Improved SPH Approach to Branched Polymer Free Surface Flows Based on the XPP Model

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1 Polymer-Plastcs Technology and Engneerng, 50: , 2011 Copyrght # Taylor & Francs Group, LLC ISSN: prnt= onlne DOI: / Improved SPH Approach to Branched Polymer Free Surface Flows Based on the PP Model Tao Jang, Je Ouyang, Ln Zhang, and ueuan L Department of Appled Mathematcs, Northwestern Polytechncal Unversty, an, Chna In ths work, an mproved smoothed partcle hydrodynamcs (IMSPH) method s presented to smulate a polymer-free surface based on PP model. The IMSPH s a coupled approach between smoothed partcle hydrodynamcs (SPH) method and the fnte partcle method (FPM), whch possesses hgher accuracy and better stablty than SPH, especally on boundary. An artfcal stress term s added to IMSPH for removng the unphyscal phenomenon. The examples of mpactng drop and et bucklng are nvestgated, and the nfluence of the physcal parameters on movng free surface wth hgh Wessenberg number s consdered. All the numercal results agree well wth the avalable data. Keywords Artfcal stress; Polymer; SPH; Surface; Vscoelastc; PP model INTRODUCTION The problems of free surface hydrodynamc flows for polymers are mportant n today s ndustry, such as extruson, contaner fllng n the food and pharmaceutcal ndustres of polymers. In these ndustres, all the flows nvolved almost exhbt vscoelastc behavor, whch ncludng the lnear or non-lnear vscoelastc behavors. To llustrate the vscoelastc behavors of branched polymer melts [1], a sutable polymer model s needed. In general, the lnear vcoelastcty corresponds to small stran or deformaton, whle the stuaton s opposte to the non-lnear vscoelastcty. Here, based on the PP model, whch s one of the extended Pom-Pom models for descrbng branched polymer melts [1], the non-lnear vscoelastc concentrated polymer materals are studed. In the non-newtonan flud mechancs communty, vscoelastc free surface flows have been studed for more than 30 years. Many methods are presented manly to capture the free surface, whch nclude partcles n cell (PIC) [2], marker and cell (MAC) [3], volume of flud (VOF) [4] and level set [5] methods. Although they have been successfully appled to varous flow problems nvolvng free surfaces Address correspondence to Je Ouyang, Department of Appled Mathematcs, Northwestern Polytechncal Unversty, an , Chna. E-mal: eouyang@nwpu.edu.cn such as reported n [6,7], PIC and MAC are complcated n programmng especally for three-dmensonal case. The VOF method s used to ascertan the free surface by the flled part of each control volume whch s decded by solvng an addtonal partal dfferental equaton. The methods mentoned above are based upon grd-based numercal methods such as fnte dfference methods (FDM) and fnte element methods (FEM) that are commonly used to solve the equatons of Naver Stokes. However, t s usually dffcult for the smulaton of large deformaton wth the grd-based methods. Recently, the varous partcle methods have been proposed n a Lagrangan framework. Among the varous meshless methods, smoothed partcle hydrodynamcs (SPH) method [8,9] s one of the earlest meshless methods employng Lagrangan descrpton of moton. The SPH has several advantages over grd-based methods: It handles convecton domnated flows and large deformaton problems wthout any numercal dffuson; It handles easly complex geometres especally for three dmensons. Complex free surfaces are modeled naturally wthout the need of any form of explct surface trackng technque. Programmng for complex problems s easy to mplement compared wth grd-based methods. In 1994, the SPH method was frst used to deal wth flud mechancs problems [10]. Snce ts nventon, t has been extensvely studed n many areas such as vscous flows [11,12], ncompressble fluds [13], heat transfer [14], mult-phase flows [15,16], vscoelastc flows [17]. However, the SPH suffers from several drawbacks ncludng low accuracy, tensle nstablty, and dffculty n enforcng essental boundary condtons. Therefore, varous methods have been developed to mprove the accuracy of the conventonal SPH. In 2000, Chen and Beraun [18] developed a corrected smoothed partcle hydrodynamcs method for non-lnear dynamc problems usng Taylor seres expanson. Lu et al. [19] appled a smlar dea to vscous flows, whch was called as fnte partcle method (FPM). Followng, an mproved SPH method (IMSPH) couplng SPH and FPM was presented by Jannong Fang et al. [20] and appled to free surface flows of vscous fluds. 203

2 204 T. JIANG ET AL. It s not untl 2006 that the SPH method s frstly appled to transent vscoelastc free surface flows by Fang et al. [21], where a drop of an Oldroyd B flud mpactng a rgd plate s smulated by usng SPH. Later, Rafee et al. [22] proposed the ncompressble SPH method of usng a Posson pressure equaton to satsfy the ncompressblty constrants, n whch vscoelastc free surface flows for an Oldroyd B and Maxwell models was smulated. To our knowledge, no meshless approaches to vscoelastc free surface flows for concentrated polymer solutons and melts have appeared so far. Ths s because that the rheologcal behavors of branched polymer melts are more complcated than those of the general vscoelastc flows such as Oldryod-B flud. Smlarly, t has not appeared that the IMSPH s used to smulate the free surface flows of concentrated polymer solutons. In ths work, the IMSPH s extended and tested for smulatng blanched polymer free surface flows usng the PP model, whch s one of the extended Pom-Pom models for descrbng concentrated polymer solutons and melts [23]. Thus, the PP model s more general than the Oldroyd-B model n the descrpton of polymer melts, and the former model can degrade nto the latter model. Ths may be another reason why we use the PP model. Moreover, the phenomenon of free surface based on the PP model s dfferent from the correspondng Oldroyd-B model case. Ths paper s organzed as follows: next, the governng equatons for the PP model are ntroduced; then, we descrbe the IMSPH dscretzaton of the Naver-Stokes equatons, ncludng artfcal vscosty, boundary condtons, temporal dscretzaton of the governng equatons. Specally, the tensle nstablty and artfcal stress are also dscussed. The valdty of IMSPH for capturng polymer free flow s frst tested by smulatng the combned Poseulle and Couette flow. Subsequently, two challengng numercal examples, ncludng a drop mpactng onto a dry surface and the et bucklng are solved to demonstrate the capablty of the IMSPH method n handlng vscoelastc free surface flows for concentrated polymer solutons and melts, especally for hgh Wessenberg number. GOVERNING EQUATIONS FOR THE PP MODEL The flow of concentrated polymer solutons and melts s governed by the conservaton of mass and momentum equatons, together wth a consttutve equaton. In a Lagrangan frame, the flow of an sothermal flud s descrbed by the followng equatons of moton and contnuty: Dq Dt ¼ qru ð1þ where u ¼ (u, v) s the velocty vector, q s the flud densty, t s tme, r s Cauchy stress tensor, g s the gravtatonal acceleraton and D=Dt s the materal dervatve D=Dt þ u r. The Cauchy stress tensor n Eq. (2) s decomposed nto the ordnary sotropc pressure p, vscous 2g s d and polymerc s contrbutons r ¼ pi þ 2g s d þ s; where I refers to dentty matrx, d s the rate of deformaton tensor, whch s gven by d ¼ 1 ru þðruþt ð4þ 2 PP Model In ths paper, the sngle equaton verson of the extended pom-pom model (PP) n mult-mode form [24] s employed. The PP model has two mportant features: The frst feature les n recognzng the dependence of melt rheology upon the polymer molecular structure. The second feature s that the spectrum of relaxaton tme to be taken nto account leads to two partal dfferental equatons, one for orentaton and one for stretch. The consttutve equaton for the PP model s f ðk; sþs þ k r 0b s þg 0 ½f ðk; sþ 1ŠIþ a 0 s s ¼ 2k 0b G 0 d G 0 where the functon f(k, s) s gven by f ðk; sþ ¼2 k 0b e nðk 1Þ 1 1 þ 1 k 0s k k 2 1 a 0I ss 3G0 2 and the symbol r represents the followng upperconvected dervatve r Ds s ¼ s ru Dt ð Þ ð ru ÞT s ð7þ k 0b and k 0s are the orentaton and backbone stretch relaxaton tme respectvely, G 0 s the lnear relaxaton modulus, I refers to the trace of a tensor. The consttutve equaton possesses the features of Gesekus model snce a non-zero second normal stress dfference s predcted when the ansotropy parameter a 0 6¼ 0. In the PP model, the backbone stretch k s related to the vscoelastc stress tensor sffffffffffffffffffffffffffffffff k ¼ 1 þ I s ð8þ 3G 0 ð3þ ð5þ ð6þ Du Dt ¼ 1 q rr þ g ð2þ where the symbol represents the absolute value. The parameter v n the exponental term n Eq. (6) s

3 SMOOTH PARTICLE HYDRODYNAMICS AND POLYMER FLOW 205 ncorporated nto the stretch relaxaton tme to remove the dscontnuty from the gradent of the extensonal vscosty. Its value s found to be nversely proportonal to the number of arms q, n ¼ 2=q. Here, the followng parameters are ntroduced, namely the total vscosty g ¼ g s þ g p, g p ¼ G 0 k 0b, e ¼ k 0s =k 0b, b 0 ¼ g s =(g s þ G 0 k 0b ) where e s the rato of the stretch to orentaton relaxaton tme that shows that small values of e correspond to hghly entangled backbones [1,23]. Specally, Eq. (2) degenerates the equaton of moton for Oldroyd-B flud when a 0 ¼ 0 and f(k, s) ¼ 1 n Eq. (5). In addton, f b ¼ 0 then the UCM model s obtaned. Equaton of State Sometme, the ncompressble flows are treated as a slghtly compressble by adoptng a sutable equaton of state n many prevous works (see Monaghan [10] ). Here, the ncompressble flows are also treated as slghtly compressble flows by usng the followng equaton of state [17] rather than the method of pseudocompressblty [25]. pðqþ ¼ c 2 q 2 = 2q 0 ð9þ where c s the speed of sound and q 0 a reference densty. It can be shown that the densty varaton s proportonal to the Mach number M 2 (M V=c, where V s a typcal reference velocty) [10]. IMSPH DICRETIZATION To mprove the effcency of the hgher-order varant, a coupled method wth the dea of usng the SPH approxmaton for the nteror partcles and the FPM [19] approxmaton for the exteror partcles was proposed by Fang et al. [20] and appled to smulate vscous fluds. Here, the IMSPH s extended and tested for smulatng non-lnear vscoelatc free surface flows of concentrated polymer solutons and melts. Next, we brefly descrbe the SPH, FPM and IMSPH method for dscretzng the Eqs. (1) (2) based on the PP model, respectvely. SPH Method The SPH method [8,9] s based on the nterpolaton theory, whch s the theory of ntegral nterpolates usng a kernel functon. In the SPH method, the flud doman s dscretsed nto a fnte number of partcles, where all the relevant physcal quanttes are approxmated n terms of the ntegral representaton over neghborng partcles. Each partcle carres a mass m, velocty u, and other physcal quanttes, dependng on the problem. Any functon f defned at the poston r ¼ (x, y) can be expressed by the followng ntegral Z hf ðrþ ¼ f ðr 0 ÞWðr r 0 ; hþdr 0 ð10þ where W s the so-called kernel functon and h s the smoothng length defnng the nfluence area of W. The kernel functon usually satsfes the followng propertes: and lm h!0 Z Wðr r 0 ; hþdr 0 ¼ 1 ð11þ Wðr r0; hþ ¼ dðr r 0 Þ ð12þ To be vald, t s often requred that Wðr r 0 ; hþ > 0 over ð13þ Wðr r 0 ; hþ ¼ 0 when r r 0 > kh ð14þ where k s a constant that s usually chosen by the gven kernel functon. If the smoothng functon W s an even functon over, by usng the Taylor seres expanson of f ðr 0 Þ around r, t can be shown that the ntegral representaton of f ðr 0 Þ s of second order accuracy. However, ths s true only for nteror regons. We can get the followng partcle approxmaton formula from Eq. (10) hf ðþ r m f W r r ; h q ð15þ where m and q are the mass and densty of the th partcle, and f ¼ f(r ) m =q represents the occuped volume by the th partcle. In order to have an accurate nterpolaton, the smoothng length h should be chosen bgger than the mean nter-partcle dstance. The partcle approxmaton for a functon and ts frst dervatve at partcle can be wrtten n condensed form as f m q f W m q ðf ð16þ ð17þ where W ¼ W(r r, =@r r, h)=@r =@r =@r. The smoothng functon s related not only wth the accuracy but also wth the effcency and stablty of the resultng algorthm. In order to ncrease the accuracy of the SPH, the quntc splne functon s chosen as the smoothng functon whch s the functon about r ¼r r and h. Let k ¼ 3n Eq. (14), then t reads

4 206 T. JIANG ET AL. W ¼ Wðr; hþ 8 ½3 ðr=hþš 5 6½2 ðr=hþš 5 ; 0 r=h< 1 >< þ 15½1 ðr=hþš 5 ¼ w 0 ½3 ðr=hþš 5 6½2 ðr=hþš 5 ; 1 r=h< 2 ½3 ðr=hþš 5 ; 2 r=h< 3 >: 0; r=h 3 ð18þ where the normalzaton factor w 0 s chosen as 7=(478ph 2 )n two-dmensonal problems. In ths work, the velocty and total stress gradent are calculated by Eq. (17). Then, the dscretzaton schemes of the governng equatons for PP model can be obtaned at the partcle by the followng expressons ðruþ ¼ 1 q rrab ¼ m q ðu b u b m r ab q 2! þ q b ð19þ ð20þ where u b s the b th component of the flud velocty, r ab the (a, b) th component of the total stress tensor, x b the spatal coordnate. Introducng the velocty gradent b ¼ m q ðu a u b ð21þ and the dscretzaton schemes of consttutve equaton for the PP model can be defned as r ab ¼ pd ab þb 0 g ab þ ba þ s ab ð22þ ds ab ¼ ac s cb þ bc s ca dt ð Þ f k; s k 0b a 0 s ab ð1 bþg sac ð1 b 0Þg ðk 0b Þ 2 ½f ðk; s Þ 1 þ ð1 b 0Þg ab k 0b s cb where f ðk; s Þ¼ 2 e enðk 1Þ 1 1 k þ 1 k 2 0b k 2 1 a 0 s ab 3ð1 b 0 Þg sffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff k ¼ 1 þ k 0b s aa 3ð1 b 0 Þg d ab ¼ 1fa ¼ b and d ab ¼ 0fa 6¼ b. Šd ab þ ba s ab! ; ð23þ ð24þ Fnte Partcle Method (FPM) By the Taylor seres expanson s concept, Lu et al. [19] proposed a correcton formula for both the SPH kernel functon and the dervatves of kernel functon, whch was the so-called fnte partcle method. The FPM conserves both lnear and angular momentums (see [20] ), and possesses the hgher accuracy than that of the SPH method. For a suffcently smooth functon f(r) at pont r ¼ (x, y ), by a Taylor seres expanson and to retan only three terms n the seres, t can be expressed as f ðþ¼f r þ x a x a f;a ð25þ where a s the dmenson ndex repeated from x to y, f and f,a are defned as f ¼ f(r ), f ;a ð r respectvely. a Multplyng both sdes of Eq. (25) wth a weght functon w(r r, h), and ntegratng over the doman, t can yeld Z Z f ðþw r ðr r ; hþdx ¼ f wðr r ; hþdx þ f ;a Z x a x a w ð r r ; hþdx ð26þ By the concept of SPH method, Eq. (26) can be numercally approxmated by summaton over the partcles surroundng pont r as f r w r r ; h DV ¼ f w r r ; h DV þ f ;a x a x a w r r ; h DV ð27þ Equaton (27) can be rewrtten n the followng form of matrx equaton where S ;m ¼ A ;mn F ;n ð28þ T F ;n ¼ f ; f ;x ; f ;y ð29þ S ;m ¼ f r w m r r ; h DV ð30þ A ;mn ¼ w m ðr r ; hþdv ; ðx a x a Þwm ðr r ; hþdv ð31þ w 1 ¼ w, w w

5 SMOOTH PARTICLE HYDRODYNAMICS AND POLYMER FLOW 207 Here, the weght functon w s chosen the same as kernel functon W of SPH method, so w 1 ¼ W ; w 2 ; w 3 ð32þ Solvng the matrx equaton (28), the functon value as well as the frst dervatve at partcle can be calculated as f m q f ^W m q ð33þ ð34þ where DV s replaced by m q and ^W s called the modfed kernel functon. Rearrangng Eqs. (28) (31), ^W and ts dervatve can be obtaned as A 11 A 12 A 13 ^W A 21 A 22 A 23 ^W A ð35þ A 31 A 32 A where A l1 ¼ P m q w l, A l2 ¼ P m q / x w l, A l3 ¼ P m q / y w l, l ¼ 1, 2, 3, /(x ) ¼ x x, /(y ) ¼ y y. In the FPM, a local matrx should be solved at partcle. When the matrx A s not sngular, Eq. (35) can determne a unque set of soluton at partcle for the w l. Accordng to ref. [20], we ðruþ ¼ 1 q rrab ¼ m q q b ¼ and get m q ðu b u b b r b r b m q ðu a u a b! ð36þ ð37þ ð38þ IMSPH Method The IMSPH method s based on the hgher accuracy of FPM than SPH and the lower computatonal effcency of SPH than FPM. Its dea s n the nteror feld of usng the SPH method and n the exteror feld of usng the FPM. Therefore, the IMSPH method possesses the merts of SPH and FPM. Accordng to ref. [15], free surface partcles are dentfed by partcle denstes. Snce there s no partcle outsde the surface partcles, the partcle densty wll drop on the surface. A partcle s regarded as a surface partcle f ts densty wll be less than a predefned tolerance value. Accordng to prevous numercal experments, we wll set partcle as a surface=exteror partcle f q < 0.96q 0, otherwse, as an nteror partcle. The IMSPH approxmate schemes of the PP model can be obtaned by usng Eqs. (19) (22) for the nteror partcles and usng Eqs. (35) (38) for the exteror partcles, that s dq ¼ q dt ðu b u b ÞHb ð39þ du a ¼ 1 dt q r ab H b b ¼! r ab H b þ g a ðu a u a ÞHb where ( H b ¼ ðm =q Þð@W =@x b Þ; 2 nteror partcles ðm =q Þð@ ^W =@x b Þ; 2 exteror partcles ( H b ¼ ðm =q Þð@W =@x b Þ; 2 nteror partcles ðm =q Þð@ ^W =@x b Þ; 2 exteror partcles ð40þ ð41þ ð42þ ð43þ Artfcal Vscosty and Artfcal Stress Artfcal Vscosty. For ncreasng the stablty of SPH method, many forms of the artfcal vscosty have been proposed n prevous works. Here, the artfcal vscosty term s also added to the dscrete momentum equaton of IMSPH method. The artfcal vscosty term, [26] whch s usually used s where P ¼ ( a P c / þb P / 2 q ; u r < 0 0; u r 0 / ¼ hu r 2 þ0:01h ; 2 r c ¼ c þ c ; 2 q ¼ q þ q ; u ¼ u u ; r ¼ r r 2 ð44þ In Eqs. (44), a P and b P are constants, whch are usually chosen approxmately equal to 1. In the artfcal vscosty, the frst term assocated wth a P produces a shear and bulk vscosty, whle the second term assocated wth b P s necessary to handle hgh Mach number shocks and s smlar to the Von-Neumann-Rchtmeyer vscosty used n fnte dfference methods. Here, the values 1.0 and 2.0 are chosen for a P and b P, respectvely. The 0.01h 2 term s ncluded to prevent numercal dvergence when two partcles are approachng each other.

6 208 T. JIANG ET AL. Artfcal Stress. It s well known that when the SPH method s appled to solds, the phenomenon of unphyscal clusterng of partcles arses, whch s known as tensle nstablty. It s due to that small clumps of partcles gve rse to unrealstc fracture behavor of the materal when the materal s n a state of stretchng, and was frst nvestgated n detal by Swegle et al. [27] At present, a number of methods have been proposed to remove the tensle nstablty n elastc dynamcs of sold materals. The artfcal stress method [28,29] s one of the most successful approaches, whch was successfully appled to smulate vscoelastc free surface flows for Oldroyd-B model by Fang et al. [20] and Rafee [21]. In ths work, the same artfcal term s also adopted when the IMSPH method s appled to smulate the non-lnear vscoelastc flows based on PP model. The effect of the artfcal stress term s to prevent two partcles from gettng too close when they are n a state of tensle stress. Specally, we can modfy the momentum equaton by smplfyng the artfcal stress term [28,29] f n R ab þ R ab ð45þ where n ¼ W(0, h)=w(dd, h), f ¼ W (r r, h)=w (Dd, h) wth Dd beng the ntal dstance between two neghborng partcles along the coordnate axs. Accordng to ref. [28], the expresson for R ab s gven as R ab ¼ brab =q 2 ; r ab > 0 0; r ab 0 ð46þ where b s a postve parameter (0 < b < 1). Numercal experments show that b ¼ 0.1 and b ¼ 0.3 are sutable for smulatng the Newtonan flud and non-newtonan flud, respectvely. The valdaton of smplfed artfcal stress term (45) applyng to smulate vscoelastc free surface flow problem has been successfully tested n ref. [30]. The artfcal vscosty term (39) and the artfcal stress term (45) are added to the dscrete momentum equaton as usually done for SPH methods; then the dscretzaton of momentum equaton of usng the IMSPH method can be modfed. Introducng the K ¼ f n R ab þ R ab, the artfcal vscosty term Eq. (44) and the artfcal stress term Eq. (45) are added to the dscrete momentum equaton as usually done for SPH methods, then the dscretzaton of momentum equaton of usng the IMSPH method can be obtaned Usually, the partcle postons are updated by the followng equaton Dx a Dt ¼ va ð48þ Tme Integraton Scheme To solve the system of ordnary dfferental Eq. (23), Eq. (39), Eq. (40), Eq. (47) and Eq. (48), a smple leapfrog scheme [19] s adopted for ts second-order accuracy and hgh computatonal effcency. For the non-newtonan flud, the leapfrog scheme can refer to ref. [30]. Boundary Treatment In ths work, two types of boundary condtons are taken nto account, whch are a rgd wall and a free surface. Wall Boundares. There are several methods for treatng boundary condtons n prevous work. Generally, the sold walls are also smulated by partcles, whch exert repulsve force on nner flud partcles to prevent them from penetratng the wall. The wall boundary condtons can be modeled ether by fxed partcles or by mage partcles [13] that mrror the physcal propertes of nner flud partcles. As shown Fgure 1, two types of vrtual partcles [21] are used to mplement the boundary condtons on a rgd wall n ths work. The frst type vrtual partcles are located rght on the sold wall, namely wall partcles. The densty of wall partcles s not evolved. Meanwhle, the non-slp condton s enforced on the sold wall and the wall partcle postons reman fxed n tme. The pressure and components of the vscoelastc stress on the wall partcles are calculated accordng to the followng normalzed nterpolaton formula f ¼ m q f W, m q W ð49þ The normal elastc stress n the drecton perpendcular to the sold wall surface s set to zero. du a ¼ 1 dt q r ab H b q P d ab K r ab H b! H b þ ga ð47þ FIG. 1. The placement of wall partcle and dummy partcle. (Fgure s provded n color onlne.)

7 SMOOTH PARTICLE HYDRODYNAMICS AND POLYMER FLOW 209 The second type vrtual partcles are placed ust outsde the sold wall and fll a doman wth at least a range of depth comparable wth the support length kh, whch are called dummy partcles and have fxed densty and postons. The velocty and the vscoelstc stress tensor on the dummy partcles are computed n the followng way: For each dummy partcle, D, a correspondng pont W and pont F are ust on the sold wall and nsde the flud doman, respectvely. In order to calculate convenently we can make the normal dstances of the ponts D and F to the sold wall equal. So, the velocty u a D, vscoelastc stress s ab D and pressure p D on the dummy partcles are obtaned through the followng lnear extrapolatons: S D ¼ 2S W S F ; ð50þ where S represents the vector of varables (u a, s ab, p). To specfy the values for S F, the nterpolaton formula (49) s appled agan. Free Surface. For surface partcles, the followng total stress-free condton must be satsfed n the computatonal doman r n ¼ 0 ð51þ where n denotes a unt normal vector to the surface. In ths work, the surface tensor s neglected and a Drchlet boundary condton of zero pressure s gven to the surface partcles. Ths condton Eq. (51) s satsfed naturally by the SPH and IMSPH method, whch s based on the analyss n ref. [21]. Moreover, an nflow boundary condton should be defned for the et bucklng problem. In [22], a technque of mplementng nflow boundary condton for SPH method was proposed, whch was acheved by settng the unform necton velocty at the nozzle ext secton and placng a set of partcles whose veloctes were equal to those of the ncomng et. NUMERICAL SIMULATION FOR THE TRANSIENT FREE SURFACE FLOWS BASED ON PP MODEL In ths secton, some numercal results of smulatng the unsteady vscoelastc free surface flows are presented by usng the IMSPH method. Some problems are smulated to demonstrate the ablty of IMSPH method and dsplay non-lnear vscoelastc for concentrated polymer solutons and melts, whch contan mpactng drop and et bucklng based on the PP model. Valdaton of IMSPH Method To verfy the capacty of IMSPH method for capturng vscoelastc free flow, the IMSPH method s tested for smulatng the combned Poseulle and Couette flow of Oldroyd-B flud and compared wth SPH. Addtonally, ther numercal results and the theory soluton are also compared. Meanwhle, the problems of mpactng drop of Newtonan and Oldroyd-B flud are smulated respectvely by the IMSPH method for the purpose of comparng wth the results of SPH n ths subsecton. The Oldroyd-B model s the reduced model of PP model, when a 0 ¼ 0, f(k, s) ¼ 1 n Eq. (5), and k 0b represents the relaxaton tme. The dmensonless form of governng equatons of Oldroyd-B flud s adopted n smulatng the combned Poseulle and Couette flow and the combned Poseulle and Couette flow s analytcal soluton can be obtaned n refs. [17,31]. The technque of dmensonless form can refer to ref. [31], and the dmensonless parameters Reynolds number and Wessenberg number are ntroduced as Re ¼ qdu 0 =g, We ¼ k 0b U 0 =D. Case 1. The combned Poseulle and Couette flow s that the ntally statonary flud s drven by a body force F parallel to the x-axs, whle the upper plate suddenly moves at a certan constant velocty U 0 horzontally. For the smulaton presented here, a rectangular problem doman wth L x L y ¼ (L y ¼ D) s modeled wth real partcles, 2 10 wall partcles, 5 10 dummy partcles, and the perodc boundary case s the same case [17] s adopted. The used parameters are that q 0 ¼ 1, F ¼ 1, U 0 ¼ 0.01, k 0b ¼ 0.01, b 0 ¼ 3=9, sound speed c ¼ 0.1m=s and the tme step s set to Dt ¼ 10 5, whch corresponds to Reynolds number of Re ¼ 0.05, Wessenberg number of We ¼ 1. Fgure 2 shows the quanttatve comparsons between the velocty profles obtaned usng IMSPH, SPH and the analytcal soluton (see refs. [17,31] ). The IMSPH results are much closer to the analytcal soluton than the SPH results. The fnal steady s acheved after about steps (or dmensonless tme 2), and the velocty reaches peak value after about The velocty overshootng behavor s evdent for the Oldroyd-B flud wth the We ¼ 1. The accuracy of IMSPH s hgher than that of SPH. Case 2. The problems of mpactng drop of Newtonan and Oldroyd-B flud are also smulated, respectvely, by the IMSPH method n ths subsecton. Ths problem has been nvestgated by usng the SPH and fnte dfference methods n refs. [21,31], respectvely. In ths case, the dmensonal parameters are chosen as the same as those n ref. [21], namely, the rato between Newtonan vscosty and total vscosty b 0 ¼ 0.1, the relaxaton tme k 0b ¼ 0.02s, and the dmensonless parameters Re ¼ 5, We ¼ 1. The comparsons of the wdths for a Newtonan drop and an Oldroyd-B flud drop obtaned respectvely by IMSPH as well as the presented methods [21,31] are shown n Fgure 3. The IMSPH results agree well wth those of the methods n refs. [21,31]. Though the dfference between

8 210 T. JIANG ET AL. (t > 3.75), the elastc effects becomes weak and the drop spreads out slowly lke ts Newtonan counterpart. In a word, the IMSPH method s very effectve for smulatng the vscoelastc free surface problem of polymer melts and possesses hgher accuracy than the SPH method. FIG. 2. The comparsons of Velocty profles at dfferent nstants for the combned Poseulle and Couette flow usng IMSPH and SPH. Re ¼ 0.05, We ¼ 1. (Fgure s provded n color onlne.) the results of IMSPH and SPH s small, the numercal results of IMSPH are more relable than those of SPH. The Newtonan drop hts the wall and spreads out evenly whle retanng ts convex shape. For the vsoelastc drop, the flow process may be dvded nto three phases. At the dmensonless tme t ¼ 1.33, the drop ust touches the rgd plate. The perod from t ¼ 1.33 to t ¼ 2.3 s the frst phase whch may be assocated wth a negatve vertcal velocty. In ths phase the Oldroyd-B flow dsplays a greater tendency to spread horzontally than the case of Newtonan flud. In the second perod (2.3 < t < 3.75), the vscoelastc drop contracts because of the elastcty of the flud and has a postve vertcal velocty. In the fnal perod Numercal Smulaton of Impactng Drop of PP Flud The PP model can provde a good fttng to the rheologcal behavors of branched polymer melts. Moreover, the PP model reduces the nfluences of stress sngularty to some extent so that Wessenberg number can reach a hgher value than some other models developed based on phenomenologcal theory [1]. In ths subsecton, the numercal smulaton of mpactng drop of a PP flud s consdered. In partcular, the IMSPH method s appled to smulate the PP flud drop wth hgh vscosty and hgh Wessenberg number. In order to obtan an approprate smulaton, we set a 0 ¼ 0.15, q ¼ 2, e ¼ 1=3 and the other parameters keep the same as those n Fg. 3 except for the parameter g and k 0b. As can be seen n Fgure 4, the shapes of the PP flud drop at dfferent dmensonless tme are dfference wth the case of Oldroyd-B (see Fg. 3 of ref. [21]) under the same total vscosty and Wessenberg number. By comparng Fgure 4 and the Fgure 4 of ref. [21], we can observe that the man dfference s that of the smaller contractng trend of the PP flud drop than that of Oldroyd-B flud drop n the second phase (2.6 < t < 3.9, see Fg. 5). The phenomenon of concave shape occurs for the Oldroyd-B flud case, but t does not occur for the PP model n the second phase. Meanwhle, we can know that the tensle nstablty s more severe for the Oldroyd-B flud than the PP flud n the numercal smulaton of mpactng drop. In Fgure 5, the evoluton of the wdth for a PP flud drop wth the same total vscosty g ¼ 4Pa s and dfferent Wessenberg numbers are shown. The larger the Wessenberg number s, the smaller the contractng trend s. FIG. 3. Comparsons of the numercal results obtaned usng varous numercal methods for the wdth of a Newtonan drop (left) and Oldroyd-B drop (rght) wth dmensonless tme. (Fgure s provded n color onlne.)

9 SMOOTH PARTICLE HYDRODYNAMICS AND POLYMER FLOW 211 FIG. 4. IMSPH smulaton of fallng drop for a PP flud wth Re ¼ 5 and We ¼ 1. Vsualzaton of the flud at dfferent dmensonless tmes. (Fgure s provded n color onlne.) The bgger Wessenberg numbers does not adopted for the Oldroyd-B flud drop [21,31] when g ¼ 40Pa s, but whch s consdered here. The vscoelastc flud drop wll bounce n the contractng perod, when the materal parameters are chosen sutably. The phenomenon can be seen n Fgure 6 wth hgh vscosty g ¼ 40Pa s correspondng to Re ¼ 0.5, and the phenomenon of bouncng dsappears when the Wessenberg number ncreases to 10. The parameter e ¼ 1=3 s fxed, and the orentaton relaxaton tme k 0b ¼ 0.2 s and k 0b ¼ 2 s correspond to We ¼ 10and We ¼ 100, respectvely. The nfluence of the ncreased Wessenberg number s evdent, and the phenomenon of contractng dsappears when the Wessenberg number s equal to 100, whch s shown n Fgure 6. Fgure 7 shows the evoluton of the wdth for a PP flud drop obtaned usng the IMSPH method wth dfferent Wessenberg numbers. At the moment of droplet mpact, the mechancal energy rapdly reduces. However, all the physcal quanttes for a PP flud drop wth We ¼ 1 become large at short tme frstly and then decrease wth the ncreasng dmensonless tme except for the velocty, whch s named as the relaxaton phenomenon of vscoelastc stress. The smaller the values of vscoelastc stress and the frst normal stress FIG. 5. Evoluton of the wdth of a PP drop wth Re ¼ 5 and dfferent Wessenberg numbers We ¼ 1, 2, 3. (Fgure s provded n color onlne.) FIG. 6. IMSPH smulaton of fallng drop for a PP flud wth Re ¼ 0.5. Vsualzaton of the flud at dfferent dmensonless tmes. Wth dfferent Wessenberg numbers: We ¼ 1 (frst column), We ¼ 10 (frst column), We ¼ 100 (thrd column). (Fgure s provded n color onlne.)

10 212 T. JIANG ET AL. surface. Ths problem has been nvestgated by several researchers [32 35], and a theory of et bucklng has not yet been developed. However, the problem of a twodmensonal Newtonan et bucklng was predcted by expermental and theoretcal estmates n 1988 [33]. These estmates were based upon the et wdth D, the heght of the nlet to the rgd plate H and the Reynolds number, whch were that f both Re < 0.56 and H=D > 3p were stratfed then a two-dmensonal et would buckle. Bonto et al. [35] have shown that a Newtonan et buckles f Re < 0.53 and H=D ¼ 20, and Tome et al. [36] have proposed the consstent condton Re 2 1 ðh=dþ 2:6 8:8 2:6 p ðh=dþ 2:6 ð52þ FIG. 7. Evoluton of the wdth of a PP drop wth Re ¼ 0.5 and dfferent Wessenberg numbers We ¼ 1, 10, 100. (Fgure s provded n color onlne.) dfference are, the larger the Wessenberg number s. Wth ncreasng Wessenberg number, the slghted larger value of stretch s ganed. Fgure 8 shows the streamlnes of a PP flud drop wth dfferent Wessenberg numbers We ¼ 1, 10, 100, at three dfferent dmensonless tmes. The movement of the vscoelastc drop wth dfferent Wessenberg numbers durng the evoluton can be seen. Numercal Smulaton of Jet Bucklng of PP Flud The phenomenon of a et flow onto a rgd plate known as et bucklng can occur, whch nvolves complex free More recently, a two-dmensonal vscoelastc et flows modeled by the Oldroyd-B consttutve equaton was studed by usng fnte dfference method (FDM) [31] and SPH method [22]. Accordng to the numercal results n refs. [22,31], we can know that the vscoelastc et wll buckle when the macro-physcal parameters Re and We are chosen sutably, whle the other parameters are fxed. In order to further study the problem of a two-dmensonal non-newtonan et bucklng, the et bucklng phenomenon of polymer melts based on PP model s consdered wth a wde range of Wessenberg numbers by usng the IMSPH method. The transent et bucklng problem nvolves a flow of a et nected downwards nto a rectangular cavty. In ths paper, the et wdth s set D ¼ 0.005m and the heght of the cavty s chosen as H ¼ 0.1 m, whch corresponds to H=D ¼ 20. A unform necton velocty s set U 0 ¼ 1m=s. The reference densty q 0 ¼ 1030 kg=m 3, and the gravtatonal force acts FIG. 8. The streamlnes of fallng drop for a PP flud wth Re ¼ 0.5 and dfferent Wessenberg numbers We ¼ 1 (frst column), We ¼ 10 (second column), We ¼ 100 (thrd column). Vsualzaton of the flud at dfferent dmensonless tmes. (Fgure s provded n color onlne.)

11 SMOOTH PARTICLE HYDRODYNAMICS AND POLYMER FLOW 213 downwards wth g ¼ 9:81 m=s 2. The ntal partcle spacng s Dd ¼ m, correspondng to the stable tme-step s , and the other parameters reman the same as those prevously studed, except for the total vscosty g,theratob 0 and the orentaton relaxaton tme k 0b. The dmensonless parameters Re ¼ qdu 0 =g and We ¼ k 0b U 0 =D are also adopted n ths subsecton. However, the artfcal stress term s neglected for smulatng the et bucklng problem. Fgure 9 shows the shapes of a Newtonan et and a PP flud et at dmensonless tme 40. In ths case, b 0 ¼ 0.1, k 0b ¼ s and the total vscosty g ¼ 5.15Pa s, correspondng to We ¼ 1 and Re ¼ 1 > The phenomenon of et bucklng s not observed for the Newtonan case whle the bucklng occurs weakly for the PP flud case n Fgure 10. For the same rato b 0, we adopt total vscosty g ¼ 10.3Pa s, and the orentaton relaxaton tme varyng from 0.005s (We ¼ 1) to 0.5 s (We ¼ 100). The comparson of the results obtaned for the Newtonan et and PP flud et wth Re ¼ 0.5 and dfferent Wessenberg numbers We ¼ 1, 10, 100 s shown n Fgure 10. The et behavors of polymer melts are entrely dfferent wth the Newtonan fluds, whch can be seen from Fgure 10. It s observed that the phenomenon of et bucklng for polymer melts has great dfference wth the dfferent Wessenberg numbers lkes ts Oldroyd-B flud [22] counterpart. In ths case, the Newtonan et foldng lags n comparson wth the PP flud et wth We ¼ 1. It s also noted that the et has a greater tendency wth ncreasng Wessenberg number. The et bucklng phenomenon becomes more clear wth decreasng Wessenberg number, but the bucklng phenomenon has not occurred when We ¼ 100. The et bucklng phenomenon s llustrated wth the ncreasng rato b 0 ¼ 0.3 n Fgure 11 and Fgure 12. Fgure 11 shows the shapes of PP flud et wth- Re ¼ 0.5 < 0.53 and dfferent Wessenberg numbers 10 and 100. It s observed that the phenomenon of vscoelastc et bucklng wth We ¼ 10 s more volent than the case n Fgure 10 wth b 0 ¼ 0.1, at the dmensonless tme 40, whle FIG. 10. IMSPH smulaton of et bucklng wth Re ¼ 0.5 and dfferent Wessenberg numbers. Vsualzaton of the flud at dfferent dmensonless tmes. Newtonan et (frst column) and vscoleastc et: We ¼ 1 (second column), We ¼ 10 (thrd column), We ¼ 100 (fourth column). (Fgure s provded n color onlne.) FIG. 9. IMSPH smulaton of et bucklng for a Newtonan et (left column) and a PP et (rght column) at dmensonless tme 40. (Fgure s provded n color onlne.) FIG. 11. IMSPH smulaton of et bucklng for a PP et at dmensonless tme 40. Re ¼ 0.5, b 0 ¼ 0.3, We ¼ 10 (frst column), We ¼ 100 (second column). (Fgure s provded n color onlne.)

12 214 T. JIANG ET AL. the vsoelastc et bucklng has yet not occurred wth We ¼ 100. As t s antcpated, by decreasng the Reynolds number and choosng approprate materal parameters, the phenomenon of vsocelastc et bucklng has yet occurred wth the hgher Wessenberg number (We ¼ 100). Ths phenomenon can be seen n Fgure 12 where the Re ¼ The free surface between the Newtonan et and the vscoelastc et wth We ¼ 1 becomes smlar n Fgure 12. From Fgure 10 and Fgure 12, we can see that the phenomenon of et thnnng s evdent for the bgger Wessenberg number. CONCLUDING REMARKS In ths work, an mproved SPH method (IMSPH) s ntroduced to smulate transent branched polmer free surface flows of a concentrated polymer solutons and melts based on PP model. In order to allevate the tensle nstablty, the stress term s successfully added to the momentum equaton whch s approxmated by IMSPH method. The numercal results obtaned usng the IMSPH show that t s a powerful tool to smulate the complex free surface for concentrated polymer solutons and melts. Two challengng test examples are solved for a wde range of Wessenberg numbers, whch demonstrate the capablty and stablty of the proposed scheme. The bggest advantage s that the IMSPH has the hgher accuracy and better stablty than those of the SPH. The extended IMSPH method s equally applcable to study not only polymer free surface flows but also mult-phase flows and mold fllng. ACKNOWLEDGMENTS The support from the Natonal Natural Scence Foundaton of Chna (NSFC) (No ) and Natonal Basc Research Program of Chna (No. 2005CB321704) are fully acknowledged. FIG. 12. IMSPH smulaton of et bucklng wth Re ¼ 0.25, b 0 ¼ 0.3 and dfferent Wessenberg numbers. Vsualzaton of the flud at dfferent dmensonless tmes. Newtonan et (frst column) and vscoleastc et: We ¼ 1 (second column), We ¼ 10 (thrd column), We ¼ 100 (fourth column). (Fgure s provded n color onlne.) REFERENCES 1. Wang, W.; L,.K.; Han,.H. Equal low-order fnte element smulaton of the planar contracton flow for branched polymer melts. Polym. Plast. Technol. Eng. 2009, 48, Harlow, F.H. The partcle-n-cell method for numercal soluton of problems n flud dynamcs. Proc. Symp. Appl. Math. 1963, 15, Harlow, F.H.; Welch, E. Numercal calculaton of tme-dependent vscous ncompressble flow of fluds wth free surface. Phys. Fluds 1965, 8, Hrt, C.W.; Ncholls, B.D. Volume of flud (VOF) method for dynamcs of free boundares. J. Comput. Phys. 1981, 39, Osher, S.; Sethan, J.A. Fronts propagatng wth curvature-dependent speed: algorthms based on Hamlton Jacob formulatons. J. Comput. Phys. 1988, 79, Tome, M.F.; Gross, L.; Castelo, A.; Cumnato, J.A.; McKee, S.; Walters, K. De-swell, splashng drop and a numercal technque for solvng the Oldroyd B model for axsymmetrc free surface flows. J. Non-Newtonan Flud Mech. 2007, 141,

13 SMOOTH PARTICLE HYDRODYNAMICS AND POLYMER FLOW De Paulo, G.S.; Tome, M.F.; McKee, S. A. Marker and cell approach to vscoelastc free surface flows usng the PTT model. J. Non-Newtonan Flud Mech. 2007, 147, Gngold, R.A.; Monaghan, J.J. Smoothed partcle hydrodynamcs theory and applcaton to non-sphercal stars. Royal Astronomcal Socety, Monthly Notces 1977, 181, Lucy, L.B. A numercal approach to the testng of the fsson hypothess. Astron. J. 1977, 83, Monaghan, J.J. Smulatng free surface flows wth SPH. J. Comput. Phys. 1994, 110, Flebbe, O.; Munzel, S.; Herold, H.; Rffert, H.; Ruder, H. Smoothed partcle hydrodynamcs physcal vscosty and the smulaton of accreton dsks. Astrophys. J. 1994, 431, Morrs, J.P.; Fox, P.J.; Zhu, Y. Modelng low Reynolds number ncompressble flows usng SPH. J. Comput. Phys. 1997, 136, Cummns, S.J.; Rudman, M. An SPH proecton method. J. Comput. Phys. 1999, 152, Cleary, P.W.; Monaghan, J.J. Conducton modelng usng smoothed partcle hydrodynamcs. J. Comput. Phys. 1999, 148, Monaghan, J.J.; Kocharyan, A. SPH smulaton of mult-phase flow. Comput. Phys. Commun. 1995, 87, Morrs, J.P. Smulatng surface tenson wth smoothed partcle hydrodynamcs. Int. J. Numer. Meth. Fluds 2000, 33, Ellero, M.; Tanner, R.I. SPH smulatons of transent vscoelastc flows at low Reynolds number. J. Non-Newton. Flud Mech. 2005, 132, Chen, J.K.; Beraun, J.E.; A generalzed smoothed partcle hydrodynamcs method for nonlnear dynamc problems. Comput. Methods Appl. Mech. Engrg. 2000, 190, Lu, M.B.; e, W.P.; Lu, G.R.; Modelng ncompressble flows usng a fnte partcle method. Appl. Math. Model. 2005, 29, Fang, J.; Parraux, A.; Rentschler, M.; Ancey, C. Improved SPH methods for smulatng free surface flows of vscous fluds. Appl. Numer. Math. 2009, 59, Fang, J.; Owens, R.G.; Tacher, L.; Parraux, A. A numercal study of the SPH method for smulatng transent vscoelastc free surface flows. J. Non-Newtonan Flud Mech. 2006, 139, Rafee, A.; Manzar, M.T.; Hossen, M. An ncompressble SPH method for smulaton of unsteady vscoelastc free surface flows. Inter. J. Non-Ln. Mech. 2007, 42, Vebeeten, W.M.H.; Peters, G.W.M.; Baaens, F.P.T. Dfferental consttutve equatons for polymer melts: The extended Pom-Pom model. J. Rheol. 2001, 45, McLesh, T.C.B.; Larson, R.G. Molecular consttutve equatons for a class of branched polymers: the pom-pom polymer. J. Rheol. 1998, 42, Rogers, S.E.; Kwak, D.; Kaul, U.K. On the accuracy of the pseudocompressblty method n solvng the ncompressble Naver Stokes equatons. Appl. Math. Model. 1987, 11, Monaghan, J.J. Smoothed partcle hydrodynamcs, Annu. Rev. Astron. Astrophys. 1992, 30, Swegle, J.W.; Hcks, D.L.; Attaway, S.W. Smoothed partcle hydrodynamcs stablty analyss. J. Comput. Phys. 1995, 116, Monaghan, J.J. SPH wthout a tensle nstablty. J. Comput. Phys. 2000, 159, Gray, J.P.; Monaghan, J.J.; Swft, R.P. SPH elastc dynamcs. Comput. Meth. Appl. Mech. Eng. 2001, 190, Jang, T.; Ouyang, J.; Yang, B..; Ren, J.L. The SPH method for smulatng a vscoelastc drop mpact and spreadng on an nclned plate. Comput. Mech. 2010, 45, Tome, M.F.; Mangavacch, N.; Cumnato, J.A.; Castelo, A.; Mckee, S. A fnte dfference technque for smulatng unsteady vscoelastc free surface flows. J. Non-Newton. Flud Mech. 2002, 106, Cruckshank, J.O.; Munson, B.R. Vscous-flud bucklng of plane axsymmetrc ets. J. Flud Mech. 1981, 113, Cruckshank, J.O. Low-Reynolds-number nstabltes n stagnatng et flows. J. Flud Mech. 1988, 193, Rbe, N.M. Perodc foldng of vscous sheets. Phys. Rev. E 2003, 68, Art. No Part Bonto, A.; Pcasso, M.; Laso, M. Numercal smulaton of 3D vscoelastc flows wth free surfaces. J. Comput. Phys. 2006, 215, Tome, M.F.; McKee, S. Numercal smulaton of vscous flow: Bucklng of planar ets. Int. J. Numer. Methods Fluds 1999, 29,