Introduction
Hydromagnetic boundary layer flow of an electrically conducting viscous
incompressible fluid with a convective surface boundary condition is
frequently encountered in many biological, industrial and technological
applications such as extrusion of plastics in the manufacture of Rayon
and Nylon, MHD generators, the cooling of nuclear reactors, geothermal
energy extraction, purification of crude oil, textile industry, polymer
technology, metallurgy, drag reduction in aerodynamics, MHD blood flow
meters among others. Since the pioneering work on MHD boundary layer
flow by Sakiadis \(\mathbf{[1]}\), numerous aspects of
steady and unsteady boundary layer flow of convectional fluids as well
as nanofluids have been investigated by various authors\(\mathbf{[2-6]}\).
Stagnation-point flow, describing fluid motion over a continuously
stretching surface in the presence of electromagnetic fields are
significant in many engineering processes with various industrial
applications such as extraction of polymer sheet, paper production,
metallurgy, polymer processing, glass blowing, glass-fibre production,
plastic films drawing, filaments drawn through a quiescent electrically
conducting fluid subject to a magnetic field and the purification of
molten metals from non-metallic inclusions. The quality of the final
product depends to a great extent on the rate of cooling at the
stretching surface, thus for superior products the heat transfer should
be controlled. Since the pioneering work in this area by Crane\(\mathbf{[7]}\), who investigated steady boundary layer
flow of a viscous fluid over a linearly stretching plate, many aspects
of this problem have been investigated by other authors\(\mathbf{[8-}\mathbf{12}\mathbf{]}\).
Non-Newtonian fluid flows are encountered in chemical, material and
industrial processing engineering. In particular, such materials are
involved in geophysics, oil reservoir engineering, bioengineering,
chemical and nuclear industries, polymer solution, cosmetic processes,
paper production, design of thrust bearings and radial diffusers etc.
These fluids exhibit a non-linear relationship between shear and the
rate of strain which deviate significantly from the Newtonian fluids
(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, various models have been used for non-Newtonian
fluids, with their constitutive equations varying greatly in complexity\(\mathbf{[13-16]}\). The different types of non-Newton
fluids are viscoelastic fluid, couple stress fluid, micropolar fluid,
power-law flow, Casson fluid, among other types.
Casson fluid behaves like an elastic solid, with a yield shear stress
existing in the constitutive equation. It is a is a shear thinning
liquid which is 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 of Casson fluids are fluids such as
yoghurt, molten chocolate, cosmetics, nail polish, tomato puree, jelly,
honey, soup, concentrated fruit juices, human blood. The first model for
such fluids was formulated by Casson \(\mathbf{[17]}\)whose objective was to investigate the flow behaviour of pigment oil
suspensions of the printing ink type. Thereafter, several researchers
studied Casson fluid pertaining to different flow situations\(\mathbf{[18-}\mathbf{2}\mathbf{6}\mathbf{]}\). Medikareet al . \(\mathbf{[22]}\) investigated MHD
stagnation-point flow of a Casson fluid over a nonlinearly stretching
sheet with viscous dissipation.
Owing to the numerous application of non-Newtonian fluids in industrial
processes and the literature gap in hydromagnetic Casson fluid flow has
given a strong motivation to understand their behaviour in several
transport processes. The current study extends the work of Medikareet al . \(\mathbf{[22]}\) by incorporating buoyancy
force and considering a convective boundary layer in the numerical
analysis of the hydromagnetic stagnation-point flow of a steady,
incompressible Casson fluid towards a shrinking/stretching sheet.