A one-dimensional non-linear diusion wave equation is derived from the Saint Venant equations with neglect of the inertia terms. This non-linear equation has no general analytical solution. Numerical schemes are therefore employed to discretize the space and time axes and convert the dierential equation to dierence form. In this study, the mixing cell method is used to convert the diusion wave equation to dierence form, in which the dierence term can be eliminated by selecting an optimal space step size Dx when time step size Dt is given. When the time step size Dt 3 0, the space step size Dx Qa2S 0 BC k where Q is discharge, S 0 is bed slope, B is channel width and C k is kinematic wave celerity, which is the same as the characteristic length proposed by Kalinin and Milyukov. The results of application to two cases show that the mixing cell and linear channel¯ow routing methods produce hydrographs that are in agreement with the observed¯ood hydrographs.
A numerical model is developed to simulate the simultaneous bubble nucleation and growth during depressurization of thermoplastic polymers saturated with supercritical blowing agents. Of particular importance is the ability of the model to predict the formation of nano-cellular foams, including the cell size distribution within the foam, based on the specific process conditions and polymer properties. Additionally the model differentiates between the "Free" and "Limited" expansion phases in the growth of a single bubble. Classical nucleation theory is used to predict nucleation rate and the popular "Influence Volume Approach" is used to determine the end of nucleation phase. By solving the mass, momentum and species conservation equations for each bubble, the model is capable of predicting bubble size distribution and bulk porosity.
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