Description
The preparation of microporous materials, often in the form of membranes or hollow fibers by phase inversion is an industrially important process. In this process, a polymer / solvent mixture with an anti-solvent (usually water) is brought into contact. The aqueous phase is enriched by solvent, forming a polymer-rich (solid) phase and a liquid phase consisting of water and solvent during the phase inversion. At the beginning only a basic study on the phase separation of a binary polymer blend with SPH was present [1] in the literature. This approach, which was developed for WCSPH has proved to be inapplicable to ISPH because the method of numerical initialization of the velocity field cannot be implemented for the ISPH method. To calculate the structure formation the theory of Cahn and Hilliard was picked up, with which it is possible to describe the dynamics of phase separation [2]. To discretize the equation corresponding to the SPH method, it is necessary to approximate a gradient of 4th order. This new approximation has been validated for both 2-dimensional [3] as well as 3-dimensional decay processes [4] with the Lifshitz-Slyosov Growth Law of the dynamics of diffusive segregation. Further investigations have shown that the model can also be a simplified description of the manufacturing process of polymer membranes with foam pore structure. By the end of the current funding period, a more advanced parametric study will be completed by the phase inversion model. The equation of state has been also chosen because of its simple structure and is not sufficient for a quantitative description. They should therefore be replaced by the calculation of the chemical potential with the help of PC-SAFT equation of state (see [5]), which can expect a significant improvement in the quantitative description of realistic polymer systems [6].
The Smoothed Particle Hydrodynamics (SPH) method has advantages for complex structural changes. Simple geometries and homogeneous flow regimes are, however, very compute- and memory-intensive for SPH. Grid-based methods such as the finite volume method (FVM) or the Lattice Boltzmann Method (LBM), owing to their efficient parallelization are very good for large, regular areas. To simulate realistic applications, a coupling of the SPH method is sought with an appropriate grid-based method. In the literature, the first steps for coupling of the SPH method is already established with various grid-based methods. Examples are models for the coupling of SPH with the finite element method (FEM) [7] and finite volume method (FVM) [8]. The FEM is used here mainly for the description of solid deformation. The FVM is used for discretization of flow fields. In order to allow coupling between SPH and a grid method, all models are based on the use of a so-called buffer zone, either in the form of ghost particles or as an overlapping area. The use of ghost particles, which is a sort of the discretization of solid wall particles, is particularly suitable for the fluid-structure interaction. The use of an overlapping region in the discretization, as proposed in [8], allows the development of a general interface between SPH and any other method. Calculations using this method for coupling the weakly compressible SPH with FVM performed well. It remains open whether this approach can be transferred directly to the incompressible SPH. The development of a general interface for coupling with established CFD packets is of interest, showed Kumar et al. [9]. The methods of coupling WCSPH with lattice methods are also found in the computer graphics application. Models for coupling ISPH and FEM are rare and mainly focused on solid deformation [10]. So far, no models for coupling of SPH and LBM are available.
Sources
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[5] B. Plankova, J. Hruby, and V. Vins. “Prediction of the homogeneous droplet nucleation by the
density gradient theory and PC-SAFT equation of state”. In: AIP Conference Proceedings 1527 (1)
(2013), pp. 101–104.
[6] J. Gross and G. Sadowski. “Perturbed-Chain SAFT: An Equation of State Based on a Perturbation
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[7] G. Fourey, G. Oger, D. L. Touzé, and B. Alessandrini. “SPH/FEM coupling to simulate Fluid-
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[8] B. Bouscasse, S. Marrone, A. Colagrossi, and A. D. Mascio. “Multi-purpose interfaces for coupling
SPH with other solvers”. In: 8th International SPHERIC workshop. 2013.
[9] P. Kumar, Q. Yang, V. Jones, and L. McCue. “Coupled SPH-FVM simulation within the OpenFOAM
framework”. In: IUTAM Symposium on Particle Methods in Fluid Mechanics. 2012.
[10] M. Asai, A. M. Aly, and Y. Sonoda. “ISPH-FEM coupling simulator for the FSI problems”. In: 8th
International SPHERIC workshop. 2013.