## Appendix A  
The kinematic equation of the melt flow in the \(z\) direction can be expressed as follows:  
\[\rho\left(\frac{\partial v_{z}}{\partial t}+v_{r}\frac{\partial v_{z}}{ \partial r}+\frac{v_{\theta}}{r}\frac{\partial v_{z}}{\partial\theta}+v_{z} \frac{\partial v_{z}}{\partial z}\right)=-\frac{\partial p}{\partial z}+ \left[\frac{1}{r}\frac{\partial}{\partial r}(r\tau_{rz})+\frac{1}{r}\frac{ \partial\tau_{6z}}{\partial\theta}+\frac{\partial\tau_{zz}}{\partial z}\right] +\rho g_{z} \tag{1}\]  
where, \(\rho\) is the density of the polymer melt, \(v_{z}\) is the velocity vector of \(z\) direction, \(t\) is time, \(p\) is the pressure, \(\tau_{rz}\), \(\tau_{\theta z}\) and \(\tau_{zz}\) are stress tensors and \(g_{z}\) is the gravitational acceleration vector of \(z\) direction. According to the assumption mentioned above, \(v_{r}\), \(\partial v_{z}/\partial\theta\), \(\partial v_{z}/\partial z\), \(\tau_{rz}\), \(\tau_{\theta z}\) and \(g_{z}\) can be regarded as 0, thus Equation (1) becomes  
\[\rho\frac{\partial v}{\partial t}=-\frac{\partial p}{\partial z}+\frac{1}{r} \frac{\partial}{\partial r}(r\tau) \tag{2}\]  
Integrating both sides of Equation (2), the shear stress can be obtained by  
\[\tau=\frac{r}{2}\frac{\partial p}{\partial z}+\frac{\rho}{r}\frac{\partial}{ \partial t}\int_{0}^{r}vrdr=\frac{r}{2}\frac{\partial p}{\partial z}+\frac{ \rho r}{2}\frac{\partial[v(t)]}{\partial t} \tag{3}\]  
Applying vertical vibration to the extrusion liquefier, the velocity variation inside the liquefier is  
\[v(t)=v_{0}+v_{1}=v_{0}-A\cos\omega t \tag{4}\]  
where, \(v_{0}\) is the material wire feed speed, \(v_{1}\) is the speed of applied vibration, \(A=A_{0}\cdot\omega\cdot 1/g\), \(A_{0}\) is the vibration amplitude, \(g\) is the gravitational acceleration and \(\omega\) is the angular frequency of vibration. Substituting Equation (4) into (3), the internal shear stress of the melt in the channel is  
\[\tau=\frac{r}{2}\frac{\Delta p}{L}+\frac{\rho r}{2}A\omega\sin\omega t \tag{5}\]  
Taking into account the material properties, the melt is regarded as Newtonian fluid and its flow behavior follows the power law [29], which is expressed by  
\[\mu=K\cdot\dot{\gamma}^{n-1} \tag{6}\]  
where, \(\mu\) is the apparent viscosity, \(K\) is the consistency coefficient, \(n\) is the power law index and \(\dot{\gamma}\) is the shear rate. According to [PERSON]'s power law, the shear rate and shear stress have the following relationship:  
\[\dot{\gamma}=K^{-\frac{1}{n}}\cdot\tau^{\frac{1}{n}} \tag{7}\]  
Substituting Equation (7) into (6), the relationship between apparent viscosity and shear stress can be obtained by  
\[\mu=K^{\frac{1}{n}}\cdot\tau^{1-\frac{1}{n}} \tag{8}\]  
Considering the influence of vibration field applied, the Tanner nonlinear constitutive equation modified from [PERSON] linear constitutive equation is used to accurately reflect the time-varying and viscoelastic behavior of the melt [20]. The expression is \[\tau+\lambda\frac{\partial\tau}{\partial t}=\eta\big{(}\dot{y}\big{)}\dot{y}^{ \prime}=K\bigg{(}-\frac{\partial v}{\partial r}\bigg{)}^{n} \tag{10}\]  
where, \(\lambda\) is the relaxation time of the polymer. Based on Assumption 1 and 3, the pressure gradient can be expressed by  
\[\frac{\partial p}{\partial z}=\frac{\Delta p}{L^{\prime}} \tag{11}\]  
where, \(L^{\prime}\) is the length of the fully developed flow region, \(\Delta p\) is the pressure difference of two extremities of the fully developed flow region. Since Region I (shown in Figure 2) is the upper end of the liquefier, the material cannot completely melt into liquid and the melt does not fully flow here, thus the pressure gradient is corrected by Bagley correction method. The corrected pressure gradient is given by  
\[\rho\frac{\partial v}{\partial t}=-\frac{\partial p}{\partial z}+\frac{1}{r} \frac{\partial}{\partial r}(r\tau) \tag{2}\]0  
where, \(N_{B}\) is the revised factor of pressure gradient under the vibration field and \(R\) is the pipe radius of Region I.