Steady viscous flow through long eccentric annuli is examined theoretically in some detail. On the restricted side of the channel the maximum velocity is shown to be notably less than on the opposite side. A small variable eccentricity being difficult to avoid in experimental annuli, lateral disturbances of the stream may commonly be expected. These are investigated visually in the case of a comparatively short annulus. They are suggested to be associated with an early failure to produce rectilinear flow in long annuli.