The spinning flow of Boger fluids and the gravity-drawing extrusion now of a Newtonian as well as a Boger fluid have been simulated by using the stream-line finite element method and the technique of matching the finite element solutions with those of one-dimensional spinning equations. The recoverable shear strain is proved not to be a basic parameter in characterising the spinning flow of Boger fluids. For Newtonian fluids this technique predicts the experimental jet shape accurately. For Boger fluids, the numerical simulation agrees with the experimental data of spinning flow reported by Sridhar et al., but seems to give an insufficient swelling and over contraction of the jets when drawn by its own weight, compared with the experimental results of Trang and Yeow. It implies that the Oldroyd-B model fitting the viscometric-flow data fails to describe accurately the elasticity and extensional viscosity in the extrusion flow of Boger fluids with gravity-drawing.