CASSON FLUID OF A STAGNATION-POINT FLOW (SPF) TOWARDS A VERTICAL SHRINKING/STRETCHING SHEET

This study presents a convectively heated hydromagnetic Stagnation-Point Flow (SPF) of an electrically conducting Casson fluid towards a vertically stretching/shrinking sheet. The Casson fluid model is used to characterize the non-Newtonian fluid behaviour and using similarity variables, the governing partial differential equations are transformed into coupled nonlinear ordinary differential equations. The dimensionless nonlinear equations are solved numerically by Runge-Kutta Fehlberg integration scheme with shooting technique. The effects of the thermophysical parameters on velocity and temperature profiles are presented graphically and discussed quantitatively. The result shows that the flow field velocity decreases with increase in magnetic field parameter and Casson fluid parameter β.


INTRODUCTION
An electrically conducting hydromagnetic viscous incompressible boundary layer flow fluid with a convective surface boundary condition is frequently used in many areas of biological, industrial and technological applications (Mutuku, 2014). Some of these applications includes extrusion of plastics in the manufacture of rayon and nylon, MHD (magnetohydrodynamic) blood flow meters and generators, cooling of nuclear reactors, geothermal energy extraction, and drag reduction in aerodynamics, purification of crude oil, textile, polymer technology, and metallurgy, among others. Since inception on MHD boundary layer flows research (Sakiadis, 1961), various authors have investigated numerous aspects of steady and unsteady boundary layer flow of a convective fluids as well as nanofluids (Makinde and Aziz, 2010;Bachok et al., 2012;Mutuku and Makinde, 2014;Khan and Khan, 2016;Makinde, 2012).
Stagnation-point fluid (SPF) flow over a continuously stretching/shrinking surface is significantly relevant in many engineering and industrial processes such as extraction of polymer sheet, polymer processing, paper production, glass blowing, glass-fibre production, plastic films drawing, filaments drawn through a quiescent electrically conducting fluid and the purification of molten metals from non-metallic inclusions. The stretching surface and heat transfer is controlled for superior products since the final product quality depends on the rate of cooling and many aspects of this problem have been investigated by several other authors (Chen et al., 1970;Gupta and Gupta, 1977;Chiam, 1994;Layek et al., 2007;Makinde and Aziz, 2011;Crane, 2018).
Non-Newtonian fluid flows are expressed in several engineering processes (oil reservoir engineering, bioengineering), geophysics, chemical and nuclear industries, polymer solution, cosmetic processes, paper production, design of thrust bearings and radial diffusers among others. These fluids exhibit a nonlinear relationship between shear stress and rate of strain which deviate significantly from the Newtonian fluid (Navier-Stokes) model making it difficult to express these properties in a single constitutive equation. Owing to the complexity of these fluids, there is not a single constitutive equation which exhibits all their properties thus, amongst the different types of non-Newtonian fluids namely; viscoelastic fluid, couple stress fluid, micropolar fluid, power-law flow and Casson fluid, various models have been used for non-Newtonian fluids, with their constitutive equations varying greatly in complexity (Fox et al., 1969;Lun-Shin and Manun, 2008;Xu and Shi-Jun, 2009;Reddy et al., 2012).
Casson fluids behave like an elastic solid, with a yield shear stress existing in the constitutive equation. It is a shear thinning liquid assumed to have an infinite viscosity at zero rate of shear, a yield stress below which no flow occurs, and a zero viscosity at an infinite rate of shear. This implies that if a shear stress greater than yield stress is applied, it starts to move whereas if a shear stress less than the yield stress is applied to the fluid, it behaves like a solid. Examples include yoghurt, molten chocolate, cosmetics, nail polish, tomato puree, jelly, honey, soup, concentrated fruit juices, human blood, amongst others. Casson (1959) investigated the flow behaviour of pigment oil suspensions of the printing ink type.  looked at MHD stagnation-point flow of a Casson fluid over a nonlinearly stretching sheet with viscous dissipation.

MATHEMATICAL FORMULATION
where is the product of the component of deformation rate with itself, = ; are the ( , ) ℎ components of the deformation rate and is a critical value of this product based on the non-Newtonian model, is plastic dynamic velocity of the non-Newtonian fluid and is the yield stress of the fluid.
The MHD boundary layer equations for the steady incompressible SPF is given by The boundary conditions at the sheet surface and free stream are: Using equation (7), equations (3) -(6) are transformed to a set of couple nonlinear ordinary differential equations with dimensionless boundary conditions From equations (8), (9) and (10), prime denotes differentiation with respect to , is the velocity ratio parameter, is Grashof number, is magnetic field parameter, is Prandtl number, is Eckert number and is the Biot number respectively defined as follows: The physical quantities of practical interest are the skin friction coefficient and local Nusselt number defined as where, is the shear stress or skin friction along the stretching sheet and is the heat flux from the sheet and defined as

RESULTS AND DISCUSSION
The numerical computations are carried out for the various values of the physical parameter with Runge-Kutta Fehlberg integration scheme. The effects of the varying physical parameters: magnetic field parameter ( ), Casson parameter ( ), velocity ratio parameter ( ), Grashof number ( ), Biot number ( ), Eckert number ( ) and Prandtl number ( ) on velocity and temperature profiles has been analyzed.
The obtained computation results are presented graphically in Fig. 1 -Fig. 8 and discussed.
The effects of various values of magnetic field parameter on the flow field velocity and temperature profiles are displayed in  due to the increase in , the yield stress reduces and consequently, the velocity boundary layer thickness reduces. Fig. 4 shows the influences of Casson parameters on the temperature profiles. It shows that temperature decreases with increasing values in . This implies that thermal boundary layer decreases . Fig. 5 presents the effects of temperature profiles for varying values of Biot number . It depicts that increasing values of , decreases in temperature profiles. However, as the flow field moves far away from the sheet within the thermal boundary layer, Biot number varnishes.
The velocity profile for various values of velocity ratio parameter is shown in Fig. 6. As described by Mastapha and Gupta (2001) and Bhattacharyya (2013b) for Newtonian fluid, the velocity of fluid inside the boundary layer decreases from the surface towards the edge of the layer for the first kind ( < 1) and the fluid velocity increases from the surface towards the edge for the second kind ( > 1). Similarly, it is important to note that the stretching velocity and straining velocity are equal as such there is no boundary layer of Casson fluid flow near the sheet (Chaim, 1994). The velocity profiles for different values of Grashof number is displayed in Fig. 7. It revealed that the flow field velocity decreases with increasing values of Grashof number thereby reducing the thermal boundary layer along the sheet. The viscous dissipation effect on temperature profiles is shown in Fig. 8. It illustrates that temperature increases with increase in Eckert number (viscous dissipation parameter). The Eckert number produces heat due to drag between the fluid particles causing an increase of the initial fluid temperature due to the extra heat. However, may not only cause thermal reversal but also increases the thermal boundary layer ). Fig. 9 shows the effects of Prandtl number of temperature profiles. It depicts that temperature initially increases with increasing values of Prandtl number and later decreases with increased values of towards the thermal boundary layer. The use of Prandtl number in heat transfer problems reduces the relative thickening of the momentum and the thermal boundary layer . Thus, the rate of heat transfer is enhanced with causing the reduction of the thermal boundary layer thickness.

CONCLUSION
The magnetohydrodynamic SPF of a Casson fluid towards a convectively heated stretching/shrinking sheet is investigated taking the buoyancy force into account. Using the similarity variables, the governing differential equations are transformed to ordinary differential equations and solved numerically by shooting method with Runge-Kutta Fehlberg integration scheme. The effects of the various governing physical parameters were analysed and the following conclusions are drawn: a. The velocity boundary layer thickness reduces with increasing values of the magnetic field parameter. b. The flow field velocity decreases with increase in Casson parameter as well as the thermal boundary layer thickness.

ACKNOWLEDGMENT
The Authors wish to acknowledge all reviewers of the manuscript for their time in evaluating the content towards a better output.