Velocity and surface density distribution of the outflow calculated by solving Eqs. (18). The blue lines are the numerical results and the red lines are results calculated by assuming the local conservation of energy and momentum (Sect. 2.5). The upper panels are the results for a 10⁴yr outflow. The lower panels are the results for an unrealistically old (10⁷yr) outflow. At smaller radii, the deviation of the numerical solutions from the analytical solutions are caused by the gravity term in Eq. (22).