A solution is presented for the flow of a viscous incompressible fluid in the entrance region of an annulus with slip occurring at the walls. The condition is imposed that the pressure drop found from momentum considerations should be the same as that from mechanical energy considerations. Results are given for radius ratios of 1.05, 2, 5 and 10 for Knudsen numbers 0, 0.010, 0.050 and 0.1.