Indian Journal of Pure & Applied Physics Vol. 51, October 2013, pp. 683-689
MHD flow and heat transfer near the stagnation point of a micropolar fluid over a stretching surface with 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 E-mail:
[email protected],
[email protected] *E-mail:
[email protected] Received 31 December 2012; revised 17 June 2013; accepted 25 July 2013 The steady laminar two dimensional stagnation point flow of an incompressible electrically conducting micropolar fluid impinging on a permeable stretching surface with 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, which 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 flow and heat transfer are presented and discussed graphically. Keywords: Micropolar fluid, MHD, Heat transfer, Stagnation point, Stretching surface
1 Introduction Micropolar fluids are those fluids which 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 skewsymmetric gyration tensor. Polymer fluids, fluids with certain additives such as animal blood, milk etc. are some examples of micropolar fluids. Eringen1,2 probably is the first researcher who 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 review of the subject and the application of micropolar fluid, mechanics has been given by Ariman et al3,4. Ahmadi5 obtained a self-similarity solution for micropolar boundary layer flow over a semi-infinite plate. Jena and Mathur6,7 have studied the laminar free-convection flows of thermomicropolar fluids past a non-isothermal vertical flat plates. Gorla8 has investigated the combined forced and free-convection in micropolar boundary layer flow on a vertical flat plate. Yucel9 and Hossain et al10. Have studied the mixed convection micropolar fluid over horizontal and vertical plates, respectively.
Further Chiu and Chou11, Char and Chang12 and Rees and Pop13 and Prakash et al.14 have investigated the free-convection in boundary layer flow of a micropolar fluid of different surfaces. Boundary layer on continuous surface is an important type of flow occurring in fluid flow problems. Sakiadis14 initiated the theoretical study of boundary layer on a continuous semi-infinite sheet moving steadily through a quiescent fluid environment. Ebert15 studied the similarity solution for boundary layer flow near a stagnation point. Hassanien and Gorla16 have investigated the mixed convection in stagnation flows of micropolar fluids with arbitrary variation of surface temperature. Boundary layer flow of a micropolar fluid towards a stretching sheet has been investigated by several researchers such as Desseaux17, Takhar et al18., Nazar et al19., Lok et al20., Ishak et al21,22., Attia23,24, Ishak and Nazar25. The interaction of a magnetic field with 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 Gorla26 investigated the effects of transverse magnetic field on a mixed convection in a micropolar fluid on a horizontal plate. Further, Eldabe and Ouaf27, Mahmoud28, Ishaq et al29., Kumar30, Khedr et al31.,
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Hayat et al32., Rawat et al33., Mahmoud and Waheed34, Ashraf et al35., Ahmad et al36., Kumar et al37. and more recently Ashraf and Rashid38 studied the MHD boundary layer flow of a micropolar fluid towards a heated shrinking sheet with radiation and heat generation. In the present paper, we have studied the flow and heat transfer near the stagnation point of an electrically conducting micropolar fluid over a stretching surface with heat generation/absorption in the presence of applied normal magnetic field. 2 Mathematical Formulation Consider a steady two-dimensional flow of an incompressible electrically conducting micropolar fluid (in the region y>0) with heat generation /absorption near the stagnation point at origin over a flat sheet coinciding with the x-axis such that the sheet is stretched in its own plane with uniform velocity proportional to the distance from the stagnation point in the presence of an externally applied normal magnetic field of constant strength B0 (Fig. 1). The magnetic Reynolds number is assumed to be small so that the induced magnetic field is negligible. The stretching sheet has uniform temperature Tw and a linear velocity Uw whereas the temperature of the micropolar fluid far away from the sheet is T and the velocity of the flow 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 19 generation/absorption are (Nazar et al .): ∂u ∂v + =0 ∂x ∂y .
u
…(1)
∂u ∂u dU ( µ + h) ∂ 2u h ∂N 1 +v =U + + − σ B 2u ∂x ∂y dx ρ ∂y 2 ρ ∂y ρ 0 …(2) § ∂N ∂N +v ∂ x ∂y ©
ρ ¨u
· γ ∂2 N h § ∂u · − ¨ 2N + ¸= ¸ 2 j j ∂y ¹ ∂y ¹ ©
§ ∂T ∂T +v ∂y © ∂x
ρcp ¨ u
· ∂ 2T ¸ = κ 2 + Q (T − T∞ ) ∂y ¹ 2
§ ∂u · + ( µ + h) ¨ ¸ + σ B02u 2 © ∂y ¹
...(3)
…(4)
Fig. 1 — Physical model and coordinate system
The boundary conditions are: y = 0 : u = Uw = ax , v = 0 , T = Tw , N = − m y → ∞ : u = U ( x) = bx , T = T∞ , N = 0
∂u ∂y …(5)
where u and v are the velocity components along xand y-axes, respectively, T is fluid temperature, N is the micro-rotation or the angular velocity whose direction of rotation is in the xy-plane, Q the volumetric rate of heat generation /absorption, µ the viscosity, ȡ the density, ı the electrical conductivity, ț the thermal conductivity, cp the specific heat at constant pressure, j the micro-inertia density, the spin gradient viscosity, h the vortex viscosity, a(>0) and b (0) are constants and m (0 m 1) is the boundary parameter. Here , j and h are assumed to be constants and is assumed to be given by Nazar et al19. § ©
h·
γ =¨µ + ¸ j 2 ¹
...(6)
υ
as a reference length, where υ is the a kinematic viscosity. We take j =
3 Analysis The equation of continuity given in Eq. (1) is identically satisfied by stream function Ψ (x,y) defined as:
JAT et al.: MHD FLOW AND HEAT TRANSFER OF A MICROPOLAR FLUID
u=
∂ψ ∂ψ , v=− ∂x ∂y
…(7)
For the solution of momentum, micro-rotation (spin) and the energy Eqs (2) to (4), the following similarity transformations in order to convert the partial differential equations into the ordinary differential equations are defined:
ψ ( x, y ) = x ( aυ )
1
2
§a· f (η ) , N ( x, y) = ax ¨ ¸ ©υ ¹
T − T∞ §a· , η = y¨ ¸ θ (η ) = Tw − T∞ ©υ ¹
1
1 2
g(η ) ,
2
…(8)
Using transformations given in Eq.(8), the Eqs (2) to (4) reduce to (after some simplifications):
(1 + K ) f '''+ ff ''− f '2 + Kg '+ C 2 − Mf ' = 0 K· § ¨ 1 + 2 ¸ g ''+ fg '− f ' g − K (2 g + f '') = 0 © ¹
…(9) …(10)
θ ''+ Pr f θ '+ Pr Bθ + (1 + K ) Pr Ecf ''2
…(11)
+ MEc Pr f '2 = 0 The corresponding boundary conditions are:
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η = 0: f = 0, f ′ = 1, g = −mf ″, θ = 1 η ĺ ∞: f ′ ĺ C, g ĺ 0, θ ĺ 0
…(12)
where primes denote differentiation with respect to Ș, b h is the K = ( ≥ 0 ) is the material parameter, C = a µ velocity ratio parameter (Stretching parameter), µcp σ B02 is the magnetic parameter, Pr = is M= κ ρa Q is the heat generation the Prandtl number, B = aρcp (>0)/absorption (
