This work extends the application of the momentum integral method to entrance flow problems with axial transport of vorticity. Laminar flow development of an incompressible Newtonian fluid in the entrance region of a circular tube and a parallel-plate channel at various Reynolds numbers is analyzed. Center-line velocity development and excess pressure drop in the entrance region are presented. Approximate solutions in closed form are also obtained. Results are compared with existing theoretical and experimental investigations and gives satisfactory accuracy. The effect of density variation on the flow is also investigated with the assumption of isothermal flow.