Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 51, 1, pp. 25-38, Warsaw 2013 MHD FLOW AND HEAT TRANSFER OF A MICROPOLAR FLUID OVER A STRETCHABLE DISK Muhammad Ashraf, Kiran Batool Bahauddin Zakariya University, Centre for Advanced Studies in Pure and Applied Mathematics, Multan, Pakistan e-mail: mashraf mul@yahoo.com Anumerical study of an axisymmetric steady laminar incompressible flow of an electrically conductingmicropolar fluid over a stretchable disk is carried outwhen the fluid is subjected to an external transverse magnetic field. The governing nonlinear equations of motion are transformed into a dimensionless form throughVonKarman’s logic similarity functions. An algorithmbased on a finite difference scheme is used to solve the reduced coupled nonlinear ordinary differential equations with the associated boundary conditions. Effects of the mi- cropolar parameters, the magnetic parameter and the Prandtl number on the flow velocity and temperature distribution arediscussed. Investigationspredict that the heat transfer rate at the surface of the disk increases with an increase in the values of micropolar parameters. The magnetic field enhances the shear and couple stresses. The shear stress factor is lower for micropolar fluids as compared to Newtonian fluids, whichmay be beneficial in flow and heat control of polymeric processing. Keywords:magnetohydrodynamics (MHD),micropolarfluids, heat transfer, stretchabledisk 1. Introduction The study of flow and heat transfer over a stretching surface has generated much interest in recent years in view of its numerous industrial applications such as extrusion of polymer sheets, extrusion of plastic sheet, rolling and manufacturing plastic films, artificial fibres, extrusion paper production, glass blowing, metal spinning, metal industries and drawing plastic films. The rate of heat transfer at the stretching surface determines the quality of the final product. The flow of an incompressible viscous fluid over a stretching surface has an important bearing on several technological applications in the field of metallurgy and chemical engineering. Nature is abundant with examples of flows involving non-Newtonian fluids. The flow of non-Newtonian fluids occurs in a wide variety of applications: from oil and gas well drilling, to well completion operations, from industrial processes involving waste fluids, synthetic fibres, foodstuffs to extrusion of molten plastic, and as well as in some flows of polymer solutions. This class of fluids represents mathematically many industrially important fluids, as paints, blood, body fluids, polymers, colloidal fluids and suspension fluid and is expected to provide a mathematical model for the non-Newtonian fluid behavior. Such flows have been attracting the attention of investigators for a long time. This attraction has been considerably growing especially during the last few decades. The resulting equations of non-Newtonian fluids are nonlinear, high order, and much more complicated than the Navier Stokes (NS) equation. The flow characteristics of non-Newtonian fluids are quite different in comparison to Newtonian fluids. The theory of micropolar fluids introduced by Eringen (1964, 1966) takes into account the microscopic effect arising fromthe localmicrostructureand intrinsicmotionoffluidparticles, and is expected to provide a mathematical model for the non-Newtonian fluid behavior. Such fluids can be subjected to surface and body couples in which thematerial points in a volume element 26 M. Ashraf, K. Batool can undergo motions about the centre of masses along with the deformation. In practice, the theory requires that onemust solve an additional transport equation representing the principle of conservation of local angular momentum as well as the usual transport equation for the conservation of mass and linear momentum, and additional constitutive parameters are also introduced. The surface stretching problem was first proposed and analyzed by Sakiadis (1961) based on the boundary layer approximation. Crane (1970) presented an exact solution of the two di- mensional NS equations for a stretching sheet problem with the closed analytical form, where the surface stretching velocity was proportional to the distance from the slot. This problem was later generalized to a power law stretching velocity (Banks, 1983). However, for the power law stretching velocity, the solution was not exact any more. The Crane (1970) problem with suction/injection at thewallwas investigated byGuptaandGupta (1977). Recently, the axisym- metric stretching surface problem was extended to a stretchable disk with both disk stretching and rotation by Fang (2007). Fang and Zhang (2008) presented an exact solution for the steady state NS equations in cylindrical polar coordinates by a similarity transformation technique. The solution involves the flow between two stretchable infinite disks with accelerated stretching velocity. The transition effect of the boundary layer flow due to a suddenly imposed magnetic field over a viscous flow past a stretching sheet and due to sudden withdrawal of the magnetic field over a viscous flow past a stretching sheet under a magnetic field was analyzed by Kumaran et al. (2010). Non dimensionalised equations were solved numerically using the implicit finite difference method of Crank Nicholson type. Ezzat et al. (2004) considered perfectly electrically conducting fluid past a non isothermal stretching sheet in the presence of a transverse magnetic field acting perpendicularly to the direction of motion of the fluid. Attia (2007) analyzed the steady MHD laminar three dimensional stagnation point flow of a viscous fluid impinging on a permeable stretching surface with heat generation or absorption. The theoretical analysis of the laminar boundary layer flow and heat transfer of power law non-Newtonian fluids over a stretching sheet was considered by Xu and Liao (2009). Takhar et al. (2000) analyzed the flow and mass transfer characteristics of a viscous electrically conducting fluid on a continuously stretching surface having non-zero slot velocity. The implicit finite difference scheme was used to solve the governing partial differential equations. A numerical study of the two dimensional boundary layer stagnation point flowover a stretching sheet in the case of injection/suction throughporous mediumwith heat transfer was considered by Layek et al. (2007). Hayat et al. (2009) analyzed the steady two dimensional MHD stagnation point flow of an upper convected Maxwell fluid over the stretching surface. The governing nonlinear partial differential equations were reduced to ordinary ones using the similarity transformation. The homotopy analysis method (HAM) was used to solve these equations. The numerical study of a steady laminar viscous boundary layer flow of an electrically conducting fluid over a heated stretching sheet in the presence of a uniform transverse magnetic field was analyzed by Panto- kratoras (2006). Kumaran et al. (2009) studied the problem of an MHD boundary layer flow of an electrically conducting fluid over a stretching and permeable sheet with injection/suction through the sheet. The study of a steady two dimensional stagnation point flow of amicropolar fluid over a stretching sheet when the sheet was stretched in its own plane and the stretching velocity was proportional to the distance from the stagnation point was examined by Nazar et al. (2004). The resulting coupled equations of nonlinear ordinary differential equations were solved numerically. Desseaux andKelson (2000) considered a boundary layer flow of amicropo- lar fluid driven by a porous stretching sheet. A similarity solution was defined, and numerical solutions were obtained using Runge Kutta and quasilinearisation schemes. The analysis of the boundary layer flow of a micropolar fluid on a fixed or continuously moving permeable surface MHD flow and heat transfer of a micropolar fluid... 27 was presented by Ishak et al. (2007). They considered both parallel and reversemoving surfaces to the free stream. The nonlinear ordinary differential equations were solved numerically. The purpose of the present work is to present a comprehensive parametric study of anMHD flow and heat transfer of a steady incompressible viscous electrically conducting micropolar fluid over a stretching disk in the presence of a uniformmagnetic field. A numerical solution is obtained for governingmomentum, angularmomentumand energy equations using an algorithm based on the finite difference approximation. 2. Problem formulation Consider an axisymmetric laminar incompressible flow of an electrically conducting micropolar fluid over a stretchable disk. A uniform transversemagnetic field B0 is applied at the disk. The governing equations ofmotion for theMHD laminar viscous flowof amicropolar fluid introduced by Eringen (1964, 1966) are: — continuity equation ∂ρ ∂t +(∇·ρV) = 0 (2.1) —momentum equation (λ+2µ+κ)∇(∇·V)+κ∇×v−∇p− (µ+κ)[∇(∇·V)−∇2V]+J×B= ρV̇ (2.2) — angular momentum equation (α+β+γ)∇(∇·v)−γ[∇(∇·v)−∇2v]+κ∇×V−2κv+ρl= ρjv̇ (2.3) —Gauss’s law for magnetism ∇·B=0 (2.4) —Ampere’s law ∇×B=µmJ (2.5) —Faraday’s law of induction ∇×E=0 (2.6) —Ohm’s law J=σe(E+V×B) (2.7) where V is the fluid velocity vector, v the microrotation, ρ the density, p the pressure, l the body couple per unit mass, j the microinertia, J the current density, ∇ the gradient operator, µm themagneticpermeability, E the electric field, B the totalmagnetic field so that B=B0+b, b the inducedmagnetic field and σe the electrical conductivity of the fluid, λ, µ,α, β, γ, κ the material constants ofmicropolar fluids (or viscosity coefficients), where the dot signifiesmaterial derivatives. From equation (2.4) and (2.5), we see that ∇·J = 0. The velocity vector V and the microrotation vector v are unknown. The induced magnetic field b is negligible as compared with the imposed field so that the magnetic Reynolds number is small (Shereliff, 1965). The applied polarization voltage is also assumed to be zero, which implies that the electric field E = 0. The electrical current flowing 28 M. Ashraf, K. Batool in the fluid will give rise to an induced magnetic field which would exist if the fluid was an electrical insulator.We have considered the fluid as electrically conducting, for our problem. In view of the above assumptions, the electromagnetic body force acting in equation (2.2) has the following linearized form (Rossow, 1958) J×B=σe[(V×B0)×B0] =−σeB20V (2.8) A suitable coordinate system for the present problem is a cylindrical polar coordinate system. The components of velocity (u,v,w) andmicrorotation (v1,v2,v3) along the radial, transverse and axial directions can be written as u=u(r,z) v=0 w=w(r,z) v1 =0 v2 = v2(r,z) v3 =0 (2.9) For the problem under consideration, equation (2.8) can be written as J×B=(−σeB20u,0,0) (2.10) For axisymmetric steady viscous incompressible fluid in the cylindrical polar coordinate system, governing equations of motion (2.1)-(2.3) for the boundary layer approximation, in view of equations (2.8)-(2.10), are reduced to the following dimensionless form: u r + ∂u ∂r + √ ω ν ∂w ∂η =0 ρ ( u ∂u ∂r +w √ ω ν ∂u ∂η ) =− ∂p ∂r −κ √ ω ν ∂v2 ∂η −σeB20u+(µ+κ) (∂2u ∂r2 + 1 r ∂u ∂r + ω ν ∂2u ∂η2 − u r2 ) ρj ( u ∂ν2 ∂r +w √ ω ν ∂ν2 ∂η ) =κ ( √ ω ν ∂u ∂η − ∂w ∂r ) −2κv2+γ (∂2v2 ∂r2 + 1 r ∂v2 ∂r + ω ν ∂2v2 ∂η2 − v2 r2 ) (2.11) And the equation for temperature field, neglecting the viscous dissipation, can be written as ρcp ( u ∂T ∂r +w √ ω ν ∂T ∂η ) −κ0 (∂2T ∂r2 + 1 r ∂T ∂r + ω ν ∂2T ∂η2 ) =0 (2.12) where T is the temperature, cp the specific heat at a constant pressure, κ0 is the thermal conductivity of the fluid, and η= z √ ω/ν is the similarity variable. Here, the quantity ω is a pseudoangular velocity corresponding to the disk stretching and its unit is 1/s. The boundary conditions over the stretching disk for the velocity field for the present problem are u(r,0)= rω w(r,0)= 0 u(r,∞) = 0 (2.13) The no-spin boundary conditions at the boundaries for microrotation are given by (v1,v2,v3)= (0,0,0) at η=0 and η=∞ (2.14) The boundary conditions for the temperature field can be written as T = { T0 at η=0 T∞ at η=∞ (2.15) where T0 is the constant temperature at thedisk, and T∞ is the constant temperature at infinity (with T0 >T∞). MHD flow and heat transfer of a micropolar fluid... 29 In order to obtain the velocity, microrotation and temperature fields for the present pro- blem, we have to solve equations (2.11) and (2.12) subject to the boundary conditions given in equations (2.13) and (2.14). For this purpose, we use the following similarity transformation u=− rωH′(η) 2 w= √ ωνH(η) p= ρωνP(η) v2 = √ ω ν rωg(η) θ(η)= T −T∞ T0−T∞ (2.16) Using equations (2.16)1,2, we see that continuity equation (2.11)1 is identically satisfied, and hence velocity components represent a possible fluidmotion. Now by using equations (2.16)1,2,3 in momentum equation (2.11)2, we get (H′)2 2 −HH′′+2c1g′+(1+ c1)H′′′−M2H′ =0 (2.17) where c1 = κ µ M2 = σeB 2 0 ρω are the vortex viscosity parameter and the magnetic parameter, respectively. M is also known as Hartmann number. Using equations (2.16)1−4 in the angular momentum equation (2.11)3, we get c2g ′′− c1 (H′′ 2 +2g ) − c3 ( Hg′−H ′ 2 g ) =0 (2.18) where c2 = γω νµ c3 = ρωj µ are the spin gradient viscosity parameter and themicroinertia density parameter, respectively. Energy equation (2.12), in view of equations (2.16)1,2 and (2.16)5, takes the form θ′′−PrHθ′ =0 (2.19) where Pr=µcp/κ0 is the Prandtl number. Boundaryconditions (2.13)-(2.15) inviewof transformation equations (2.16) indimensionless the form can be written as H(0)= 0 H′(0)=−2 H′(∞)= 0 g(0)= 0 g′(∞)= 0 θ(0)= 1 θ(∞)= 0 (2.20) The quantities of physical interest, the shear and couple stresses on the disk are defined respec- tively as τω =−(µ+κ) ∂u ∂z ∣ ∣ ∣ ∣ z=0 =µ(1+ c1)rω √ ω ν (H′′ 2 ) ∣ ∣ ∣ ∣ η=0 mω =−γ ∂v ∂z ∣ ∣ ∣ ∣ z=0 =−γrω ω ν g′ ∣ ∣ ∣ ∣ η=0 (2.21) We have to solve the system of equations (2.17), (2.18) and (2.19) subject to the boundary conditions given in equation (2.20). 30 M. Ashraf, K. Batool 3. Numerical solution Governing ordinary differential equations (2.17), (2.18) and (2.19) being highly nonlinear are difficult to solve analytically. We seek for a numerical solution to these equations subject to boundary conditions (2.20) using a finite difference scheme. The order of governing equations (2.17) and (2.18) can be reducedby one as donebyChamkhaand Issa (2000), Ashraf andAshraf (2011) and Ashraf and Bashir (2011), by substituting q=H′ = dH dη (3.1) Equations (2.17) and (2.18) in view of equation (3.1) can be written as q2 2 −Hq′+2c1g′+(1+ c1)q′′−M2q=0 c2g ′′−c1 (q′ 2 +2g ) − c3 ( Hg′− q 2 g ) =0 (3.2) Boundary conditions (2.20) are reduced as H(0)= 0 q(0)=−2 q(∞)= 0 g(0)= 0 g′(∞)= 0 θ(0)= 1 θ(∞)= 0 (3.3) For the numerical solution to the above boundary value problem, consisting of equations (3.2) and (2.19), wediscretize the domain [0,∞) uniformlywith a step size h. Simpson’s rule (Gerald, 1974) with the formula given inMilne (1953) is applied to integrate equation (3.1). The central difference approximations are used to discretize equations (2.19) and (3.2) at a typical grid point η= ηn of the interval [0,∞). The Successive Over Relaxation (SOR) method is used to solve iteratively the obtained system of algebraic equations subject to the associated boundary conditions given in equation (3.3). The solution procedure, which is mainly based on the algorithm described in Syed et al. (1997) is used to accelerate the iterative procedure and to improve the accuracy of the solution. The iterative procedure is stopped if the following criteria is satisfied for four consecutive iterations max ( ‖q(i+1)−q(i)‖2,‖g(i+1)−g(i)‖2,‖H(i+1)−H(i)‖2,‖θ(i+1)−θ(i)‖2 )