MHD flow and heat transfer near the stagnation point of a micropolar fluid over a stretching surface with heat generation/absorption

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1 Indian Journal of Pure & Applied Physics Vol. 5, October 3, pp MHD flo and heat transfer near the stagnation point of a micropolar fluid over a stretching surface ith heat generation/absorption R N Jat, Vishal Saxena* & Dinesh Rajotia Department of Mathematics, University of Rajasthan, Jaipur, India *Department of Mathematics, Yagyavalkya Institute of Technology, Jaipur, India Jat.rn_jp@yahoo.com, vishaljpr.raj@gmail.com * vishaljpr.raj@gmail.com Received 3 December ; revised 7 June 3; accepted 5 July 3 The steady laminar to dimensional stagnation point flo of an incompressible electrically conducting micropolar fluid impinging on a permeable stretching surface ith heat generation or absorption in the presence of transverse magnetic field has been studied. The viscous dissipation effect is taken into account. By taking suitable similarity variables, the governing system of partial differential equations are transformed into ordinary differential equations, hich are then solved numerically. The effects of various parameters such as the magnetic parameter, the surface stretching parameter, heat generation/absorption coefficient, material parameter, Eckert number and Prandtl number on the flo and heat transfer are presented and discussed graphically. Keyords: Micropolar fluid, MHD, Heat transfer, Stagnation point, Stretching surface Introduction Micropolar fluids are those fluids hich are consisting of randomly oriented particles suspended in a viscous medium. For micropolar fluids, in addition to their usual motion, fluid particles possess the ability to rotate about the centroid of the volume element in an average sense described by the skesymmetric gyration tensor. Polymer fluids, fluids ith certain additives such as animal blood, milk etc. are some examples of micropolar fluids. Eringen, probably is the first researcher ho introduced the concept of a micropolar fluid in an attempt to explain the behaviour of a certain fluid containing polymeric additives and naturally occurring fluids. A thorough revie of the subject and the application of micropolar fluid, mechanics has been given by Ariman et al 3,4. Ahmadi 5 obtained a self-similarity solution for micropolar boundary layer flo over a semi-infinite plate. Jena and Mathur 6,7 have studied the laminar free-convection flos of thermomicropolar fluids past a non-isothermal vertical flat plates. Gorla 8 has investigated the combined forced and free-convection in micropolar boundary layer flo on a vertical flat plate. Yucel 9 and Hossain et al. Have studied the mixed convection micropolar fluid over horizontal and vertical plates, respectively. Further Chiu and Chou, Char and Chang and Rees and Pop 3 and Prakash et al. 4 have investigated the free-convection in boundary layer flo of a micropolar fluid of different surfaces. Boundary layer on continuous surface is an important type of flo occurring in fluid flo problems. Sakiadis 4 initiated the theoretical study of boundary layer on a continuous semi-infinite sheet moving steadily through a quiescent fluid environment. Ebert 5 studied the similarity solution for boundary layer flo near a stagnation point. Hassanien and Gorla 6 have investigated the mixed convection in stagnation flos of micropolar fluids ith arbitrary variation of surface temperature. Boundary layer flo of a micropolar fluid toards a stretching sheet has been investigated by several researchers such as Desseaux 7, Takhar et al 8., Nazar et al 9., Lok et al., Ishak et al,., Attia 3,4, Ishak and Nazar 5. The interaction of a magnetic field ith the electrically conducting micropolar fluid is of more recent origin and has received considerable interest due to the increasing technical applications of the magnetohydrodynamic effects. Mohammadien and Gorla 6 investigated the effects of transverse magnetic field on a mixed convection in a micropolar fluid on a horizontal plate. Further, Eldabe and Ouaf 7, Mahmoud 8, Ishaq et al 9., Kumar 3, Khedr et al 3.,

2 684 INDIAN J PURE & APPL PHYS, VOL 5, OCTOBER 3 Hayat et al 3., Raat et al 33., Mahmoud and Waheed 34, Ashraf et al 35., Ahmad et al 36., Kumar et al 37. and more recently Ashraf and Rashid 38 studied the MHD boundary layer flo of a micropolar fluid toards a heated shrinking sheet ith radiation and heat generation. In the present paper, e have studied the flo and heat transfer near the stagnation point of an electrically conducting micropolar fluid over a stretching surface ith heat generation/absorption in the presence of applied normal magnetic field. Mathematical Formulation Consider a steady to-dimensional flo of an incompressible electrically conducting micropolar fluid (in the region y>) ith heat generation /absorption near the stagnation point at origin over a flat sheet coinciding ith the x-axis such that the sheet is stretched in its on plane ith uniform velocity proportional to the distance from the stagnation point in the presence of an externally applied normal magnetic field of constant strength B (Fig. ). The magnetic Reynolds number is assumed to be small so that the induced magnetic field is negligible. The stretching sheet has uniform temperature T and a linear velocity U hereas the temperature of the micropolar fluid far aay from the sheet is T and the velocity of the flo external to the boundary layer is U(x). All the fluid properties are assumed to be constant throughout the motion. Under usual boundary layer and Boussinesq approximations the governing boundary layer equations by considering viscous dissipation and heat generation/absorption are (Nazar et al 9.): u v + = x y. () u u du ( µ + h) u h N u + v = U + + σ B u x y dx ρ y ρ y ρ () N N γ N h u u + v = N + ρ x y j y j y T T T ρcpu + v = κ + Q T T x y y ( ) u + ( µ + h) + σ B u y...(3) (4) Fig. Physical model and coordinate system The boundary conditions are: y = : u = U = ax, v =, T = T, u N = m y y : u = U( x) = bx, T = T, N = (5) here u and v are the velocity components along x- and y-axes, respectively, T is fluid temperature, N is the micro-rotation or the angular velocity hose direction of rotation is in the xy-plane, Q the volumetric rate of heat generation /absorption, µ the viscosity, the density, the electrical conductivity, the thermal conductivity, c p the specific heat at constant pressure, j the micro-inertia density, the spin gradient viscosity, h the vortex viscosity, a(>) and b () are constants and m ( m ) is the boundary parameter. Here, j and h are assumed to be constants and is assumed to be given by Nazar et al 9. h γ = µ + j...(6) υ We take j = as a reference length, here υ is the a kinematic viscosity. 3 Analysis The equation of continuity given in Eq. () is identically satisfied by stream function Ψ (x,y) defined as:

3 JAT et al.: MHD FLOW AND HEAT TRANSFER OF A MICROPOLAR FLUID 685 ψ ψ u =, v = y x (7) For the solution of momentum, micro-rotation (spin) and the energy Eqs () to (4), the folloing similarity transformations in order to convert the partial differential equations into the ordinary differential equations are defined: ( ) ψ ( x, y) = x aυ f ( ), T T ( ) = T T a, = y υ a N( x, y) = ax g( ), υ (8) Using transformations given in Eq.(8), the Eqs () to (4) reduce to (after some simplifications): ( + K) f ''' + ff '' f ' + Kg ' + C Mf ' = (9) K + g '' + fg ' f ' g K ( g + f '') = '' + Pr f ' + Pr B + ( + K)Pr Ecf '' + MEc Pr f ' = The corresponding boundary conditions are: () () = : f =, f =, g = mf, = : f C, g, () here primes denote differentiation ith respect to, h b K = ( ) is the material parameter, C = is the µ a velocity ratio parameter (Stretching parameter), σ B M µc p = is the magnetic parameter, Pr = is ρa κ Q the Prandtl number, B = is the heat generation aρc (>)/absorption (<) coefficient and U Ec = is the Eckert number. cp ( T T ) Eq. () is obtained by Attia 4 for non-magnetic case hereas the Eqs (9) and () are differential equations ith values prescribed at to boundaries hich can be converted to initial value problem by knon technique. These equations are solved numerically on the computer for different values of the various parameters. The physical quantities of interest are the local skin-friction coefficient C f and the local Nusselt number Nu hich are defined, respectively as: C f τ =, ρu xq Nu = κ( T T ) p (3). C=.8, K= ; C=.8, K= ; C=.5, K= ; C=.5, K= ; C=., K= ; C=., K=.8 f' Fig. Velocity profile against for various values of parameters C and K hen M=.

4 686 INDIAN J PURE & APPL PHYS, VOL 5, OCTOBER 3 here τ is the all shear stress and is given by: u τ = ( µ + h) + hn y and q T = κ y sheet. Thus, e get: K + '' ( ) f C f ( Re) y= y= =, ( ) is the heat transfer from the Nu = Re '() (4) U x here Re = is the local Reynolds number. υ. 4 Results and Discussion The non-linear ordinary differential Eqs (9) and () subject to the boundary conditions given in Eq. () ere solved numerically. The computations ere done by a programme hich uses a symbolic and computational computer language Matlab. Figures to 4 sho the velocity profiles for various values of different parameters. It is observed from these graphs of velocity profiles that the velocity increases ith the increasing values of the parameters C and K hile it decreases ith the increase of the parameter M. Figures 5 to 9 sho the profiles of temperature distribution for various parameters. It is evident from Figs 5-9 that the temperature distribution shos reverse phenomenon ith the parameters C, K and Pr i.e. the temperature decreases ith the increasing values of these parameters hile it M =,.,.,.5,.8 f' Fig. 3 Velocity profile against for various values of parameter M hen C=.5 and K=. K = 5, 3,,,.8 f' Fig. 4 Velocity profile against for various values of parameter K hen M=. and C=.5

5 JAT et al.: MHD FLOW AND HEAT TRANSFER OF A MICROPOLAR FLUID 687. C=., K= ; C=., K= ; C=.5, K= ; C=.5, K= ; C=, K= ; C=, K= Fig. 5 Temperature profile against for various values of parameters C and K hen Pr=.7, B=., M=. and Ec=.. C=., B=. ; C=., B= -. ; C=.5, B=. ; C=.5, B= -. ; C=, B=. ; C=, B= Fig. 6 Temperature profile against for various values of parameters C and B hen Ec=., M=., Pr=.7 and K=. M =,.5,.,., Fig. 7 Temperature profile against for various values of parameter M hen Pr=.7, Ec=., C=.5, B=. and K=

6 688 INDIAN J PURE & APPL PHYS, VOL 5, OCTOBER 3. C=., Pr=. ; C=.5, Pr=. ; C=, Pr=. ; C=., Pr= ; C=.5, Pr= ; C=, Pr= Fig. 8 Temperature profile against for various values of parameters C and Pr hen K=, Ec=., M=. and B=.. Ec=, B=. ; Ec=.5, B=. ; Ec=, B= -. ; Ec=., B=. ; Ec=.5, B= -. ; Ec=., B = Fig. 9 Temperature profile agaist for various values of parameters Ec and B hen Pr=.7, K=, M=. and C=.5 Table Numerical values of f () for various values of parameters M and C hen K= M C=. C=. C=.3 C=.5 C= Table Numerical values of () for various values of parameters M and B hen C=.5, K=, Pr=.7 and Ec=. M B=-. B=-.5 B= B=.5 B= Table 3 Numerical values of () for various values of parameters Pr and C hen K=, B=., Ec=. and M=. Pr C=. C=. C=.3 C=.5 C= increases ith the parameter M, B and Ec. The variations of skin friction coefficient and heat transfer rate at the surface ere obtained and shon in Tables -3. References Eringen A C, Int J Eng Sci, (964) 5. Eringen A C, J Math Anal Appl, 38 (97) 48.

7 JAT et al.: MHD FLOW AND HEAT TRANSFER OF A MICROPOLAR FLUID Ariman T, Turk M A & Sylvester N D, Int J Eng Sci, (973) Ariman T, Turk M A & Sylvester N D, Int J Eng Sci, (974) Ahmadi G, Int J Eng Sci, 4 (976) Jena S K & Mathur M N, Int J Eng Sci, 9 (98) Jena S K & Mathur M N, Acta Mech, 4 (98) 7. 8 Gorla R S R, Int J Eng Sci, 6 (988) Yucel A, Int J Eng Sci, 7 (989) 593. Hossain M A, Chodhury M K & Takhar H S, J Theor Appl Fluid Mech, (995) 64. Chiu C P & Chou H M, Acta Mech, (993) 6. Char M I & Chang C L, J Phys D: Appl Phys, 8 (995) Rees D A S & Pop I, IMA J Appl Math, 6 (998) 7. 4 Prakash J, Ogulu A & Zhandire E, Indian J Pure & Appl Phys, 46 (8) Sakiadis B C, A I Ch E J, 7 (96) 6. 6 Ebert F, Chem Eng J, 5 (973) Hassanien I A & Gorla R S R, Int J Eng Sci, 8 (99) Desseaux A, The boundary layer flo of a micropolar fluid over a stretching porous sheet, 3 th Australasian Fluid Mechanics Conference, Manash University, Melbourne, Australia (998). 9 Takhar H S, Agaral R S, Bhargava R & Jain S, Heat Mass Transfer, 34 (998) 3. Nazar R, Amin N, Filip D & Pop I, Int J Non-Linear Mech, 39 (4) 7. Lok Y Y, Amin N, Campean D & Pop I, Int J Num Methods Heat Fluid Flo, 5 (5) 654. Ishak A, Nazar R & Pop I, Meccanica, 4 (6) Ishak A, Nazar R & Pop I, Meccanica, 43 (8) 4. 4 Attia H A, Tamkang J Sci Eng, 9 (6) Attia H A, J Braz Soc Mech Sci Eng XXX, (8) 5. 6 Ishak A & Nazar R, I C M S, (). 7 Mohammadien A A & Gorla R S R, Acta Mech, 8 (996). 8 Eldabe N T & Ouaf M E M, Appl Math Comput, 77 (6) Mahmoud M A A, Physica A, 375 (7) 4. 3 Ishaq A, Roslinda N & Pop I, Computers Mathematics Applications, 56 (8) Kumar L, Comp Mater Sci 46 (9) Khedr M E M, Chamkha A J & Bayomi M, Nonlinear Analysis: Modelling and Control, 4 (9) Hayat T, Javed T & Abbas Z, Nonlinear Analysis: Real World Applications, (9) Raat S, Bhargava R & Anar Beg O, Int J Appl Math Mech, 6 () Mahmoud M & Waheed S, Hindai Publishing Corporation, Mathematical Problems in Engineering, Article ID 5796 (). 36 Muhammad Ashraf, Anar Kamal M & Ashraf M, Middle- East Journal of Scientific Research, () Ahmad S, Ashraf M & Syed K S, World Applied Sciences Journal, 5 () Kumar L, Singh B, Kumar Lokesh & Bhargava R, Int J Appl Math Mech, 7 () Ashraf M & Rashid M, World Applied Sciences Journal, 6 () 338.