Keywords: Stretching sheet, Biomagnetic fluid, Magnetohydrodynamic, Ferrohydrodynamic, Magnetic dipole, Magnetization, Variable viscosity, Thermal conductivity.

Introduction

Over the last few decades, research on biological fluid (which is also part of BFD) in presence of applied magnetic field has been adoption serious attention from research due to its numerous applications in medical and bio-engineering, for example: magnetic resonance imaging (MRI), in cancer tumor treatment (elctromagnetic hypothermia), magnetic particles used as drug delivery, development of magnetic devices for cell separation etc as early mentioned   \cite{pai1996,tzirtzilakis2005,kafoussias2003}.  It is an in disciplinary field of BFD which directly connected to finding and developing accomplishment of human body related diseases and disorders. Blood is one of the common peculiarities of BFD because blood behaves as a magnetic fluid due to presence of ions in plasma which slightly produce current. 
For this reason, several mathematical model of BFD has been proposed by researchers which incorporate with principles of MHD and FHD.  \citep{pai1996} was the first developed a mathematical model of BFD which consist principle of ferrohydrodynamic (FHD) and the dominant force in flow field is that of magnetization. Later on,  \cite{tzirtzilakis2005} explore this model with combining principle of ferrohydrodynamic (FHD) and magnetohydrodynamic (MHD). In that study he proposed that blood flow can be reduce up to 40% under the influence of strong magnetic field. A mathematical analysis of heated ferrofluid under the influence of magnetic dipole through a two dimensional linear stretching sheet perused  \citep{kafoussias2003} . Studies on arterial blood flow with composited stenosis are mathematically presented  \citep{2014} . To see flow feature of blood,  \citep{2014a} developed a arterial stenosis model. 
The behavior of boundary layer over a two dimensional stretched surface was conducted by the mathematician  mathematician  \cite{sakiadis1961} . Further,  Further,  \cite{crane1970} elongated the idea of  of  \cite{sakiadis1961} and the problem possesses to an exact solution with considering stretched velocity proportional to distance of origin. An incompressible two dimensional boundary layer flow over a stretching sheet under the influence of variable heat flux and variable wall temperature presented by  by  \cite{pop2006} and numerically solved by using Keller-box method. The effect of variable fluid properties on continuous moving stretched surface examined  by  \cite{ljcrane1982}   and  \cite{bobba1985} .  \cite{acrivos1981} presented an exact solution of the Navier-Stokes equations flow problem through a channel or tube with considering surface accelerated with velocity using similarity transformation.
Studies on MHD flow and heat transfer problem in a boundary layer has gained a tremendous attraction from researcher in last few decades owing to its wide range of applications especially in the area of chemical engineering, thermal insulation, power generation, metallurgy etc. The effect of fluid viscosity and thermal conductivity of an electrically conducting fluid through a continuous stretched sheet in presence of a magnetic field carried out numerically  numerically  \cite{2007} . \cite{ganga2009} studied the MHD flow, heat and mass transfer over a two dimensional stretched surface in porous media under the influence of viscous dissipation. The effects of variable fluid properties like fluid viscosity, thermal conductivity, thermal radiation etc. on MHD flow over a steady/unsteady stretched surface including cylindrical surface/ oscillating surface under several boundary conditions has been numerically investigated by several researchers such as  as  \cite{layek2008,2013,agarwal2013,ali2015,abbas2015,2015,mahmoud2007} and found that the fluid velocity and temperature profile as well as skin friction coefficient and rate of heat transfer are significantly changed under the influence of above parameters.  parameters.  
The ultimate aim of the present analysis is to seek the effect of variable fluid properties on biomagnetic fluid flow over a two dimensional stretched sheet under the influence of magnetic dipole. The governing partial differential equations are converted into ordinary differential equations using suitable similarity transformations and numerically solved in MATLAB software by employing bvp4c function technique and numerical results are shown in graphical and tabular form. Numerical code also validate with some existing work of previous literature in order to check the accuracy of the solution.