Hydrodynamically developing and simultaneously (that is, hydrodynamically and thermally) developing laminar flows of a Bingham plastic in a circular pipe have been investigated numerically. Solutions have been obtained by using a four-step fractional method combined with an equal order bilinear finite element method. For hydrodynamically developing flow, shorter entrance length is required to reach fully developed velocity field and thicker unyielded region appears closer to the inlet as the yield stress becomes larger. For simultaneously developing flow, the heat transfer characteristics show the same trends as those predicted from the thermally developing flow.