Chemical Reaction, Radiation and Dufour Effects on Casson Magneto Hydro Dynamics Fluid Flow over A Vertical Plate with Heat Source/Sink
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1 Global Journal of Pure and Applied Mathematics. ISSN Volume, Number (6), pp. 9- Research India Publications Chemical Reaction, Radiation and Dufour Effects on Casson Magneto Hdro Dnamics Fluid Flow over A Vertical Plate with Heat Source/Sink N.Vedavathi Assistant Professor, Department of Mathematics, K.L.Universit, Guntur Dist, A.P. India. G.Dharmaiah Assistant Professor, Department of Mathematics, Narasaraopeta Engineering College, NarasaraoPet, Guntur Dist, A.P., India. K.S.Balamurugan Associate Professor, Department Of Mathematics, R.V.R& J.C College of Engineering, Guntur, A.P, India. G. Charan Kumar Assistant Professor, Department of Mathematics, K.L.Universit, Guntur Dist, A.P., India. Abstract In this article it is examined Chemical reaction, Radiation and Dufour effects on casson MHD fluid flow over a vertical plate with heat source/sink. This Problem is solved numericall using the perturbation technique for the velocit, the temperature and the concentration species. The skin friction, Nusselt number and Sherwood number are also obtained and are shown in tabular form. The effects of various phsical parameters like as Chemical reaction parameter, Dufour parameter, Radiation parameter, Heat parameter, Casson parameter, Schmidt number, Grashof number, Prandtl number, Hartmann Parameter, and modified Grashof number has been discussed in detailed. Kewords: Casson fluid, Dufour effect, MHD, Chemical reaction.
2 9 N.Vedavathi et al Introduction The non-newtonian fluids plas a vital role in various fields such as pharmaceutical, biological, chemical, food, and personal care processing industries. complex fluids(non-linear) are power-law fluid, nano fluid, casson fluid [tomato sauce, hone, concentrated fruit juices, Human blood, etc.], micro polar fluid etc. These include mixtures of fluids, blood fluids and bio-fluids etc. Eldate.N.T.M. [] studied Heat Transfer of MHD non-newtoniancasson fluid flow between two rotating clinder. Mustafa.M., et al [] analsed Stagnation-point flow and Heat Transfer of a casson fluid towards a stretching sheet.rama SubbareddGorla.,et al., [3] discussed Mixed Convection in Non-Newtonian Fluids along a Vertical Plate in Porous Media with Constant Surface Heat Flux. KerehalliVinaaka Prasad., et al [4] studied Non- Newtonian Power-Law Fluid Flow and Heat Transfer over a Non-Linearl Stretching Surface. Kenneth walters., [5] used non-newtonian fluid Mechanics. RobensonCherizol.,et al [6] introduced Review of Non-Newtonian Mathematical Models for Rheological Characteristics of Viscoelastic Composites.Schowalter.w.r. [7] Mechanics of non-newtonian Fluids.Acrivosa.,shah.M.J. [8] explained Momentum and Heat Transfer in Laminar Boundar Laer Flows of Non-Newtonian Fluids Past External Surfaces. Authors [9-4] have been analsed on Non-Newtonian Fluids and other aspects. In this article it is examined Chemical reaction, Radiation and Dufour effects on Casson MHD fluid flow over a vertical plate with heat source/sink. This Problem is solved numericall using the perturbation technique for the velocit, the temperature and the concentration species. The skin friction, Nusselt number and Sherwood number are also obtained and are shown in tabular form. The results are made in this article are good agreement with previous work[5]. Formulation of the problem MHD Casson fluid of incompressible, viscous, electricall- conducting fluid over a vertical plate moving with constant velocit with radiation and chemical reaction in the presence of Dufour effect is considered. The rheological equation of state for an isotropic and incompressible flow of Casson fluid [,] is B p eij, c ij B p c eij, c A p Where B is plastic dnamic viscosit, is ield stress, c is critical value of,and is the product of the component of deformation rate with itself, namel, eijeij, eij is the ( i, j ) th component of deformation rate. The x - axis is taken along
3 Chemical Reaction, Radiation And Dufour Effects On Casson Magneto Hdro 93 the plate in the vertical upward direction and the - axis is taken normal to the plate. The surface temperature of the plate oscillates with small amplitude about a nonuniform mean temperature. The fluid is assumed to have constant properties except for the influence of the densit variations with temperature and concentration which are considered onl in the bod force term. The temperature of the plate oscillates with little amplitude about a non-uniform temperature. B usual Boussinesq s approximation, the flow is governed b the following equations. u u g TT t ' g CC B u...() () T k T q r t C C Q C p p C ' ( TT ) Du...() C C D K C C t p...(3) r (3) Equations (), () and (3) refers Momentum equation, Energ Equation and Species Equation respectivel. Where u is the velocit of the fluid, is Casson parameter, Q is the heat source/sink parameter, D is the molecular diffusivit, k is thermal conductivit, C is mass concentration, t is time, υ is the kinematics viscosit, g is the gravitational constant, * and are the thermal expansions of fluid and concentration, T is temperature of fluid, ρ is densit, cp is the specific heat capacit at constant pressure, is distance, qr is the radiative flux, is the magnetic field, kr is the chemical reaction rate constant. R.H.S. of equation (), third term is thermal concentration effect, fourth term is magnetic effect, second term is thermal buoanc effect. R.H.S. of equation () second term is thermal radiation flux and third term is thermal radiation and fourth term is Dufour effect. R.H.S. of equation (3), second term is chemical reaction and third term Dufour (Diffusion Thermo) effect. Under the above assumptions the phsical variables are functions of and t. The boundar conditions for the velocit, temperature and concentration fields are: ()
4 94 N.Vedavathi et al it u U, T T ( T T) e, it C C ( C C ) e at u, T, C, as...(4) (4) Introducing the dimensionless quantities with thermal radiation flux gradient expressed andwe assume that the temperature differences within the flow are 4 sufficientl small so that T can be expressed as a linear function of T after using 4 Talor s series to expand T about the free stream temperature T and neglecting higher-order terms. This results in the following approximation: q r 4 4 4a T T and T 4T T 3T ; u U tu u,, t, U v v T T C C Q, C, Q T T C C CU p w w Ku C p v K,Pr, Sc, v k D Bv v g TT w...(5) M, Gr, 3 U U v g C C w Kv Gm, K, 3 r U U 3 6 a * T Du R, v, D C U v T w T p (5) The thermal radiation flux gradient ma be expressed as follows q 4 r 4 a *( T ' T ' ) ' (6) Considering the temperature difference b assumption within the flow are sufficientl 4 small such that T ' ma be expressed as a linear function of the temperature. This is 4 attained b expanding in T ' T ' talor s series about and ignoring higher orders terms T ' 4 T ' T ' 3 T ' (7)
5 Chemical Reaction, Radiation And Dufour Effects On Casson Magneto Hdro 95 Substituting the dimensionless variables (5) into () to (3) and using equations (6) and (7), reduce to the following dimensionless form. Substituing the dimentionless variables(5) into above all equations and reduces to the following. u u Gr Gc Mu t (8) C R Q D t Pr (9) C t Sc C KrC The corresponding boundar conditions of (4) in dimensionless form are u, e, C e at u,, C as () Where Gr is thermal Grashof number, Pr is the prandtl number, kr is the chemical reaction parameter, R is the thermal radiation conduction number, M is Hartmann number, Gc is the mass Grashof number, Q is the heat source/sink parameter. () Method of Solution Equations (8),(9) and () represents a set of partial differential equations that cannot be solved in closed form. However, it can be reduced to a set of ordinar differential equations in dimensionless form that can be solved analticall. This can be done b representing the velocit, temperature and concentration as u u e u O e O C C e C O ( )..., ( )..., ( )... u ( ), u ( ), ( ), ( ), C ( ), andc ( ) have to be determined. u" Mu Gr GcC A u" ( M iw) u Gr GcC Where 4 " ( R Q) DC " Pr " ( R Q iw) Pr D C " Pr C " Sck C r C " Sc( k iw) C r All primes denote differentiation with respect to. () (3)
6 96 N.Vedavathi et al The boundar conditions are u,, C at, u,, C at, u,, C as, u,, C as. Solving the sstem () subject to the boundar conditions (), We obtain A5 A 6 A u ge e A A5 A Sckr A 5 e Sckr g u B5e e A A8 A8 A g g e e A A A A 3 A3 4 A A Be Be B e B e A 4 3 Sckr A3 Sckr A3,...(5),, C e C e (5) In view of the above solutions, the velocit, temperature and concentration distributions in the boundar laer becomes u u e u, e, C C e C (6) The skin friction(cf), Nusselt number(nu) and Sherwood number(sh) are obtained from equation(4) when differentiated at=. A6 g A5 A A A 5 C f A Sckr Sckr A5 g B5 A8 A A A 8 e, ( g3 g4) A3 A3 A8 ( B A Sckr. B) Nu e ( B A B A Sh Sckr e A3, Using equation (6),(7) the results analsed and determined. (4) (7)
7 Chemical Reaction, Radiation And Dufour Effects On Casson Magneto Hdro 97 Results and Discussion Casson MHD flow over a vertical plate with dufour parameter has been formulated and analsed analticall. Onl three computations are performed for Variation of the velocit with thermal coefficient, Variation of the temperature with thermal radiation conduction, Variation of the concentration with chemical reaction parameter. velocit.5 =.5 = = = Figure : Variation of the velocit with thermal coefficient The dimensional governing equations are solved b two term perturbation technique in this article with Pr =, Gr =., GC =., ε =., M =.5, t =.,Sc=., kr =.5,R =.,Q=.,ω=.,Du=.3. All graphs therefore correspond to these unless specificall indicated on the appropriate graph.the influence of Casson parameter in velocit is shown in Fig.. It is recognised that velocit decreases far wa the plate and increases near the plate while increase in beta. Fig.. tells the influence of thermal radiation conduction on the temperature. It is cleared that temperature is decrease when R is increase. The influence of kr on the concentration is illustrated in Fig.3. The concentration is decreases as the chemical reaction parameter increases..5 temperature.5 R =. R =.5 R =.8 R = Figure : Variation of the temperature with thermal radiation conduction
8 98 N.Vedavathi et al.5 concentration.5 kr =. kr =.5 kr =. kr = Figure 3: Variation of the concentration with chemical reaction parameter. Table : Variation of the the Skin-friction coefficient, Nusselt number and Sherwood number. β M R Cf Nu Sh Table tells the effects of Casson parameter, Hartmann number and thermal radiation conduction with Pr =, Gr =., GC =., ε =., M =.5, t =.,Sc=., kr =.5,R =.,Q=.,ω=., Du=.3. Conclusion In this article it is examined Chemical reaction, Radiation and Dufour effects on Casson MHD fluid flow over a vertical plate with heat source/sink. This Problem is solved numericall using the perturbation technique for the velocit, the temperature and the concentration species. The skin friction, Nusselt number and Sherwood number are also obtained and are shown in tabular form. The skin-friction is increases with the effect of casson parameter, at the plate. As perturbation parameter increases, the Skin-friction coefficient, Nusselt number and Sherwood number are decreases at the plate and hence some will not discuss an further due to brevit.
9 Chemical Reaction, Radiation And Dufour Effects On Casson Magneto Hdro 99 Acknowledgements Firstl, I would like to express m sincere gratitude to our advisor, Associate Professor Dr.K.S.Bala Murugan, R.V.R.College Of Engineening,for the continuous support of our stud and related research, for his patience, motivation, and immense knowledge. His guidance helped us in all the time. The authors are grateful to Dr.K.S.BalaMurugan for his valuable suggestions, discussions and guidance on this work. References [] Eldate.N.T.M. Heat Transfer of MHD non-newtoniancasson fluid flow between two rotating clinder. Journal of Phs SocJpn64,(995): [] Mustafa.M.,Haat.T., Pop.I., Hendi. A. Stagnation-point flow and Heat Transfer of a casson fluid towards a stretching sheet. Z naturforsch,67():7-76. [3] Rama SubbareddGorla., Ali J. Chamkha., HarmindarTakhar. Mixed Convection in Non-Newtonian Fluids along a Vertical Plate in Porous Media with Constant Surface Heat Flux. Thermal Energ and Power Engineering - (3): [4] KerehalliVinaaka Prasad., SeetharamanRajeswariSanthi., PampannaSomannaDatti., Non-Newtonian Power-Law Fluid Flow and Heat Transfer over a Non-Linearl Stretching Surface. Applied Mathematics, 3, (): [5] Kenneth walters., non-newtonian fluid Mechanics. Rheolog,. [6] RobensonCherizol., MohiniSain., Jimi Tjong., Review of Non-Newtonian Mathematical Models for Rheological Characteristics of Viscoelastic Composites Green and Sustainable Chemistr 5, (5): 6-4. [7] Schowalter.w.r. Mechanics of non-newtonian Fluids. Pergamum press oxford. [8] Acrivosa.,shah.M.J. andpetersen.e.e. Momentum and Heat Transfer in Laminar Boundar Laer Flows of Non-Newtonian Fluids Past External Surfaces. AICHE Journal 6(96):3-37. [9] Chen HT., Chen. CK., Natural convection of a non-newtonian fluid about a horizontal clinder and sphere in a porous medium. International Communications in Heat and Mass Transfer 5,(988): [] Nakaama A., Koama H., Buoanc induced flow of non-newtonian fluids over a non-isothermal bod of arbitrar shape in a fluid-saturated porous medium. Applied Scientific Research 48 (99): [] Yang.Y.T., Wang. S.J., Free convection heat transfer of non-newtonian fluids over axismmetric and two-dimensional bodies of arbitrar shape embedded in a fluid-saturated porous medium. International Journal of Heat and Mass Transfer 39 (996): 3-.
10 N.Vedavathi et al [] K.N. Mehta., K.N. and K.N. Rao., K.N. Buoanc-induced flow of non- Newtonian fluids in a porous medium past a vertical plate with nonuniform surface heat flux. Int. J. Eng. Sci., 3(994): [3] Gorla. R.S.R., Shanmugam. K., and Kumari. M., Mixed convection in non- Newtonian fluids along nonisothermal horizontal surfaces in porous media. Heat and Mass Transfer, 33 (998): [4] Chamkha, A.J. and Al-Humoud, J. Mixed convection heat and mass transfer of non-newtonian fluids from a permeable surface embedded in a porous medium. Int. J. Numer.Meth. Heat & Fluid Flow, 7 (7): 95-. [5] M.J.Subhakar,T.PrasannaKumar,K.Kezia,K.Gangadhar, Effect of MHD Casson Fluid flow over a vertical Plate with Heat Source International journal of Scientific and Innovative Mathematical Research.,3(5),(5):- 38.
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