NUMERICAL SOLUTION ON HEAT TRANSFER MAGNETOHYDRODYNAMIC FLOW OF MICROPOLAR CASSON FLUID OVER A HORIZONTAL CIRCULAR CYLINDER WITH THERMAL RADIATION

This paper focuses on the numerical solution for magnetohydrodynamic (MHD) flow of micropolar Casson fluid with thermal radiation over a horizontal circular cylinder. The nonlinear partial differential equations of the boundary layer are first transformed into a non-dimensional form and then solved numerically using an implicit finite difference scheme known as Keller-box method. The The effects of the emerging parameters, namely Casson fluid parameter, magnetic parameter, radiation parameter and micropolar parameter on the local Nusselt number and the local skin friction coefficient, as well as the temperature, velocity and angular velocity profiles are shown graphically and discussed. The present results of local Nusselt number and the local skin friction for viscous fluid are found to be in good agreement with the literature.


INTRODUCTION
Casson fluids in the presence of heat transfer is widely used in the processing of chocolate, foams, syrups, nail, toffee and many other foodstuffs (Ramachandra et al. (2013). Casson (1959), in his pioneering work introduced this model to simulate industrial inks. Later on, a substantial study has been done on the Casson fluid flow because of its important engineering applications. Mustafa et al. (2011) have studied the heat transfer flow of a Casson fluid over an impulsive motion of the plate using the homotopy method. The exact solution of forced convection boundary layer Casson fluid flow toward a linearly stretching surface with transpiration effects are reported by Mukhopadhyay et al. (2013). In the same year, Subba et al. (2015) considered the velocity and thermal slip conditions on the laminar boundary layer heat transfer flow of a Casson fluid past a vertical plate. Mahdy and Ahmed (2017) studied the effect of magnetohydrodynamic on a mixed convection boundary flow of an incompressible Casson fluid in the stagnation point of an impulsively rotating sphere. The convective boundary layer flow of Casson nanofluid from an isothermal sphere surface is presented by Nagendra et al. (2017). Mehmood et al. (2017) investigated the micropolar Casson fluid on mixed convection flow induced by a stretching sheet. Shehzad et al. (2013) discussed the viscous chemical reaction effects on the MHD flow of a Casson fluid over a porous stretching sheet. Recently, Khalid et al. (2015) developed exact solutions for unsteady MHD free convection flow of a Casson fluid past an oscillating plate. Amongst the various investigations on Casson fluid, the reader is referred to some new attempts made in Qasim and Noreen (2014;Hussanan et al. (2014) and Haq et al. (2014), and the references therein.
Among the class of several other non-Newtonian fluid models namely micropolar fluids, this fluid flow lies in the extension of the constituent equation for Newtonian fluid, so that more complex fluids such as liquid crystal, particle suspensions, animal blood, lubrication and turbulent shear flows can be described by this theory Lukaszewicz (1999). The theory of micrpolar fluids was first introduced by Eringen (1966). Ariman et al. (1973) investigated the application of micropolar fluid mechanics as review paper. The recent book by Eringen (2001) presented a useful account of the theory and extensive surveys of literature of micropolar fluid theory.
The study of boundary layer flow on a horizontal circular cylinder was first studied by Blasius (1908), who successfully solved the momentum equation of forced convection boundary layer flow. Merkin (1976) considered the free convection boundary layer on an isothermal horizontal cylinder with constant wall temperature and became the first who obtained the exact solution for this problem. Ingham (1978) developed the numerical method to solved free convective boundary layer flow on an isothermal horizontal cylinder. Merkin and Pop (1988) presented the numerical solution of the free convection boundary layer flow on a horizontal circular cylinder with constant heat flux using the Keller-box method. Next, the extended by the work of Merkin (1976) and Merkin and Pop (1988) for free convection boundary layer flow on a horizontal circular cylinder in viscous fluid to a micropolar fluid was investigated by Nazar et al. (2002). Moreover Salleh and Nazar (2010) and Alkasasbeh et al (2015; work with Newtonian heating. Gaffar et al. (2015) investigated the laminar boundary layer flow and heat transfer of a Tangent Hyperbolic non-Newtonian fluid from horizontal circular cylinder with slip condition. Recently, Gaffar et al. (2017) studied the magnetohydrodynamic (MHD) free convection flow and heat transfer of non-Newtonian tangent hyperbolic fluid from horizontal circular cylinder with convective boundary conditions. Based on the above contribution, the aim of present study is to investigate the effect of MHD on free convective boundary layer flow about a horizontal circular cylinder in a micropolar Casson fluid with thermal radiation and this problem has to the author knowledge not appeared thus far in the scientific literature.

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MATHEMATICAL MODELING
Consider the steady, laminar, two-dimensional, viscous, incompressible, buoyancy-driven convection heat transfer flow from a horizontal permeable circular cylinder embedded in a micropolar Casson fluid. For many actual fluids and flow conditions, a simple and convenient way to express the density difference ( ) ρ ρ ∞ − in the buoyancy term of the momentum equations is given by the Boussinesq approximation where ρ ∞ is the constant local density, T is the local temperature, T ∞ is the temperature of the ambient medium. ρ is the fluid density and B is the thermal expansion coefficient,. Figure 1 shows the flow model and physical coordinate system. The x − coordinate is measured along the circumference of the horizontal cylinder from the lowest point and the y − coordinate is measured normal to the surface, with a denoting the radius of the horizontal cylinder. / x a is the angle of the y − axis with respect to the vertical 0 / x a π ≤ ≤ . The gravitational acceleration, g acts downwards. Both the horizontal cylinder and the fluid are maintained initially at the same temperature. Instantaneously they are raised to a temperature w T T ∞ > the ambient temperature of the fluid which remains unchanged.
The constitutive relationship for an incompressible Casson fluid flow, reported by Mukhopadhyay et al. (2013).
, ij e is the ( , ) i j − th component of the deformation rate, B µ is the plastic dynamic viscosity of the non-Newtonian fluid, c π is a critical value of this product based on the non-Newtonian model and y p is the yield stress of the fluid.

Fig. 1 Physical model and coordinate system
Introducing the boundary layer approximations, the continuity, momentum, microrotation and energy equations, respectively can be written as follows: these equations are subjected to the boundary conditions Nazar et al. (2002), where u and v are the velocity components along the x and y directions, respectively, H is the angular velocity of micropolar fluid, κ is the vortex viscosity, g is the gravity acceleration, k is the thermal conductivity, σ is the electric conductivity, α is the thermal diffusivity, ν is the kinematic viscosity, µ is the dynamic viscosity, ρ c is the is the parameter of the Casson fluid and the spin gradient viscosity We introduce now the following non-dimensional variables Nazar et al. (2002), is the Grashof number. Using the Rosseland approximation for radiation, the radiative heat flux is simplified as (Bataller (2008) where * σ and * k are the Stefan-Boltzmann constant and the mean absorption coefficient, respectively. We assume that the temperature differences within the flow through the micropolar fluid such as that the term 4 T may be expressed as a linear function of temperature. Hence, expanding 4 T in a Taylor series about T ∞ and neglecting higher-order terms, we get (6)-(8) into equations (1)-(4), we obtain the following non-dimensional equations of the problem under consideration: To solve equations (9) to (12), subjected to the boundary conditions (13), we assume the following variables: where ψ is the stream function defined as so that f u x y which satisfies the continuity equation (9). Thus, (10) to (12)  , It can be seen that at the lower stagnation point of the cylinder, 0, x ≈ equations (16) to (18) reduce to the following nonlinear system of ordinary differential equations: the boundary conditions (19) become using the non-dimensional variables (6)

SOLUTION PROCEDURES
Equations (16) to (18) subject to boundary conditions (19) are solved numerically using the Keller-box method as described in the book by Cebeci and Bradshaw Cebeci and Bradshaw (1984). The solution is obtained by the following four steps:  reduce (16) to (18) to a first-order system,  write the difference equations using central differences,  linearize the resulting algebraic equations by Newton's method, and write them in the matrix-vector form,  solve the linear system by the block tridiagonal elimination technique. The details of this method can be found in Nazar et al. (2002)

RESULTS AND DISCUSSION
The numerical solutions of the nonlinear system of partial differential equations (16) to (18) with boundary conditions (19) are solved by the Keller-box method (KBM) with four parameters considered, namely the Prandtl number Pr, the magnetic parameter M, the micropolar parameter K and Casson parameter β . This method is an implicit finitedifference method in conjunction with Newton's method for linearization. This is a suitable method to solve parabolic partial differential equations. The boundary layer thickness 16 y ∞ = and step size 0.01, y ∆ = 0.005 x ∆ = are used in obtaining the numerical results. The numerical solutions start at the lower stagnation point of the cylinder 0 x ≈ , with initial profiles as given by equations (20) to (22) and proceed round the cylinder up to x π = .
In order to verify the accuracy of the present applied numerical scheme, a comparison with previously published results has been made. It is noticed from Table 1 that when Pr = 7, K = 0, 2, M = 0 and , β → ∞ the results under consideration for local Nusselt number u N reduce to the results reported by Merkin (1976) and Nazar et al. (2002) for the case of viscous and micropolar fluids respectively. It is found that the results are a good agreement. Furthermore I believe that Kellerbox method is proven to be very efficient to solve this problems.      h y decreases. This is in accordance to the physics of the problem, since the application of a transverse magnetic field results in a resistive type force (Lorentz force) similar to drag force which tends to resist the fluid flow and thus reducing its velocity and angular velocity. Figures 14-16 present the effect of radiation parameter R on temperature, velocity and angular velocity profiles. The observation shows that the temperature, velocity and angular velocity profiles increases with an increase in R because increase the value of radiation parameter provides more heat to fluid that causes an enhancement in the temperature, velocity, angular velocity profiles and the thickness of thermal boundary layer.

CONCLUSIONS
In this paper we have theoretically and numerically studied the problem of the effect of MHD free convective boundary layer flow about a cylinder in a micropolar Casson fluid with thermal radiation. We can conclude that, to get a physically acceptable solution:

ACKNOWLEDGEMENTS
The author thank to the reviewers for providing valuable comments on this paper.