A mathematical model is presented for hydro-magnetic wavy (sinusoidal) flow of a non-Newtonian couple-stress fluid in the annular gap between flexible and rigid channels. This study is motivated by bio-magnetic delivery systems (of relevance to drug targeting and endoscopy) and also industrial applications such as electrically-conducting rheological waste conveyance and transport of non-Newtonian corrosive magnetized fluids where fluid contact with machinery parts is prohibited. Such systems can be controlled effectively with applied magnetic fields which generate a Lorentzian drag force in the flow. Analytical solutions for axial velocity and pressure rise per wavelength, obtained in closed form under long wavelength and low Reynolds number approximations. The influence of the pertinent control parameters such as rheological couple-stress parameter, Hartmann (magneto-hydrodynamic body force) number, and the ratio of the width of both channels on velocity profiles and pressure rise per wavelength are illustrated.