The flow of a visco-elastic fluid in a channel with stretching walls under the action of an externally applied magnetic field generated by a magnetic dipole was studied in this paper. As per an experimental report, the variation in magnetization $M$ of the fluid with temperature $T$ was approximated as a linear equation of state $M=K1T$, where $K1$ is a constant called the pyromagnetic coefficient. In this investigation the model used is that of Walter’s liquid B fluid, which includes the effect of fluid visco-elasticity. By introducing appropriate nondimensional variables, the problem is reduced to solving a coupled nonlinear system of ordinary differential equations subject to a set of boundary conditions. The problem is solved by developing a suitable numerical technique based on finite difference approach. Computational results concerning the variation in the velocity, pressure and temperature fields, skin friction and the rate of heat transfer with magnetic field strength, Prandtl number, and blood visco-elasticity are presented graphically. The results presented reveal that the velocity of blood in the normal physiological state can be lowered by applying a magnetic field of sufficient intensity. The study bears the promise of important applications in controlling the flow of blood during surgery and also during treatment of cancer by therapeutic means when it involves magnetic drug targeting (hyperthemia).