It was demonstrated by simulation in our previous study that both the normal stress and its gradient are responsible for the negative wake generation (overshoot in the axial velocity) and streamline shifting. Extensional properties of the fluids dominate the generation of the negative wake, while other factors strengthen or weaken the formation of velocity overshoot. In this study, the criteria for the negative wake generation are discussed in detail for various fluid models, including the PTT, the FENE-CR, the FENE-P, and the Giesekus models. With the FENE-CR fluid, it is easier to generate negative wake than with the FENE-P fluid. This confirms that the constant shear viscosity FENE-CR fluid enhances the velocity overshoot, and that the shear-thinning viscosity FENE-P fluid delays the negative wake generation. The Giesekus fluid has a similar behaviour to the PTT fluid with regarding to the critical conditions of negative wake generation when appropriate fluid parameters are selected. The mechanism of wall proximity in enhancing the negative wake generation is also demonstrated with the analysis for the first time.