Qiang Li,Fangcao Qu

School of Mathematics and Information Science,Henan Polytechnic University,Jiaozuo 454003,China

Keywords:LS-IB method Non-isothermal filling Non-Newtonian fluids Multiscale Numerical simulation

ABSTRACT In this work,the polymer melt filling process is simulated by using a coupled finite volume and level-set based immersed boundary (LS-IB) method.Firstly,based on a shape level set (LS) function to represent the mold boundary,a LS-IB method is developed to model the complex mold walls.Then the nonisothermal melt filling process is simulated based on non-Newtonian viscoelastic equations with different Reynolds numbers in a circular cavity with a solid core,and the effects of Reynolds number on the flow patterns of polymer melt are presented and compared with each other.And then for a true polymer melt with a small Reynolds number that varies with melt viscosity,the moving interface,the temperature distributions and the molecular deformation are shown and analyzed in detail.At last,as a commonly used application case,a socket cavity with seven inserts is investigated.The corresponding physical quantities,such as the melt velocity,molecular deformation,normal stresses,first normal stress difference,temperature distributions and frozen layer are analyzed and discussed.The results could provide some predictions and guidance for the polymer processing industry.

1.Introduction

Plastic products are processed by some specific technologies,such as pressure molding,blow molding,foam molding and injection molding,etc.And the injection molding is one of the main ways of plastic processing which includes four processes,i.e.,melt filling,packing,cooling and demoulding stages.The melt filling process is the primary and one of the most important stages.In this process,polymer melt endures the shear or stretch stress,and polymer molecules would deform and orient,which plays an important role in the mechanical properties and surface quality of the final plastic products [1].

The polymer filling process is a typical process with free surface flows.During this process,some important physical phenomena can be observed,such as corner effect,fountain and welding.There are many interface tracking methods used to deal with the moving interface problems.These methods can be divided into two types:the Lagrangian and Eulerian methods.The Lagrangian methods mainly include particle-based methods,such as smoothed particle hydrodynamics (SPH) method [2–4] and finite pointset method[5,6].While the Eulerian mehods include Marker and Cell method[7],volume-of-fluid (VOF) [8–10] method and level-set (LS)[11–13] method.Among these Eulerian methods,the VOF and LS methods are the most commonly used methods.However,it is known that LS and VOF methods have both advantages and disadvantages.LS method can handle merging and breaking of the interface automatically,but it cannot preserve the mass of reference fluid in the whole computational process [11].Compared with LS method,VOF method could ensure mass conservation naturally,but has the disadvantage that it cannot calculate accurately the parameters such as normal direction and curvature.So it is a natural trend to develop the hybrid methods in capturing free surface flow problems,which can combine both advantages and compensate for the disadvantages of individual methods,such as CLSVOF(coupled level set and volume of fluid)[14–18]and VOSET(volume of fluid and level set)[19–21]methods.Recently,a simple CLSVOF(S-CLSVOF)[22,23]method was used to track the moving interface in melt filling process.In S-CLSVOF method,only the VOF advection equation is solved,and then the exact signed distance field is obtained by using a relational formula and solving the reinitialization equation.

In addition,another important issue is how to handle the arbitrary boundary of mold cavity in the Cartesian coordinate system.To address this problem,the immersed boundary method (IBM)is usually used to treat the complex boundary problems [15,24–31].From the point of view of implementation,there are two kinds of IBMs,continuous forcing and discrete forcing methods [24–31].In the former,a force term is added to the fluid control equations before discretization[24–26].While in the latter,the control equations are first discretized on the grids without consideration of the immersed solid.Then the interpolation schemes are applied for the grid points at the vicinity of the immersed boundary [27–31].In such a way,the required boundary condition is satisfied at the interface.Besides,the computation is carried out on the Cartesian grids,while the solid boundary is represented by Lagrangian points.But it is a little difficult to exchange information on the two different grids.Recently,Cui et al.[31] proposed a level-set based immersed boundary(LS-IB)method,which was used to simulate the flow motions with arbitrarily deforming smooth bounderies.And the algorithm has been shown to perform very well in fluid–structure interaction problems.In their simulation,the solid boundary was created by several piecewise functions,and then the LS function became an accurate signed distance function by solving the re-initialization equation.But it is a little tedious to solve the re-initialization equation.In this work,a shape LS function is employed to represent the complex boundary of mold cavity,which is achieved by R-function based on Boolean operations of some basic geometries.This algorithm works well in producing an exact signed distance field.

This work focuses on the macroscopic and mesoscopic numerical simulation of melt filling process by using the S-CLSVOF,finite volume and LS-IB methods.To the best of our knowledge,it is the first time that the coupled finite volume and LS-IB method is applied to the simulation of viscoelastic melt flow.The governing equations are discretized by the finite-volume method and SIMPLEC (semi-implicit method for pressure-linked equationsconsistent) algorithm on the collocated Cartesian meshes.And the momentum interpolation method is used to solve the problem of the pressure–velocity decoupling.In order to obtain the correct velocity of polymer melt with complex boundary,the LS-IB method is used to obtain the virtual force field and corrected velocity near the solid–fluid interface.The improved S-CLSVOF method is employed to simulate the moving melt-front interface.And the oriented ellipses are employed to describe the molecular orientation and deformation based on the FENE-P constitutive model.Moreover,the visualization of polymer molecular deformation is achieved in melt filling process.

2.Numerical Methods

2.1.An improved S-CLSVOF method

In [23] we proposed an improved simple CLSVOF (S-CLSVOF)method,in which the VOF function is transported only,and the moving interface is constructed by using PLIC (piecewise linear interface construction) scheme.The VOF transport equation is,

where F is the VOF function.

Based on the interface,the LS function can be obtained from the volume fraction F on the interface grids (0

where ε0=0.2Δl,and Δl(Δx=Δy=Δl)is the uniform grid size.The value φ is then re-distanced by solving the re-initialization equation,

where τ is the artificial time step,x is the position vector.

Since the Eqs.(2) and (3) cannot conserve the liquid mass correctly,a mass correction formula is calculated for the interface cells according to their local interfacial curvatures,

where Ftis the actual liquid volume at time t,while F(φ) is liquid volume calculated by LS function value φ.κ is the interface curvature.Eq.(4) is discretized by the forward difference scheme.While the liquid volume fraction F(φ) is calculated by LS function [32],

2.2.Non-Newtonian fluid model

On the Cartesian grids,the immersed boundary method is used to treat the complex boundary problem [31].The governing equation for polymer melt can be expressed as [15],

Continuity equation:

Momentum equation:

where Reynolds number Re=ρmUL/?m,Peclet number Pe=ρmCm﹒UL/κm,Brinkman number Br=?mU2/κmT0(L and U are characteristic length and characteristic velocity,respectively).Density ρ=ρ(φ)=ρa+(ρm-ρa)Hε(φ),viscosity ?=?(φ)=?a+(?m-?a)Hε(φ),β is the ratio of Newtonian viscosity and total viscosity,subscript m and a represent melt and air,respectively.f is the volume force.μ is the solid volume fraction on each computational grid,i.e.,μ=0 for fluid grids,μ=1 for solid grids,and 0<μ<1 for the interface grids.The solid volume fraction μ is calculated by the shape LS function φ(refer to Eq.(5)).

In Eq.(7) the smooth Heaviside function Hεφ() is,

where ε=1.5Δl.

In this work the FENE-P model is used to describe the constitutive equations of polymer viscoelastic melt [33],

where Denorah number De=λ0bU/L,λ0bis the orientation relaxation time of polymer molecular chain.C=H0〈Q Q〉/(kT1) is the conformation tensor,H0is spring constant,Q is end-to-end distance of dumbbell,T1is absolute temperature,k is parameter,〈﹒〉 is the ensemble average of function.b=is the maximum tensile capacity of dumbbell;δ is the unit tensor.is the upper derivative of C:

2.3.Cross-WLF viscosity model and Tait state equation

In the mold filling process,the Cross-WLF model is used to describe the polymer melt viscosity ? [34],

where τ*is the model constant,n non-Newtonian index;shear strain rate,?0(T,p) the melt viscosity under zero-shear-rate conditions,and ?0(T,p) is,

where T*=D2+D3﹒P;A2=+D3﹒P;D1,D2,D3,A1andare all material parameters.

In melt filling process,in order to describe the change of viscoelastic melt with the temperature and pressure,the double domain Tait state equation is adopted to describe the melt density change [35],

where C0=0.0894 is a universal constant;b1,m,b1,s,b2,m,b2,s,b3,m,b3,s,b4,m,b4,s,b5,b6,b7,b8,b9are material constants.The subscript m and s represent the melt and solid,respectively.Tt(P)=b5+b6P is glass transition temperature.V0(T)is the isothermal specific volume under zero pressure conditions.B(T)expresses the degree of influence of pressure on the specific volume.Vt(p,T)is introduced to account for the volume decrease due to crystallization.

2.4.Implicit mold construction method

In order to construct the mold cavity more effectively,a kind of excellent implicit modeling tool—R-function is used to construct the implicit mold cavities accurately [36].The R-function relates to Boolean operations of intersection (∩) and union (∪) between two geometries.In particular,a system Rαis obtained as follows,

where -1<α ≤1,φ1and φ2are two implicit functions,‘‘∧α”and‘‘?α”are R-conjunction ∧and R-disjunction ?.These are equivalent to Boolean operations ∩and ∪widely used in CAD.For the sake of simplicity,α takes 0,and the common used system is obtained,

The R-function is not differentiable only when φ1=φ2=0,i.e.,only at the intersection point of geometry boundary it is not differentiable [36].Thereafter,Eq.(18) is used to construct the implicit shape LS function φ representing the mold cavity.Take a circular mold cavity with a core as an example to illustrate the process.Firstly a disk Ω1at point A(6.75,11.5) with a radius of 6.75 can be constructed easily(Fig.1(a)).Then a rectangle Ω2with the size of 9.0×4.5 shown in Fig.1(b).And we can get a circular mold cavity Ω3from Ω1and Ω2by R-disjunction ?,shown in Fig.1(c).Fig.1(d) provides a circular insert Ω4at piont B(6.75,11.5)with a radius of 2.25.At last,the circular mold cavity Ω5with a core is obtained based on Ω3and Ω4by R-conjunction ∧.And the overall size of mold cavity is 18.3×13.5.

Fig.1.The combination of a circular mold cavity with a core by using R-function.

Fig.2.Sketch of the level-set based immersed boundary method.(The blue square represents solid node,red circle is virtual node and green triangle is fluid node).

2.5.Level set based immersed boundary method

In[31],the initial shape LS function φ is constructed by several piecewise functions,but overall it is not a distance field.Then by solving the re-initialization equation the shape LS function become an accurate signed distance function.But it is a little complicated to solve the re-initialization equation.Instead we employ Rfunction to create the signed distance function from some basic geometries.Here a shape LS function is used to represent the solid boundary of mold cavity.The nodes whose LS values are negative are solid nodes.The nodes whose LS values and those of the four adjacent nodes are both positive,then the nodes are fluid nodes.Otherwise the nodes are virtual nodes.Because the solid is stationary,the velocities of solid nodes are zero in the whole computational process.The velocities for fluid nodes can be obtained by solving the non-Newtonian equations.Due to the existence of solid boundary,the velocities for the virtual nodes near the solid boundary are usually not correct,which should be adjusted.Here the bilinear interpolation is used to obtain the velocities for virtual nodes [31].

Fig.2 shows the triangle template of bilinear interpolation scheme which is used to reconstruct the velocity ubat the virtual node b.For the variable α (here refers to velocity) in the twodimensional space,the coefficients for the linear interpolation can be written as:

Fig.3.Result of melt filling process in a circular cavity at different time (left:numerical result in [38];right:numerical result in this work).

Fig.6.The weldline at the end of melt filling (Left:experimental result;Right:numerical result).

The coefficients b1,b2and b3in Eq.(19) are obtained by:

where(xi,yi)with i=1,2 and 3 in matrix A are the node coordinates of the interpolation stencil,i.e.,the triangle vertexes 1,2 and 3.And the coordinates of triangle vertexes are easy to obtained by the shape LS function φ [31].In this simulation,because the solid boundaries are not moving with time,the inversion of matrix A is calculated in advance,and we could use it directly at each time step.Once the coefficients b1,b2and b3are obtained,substituting the node coordinates (xi,yi) into Eq.(19),we can get the interpolation velocity,and then the virtual force on the virtual points is calculated as:

where RHSnrepresents the source term,i.e.,the sum of the convection term,the viscous term,the pressure gradient and viscoelastic term.

The time step Δt is determined by restrictions due to CFL condition and viscosity [37]

Then the eventual restriction on the time step is chosen as Δtn+1=min(Δtc,Δt?).

3.Numerical Results

In this section,the present numerical model is validate by two viscoelastic flow problems:flow around a circular cylinder and melt filling in a circular mold cavity with a core.Then the melt filling processes for the polymer ABS780 are simulated in two different complex mold cavities,i.e.,a circular mold cavity with a core and a socket cavity with seven inserts.Some important physical quantities,such as velocity,normal stresses,first normal stress difference,are provided and analysed in detail.Meanwhile,the phenomena of melt molecular conformation and solidification are also shown and discussed.And the whole computation module is developed by in-house FORTRAN code.

3.1.Validation test for the LS-IB method

The LS-IB method was used to simulate incompressible flows with arbitrarily deforming smooth boundaries [31].And the accuracy and robustness of LS-IB method were tested systematically by several benchmark cases.It was found that the algorithmperformed well in simulating the flow motions surrounding the deforming and moving bodies [31].In this work,the R-function is employed to construct the implicit boundaries of mold cavities accurately,which is more simple and efficient than the method used in [31].And the LS-IB method could be extended to threedimensional simulation easily.Hereafter,three numerical examples for viscoelastic flows of polymer melt are discussed to illustrate the efficiency of the coupled algorithm.

Fig.7.Results of melt filling process in a circular mold cavity with different Re numbers:Re=5,2 and 0.2 (from left to right).

Fig.8.Molecular orientation in a circular mold cavity with a core at t=8.2 for different Re numbers:Re=5,2 and 0.2 (from left to right).

3.1.1.Melt filling in a circular mold cavity with a core

Fig.1 shows the shape of a circular mold cavity with a core in this simulation.Ren et al.[38] simulated the melt filling process in the same cavity using SPH method.Fig.3 shows cavity filling process at different times.At t=6.2 after the polymer melt flows through the circular inset and the melt is divided into two branches,which hit the ring solid wall separately,and then flow along the circular solid wall.When the two branches of polymer melt meet each other,the melt flows back to the circular inset(Fig.3(b)).From Fig.3 it is easy to see that different methods and treatment of boundary conditions result in slightly different numerical results,but the numerical result is in qualitative agreement with that obtained by SPH method (Fig.3-left) (see Fig.4).

3.1.2.Melt filling in a rectangle cavity with a diamond insert

In order to further validate the coupled method for two-phase free surface flow problem,the melt filling process is simulated ina rectangle cavity with a diamond insert,whose size is illustrated in Fig.4.The material parameters can be found in [40].The size of the rectangle cavity with an insert is shown in Fig.5.The maximum injection velocity is 0.9 m.s-1.And the melt and cavity temperatures are 498 K and 343 K,respectively [40].

Fig.4.The size of the rectangle cavity with a diamond insert.

Fig.5.The experimental (a) and numerical (b) results of the melt fronts at different time (From left to right:t=0.4 s,0.5 s,1.0 s,1.3 s).

Fig.9.Melt front and molecular orientation in a circular mold cavity at different times.

Fig.10.Temperature distributions of melt in a circular mold cavity with a core at different times.

Fig.11.Molecular orientation and temperature distribution in a circular mold cavity with a core at the end of melt filling.

Fig.12.Sketch and size of a socket cavity.

Fig.5 shows the comparison of numerical simulation results and experimental results of melt filling process.It can be seen from Fig.5 that the numerical results by the coupled method are in good agreement with the experimental results.When the melt flows through the insert,it is divided into two streams of melt flowing towards the end of the product (Fig.5(a)).At the same time the weldline will form at the meeting point and disappear when the encounter angle is great than 423 K [40].According to this theory,the two streams of melt meet at the right vertex of the insert,resulting in the weldline (Fig.5(b)).Fig.6 shows the location of the weldline at the end of melt filling,which is in good agreement with the real position of weldline in [39].

3.2.Filling results for the polymer melt

3.2.1.Filling results for different Re numbers

As is known that the Reynolds (Re) number can affect the flow style significantly.Here the polymer of ABS780(Acrylonitrile butadiene Styrene)is chosen as simulation material.The parameters of Cross-WLF viscosity model and Tait state equation coefficients are listed in Tables 1 and 2,respectively,where,and are 0.The com-putational domain is in dimensionless form.The sizes of space-step and time-step are 0.1 and 0.025,respectively.In order to study how the Re numbers affect the flow pattern,in this simulation Re number is taken as a fixed value in Eq.(7).Herein three different Re numbers,i.e.,Re=5,2 and 0.2,are chosen as examples,respectively.

Table 1 Cross-WLF viscosity model and thermal properties parameters of ABS780

Table 2 Tait state equation parameters of ABS780

Fig.13.Melt interface fronts for different step sizes of space and time (blue:Size1,black:Size2,red:Size3) at different time ((a) t=1.7,(b) t=2.8).

Fig.14.The melt velocity field at (a) t=0.6,(b) t=1.9,(c) t=2.9 and (d) t=4.0.

Fig.7 illustrates the results of melt filling process with different Re numbers.From Fig.7,it is easy to find that the melt flow tends to be stable with the decrease of Re numbers.For Re=5,the melt flow is not stable.For Re=2,it is similar to that for Re=5,but more stable.In this case,it is still easy to cause air entrainment.For Re=0.2,the phenomenon of air entrainment also appear in the two bottom sides of annular region.But in the region where two branches of melt meet,the phenomenon of air entrainment does not occur.From these results we can find that in the mold filling process,in order to obtain qualified products,we should inject polymer melt into the mold cavity with small Re numbers.

Since in Eq.(9) is symmetric and positive definite,the eigenvalue and eigenvector of can be obtained by Jacobi method.In this work,the oriented ellipses are used to describe the deformation and orientation of polymer molecules,which is represented by[33].Fig.8 shows the melt front and molecular orientation with different Re numbers at t=8.2.From Fig.8 we can see that the macromolecules near the solid wall endure large shear stress,and orient in the flow direction.With the decrease of Re numbers,the shear stress becomes smaller.The numerical results show that the Re number can not only affect the flow pattern,but also influence the shear stresses significantly.

3.2.2.Filling results for a small Re number

In fact,in the true injection molding process,due to the large viscosity and low injection velocity,the Re number of melt flow is usually very small.In this section,the corresponding Re number varies with melt viscosity which is described by Cross-WLF viscosity model (see Eq.(11)).In this simulation,the melt temperature and mold temperature in dimensionless form are 553 and 323(the temperature scale),respectively.So the maximum value of Re number for melt temperature T=553 is about 0.01.Fig.9 shows the melt front and molecular orientation at different times.Because of the low velocity and large viscosity,the flow of polymer melt is relatively stable,and the air entrainment phenomenon does not appear.Similar to the above situations,during the polymer melt filling process,the macromolecules near the wall endure large shear stress,especially near the wall of straight tube.

Fig.10 provides the temperature distributions of polymer melt at different times.The internal melt temperature is relatively high.And the melt temperatures are low near both of the mold wall and melt front.When two branches of polymer melt meet each other the weldline would form.Fig.10 shows molecular orientation and temperature distribution at the end of melt filling.From Fig.11 it is easy to find that the lowest temperature is mainly concentrated in the encounter area.To improve the quality of plasticproducts,we should raise the temperature of this region to reduce or eliminate the weldlines.

Fig.15.Melt front and molecular orientation in a socket cavity at different time.

3.3.Melt filling in a socket cavity with seven inserts

The above mold cavity is relatively simple,here the filling process in a socket cavity with seven irregular inserts is studied.Fig.12 shows the shape and size of the socket,which is composed of one double-aperture,one triple-aperture,and two USB charging ports.In this simulation,the whole computational domain is in dimensionless form,and the blue region represents the seven inserts and two solid bodies(at the mold tail).The material parameters are the same as those in Section 3.2.The polymer entrance area is located at the left boundary,and its y coordinates belong to [2.5,5.5].

We first test the solution sensitivity to the step sizes of space and time.The step sizes for space and time are chosen as (0.2,0.05)(Size1),(0.1,0.025)(Size2)and(0.05,0.0125)(Size3),respectively.Fig.13 shows the melt front and magnification of local interfaces for different step sizes,respectively.The blue,black and redlines denote the simulation results for Size1,Size2 and Size3,respectively.From Fig.13 it is easy to find the good step size convergence of our solutions.The numerical results also demonstrate the black lines almost coincide with red lines.Thus in the following simulation,the step sizes of space and time are chosen as (0.1,0.025).

Fig.16.The normal stresses (a),(b),(c) and (d) first normal stress difference at the end of melt filling.

Fig.17.The first normal stress difference at (a) x=1.7 and (b) x=6.1 at the end of melt filling.

Fig.14 shows the melt velocity distributions at different times.From Fig.14,it can be seen that the melt flows around the insets smoothly,and the high velocity is distributed at the wide channels.Since the velocity is relatively low,there is no vortex behind the insets.Due to the obstacles of insets and the narrow middle channel,the melt velocity between the double-aperture is more slowthan that at two wide chennels.At the whole process,the velocity is almost zero at the upper-and lower-left corners.And also it can be proved that the complex boundary problems can be handled well by the proposed LS-IB method.

Fig.18.Melt front and molecular orientation in a socket cavity at different time.

Fig.15 shows melt front and molecular orientation in a socket cavity at different times.At the beginning of the melt filling,the melt flows forward while spreading to both sides (Fig.15(a)).When the polymer melt flows around the triple-aperture,due tothe obstruction of the insert to the flow,the melt flows a little faster at both sides of the channel (Fig.15(b)).And there are two air holes on the two left cavity corners.Then the melt flow becomes complex due to the USB charging ports (Fig.15(c) and (d)).When the melt flows around the double-aperture (Fig.15(d) and (e)),the three branches of melt meet each other (Fig.15(e)),and then polymer melt flows forward till fill the mold cavity (Fig.15(f)).At the whole melt filling process,the macromolecules endure large shear stress near both of the hole inserts and the solid wall,which would lead to large residual stress,and finally affect the product quality significantly.

Fig.19.Melt temperature at (a) x=1.5 and (b) x=7.0 at the end of melt filling.

Fig.20.Frozen layer (a) and weldlines (b) in the socket cavity at the end of melt filling.

Fig.16 provides the distributions of the normal stresses,,and first normal stress difference at the end of melt filling.From Fig.16,it is easy to see that the normal stresses and first normal stress difference change dramatically around the inserts.In order to better illustrate the first normal stress difference,two different positions are chosen,i.e.,x=1.7 and x=6.1(marked with two lines in Fig.16).Fig.17 shows the distributions of first normal stress difference at x=1.7 and x=6.1.It can be observed that the maxmum values are obtained near the inserts.And the phenomenon of molecular orientation and shear deformation is in agreement with the change regularity of first normal difference.

Fig.18 provides the melt temperatures during the melt filling process at different times.Similar to the above simulations in Section 3.2,the highest temperature is distributed mainly in the internal melt.Due to the cold inserts,there are some low temperature zones around the inserts.Because the double-aperture is on the middle channel,the temperature of the polymer melt in front of the two-aperture is also very low(Fig.18(e)and(f)).Fig.19 shows the temperature distribution on the lines x=1.5 and x=7.0(marked with lines in Fig.18) at the end of melt filling.From Fig.19,it can be seen that the temperature is very low near the cavity wall,and becomes more higher from the wall to the center.Because the line of x=1.5 is near the injection gate,the inner temperature is relatively high.Due to the obstacles of double-aperture,the temperature fluctuates (Fig.19(b)).Fig.20 shows the frozen layer and the weldlines in the socket cavity at the end of melt filling.From Fig.20,the frozen layer is mainly distributed at the end of mold cavity and the zone where the weldlines exist.Moreover,if the weldline appears in the solidified layer,it is a true weldline;otherwise it actually is a fusion line.Thus,we should raise the melt temperature in these regions to reduce the weldlines,and thereby improve the quality of plastic products.

4.Conclusions

The coupled S-CLSVOF,finite volume and LS-IB method is used to simulate the filling process in two complex mold cavities.Firstly,a shape LS function is used to represent the complex cavity.Then,the LS-LB method is employed to handle the complex boundary of mold cavity,and the oriented ellipses are adopted to describe the behaviors of stretch and shear.And then the visualization of molecular conformation is realized in the melt filling process.Finally,it is discussed that the influence of Reynolds number on the flow pattern and melt molecular conformation in the circular cavity with a core.As an application case,the melt filling process is simulated in a socket cavity with seven inserts.The melt velocity,normal stresses,first normal stress difference,molecular conformation and melt temperature are shown at different times.The main conclusions are as follows:

(1) The LS-LB method can deal with the arbitrary boundary of mold cavity successfully.

(2) The Re numbers could affect the flow pattern significantly.With the decrease of Re numbers,the melt flow becomes more stable,and the air entrainment disappears for small Re numbers.

(3) The oriented ellipses can well reveal the change regularity of melt molecular conformation in the melt filling process,i.e.,the behaviors of molecules stretch and orientation.

(4) Because there are seven inserts in the socket cavity,the melt flow pattern become more complex.Around these inserts the normal stresses and first normal stress difference are more larger.And at the end of melt filling,the frozen layer is mainly distributed at the mold tail and the regions where the weldlines form.Therefore,we should raise the melt temperature in this domain to improve the quality of the plastic product.

Acknowledgements

This work is supported by the National Natural Science Foundation of China(11701153),the Foundation of the Developing Center of INCTMat at Federal University of Parana,Brazil (465591/2014-0),and the Research Fund for the Doctoral Program of Henan Polytechnic University (B2013-057).

Nomenclature

A1material constant

Br Brinkman number,Br=?mU2/κmT0

C conformation tensor

De Denorah number,De=λ0bU/L

D1material constant,Pa﹒s

D2material constant,K

D3material constant,K﹒Pa-1

F volume of fluid function

f volume force,N

Hε Heaviside function

L length scale,m

n non-Newtonian index

T temperature,K

T0temperature scale,K

U velocity scale,m﹒s-1

u velocity vector,m﹒s-1

β the ratio of Newtonian viscosity and total viscosity

? dynamic viscosity,Pa﹒s

κ thermal conductivity,W﹒m-1﹒°C-1

μ solid volume fraction

ρ density,kg﹒m-3

τ*Cross-WLF model constant,Pa

φ shape level set function

φ level set function

φ0initial value of level set function

Subscripts

g gas phase

m melt phase

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.