Advances in Mathematical Physics

Volume 2018, Article ID 5838290, 10 pages

https://doi.org/10.1155/2018/5838290

## Multiscale Numerical Simulations of Branched Polymer Melt Viscoelastic Flow Based on Double-Equation XPP Model

School of Science, Xi’an University of Architecture and Technology, Xi’an 710055, China

Correspondence should be addressed to Xuejuan Li; moc.361@kz_jxl

Received 22 January 2018; Revised 17 April 2018; Accepted 22 April 2018; Published 27 May 2018

Academic Editor: Ming Mei

Copyright © 2018 Xuejuan Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The double-equation extended Pom-Pom (DXPP) constitutive model is used to study the macro and micro thermorheological behaviors of branched polymer melt. The energy equation is deduced based on a slip tensor. The flow model is constructed based on a weakly-compressible viscoelastic flow model combined with DXPP model, energy equation, and Tait state equation. A hybrid finite element method and finite volume method (FEM/FVM) are introduced to solve the above-mentioned model. The distributions of viscoelastic stress, temperature, backbone orientation, and backbone stretch are given in 4 : 1 planar contraction viscoelastic flows. The effect of Pom-Pom molecular parameters and a slip parameter on thermorheological behaviors is discussed. The numerical results show that the backbones are oriented along the direction of fluid flow in most areas and are spin-oriented state near the wall area with stronger shear of downstream channel. And the temperature along is little higher in entropy elastic case than one in energy elastic case. Results demonstrate good agreement with those given in the literatures.

#### 1. Introduction

Branched polymer becomes more and more concerned because of its unique structural characteristics and properties, now its development is one of the fastest in macromolecular materials. Branched polymer has more complex thermorheological behavior compared with other polymers, and its rheological behavior depends on its topological structure of branched molecules [1, 2]. As compared to the linear polymer, when the main chain of branched polymer introduces a certain number and length of branched chains, the viscoelasticity is significantly different. Branched polymer in shear flow shows the similar strain softening but has a longer relaxation time at the end of branched molecular chains because of the limitation of branched chain. Moreover, branched polymer in elongation flow has entirely different strain softening. Therefore, branched has a great influence on polymer viscoelastic properties.

In recent decades, some researchers have developed many viscoelastic constitutive models for describing the rheological behavior of polymer based on different theories [3]. Among them, the model based on molecular theory can more truly reflect the rheological properties of fluid and can more fully reflect the flow of the fluid [4]. And for all we know, a branched polymer melt can be considered as a melt in which a certain concentration of branched molecules is embedded in a viscous melt. In this, Mcleish and Larson [1] proposed a Pom-Pom model based on Doi-Edwards’s peristaltic tubes theory. In the Pom-Pom model, they simplified each branched molecule to a molecule with only two branched points at each end, and with a certain number of arms at each branched points. This model is not completely consistent with the topological structure of branched molecules, but it is an important breakthrough in the field of viscoelastic constitutive models. This model introduces the important branched information and distinguishes the orientation relaxation time and extension relaxation time of backbone. It can also study the relaxation time of branched molecules and their effects on the above two relaxation time. Subsequently, Verbeeten et al. [5] improved the Pom-Pom model and proposed an extended Pom-Pom (XPP) model by using the slip tensor. This model overcomes some defects of Pom-Pom model, such as the discontinuity of steady-state stretching, the unrestricted orientation under the high strain, and the unpredicted second normal stress difference. In addition, Clemeur et al. [6, 7] proposed a Double Convected Pom-Pom (DCPP) model in order to solve the problem that the solution of XPP model is not unique. However, the DCPP model suffered from numerical instability in the numerical simulation. On the basis of this, Clemeur and Debbaut [8] proposed a modified DCPP model and Wang et al. [9] given the Simplified Modified Double Convected Pom-Pom (S-MDCPP) model with good numerical stability and easy programmable ability.

Generally, there are two kinds of Pom-Pom molecule constitutive models: single-equation model and double-equation model. Due to the simple solution and easy programming of the single-equation model, many studies have used the single-equation XPP model to simulate the viscoelastic flows [10–17], but it cannot describe some micro information. Double-equation XPP (DXPP) model can describe the microscopic orientation and stretch of backbone and study the influence of microscopic molecular parameters on the rheological behavior of branched polymers. However, due to the complexity of the DXPP model, there are few reports on the numerical simulation of this model. Therefore, DXPP model is used to study the microscopic information of the orientation and stretch of branched molecules in this paper.

In the past twenty or thirty years, the numerical simulation of viscoelastic flow has been developing rapidly and the main numerical methods are finite element method, finite volume method, and meshless method. Although there are many numerical methods, they each have their own advantages and disadvantages. There is no certain method to “dominate the world.” There is only one method to solve a problem when appropriate or not. Therefore, the combination of the merits of various methods to form a hybrid algorithm will be a trend of numerical simulation [16, 18, 19]. In this paper, the hybrid finite element method and finite volume method (FEM/FVM) are proposed based on the advantages of finite element method and finite volume method and the characteristics of the solved problem.

In addition, since the actual polymer processing is often a nonisothermal viscoelastic flow problem, the effects of temperature are also considered. The slip tensor of viscoelastic fluid actually affects the energy equation; that is, the energy equation is also different for different slip tensor [20, 21]. Therefore, we will give the derivation of the energy equation based on the slip tensor and study the influence of the slip parameter on the temperature.

Above all, the DXPP model is used to study the macro- and micro-rheological information of branched polymer melt. The energy equation based on the slip tensor is deduced and used to study the influence of slip parameters on the temperature. Subsequently, based on the characteristics of weakly-compressibility and high specific heat capacity of the polymer melt, the hybrid FEM/FVM method is used to solve the above model, and the macro and micro thermorheological properties of the branched polymer are discussed according to the numerical simulation results.

#### 2. Mathematical Models

##### 2.1. DXPP Model

Through the closed approximation, the evolution equation of backbone tube orientation tensor iswhere denotes the upper convected time derivative of orientation tensor ; is the rate of deformation tensor; the slip tensor is defined as where is a material parameter, defining the amount of anisotropy; is the relaxation time of the backbone tube orientation; the exponential stretch relaxation time ensures the stretch relaxes very fast and stays bounded for high strains; is the relaxation time for the stretch, and , where is the amount of arms at the end of a backbone; is the trace; is the backbone tube stretch and its material derivative is defined as Substituting (2) into (1) gives the orientation equationViscoelastic stress equation iswhere is the plateau modulus; is the unit tensor.

In conclusion, (3) and (4) constitute a DXPP model describing the backbone tube stretch and orientation using two decoupled equations; (5) denotes the viscoelastic stress. Here, the model is extended with a second normal stress difference when . By defining as the viscosity of polymer, as the Weissenberg number, and as the relaxation time ratio, dimensionless DXPP model can be written aswhere is the ratio of Newtonian viscosity to total viscosity and and are the velocity and length of the dimensionless parameters, respectively.

##### 2.2. Governing Equations

In the polymer processing, the weakly-compressibility of polymer melt cannot be ignored. Therefore, the weakly-compressible flow conservation equation is used to describe the polymer melt flow. For weakly-compressible viscoelastic flows, the conservation equations for mass and momentum can be expressed as follows, respectively,where denotes the Reynolds number; and are the density and viscosity of the dimensionless parameters, respectively.

The energy equations of different viscoelastic fluids also vary due to the different slip tensors. The derivation of the energy equation based on the XPP fluid slip tensor is given below.

The general form of the energy equation based on the slip tensor is as follows:where is the specific heat, is the temperature, is the heat flux, is the Cauchy stress tensor, is the material parameter, and their expressions are Substituting (2) and (12) into (11) gives the energy equationThe second term in the right-hand side of (13) reflects the contribution of entropy elasticity. The last one reflects the contribution of energy elasticity and . The dimensionless energy equation iswhere is the Peclet number, is the Brinkman number, and , , and are the temperature, specific heat, and coefficient of heat transfer of the nondimensional parameters, respectively, where

In addition, a P-V-T equation of state is necessary to satisfy the completeness of governing equations because of considering the compressibility of the polymer melt. Tait state equation [18] is usually considered as the classical empirical equation and is capable of describing both the liquid and solid regions. So Tait state equation is used in this paper.

#### 3. Numerical Methods

The flow of brand polymer melts is governed by the conservation of mass, momentum, and energy equations, Tait state equation, together with a DXPP constitutive model. The numerical simulation of the above model employs hybrid FEM/FVM [18] method. The momentum equations are solved by the FEM, in which a discrete elastic viscous stress split (DEVSS) scheme is used to overcome the elastic stress instability, and an implicit scheme of iterative weakly-compressible Crank–Nicolson-based split scheme (WCNBS) is used to avoid the Ladyzhenskaya–Babuška–Brezzi (LBB) condition. The energy and DXPP equations are solved by the FVM, in which an upwind scheme is used for the strongly convection-dominated problem of the energy equation.

To analyse the accuracy of the algorithm mentioned above, we construct the DEVSS scheme based on (10) and consider its discretization in the time domain within a typical time subinterval , which give us the form of the Wilson– method as follows:where is an added variable for constructing DEVSS scheme, .

The truncation error of (16) isBased on formula (17), (16) has the second-order accuracy when , which is adopted in this study for the Crank–Nicolson scheme.

Since the energy equation is deduced based on the slip tensor and the viscoelastic stress is calculated using a DXPP model that can describe the backbone orientation and stretch of the polymer molecules, we will detail the solution of the energy equation and the DXPP model based on the nonstaggered grid under the framework of the FVM. The energy equation and the DXPP model can be normalized as follows:where , are constants; and are the physical quantities and source term which are defined in Table 1. The terms from left to right in (18) represent the time, convective, diffusive, and source contributions, respectively.