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In this work the fundamental problem of the biomagnetic (blood) fluid flow in a channel under the influence of a steady localized magnetic field is studied. For the mathematical formulation of the problem both magnetization and electrical conductivity of blood are taken into account and blood is considered as a homogeneous Newtonian fluid. For the numerical solution of the problem, which is described by a coupled, non linear system of PDEs, with appropriate boundary conditions, the stream function-vorticity formulation is adopted. The solution is obtained by the development of an efficient numerical technique based on finite differences. Results concerning the velocity and temperature field, skin friction and rate of heat transfer, indicate that the presence of the magnetic field influences considerably the flow field. It is also obtained that the electrical conductivity of blood should be taken into account at the area of the uniform magnetic field. List of symbolsH magnetic field strength (A m À1 ) B magnetic field induction ðB ¼ l o HÞ (Tesla) T temperature (K) T u temperature of upper plate T l temperature of lower plate L length of plates (m) h distance between plates (m) ðx; yÞ components of the cartesian system ðu; vÞ velocity components p pressure q fluid density (kg m À3 ) r electrical conductivity (S m À1 ) l dynamic viscosity (kg m À1 s À1 ) l o magnetic permeability of vacuum (H m À1 ) c p specific heat at constant pressure (J kg À1 K À1 ) k thermal conductivity (J m À1 s À1 K À1 )

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