Mathematical analysis and experimental study of gas-assisted injection molding.

摘要

The motion of long bubbles into Newtonian and non-Newtonian fluids confined in horizontal circular tubes, rectangular channels, and square cross-sectional channels has been studied both theoretically and experimentally. Of particular interest is the determination of residual liquid film thickness on the walls. Isothermal experiments have been conducted to measure the displacement of the gas-liquid interface as a function of the applied pressure differential. The velocity of the interface and residual liquid film thickness have been determined for both Newtonian and non-Newtonian (shear thinning and viscoelastic) fluids. The experimental results indicate that the liquid film thickness viscoelastic (Boger fluid) fluids deposited on the tube wall is thicker than that of comparable Newtonian and non-Newtonian fluids. One of the shear thinning fluids (HEC) gives a liquid film thickness lower than that of a Newtonian fluid while the other shear thinning fluid (CMC) results in a thicker liquid film thickness. The fraction of liquid deposited on the wall for an open end tube is larger than that for a corresponding valve-mounted closed tube.; A simple mathematical model was developed using a power-law expression to model the non-Newtonian fluid. The model successfully captures the gas-liquid dynamics for Newtonian and non-Newtonian fluid displacement in a tube and rectangular channel. The model is used to determine the location and velocity of the advancing bubble front for the case of a power-law fluid. The results indicate that the gas-liquid interface advances more rapidly with decreasing values of the power-law index above a certain value of dimensionless time {dollar}(t/tsb{lcub}b{rcub}approx0.75).{dollar}; The two dimensional flow in a rectangular channel containing a Newtonian fluid is also solved analytically. This eigenfunction solution predicts well the residual liquid film thickness under certain restrictions (such as {dollar}kd=1.95{dollar} and {dollar}thetato{lcub}piover2{rcub}).{dollar}; The two dimensional flow of a power-law fluid is also solved analytically using a singular perturbation method. Inner and outer expansions are developed in terms of a small parameter {dollar}Csb{lcub}A{rcub}{dollar} (modified capillary number). A differential equation for the shape of the gas bubble is solved numerically in order to determine the inner solution. The method of matched asymptotic expansions is used to match the inner and outer solutions. This approach indicates that the residual liquid film thickness increases with decreasing power-law index which is opposite to the experimental observations of previous investigators (Tallmadge (1969, 1970) and Spiers et al. (1975)).; In addition, the amount of liquid remaining inside of a circular tube and a rectangular channel when displaced by another immiscible fluid are determined by solving the full creeping-motion equations numerically. The exact continuity of stress on the free surface is employed along with a finite difference method. In order to solve the equations, the steady-state shape of the interface is guessed and the normal stress boundary condition is dropped. The equations based on a stream function-vorticity formulation are solved with the aid of elliptic grid generation. (Abstract shortened by UMI.)