By solving the full equations of an extended two-fluid model in two dimensions, we give the first numerical study revealing non-equilibrium steady states in sheared entangled polymer mixtures. This research provides answers for some fundamental questions in sheared binary mixtures of entangled polymers. Our results reveal that non-equilibrium steady states with finite domain size do exist, and apparent scaling exponents L k ; _ g À1:05 and L ? ; _ g À1 are found over six decades of shear rate. Since the wall effects get involved in our simulations, the dependence of average domain size on system size cannot be strictly eliminated. In addition, as an obvious influence of viscoelasticity, the polymer viscosity h p appears to induce linear translation of the fitted lines. Through two-dimensional numerical simulations, we show the detailed dynamic evolution of microstructure in binary polymer mixtures with asymmetric composition under shear flow. It is found that the phase patterns are significantly different from symmetric fluids studied previously. Finally, we also identify the importance of wall effects and confirm the irreplaceable role of inertia for a non-equilibrium steady state.

The phase transition of complex fluids is intrinsically a multi-scale problem. In this paper we proposed a multi-scale two-fluid model, that couples a coarse-grained microscopic method to the two-fluid framework for studying the multi-phase fluids under shear flow. In this model the macroscopic viscoelastic stress is calculated by tracking massive microscopic Brownian configuration fields in the simulation box. Both of the macroscopic and the microscopic equations are solved using a modified PISO iterative algorithm based on finite volume discretization scheme. Our 2D numerical results reproduce numerous dynamic phenomena reported in literature and show that the theoretical model presented here could be a possible multi-scale approach to numerically study the multi-phase viscoelastic fluids under flow.

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Searching is one of the most fundamental operations in many complex systems. However, the complexity of the search process would increase dramatically in high-dimensional space. K-dimensional (KD) tree, as a classical data structure, has been widely used in high-dimensional vital data search. However, at present, common methods proposed for KD tree construction are either unstable or time-consuming. This paper proposed a new algorithm to construct a balanced KD tree based on presorted results. Compared with previous similar method, the new algorithm could reduce the complexity of the construction process (excluding the presorting process) from O (KNlog2N) level to O (Nlog2N) level, where K is the number of dimensions and N is the number of data. In addition, with the help of presorted results, the performance of the new method is no longer subject to the initial conditions, which expands the application scope of KD tree.

Free surface flows arise in a variety of engineering applications. To predict the dynamic characteristics of such problems, specific numerical methods are required to accurately capture the shape of free surface. This paper proposed a new method which combined the Arbitrary Lagrangian-Eulerian (ALE) technique with the Finite Volume Method (FVM) to simulate the time-dependent viscoelastic free surface flows. Based on an open source CFD toolbox called OpenFOAM, we designed an ALE-FVM free surface simulation platform. In the meantime, the die-swell flow had been investigated with our proposed platform to make a further analysis of free surface phenomenon. The results validated the correctness and effectiveness of the proposed method for free surface simulation in both Newtonian fluid and viscoelastic fluid.