A SPH-based particle method for simulating 3D transient free surface flows of branched polymer melts

Highlights

• We presented a particle method based on smoothed particle hydrodynamics (SPH).

• We simulated 3D transient free surface flows of branched polymer melts.

• We proposed an enhanced treatment technique of 3D solid boundaries.

• The impacting droplet was investigated for a wide range of the relevant parameters.

• The challenging jet buckling and rod-climbing effect were reproduced.

Abstract

In this article we present a particle method based on smoothed particle hydrodynamics (SPH) for simulating three-dimensional (3D) transient free surface flows of branched polymer melts which are governed by the single equation version of the eXtended Pom-Pom (XPP) model. In order to remedy the so-called tensile instability which results in particle clustering and unphysical fracture in the state of fluid stretching, both artificial stress and artificial viscosity are incorporated into the momentum equation. For facilitating the implementation of the wall boundary condition in 3D space, an enhanced treatment of solid boundaries is proposed in this paper. The proposed SPH algorithm is validated by the excellent agreement between the numerical and the analytical solutions of Hagen-Poiseuille flow of an Oldroyd-B fluid. To demonstrate the ability of the numerical method in simulating free surface flows of branched polymer melts, the fall and impact on a rigid plate of a XPP fluid droplet is investigated for a wide range of the relevant parameters. Specifically, the convergence of the numerical results is discussed by three particle sizes of different levels of refinement. To provide some additional applications of 3D free surface flows of branched polymer melts, the challenging jet buckling and rod-climbing effect of viscoelastic fluids are further simulated and analyzed. All numerical results establish the capability of the SPH-based particle method to simulate 3D transient free surface flows of branched polymer melts in an entirely mesh-free framework.

Introduction

Many manufacturing and technological applications in today’s industry involve free surface flows of branched polymer melts, such as extrusion and injection molding and container fillings in the food. For such flows, analytical methods have only a very limited use and numerical simulation technique becomes an alternative tool. Traditional grid-based numerical methods such as finite difference method (FDM), finite volume method (FVM) and finite element method (FEM) are extensively employed to simulate free surface flows. However, to cope with moving free surface and large deformation, complicated techniques of capturing and/or tracking the free surface as well as regenerating the computational grid are required, for instance, volume of fluid (VOF) [1], marker and cell (MAC) [2] and level set (LS) [3] methods. In the VOF approach, tracking of dynamical interface is accomplished by solving an additional partial differential equation for the filled fraction of each control volume. The MAC technique utilizes marker to track the free surface and has been widely used to solve free surface problem. The LS method is used to locate the moving surface by employing an additional implicit level set function.

Actually, the rheology of a viscoelastic material is intermediate between that of a purely viscous fluid and that of a purely elastic solid. Viscoelastic behavior of materials can be further categorized into the linear and the non-linear characteristics. Linear viscoelasticity is concerned with very small strains or deformations while non-linear viscoelasticity deals with cases in which deformations and rates of deformation are large. For the non-linear viscoelastic materials studied in this paper, a distinct feature is the presence of the extra stress terms which need to be computed by solving additional differential equations. To alleviate the difficulties encountered in the computation of the extra stress terms, the so-called elastic viscous stress split (EVSS) scheme [4] has been employed in which the extra stress is split into the viscous and elastic parts. In 1970, Tanner [5] made an early attempt at simulating extruding swell of a Maxwell fluid. Keunings and co-workers [6], [7], [8], [9] simulated a number of viscoelastic free surface flows in the late 1980s. Yao and McKinley [10] treated the problem of transient extensional deformation. More recently, Tomé et al. [11], [12], [13], [14] used the FDM method on a staggered grid to solve a number of viscoelastic free surface flows including impacting droplet and jet buckling. In their studies, the fluid is modeled by the MAC technique and an accurate representation of the fluid surface is employed. Bonito et al. [15] presented an Eulerian model based on the VOF formulation for the simulations of viscoelastic flows with complex free surfaces in three dimensions. The numerical methods used in all the works mentioned above are mainly based upon grid-based methods and the techniques for capturing the free surfaces.

Recently, many particle methods [16], [17] have been proposed in a Lagrangian framework to deal with the flows with moving free surface and large deformation. Actually, particle methods have a variety of advantages over conventional grid-based methods. It is inherently suitable for simulating the flows with moving free surface because the evolution of fluid particles can be readily obtained due to their purely Lagrangian mesh-free nature. Also, it is comparatively easier in numerical implementation, and is more straightforward to develop three-dimensional (3D) model than grid-based methods. As a typical particle method, smoothed particle hydrodynamics (SPH) was first introduced by Lucy [16] and Gingold and Monaghan [17] in astrophysics to study the collision of galaxies. Since its invention, it has been extensively applied in a wide range of research areas, such as free surface flows [18], [19], incompressible fluids [20], [21], multi-phase flows [22], [23], [24] and non-Newtonian flows [25], [26], [27]. For more information on the SPH method, we refer the reader to the recent review of the method by Liu and Liu [28].

With regards to the early attempt of SPH at simulating viscoelastic free surface flows, the impact of a two-dimensional (2D) Oldroyd-B fluid droplet with a rigid plate was studied by Fang et al. [29] in 2006. They reported that an artificial stress term was required to remove the so-called tensile instability, generally resulting in particle clustering and unphysical fracture in the state of fluid stretching. With the employment of pressure Poisson equation to satisfy the incompressibility constraint, Rafiee et al. [30] solved the viscoelastic free surface flows based on the Oldroyd-B and Upper-Convected Maxwell (UCM) models. More recently, Vázquez-Quesada and Ellero [31] simulated the flow of Oldroyd-B liquid around a linear array of cylinders confined in a channel and compared the dimensionless drag force acting on the cylinder with the available results. And Hashemi et al. [32] studied the 2D movement of suspended solid bodies in Oldroyd-B fluid flows using an explicit weakly compressible SPH algorithm.

The research works mentioned above are mainly confined to the Oldroyd-B and the UCM models of viscoelastic flows, and not a great deal of work would appear to have been done in developing the particle method, especially the SPH method, for an important class of polymer flows characterized by the eXtended Pom-Pom (XPP) constitutive equation. Actually, the XPP model has been employed in several numerical works [33], [34], [35] since it not only possesses a non-zero second normal stress difference coefficient in steady shear flow but also removes the unbounded nature of the orientation equation at high strain rates. As for the historical review of the XPP model, it can trace back to the simple Pom-Pom model, which was first proposed by McLeish and Larson [36] in 1998. Then, Inkson et al. [37] applied it to model low density polyethylene melts in elongational and shear flows. The transient flow of branched polymer melts in a planar 4:1 contraction is numerically studied by Bishko et al. [38]. However, for overcoming the discontinuous steady state solutions and unphysical zero normal stress differences, Verbeeten et al. [39] proposed the improved Pom-Pom model, i.e., the so-called XPP model. From then on, this improved model has attracted significant attention in the non-Newtonian fluid mechanics community. In particular, Russo and Phillips [33] studied extrudate swell behavior of branched polymer melts in a planar configuration using the multi-mode XPP model. Oishi et al. [34], [35] employed an implicit FDM MAC-type method to solve the viscoelastic free surface flows governed by the single equation version of the XPP model. In their works, the 2D phenomenon of impact of liquid droplet on solid surface, extrudate swell and jet buckling are studied. For simulating the flow of K-BKZ type of fluids, one notable work that deals with 3D transient viscoelastic free surface flow was presented by Román Marín and Rasmussen [40]. They developed a new Lagrangian finite element method, and successfully solved the filament stretching between two plates with third order accuracy in space and time. The growth of non-axisymmetric disturbances on the free surface of a stretched filament was also analyzed.

Another aspect to be considered in this study is that most of the mentioned works are performed in two space dimensions, and 3D numerical simulations are very few so far. In the framework of particle methods, the main reason is the number of particles required in 3D space is very huge, which results in massive memory requirement and long computational time. However, 3D numerical simulations would be more significant to the realistic technological and industrial applications. Furthermore, to the authors’ knowledge, there are only a few published works related to the numerical investigation of 3D viscoelastic free surface flows governed by the XPP model. Therefore, the goal of this paper is to develop a SPH-based particle method and further extend it to 3D transient free surface flows of branched polymer melts, characterized by the XPP model. Specifically, the efficient parallel SPH algorithm developed by Xu et al. [45] is employed to ensure that 3D large-scale SPH simulations can be performed within an affordable computational time, even for those involving millions of particles. For convenience the 3D implementation of wall boundary condition, an enhanced treatment of solid boundaries is proposed. Both artificial stress and artificial viscosity are also incorporated into the momentum equation to alleviate the so-called tensile instability, which is unphysical in nature.

This paper is organized as follows: in Section 2, the governing equations and the XPP constitutive equations are introduced; Section 3 describes some important implementation issues of the SPH method in details, including temporal discretization of governing equations, tensile instability, boundary conditions and time integration scheme. In Section 4, the impacting droplet is simulated to demonstrate the ability of the proposed SPH method in dealing with 3D-unsteady free surface flows of branched polymer melts. Specifically, the influence of various rheological parameters that characterized the XPP model on the flow patterns is investigated. The convergence of the numerical method in simulating 3D free surface flows is also analyzed by three particle sizes of different levels of refinement. In addition, the Hagen-Poiseuille flow of an Oldroyd-B fluid is solved to validate the numerical method. Numerical results including the challenging jet buckling and rod-climbing effect of viscoelastic fluids are also displayed. The paper ends in Section 5 with some conclusions.