Mathematical modelling of flow field in 3-dimensional additive printing

Abstract

The distribution of flow field caused by squeegee movement exercises a crucial influence over the transfer of aqueous polyurethane dispersion (PUD) in the 3-dimensional (3D) additive printing for warp knitted vamp. So, the current study's goal is to develop a mathematical model for flow field related to printing parameters, where analytical solution of squeegee deformation is crucial. This paper presents an analytical procedure to deduce the deformation equation of wedge squeegee with variable section based on the semi-inverse method from elastic mechanic. Then a mathematical model of 3D additive printing is obtained using lubrication theory, leading to non-dimensional velocity as well as the integral expression of hydrodynamic pressure that is solved by the Romberg integration algorithm. Further analysis reveals the similarities and differences between flow field studied in this paper and Couette flow. More importantly, the hydrodynamic pressure near the squeegee tip plays a crucial role in PUD transfer. Then, the sensitivity analysis is carried out to quantify the impacts of process parameters on hydrodynamic pressure. Among printing parameters, the printing angle has the greatest influence on hydrodynamic pressure, which is often neglected in traditional engineering studies. The theoretical calculations are compared against the simulations of finite volume method, showing good agreement for the distribution of velocity and pressure. This work paves the way for adjustment of process parameters and mechanism study of PUD transfer.