The theoretical problem of steady laminar magnetohydrodynamic and thermal dissipation effects, on mixed convection boundary layer flow of an incompressible, viscous fluid, along an isothermal vertical thin needle, in a variable external stream has been investigated numerically. A magnetic field of variable strength is applied perpendicular to the needle walls. The similarity solutions for isothermal walls have been obtained. The transformed system of coupled second order ordinary differential equations is solved numerically using the function bvp-4c from MATLAB. The velocity and temperature profiles, as well as, the local skin friction and the local heat transfer parameters are illustrated for different values of the governing parameters. The discussion focuses on the physical interpretation of the results. It has been found that the combination of a magnetic field and thermal dissipation affects strongly the velocity, fluid temperature, rate of heat transfer and skin friction. Increasing magnetic field and thermal dissipation will reduce the rate of heat transfer and increase thermal boundary layer when flow is forced or mixed convection. However, for the free convection case, we observed that the magnetic field promotes the fluid motion, increases the skin friction and decreases considerably the rate of heat transfer. The results are presented in both graphical and table forms and are discussed in details.

thin needle, mixed convection, MHD, thermal dissipation, boundary layer.