = 0 as η were M, Pr, Ra are defined as 3 µ C 0 p σ T Ra σb 4 M =,Pr =, =, () ρu0( m+ ) k k k Te pysical quantities of engineering interest, te friction factor and te local Nusselt number are given by u T µ ( x + b) y y Cf =, Nu x =, (3) ρuw Tw ( x) T By using (5), (9) becomes m ( ) ( ) m+ + Cf ( Rex) = ( + β ) f ''(0), Nux = Re x θ '(0), UwX Were Re x = and X = ( x + b) ν 3. Discussion of te results Te set of ODEs (8)- (0) wit te boundary conditions () is solved numerically by employing te bvp4c Matlab package. Te non-dimensional parameter values cosen as Pr = 6, 0.5 Ra =, = = A = B =, λ = (9) (8) (0) () (4) m =, M =, 0.5. Te values of tese parameters remain invariable in te present discussion unless tey are allowed to vary as in graps. Figs. and 3 depict te impact of external magnetic field on te flow and temperature fields of Casson fluid. It is observed tat te rising value of te transverse magnetic field parameter boosts te termal field and decline te velocity field. Pysically, increasing values of te magnetic field parameter strengten te Lorentz force, wic acts opposite to te flow field. Tis leads to decline te velocity profiles. Te similar results as been observed for increasing values of te Casson parameter as depicted in Figs. 4 and 5. Generally, increasing values of te Casson parameter suppresses te viscous nature of te flow. Figs. 6-8 illustrate te influence of termal radiation and non-uniform eat source/sink parameters on temperature profiles of te Casson fluid. It is noticed tat te rising values of te termal radiation and nonuniform eat source/sink parameters enances te temperature profiles of te flow. Pysically, te positive values of te uneven eat source/sink parameters act like eat generators. Tese causes to increase te temperature field. An opposite trend to above as been observed for rising value of te wall tickness parameter as sown in Fig. 9. Te effect of dimensionless velocity slip parameter on flow and termal fields is displayed in Figs. 0 and. It is clear tat te increasing value of te velocity slip declines te flow field and boosts te termal field. A reverse trend in termal field is observed for increasing values of te temperature jump parameter and Prandtl number. Wic is displayed in Figs. and 3. 8

5 ISSN (Paper) ISSN (Online) Vol.54, 07 Te variations in te skin friction coefficient and te local Nusselt number at different pertinent parameters is displayed in Table. It is observed tat te rising values of te magnetic field parameter, termal radiation parameter, velocity, termal slip parameters and Casson parameter reduces te local Nusselt number. Te increasing value of te wall tickness parameter enances te eat transfer rate. Fig. Effect of M on velocity field Fig.3 Effect of M on termal field Fig.4 Effect of β on velocity field 9

6 ISSN (Paper) ISSN (Online) Vol.54, 07 Fig.5 Effect of β on termal field Fig.6 Effect of Ra on termal field Fig.7 Effect of A on termal field 0

7 ISSN (Paper) ISSN (Online) Vol.54, 07 Fig.8 Effect of B on termal field Fig.9 Effect of λ on termal field Fig.0 Effect of on velocity field

8 ISSN (Paper) ISSN (Online) Vol.54, 07 Fig. Effect of on termal field Fig. Effect of on termal field

9 ISSN (Paper) ISSN (Online) Vol.54, 07 Fig.3 Effect of Pr on termal field Table Variations in pysical quantities at different pertinent parameters M Ra λ A β C f Nu x Conclusion In tis study, we analyzed te convective eat transfer in a D magnetoydrodynamic flow of Casson fluid over a non-uniform tickness stretcing seet in te presence of termal radiation and non-uniform eat source/sink effects. Te governing equations of te flow and eat transfer are transformed as te asset of nonlinear ODEs and solved numerically using bvp4c Matlab package. Te effect of pertinent parameters on te flow and eat transfer is investigated wit te assistance of graps. Numerical results are computed for te friction factor and reduced Nusselt number. Te numerical observations are as follows: Nonuniform eat source/sink parameters regulates te termal boundary layer. Rising values of te termal radiation decline te eat transfer rate and enances te temperature field. Increasing te wall tickness enances te eat transfer rate. Casson parameter as a tendency to reduce te eat transfer rate. Slip parameters control te termal filed. 3

10 ISSN (Paper) ISSN (Online) Vol.54, 07 References. T.Hayat, N.Saleem, N.Ali, Effect of induced magnetic field on peristaltic transport of a Carreau fluid. Commun Nonlinear Sci Numeric Simul, 5, , 00.. N.Sandeep, V. Sugunamma, P Moan Krisna, Effects of radiation on an unsteady natural convection flow of an EG-Nimonic 80a nanofluid past an infinite vertical plate. Advances in Pysics Teories and Applications, 3, 36-43, C.Sulocana, N.Sandeep, Flow and eat transfer beavior of MHD dusty nanofluid past a porous stretcing/srinking cylinder at different temperatures, Journal of Applied Fluid Mecanics, 9() (06) M. Waqas, T. Hayat, M. Farooq, S.A. Sezad, A. Alsaedi, Cattaneo-Cristov eat flux model for flow of variable termal conductivity generalized Burgers fluid, Journal of Molecular Liquids. 0 (06) N.Sandeep, Effect of Aligned Magnetic field on liquid tin film flow of magnetic-nanofluid embedded wit grapene nanoparticles, Advanced Powder Tecnology, 8, , N.Sandeep, A.J.Camka, I.L.Animasaun, Numerical exploration of magnetoydrodynamic nanofluid flow suspended wit magnetite nanoparticles, J Braz. Soc. Mec. Sci. Eng. DOI 0.007/s x, M.Jayacandra Babu, N.Sandeep, UCM flow across a melting surface in te presence of double stratification and cross-diffusion effects, Journal of Molecular Liquids, 3, 7-35, O.K.Koriko, A.J.Omowaye, N.Sandeep, I.L.Animasaun, Analysis of boundary layer formed on an upper orizontal surface of a paraboloid of revolution witin nanofluid flow in te presence of termoporesis and Brownian motion of 9 nm, International Journal of Mecanical Sciences, 4-5, -36, N.Sandeep, M.Gnaneswara Reddy, MHD Oldroyd-B fluid flow across a melting surface wit cross diffusion and double stratification, European Pysical Journal Plus, 3: 47, P. Moan Krisna, N. Sandeep, Ram Prakas Sarma, Computational analysis of plane and parabolic flow of MHD Carreau fluid wit buoyancy and exponential eat source effects, European Pysical Journal Plus, 3: 0, 07.. J.V. Ramana Reddy, V. Sugunamma, N. Sandeep, Effect of frictional eating on radiative ferrofluid flow over a slendering stretcing seet wit aligned magnetic field, Europen Pysical Journal Plus, 3: 7, 07.. W. Ibraim, O. D. Makinde: Magnetoydrodynamic stagnation point flow and eat transfer of Casson nanofluid past a stretcing seet wit slip and convective boundary condition. Journal of Aerospace Engineering, 9, Issue, Article number , T. Hayat, Taseer Muammad, S.A. Sezad, A. Alsaedi, On tree-dimensional boundary layer flow of Sisko nanofluid wit magnetic field effects, Advanced Powder Tecnology 7 (06) F.M. Abbasi, S.A. Sezad, T. Hayat, B. Amad, Doubly stratified mixed convection flow of Maxwell nanofluid wit eat generation/absorption, Journal of Magnetism and Magnetic Materials 404 (06) S.A. Sezad, Z. Abdulla, A. Alsaedi, F.M. Abbasi, T. Hayat, Termally radiative tree-dimensional flow of Jeffrey nanofluid wit internal eat generation and magnetic field, Journal of Magnetism and Magnetic Materials 397 (06) T. Hayat, T. Muammad, S.A.Sezad, A. Alsaedi and F. Al-Solamy, Radiative Tree-Dimensional Flow wit Cemical Reaction, International Journal of Cemical Reactor Engineering 4 (06) M.Turkyilmazoglu, Analytic eat and mass transfer of te mixed ydrodynamic/termal slip MHD viscous flow over a stretcing seet. Int J Mec Sci 53(0) (0) N.Sandeep, C.Sulocana, Dual solutions for unsteady mixed convection flow of MHD Micropolar fluid over a stretcing/srinking seet wit non-uniform eat source/sink. Engineering Science and Tecnology, an International Journal 8 (05) P. Moan Krisna, N.Sandeep, V.Sugunamma. Effects of radiation and cemical reaction on MHD convective flow over a permeable stretcing surface wit suction and eat generation. Walaliak Journal of Science and Tecnology Walailak J Sci & Tec 05; (9): E.M.A Elbasbesy, M.A.A Bazid, Heat transfer in a porous medium over a stretcing surface wit internal eat generation and suction or injection, Applied Matematics and Computer Science 58 (004) A. Isak, R. Nazar, I. Pop, Heat transfer over an unsteady stretcing permeable surface wit prescribed wall temperature, Nonlinear Analysis: Real World Applications 0 (009) A. Subas, P. Veena, Visco-Elastic fluid flow and eat transfer in a porous medium over a stretcing seet, Int. J. Non-Linear Mecanics, 33 (3) (998)

11 ISSN (Paper) ISSN (Online) Vol.54, Rafael Cortell, A note on flow and eat transfer of a viscoelastic fluid over a stretcing seet, International Journal of Non-Linear Mecanics, 4 (006) B. Fani, M. Kalte, A. Abbassi, Investigating te effect of Brownian motion and viscous dissipation on te nanofluid eat transfer in a trapezoidal microcannel eat sink, Advanced Powder Tecnology. 6 (05) doi:0.06/j.apt O.D. Makinde, A. Aziz, Boundary layer flow of a nanofluid past a stretcing seet wit a convective boundary condition, International Journal of Termal Sciences. 50 (0) doi:0.06/j.ijtermalsci S. Nadeem, R.U. Haq, N.S. Akbar, MHD tree-dimensional boundary layer flow of Casson nanofluid past a linearly stretcing seet wit convective boundary condition, IEEE Transactions on Nanotecnology. 3 (04) G.Sucarita, P.Laksmi Narayana, N.Sandeep, Joule eating and wall flexibility effects on te peristaltic flow of magnetoydrodynamic nanofluid, Int. J. of Mecanical Sciences, 3-3, 5-6, N. Sandeep, Ram Prakas Sarma, M. Ferdows, Enanced eat transfer in unsteady magnetoydrodynamic nanofluid flow embedded wit aluminum alloy nanoparticles, Journal of Molecular Liquids 34, , M.E.Ali, N.Sandeep, Cattaneo-Cristov model for radiative eat transfer of magnetoydrodynamic Casson-ferrofluid: A numerical study, Results in Pysics, 7, -30, M.Jayacandra Babu, N.Sandeep, Tree-dimensional MHD slip flow of a nanofluid over a slendering stretcing seet wit termoporesis and Brownian motion effects, Advanced Powder Tecnology, 7, , 06. 5

Radiation and Magneticfield Effects on Unsteady Mixed Convection Flow over a Vertical Stretching/Shrinking Surface with Suction/Injection

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