Abstarct: ty are mainly focusing on two dimensional modeling of this instability with weak elastic number by aide of Oldroyd-B constitutive equation. In present research, the three dimensional modeling was done for mention instability baxsed on Giesekus constitutive equation by aid of CFT methods and OpenFOAM software. As a first time the 3D modeling gives the opportunity of investigating the non-axisymmetric second flow regimes of Taylor-Couette instability especially axial travelling wave and standing wave regime baxsed on Giesekus equation. By making a comparison between Newtonian and viscoelastic regimes, hoop stress is determined as an effective factor in growing pressure gradient and creating Taylor-Couette viscoelastic instability. Also, the variation of hoop stress has a crucial rule in crating of instabilities and appearing of other second flows patterns. Besides, relaxation time has weak effect on critical Taylor number. Against that, mobility factor has greater effect on critical condition and causes a substantial decline in Taylor number. Finally, the greatest effect is related to viscosity ratio.