Theoretical and numerical studies of die swell flo · 2016. 8. 24. · Theoretical and numerical studies of die swell flow Korea-Australia Rheology J., 28(3), 2016 231 incompressible

© 2016 The Korean Society of Rheology and Springer 229

Korea-Australia Rheology Journal, 28(3), 229-236 (August 2016)DOI: 10.1007/s13367-016-0023-6

www.springer.com/13367

pISSN 1226-119X eISSN 2093-7660

Theoretical and numerical studies of die swell flow

Alaa H. Al-Muslimawi*

Department of Mathematics, College of Science, University of Basra, Basra, Iraq

(Received December 27, 2015; final revision received June 20, 2016; accepted July 25, 2016)

This paper focuses on the theoretical and numerical predictions of die-swell flow for viscoelastic and vis-coelastoplastic fluids. The theoretical results on die swell have been obtained by Tanner for a wide classof constitutive equations, including Phan-Thien Tanner (PTT), pom-pom, and general network type models.These results are compared with numerical solutions across swelling ratio, pressure drop, state of stress, anddissipation-rate for two fluid models, exponential Phan-Thien Tanner (EPTT) and Papanastasiou-Exponen-tial Phan-Thien Tanner (Pap-EPTT). Numerically, the momentum and continuity flow equations are solvedby a semi-implicit time-stepping Taylor-Galerkin/pressure-correction finite element method, whilst the con-stitutive equation is dealt with by a cell-vertex finite volume (cv/fv) algorithm. This hybrid scheme is per-formed in a coupled fashion on the nonlinear differential equation system using discrete subcell technologyon a triangular tessellation. The hyperbolic aspects of the constitutive equation are addressed discretelythrough upwind fluctuation distribution techniques.

Keywords: hybrid finite element/volume, viscoplastic, viscoelastoplastic, die-swell, exponential Phan-Thien

Tanner model, Papanastasiou model

1. Introduction

Much research has focused attention on the die-swell

problem which naturally introduces free-surface model-

ling. In this context, viscoelastic solutions appear in the

early study of die-swell for a Maxwell fluid by Tanner

(1970), which established the streamline iterative free-sur-

face location technique (recently, the author reviewed this

study again for a wide class of constitutive equations,

including PTT, pom-pom, and general network type mod-

els (Tanner, 2005)). Collocation and Galerkin methods

were implemented by Chang et al. (1979) on slit and cir-

cular die swell flows for generalized Maxwell fluids. In

addition, Crochet and Keunings (1982) investigated the

Oldroyd-B extrudate swell problem, with a mixed Galer-

kin formulation applied to slit, circular, and annular dies,

reporting a limiting Deborah number (based on wall

shear-rate) of 4.5 for the circular case. Bush et al. (1984)

used both finite element and boundary integral methods to

investigate planar and axisymmetric extrusion flow of

Maxwell fluids. By using the finite element method, Bush

(1990) studied swelling behavior of Oldroyd-B circular

free jets. Furthermore, Ngamaramvaranggul and Webster

(2001) developed semi-implicit Taylor-Galerkin/Pressure-

correction technique for such problems, using a streamline

prediction free-surface method. To find out more details

about this problem, see references (Clermont and Nor-

mandin, 1993; Ganvir et al., 2009; Oishi et al., 2011; Tomé

et al., 2007).

A visco-elasto-plastic model is suggested by Beverly

and Tanner (1989) with the purpose of simulating some

experiments performed by Carter and Warren (1987) on

plastic-propellant dough. There, a Herschel-Bulkley model

employed for plasticity, alongside a linearised PTT model

(LPTT, shear-thinning, sustained strain-hardening - see

below for EPTT), suitable for polymer-solution represen-

tation, that displays increase in extensional viscosity at

intermediate extension-rates, followed by a terminating

plateau at high limiting rates. Subsequently, Mitsoulis et

al. (1993) reproduced these results for an extended range

of apparent shear rates with the addition of the Papanas-

tasiou model (1987) to the CEF viscoelastic model. With

a hybrid finite element-finite volume subcell scheme and

the 4:1:4 contraction-expansion problem, Belblidia et al.

(2011) applied the construction introduced by Papanasta-

siou within the viscoelastic-viscoelastoplastic context, uti-

lizing the Oldroyd-B model to introduce the viscoelastic

dimension. Moreover, Al-Muslimawi et al. (2013a) applied

a Papanastasiou-EPTT approximation to the die-swell

problem using a hybrid finite element-finite volume sub-

cell scheme. This is achieved by combining the viscoplas-

tic Papanastasiou-Bingham model with the viscoelastic

Phan-Thien Tanner-(EPTT) model, suitable for polymer

melt response, undertaking a systematic study on swelling

ratio, exit loss, and flow response as a consequence of vis-

cous, plastic, and viscoelastic material behaviour (separate

dominance thereof).

In more recent years, Tanner (2005) reviewed the posi-

tion on die-swell solutions with updated predictive theory

for a wide class of constitutive equations, including Phan-

Thien Tanner (PTT), pom-pom, and general network type

models. The present study advances upon this position by*Corresponding author; E-mail: [email protected]

Alaa H. Al-Muslimawi

230 Korea-Australia Rheology J., 28(3), 2016

comparing the numerical solution of die-swell flow with

the theoretical presentation that introduced by Tanner

(2005), for the Phan-Thien Tanner-(EPTT) model and the

Papanastasiou-Exponential Phan-Thien Tanner (Pap-EPTT)

that is extended by Al-Muslimawi (2013). To achieve that,

the hybrid finite volume/element scheme is implemented,

alongside the free surface calculations. The influence of

variation in Weissenberg number on swelling ratio is

described for the EPTT and the Pap-EPTT. Moreover, the

relationship among the swelling ratio, dissipation, D, and

first normal stress difference, N1, has been shown through-

out this study.

2. Governing Equations and Constitutive Modeling

For the general viscoelastic context, under transient,

incompressible isothermal flow conditions, the relevant

mass conservation and momentum equations may be

expressed in non-dimensional terms, as:

, (1)

(2)

where field variables u, p and τ represent the fluid veloc-

ity, hydrodynamic pressure, and polymeric stress contri-

bution. D = ( + )/2 is the rate of deformation tensor

(here superscript T denotes tensor transpose). In addition,

the non-dimensional group of the Reynolds number may

be defined as Re = (ρUl/μo), with characteristic scales of ρ

for the fluid density, U for velocity, l for length (l/U for

time), and μ0 = μp + μs is the total zero shear-rate viscosity,

for which μp is a polymeric viscosity and μs is a solvent

viscosity. The solvent fraction parameter β is defined as β

= μs/μo.

The constitutive equation for the exponential Phan-

Thien Tanner (EPTT) model may be expressed as:

(3)

Here, the dimensionless parameters are introduced in the

form of the Weissenberg number (We = λ1U/l) which is a

function of material relaxation time, λ1, characteristic

velocity scale U and length l. In this particular exponential

version of the Phan-Thien Tanner constitutive equation,

suitable for representing the hardening-softening behaviour

of typical polymer melts, the nonlinear function f is

defined as:

. (4)

The constant parameter ε is non-dimensional and gov-

erns the non-linear function f.

Outlining greater modeling detail for non-Newtonian

viscoplastic materials, stress may be considered as a non-

linear function of the second invariant (IID) of rate of

deformation tensor (Dij). Here, following the above and

non-dimensional formulation adopted, one makes use of

the Papanastasiou model (Papanastasiou, 1987), employ-

ing non-dimensional parameter m (a regularization stress

growth exponent, with original scale of time) and τ0 (base

yield stress factor, equivalent to a Bingham Number, Bn =

τ0 = τyl/μoU with scaling on the dimensional yield stress

τy). Accordingly, the visco-plastic stress is expressed in

the form:

, (5)

where

and (6)

. (7)

The set of Eqs. (5)-(7) represents the essential basis to

incorporate the combination of the Phan-Thien Tanner

(EPTT) model with the Papanastasiou-Bingham model.

For computational convenience, the extra stress tensor T*

is divided into viscous τ (1) and elastic parts τ (2), so that

T* = τ (1) + τ (2), (9)

(10)

, (11)

where, is the upper-convected derivative of τ (2) defined

as

. (12)

For further details, the behaviour the Papanastasiou-

Exponential Phan-Thien Tanner (Pap-EPTT) material func-

tions is described by Al-Muslimawi et al. (2013b).

3. Numerical Method and Discretization

The hybrid finite element/volume method employed is a

semi-implicit, time-splitting, and fractional-staged formu-

lation that invokes finite element discretisation for veloc-

ity-pressure parts of the system and finite volume for

stress (see Matallah et al., 1998; Webster et al., 2005). The

framework is cast about a Taylor-Galerkin (TG) discreti-

sation, and a two-step Lax-Wendroff time stepping pro-

cedure (predictor-corrector), alongside an incremental

pressure-correction (PC) procedure with a constant factor

and a forward time increment factor θ2 = 1/2.

Such a pressure-correction implementation takes the

scheme through to second-order temporal accuracy under

∇ u⋅ = 0

puuDt

u∇−∇⋅−+⋅∇=

∂

∂Re)2(Re βτ

∇u ∇uT

T2(1 ) ( ).We D f We u u ut

τβ τ τ τ τ

∂= − − − ⋅∇ −∇ ⋅ − ⋅∇

∂

⎥⎦

⎤⎢⎣

⎡

−= )(

)1(exp)( τ

β

ετ tr

Wef

DIID)(2ϕτ =

1/2

0

0 12

(1 )( ) ,

2

τϕ μ

−⎛ ⎞−⎜ ⎟= +⎜ ⎟⎝ ⎠

Dm II

D

D

eII

II

IID = 1

2---trace D

2( )

τ1( )

= 2ϕ IID( )βD,

fτ2( )

+ Weτ∇ 2( )

= 2ϕ IID( ) 1 β–( )D

τ∇ 2( )

τ∇ 2( )

= ∂τ

2( )

∂t---------- + u ∇⋅ τ

2( ) − ∇u( )T τ2( ) − τ 2( ) ∇u( )⋅ ⋅

101≤≤ θ


Korea-Australia Rheology J., 28(3), 2016 231

incompressible conditions. Utilising concise semi-discrete

time-discretisation, the schematic representation of the

three-stage TGPC structure may be expressed, on a single

time step Δt = [tn, tn+1] with initial values [un, τ n, pn, pn−1],

as

Stage 1a:

(13a)

(13b)

Stage 1b:

(13c)

(13d)

Stage 2:

(13e)

Stage 3:

(13f)

where FG is a body-force vector; u* and D* denote the

intermediate-variable non-solenoidal velocity and rate of

deformation tensor, respectively.

In summary, a Galerkin discretisation may be applied to

the Stokesian components of the system; the momentum

equation at Stage 1, the pressure-correction step at Stage

2, and incompressible correction constraint at Stage 3. The

diffusion term is treated in a semi-implicit manner, en-

hancing stability, whilst avoiding the computational over-

head of a fully implicit alternative. Pressure temporal

increments invoke multi-step reference across three suc-

cessive time levels [tn−1, tn, tn+1].

4. Finite Volume Cell-vertex for Stress

The concepts and rational for application of cell-vertex

finite volume techniques in the viscoelastic context have

been presented in detail elsewhere (see Matallah et al.,

1998). Hence, a brief description of the underlying theory

is provided as may be gathered from the non-conservative

extra-stress equation, with flux term ( ) and upon

absorbing remaining terms under the source (Q), viz.:

. (14)

Then, cell-vertex fv-schemes are applied to this differ-

ential equation utilizing fluctuation distribution as the

upwinding technique, to distribute control volume resid-

uals and furnish nodal solution updates (Wapperom and

Webster, 1998). Now, consider each scalar stress compo-

nent, τ, acting on an arbitrary volume , whose

variation is controlled through the corresponding fluctua-

tion components of flux (R) and source (Q),

. (15)

Such integral flux and source variations are evaluated

over each finite volume triangle (Ωl), and are allocated

proportionally by the selected cell-vertex distribution

(upwinding) scheme to its three vertices. The nodal update

is obtained, by summing all contributions from its control

volume Ωl, composed of all fv-triangles surrounding node

(l). In addition, these flux and source residuals may be

evaluated over two separate control volumes associated

with a given node (l) within the fv-cell T, generating two

contributions, one upwinded and governed over the fv-tri-

angle T, (RT, QT), and a second area-averaged and sub-

tended over the median-dual-cell zone, (RMDC, QMDC). For

reasons of temporal accuracy, this procedure demands

appropriate area-weighting to maintain consistency, with

extension to time-terms likewise. In this manner, a gen-

eralized fv-nodal update equation has been derived per

stress component (see Sizaire and Legat, 1997), by sepa-

rate treatment of individual time derivative, flux and source

terms, and integrating over associated control volumes,

yielding,

, (16)

where bT = (−RT + QT), , ΩT is the

area of the fv-triangle T, and is the area of its median-

dual-cell (MDC). The weighting parameter, ,

proportions the balance taken between the contributions

( ) ( )1

2

1

2

1

1

2

Δ

2 ( )2

( ) ,

n n nn

nn

D

n n n n

G

Reu u Re u u

t

D DII

p p p F

τ

ϕ

θ

+

+

−

⎛ ⎞− = ∇ + ⋅∇⎜ ⎟

⎝ ⎠

⎡ ⎤+⎢ ⎥+∇ ⋅ β

⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤−∇ + − +⎣ ⎦

1/2

T

2(1 ) ( )2( ) ,

Δ ( )

n

Dn nII D fWe

t We u u u

β ϕ ττ τ

τ τ τ

+− −⎡ ⎤

− = ⎢ ⎥− ⋅∇ −∇ ⋅ + ⋅∇⎣ ⎦

1 1

2 2

1

1 2

1

2Re( ) ( ) Re( )

Δ

2 ( )2

( ) ,

n nn

n

D

nn n n

G

u u u ut

D DII

p p p F

τ

βϕ

θ

+ +∗

∗

+−

− = ∇ + ⋅∇

⎡ ⎤++∇ ⋅ ⎢ ⎥

⎣ ⎦

⎡ ⎤−∇ + − +⎣ ⎦

1/2

1

T

2(1 ) ( )1( ) ,

Δ ( )

n

Dn nII D f

t We u u u

β ϕ ττ τ

τ τ τ

+

+− −⎡ ⎤

− = ⎢ ⎥− ⋅∇ −∇ ⋅ + ⋅∇⎣ ⎦

2 n 1 n

2

( ) ,+ ∗

Δ

∇ − = ∇⋅Re

p p uθ t

n 1 n 1 n

2

2( ) ( ),

Reu u p p

t

+ ∗ +

Δ

− = −θ ∇ −

R = u ∇τ⋅

∂τ

∂t----- + R = Q

Ω = Σl

Ωl

τ

Ω Ω Ω

∂Ω = − Ω+ Ω

∂∫ ∫ ∫

l l l

d Rd Qdt

( )

( )

1

T

T

1

1

l l

l l

n

T T l

T l T T l

MDC

T T MDC

T l T l

MDC

ˆ

t

b b

τδ α δ

δ α δ

+

∀ ∀

∀ ∀

⎡ ⎤ ΔΩ + − Ω⎢ ⎥

Δ⎣ ⎦

= + −

∑ ∑

∑ ∑

bl

MDC = RMDC– QMDC+( )l

T

l

ˆΩ

0 1T

≤ ≤δ



from the median-dual-cell and the fv-triangle T. The dis-

crete stencil (16) identifies fluctuation distribution and

median dual cell contributions, area weighting, and up-

winding factors ( -scheme dependent). The interconnec-

tivity of the fv-triangular cells (Ti) surrounding the sample

node (l), the blue-shaded zone of MDC, the parent trian-

gular fe-cell, and the fluctuation distribution (fv-upwind-

ing) parameters ( ), for i = l, j, k on each fv-cell, are all

features illustrated in Wapperom and Webster (1998).

5. Problem Specification

The die-swell problem may be subdivided into two flow

regions, each of different character, the shear flow within

the die and the free jet flow beyond the die. Each region

has its unique set of boundary conditions and for reasons

of symmetry, it is only necessary to consider half of the

domain, that above the central axis of symmetry (see

Fig. 1). The finite element mesh for the die swell appears

in Fig. 2, and the mesh characteristics are included in

Table 1.

To solve the governing system of partial differential

equations for the EPTT/Pap-EPTT model it is necessary to

impose suitable boundary conditions. EPTT/Pap-EPTT

inlet stress solutions are set by solving the corresponding

set of nodal-pointwise ODEs. The inlet velocity profile

(pressure-driven pure shear flow) at the equivalent set

flow-rate, is initially adopted of fully-developed analytic

Oldroyd-B form (of constant viscosity), but subsequently

is iteratively corrected to that of EPTT/Pap-EPTT, using

feedback from the internal field solution (shear-thinning).

This procedure is then equivalent to solving the one-

dimensional equivalent shear flow problem. Fully devel-

oped boundary conditions are established at the outflow

ensuring constant streamwise velocity component vz and

vanishing cross stream component ur. In addition, no-slip

boundary conditions are imposed along the stationary die-

channel walls. Along the free surface, free kinematic con-

ditions are imposed. On the free surface, four conditions

must to be satisfied: (i) zero normal velocity; (ii) zero or

prescribed shear-stress; (iii) zero or prescribed normal

stress; and (iv) surface tension may be neglected. To con-

struct the essential basis for solution and internal starting

conditions, the problem is first resolved for Newtonian

fluid properties. Then, from this position, the viscoelastic

problem is initiated with (β = 0.9, ε = 0.25) - high solvent

fraction, low Trouton ratio. Then, from the viscoelastic

results, the viscoelastoplastic problem is initiated with

additional viscoplastic parameters of yield stress (τ0 =

0.01) and exponent (m = 102). Additionally, in the present

study a finite small value of the Reynolds number is

assumed, Re = 10−4 and the time-stepping procedure is

monitored for convergence to a steady state via relative

solution increment norms, subject to satisfaction of a suit-

able tolerance criteria, taken here as 10−8 with typical ∆t

set as O(10−4).

For free surface calculation, the time dependent predic-

tion method, which is termed Phan-Thien (dh/dt) scheme,

has been selected for the current implementation, to com-

pute free-surface movement (see Sizaire and Legat, 1997).

This is equivalent to a constrained Lagrangian Eulerian

technique (ALE) (for more details see Al-Muslimawi et

al., 2014). The free-surface location of the height function,

h(z, t), is determined via solution of the following equa-

tion (see Al-Muslimawi et al., 2014; Szabo et al., 1997):

αl

T

αi

T

Fig. 1. (Color online) Die-swell schema.

Fig. 2. (Color online) Mesh pattern, die length 4, jet length 6.

Table 1. Mesh characteristic parameters.

Meshes Elements NodesDegrees of freedom

(u, p, τ)

Medium 3200 6601 41307



, (17)

where, is the velocity vector and is

the radial height. The new position of each node on the

free surface is computed using Eq. (17). Remeshing must

be performed after each time-step to avoid excessive dis-

tortion of element in the boundary zones. The swell ratio

is then defined as χ = R/Ro, where R and Ro are the final

extrudate radial and die radius, respectively (see Fig. 1).

6. Results and Discussion

6.1. Swelling ratioThe swelling ratio profiles are plotted in Figs. 3a and 3b

for EPTT and Pap-EPTT models at fixed (τ0 = 0.01, m =

102, β = 0.9, ε = 0.25) under We variation. The findings

reveal that, as anticipated, the jet swelling increases as We

level rises, so that, the maximum swell corresponds to the

instance with the largest Weissenberg number (We = 5),

for both cases of EPTT and Pap-EPTT models: at We = 5,

swell reaches levels of 1.151 for the viscoelastic EPTT

model, and slightly less in 1.149 for the viscoelastoplastic

Pap-EPTT model. Therefore, one can see the effect of yield

stress τ0 on swelling ratio (the swelling ratio decreases as

the yield stress (τ0) increases).

According to Tanner (1970; 2005), for the PTT family

of models and from a theoretical viewpoint, the extrudate

swelling ratio of an elastic fluid may be predicted as:

, (18)

where, N1 is the first normal stress difference, τ is the

shear stress, R0 is the diameter of the tube, and R is the

diameter of the extrudate emerging from the tube. This

classical theory functionally relates the swell to the ratio

between N1 and τw at the die-wall (w) in fully developed

shear flow.

Tanner established the swell ratio relationship as:

. (19)

Although N1 is often closely proportional to τ2 it is often

even closer to

. (20)

If one uses Eqs. (18) and (19) one gathers accordingly,

the correction:

, (21)

which agrees with Eq. (18) in the limiting case

(see Al-Muslimawi et al., 2013b).

In Fig. 4, swell results may be charted against the elastic

theory, as expounded by Tanner (2005) and given in Eq.

(21) for two different values of exponent mT = 2, 3.8

and accuracy of approximation discussed therein. This

classical theory functionally relates the swell to the ratio

between N1 and τw at the die-wall in fully developed flow.

Note that, the value of mT = 4 in Eq. (21) replicates the

status of a Newtonian fluid with a swelling ratio of 1.13

(see Fig. 3). The base choice of exponent from the theory

is mT = 2, (see Tanner, 2005); yet, this provides an over-

0=−∂

∂+

∂

∂rz

uz

hv

t

h

u = ur, vz( ) h = h z, t( )χ =

R

R0

----- = 11

2---

N1

2τ-------

⎝ ⎠⎛ ⎞

w

2

+

1

6---

+ 0.13

2 2 26

1 10

2 20

10

2 [1 ( 2 ) ] ( )

(2 ) ( )

w

w

N N dR

RN d

τ

τ

τ τ τ

τ τ

+⎛ ⎞=⎜ ⎟

⎝ ⎠

∫

∫

TmkN τ=1

χ = R

R0

----- = 14 mT–

mT 2+---------------

⎝ ⎠⎛ ⎞+

N1

2τ-------

⎝ ⎠⎛ ⎞

w

2

1

6---

+ 0.13

mT 2→

Fig. 3. (Color online) Swell profiles, (a) EPTT and (b) Pap-

EPTT: We-variation, τ0= 0.01, m = 102, and β = 0.9

Fig. 4. (Color online) Swelling ratio, Tanner theory (2005), We-

variation, τ0 = 0.01, m = 102, and β = 0.9.



estimate (as commented upon by Tanner), whilst a rea-

sonable fit is extracted with mT = 3.8 (strong confirmation

of the present viscoelastic data with high solvent contri-

bution β of 90%).

6.2. Dissipation-rate-theory and predictionFollowing the notation and observations of Szabo et al.

(1997), the rate of dissipation (D), which is the rate of

working against the stress (σ), in a creeping flow on a

flow domain Ω with boundary Γ, may be expressed as:

. (22)

Hence, in the case of flows with equitable upstream and

downstream stress distributions that counterbalance each

other, the rate of dissipation, D, has been shown to be

related to the product of pressure-drop and flow-rate (Q)

alone. Moreover to extend this general theory to the case

of die-well flow, one must now account for upstream

fully-developed stressing effects (taking upstream N1 =

τzz), viz.:

. (23)

Note here, that all other boundary integral contributions

vanish on no-slip walls, free-surfaces, jet-exit flow (zero-

deformation plug flow), and along the central symmetry

flow line. Identifying flows of this nature for two different

fluids at the same flow rate yields:

, (24)

by appealing to the approximate identity,

.

Then, by calibration setting, , and

assuming a constant flow-rate configuration across the

two fluid settings for simplicitya, one gathers:

(25a)

or,

. (25b)

Hence, in our present study for die-swell flow, dissipa-

tion rate changes may be related to those in pressure and

N1 solution state at inlet aloneb.

Furthermore, we wish to establish a functional relation-

ship between such energy-related change in dissipation

rate to that equivalent in swelling ratio. Here, relative dif-

ferences in swelling ratio may be extracted, say, by

appealing to the purely elastic theory of Tanner. This pro-

vides the following postulation, on relative dissipation-

rate change to that on swelling ratio, to be validated

against our predicted numerical data:

,

(26)

where, , 2 ≤ mT ≤ 4. (27)

We may also use the following approximation within the

lhs of Eq. (26):

(28)

where A is the cross-sectional area of the channel, and

constant parameters of αT and mT are taken as: αT = 3, mT =

3.7; and from Tanner elastic theory, N1 = k1*(τrz) mT; τrz =

(η)*(αT*r).

In Fig. 5, the functional relationship between swelling

ratio and Tanner elastic theory is plotted. This applies to

the two fluids, EPTT fluid and Pap-EPTT (τ0 = 0.01) fluid,

over 0 ≤ We ≤ 10 with (β = 0.9; ε = 0.25), of comparable

properties in pure shear. From this figure, the numerical

predictions are observed to adhere closely to the theoret-

ical predictions expounded above. The trend in Tanner

theory (2005) of (curve-[2]) is upheld by the trend ex-

:D u d u n dσ σ

Ω Γ

= ∇ Ω = ⋅ ⋅ Γ∫ ∫

( )10

PQ-2 *πΔ =∫R

inlet

u N dr D

ΔP N 1

in avg––[ ]fluid2Q = Dfluid2 ΔP N 1

in avg––[ ]fluid1Q = Dfluid1

( ) ( )1 1 1

0

Q2 * = N Q

Aπ

−

Γ

⎡ ⎤= Γ ⎣ ⎦∫ ∫R

inlet inlet in avgu N dr N d

Pfluid2 Pfluid1–[ ]exit

= 0

Pfluid2 Pfluid1–[ ]inlet − N1

in avg–[ ]fluid2 N1

in avg–[ ]fluid1– inlet

= Dfluid2 Dfluid1–

Q-------------------------------

2,1

2,1 1 2,1

1 1

Q[P ]- [ ]

fluidin avg

fluid fluid inlet

fluid fluid

DN

D D

−

ΔΔ Δ =

[ ]

2,1 2,1 2,1

2,1 1 2,1

1 1 1 1

Q[P ]- [ ]

Tannerfluid fluid fluidin avg

fluid fluid inlet

fluid fluid fluid fluid

DN

D D

χχ

χ χ

−

ΔΔ ΔΔ Δ = ≈ =

1

2 6

14

12 2

T

Tanner

wT

m N

mχ

τ

⎡ ⎤⎛ ⎞− ⎛ ⎞= +⎢ ⎥⎜ ⎟ ⎜ ⎟

+ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦

( ) [ ]1 1 2,10

2,1

1 1 1[ * ] ,

A A ( 1)τ

ηα

⎡ ⎤⎡ ⎤Δ = Δ⎢ ⎥⎢ ⎥ +⎣ ⎦ ⎣ ⎦

∫R

inlet

w fluid

fluid T T

N dr Nm

Fig. 5. (Color online) Die-swell, Tanner elastic theory vs.

numerical prediction.

aOtherwise fractional dissipation: flow-rate components arise

bThis also automatically accounts for the die-swell/die-exit singularity



posed in the numerical prediction of the swelling ratio

(curve-[1]); this proves to be almost constant, and hence,

independent of We-solution. While, further confirmation is

gathered from the dissipation-rate data, with its pressure-

N1 functional relationship in the LHS of Eq. (26) (curve-

[3,4]). Evaluation of Dfluid1 in Eq. (26) may be accom-

plished by recourse to Eq. (24). Conspicuously, the numeri-

cally predicted swell matches somewhat more closely to

the pressure-N1 theory, due to its broader base of theoretic

dependency (not just elastic response, but also includes

account of energy-loss due to the singularity and shear

flow contributions). One may postulate, even closer agree-

ment may be expected with an exact match in the shear

properties of the two fluids under comparison.

The individual components in Eq. (26), curve-[3] in Fig.

5, are then illustrated in Fig. 6, for 0 ≤ We ≤ 10; showing

their relative significance and hence contribution to the

arguments above. This indicates that N1 effects are rela-

tively insignificant for 0 ≤ We ≤ 10, being dominated by

pressure contributions (between 10% to 1% of the pres-

sure data). Hence, the pressure data in die-swell flow is

the dominant factor (curve-[1]). The N1-integral approxi-

mation (curve-[3]) itself, extracted from extended theory

based on propositions in the Tanner elastic theory, is

shown to be a reasonably good approximation to the full

integral for 2 ≤ We. Again, note that the N1-integral term

difference data curve in Fig. 6 proves to be relatively

insignificant against the pressure data curve, and relatively

invariant to We-change for 5 ≤ We. Hence, ultimately the

dominating factors in the LHS of Eq. (26) are the pres-

sure-difference contributions and the ratio of (Q/Dfluid1).

This state of affairs establishes an accurate estimation of

the true swelling ratio.

7. Conclusion

In this article, we have presented an analysis of steady

free-surface flows for the viscoelastic EPTT fluid and vis-

coelastoplasticity, which is extended by Al-Muslimawi et

al. (2013a). The study has been conducted under two dif-

ferent levels of (β, ε)-parameters, ε = 0.25 and β = 0.9, at

fixed yield stress setting, τ0= 0.01 and m = 102. Thereby,

significant impact has been reported on swelling ratio due

to variation in Weissenberg number (We). A comparing

between the numerical solution of die-swell flow with the

theoretical presentation that introduced by Tanner (2005)

is introduced throughout this study. In this respect, an

excellent agreement between the theoretical and numerical

solution of the swelling ratio is appeared for .

In the case of We variation, at fixed τ0= 0.01 and m =

102, swelling ratio is found to rise with increasing We,

reaching the high percentage ranges of 15.4% with 0.

The theoretical and numerical solutions have been com-

pared over a range of material parameters to establish the-

oretical relationships on energy-losses for die swell flow,

and to link swell, pressure drop, state of stress, and dis-

sipation-rate. Overall, from the results one can note an

acceptable agreement between the theoretical and numer-

ical solution.

Acknowledgment

I acknowledge financial support from mathematics depart-

ment, college of science, Basrah University.

References

Al-Muslimawi, A., 2013, Numerical Analysis of Partial Differ-

ential Equations for Viscoelastic and Free Surface Flows, Ph D

Thesis, University of Swansea.

Al-Muslimawi, A., H.R. Tamaddon-Jahromi, and M.F. Webster,

2013a, Simulation of viscoelastic and viscoelastoplastic die-

swell flows, J. Non-Newton. Fluid Mech. 191, 45-56.


8.3≥T

m

Fig. 6. (Color online) Die-swell, functional relationship to swell vs. numerical (components) prediction.



2013b, Numerical simulation of tube tooling cable-coating

with polymer melts, Korea-Aust. Rheol. J. 25, 197-216.


2014, Numerical computation of extrusion and draw-extrusion

cable-coating flows with polymer melts, Appl. Rheol. 24,

34188.

Belblidia, F., H.R. Tamaddon-Jahromi, M.F. Webster, and K.

Walters, 2011, Computations with viscoplastic and viscoelas-

toplastic fluids, Rheol. Acta 50, 343-360.

Beverly, C.R. and R.I. Tanner, 1989, Numerical analysis of extru-

date swell in viscoelastic materials with yield stress, J. Rheol.

33, 989-1009.

Bush, M.B., 1990, A numerical study of extrudate swell in very

dilute polymer solutions represented by the Oldroyd-B model,

J. Non-Newton. Fluid Mech. 34, 15-24.

Bush, M.B., J.F. Milthorpe, and R.I. Tanner, 1984, Finite element

and boundary element methods for extrusion computations, J.

Non-Newton. Fluid Mech. 16, 37-51.

Carter, R.E. and R.C. Warren, 1987, Extrusion stresses, die swell,

and viscous heating effects in double-base propellants, J.

Rheol. 31, 151-173.

Chang, P.W., T.W. Patten, and B.A. Finlayson, 1979, Collocation

and Galerkin finite element methods for viscoelastic fluid

flow-II. Die-swell problems with a free surface, Comput. Flu-

ids 7, 285-293.

Clermont, J.R. and M. Normandin, 1993, Numerical simulation

of extrudate swell for Oldroyd-B fluids using the stream-tube

analysis and streamline approximation, J. Non-Newton. Fluid

Mech. 50, 193-215.

Crochet, M.J. and R. Keunings, 1982, Finite element analysis of

die swell of a highly elastic fluid, J. Non-Newton. Fluid Mech.

10, 339-356.

Ganvir, V., A. Lele, R. Thaokar, and B.P. Gautham, 2009, Pre-

diction of extrudate swell in polymer melt extrusion using an

Arbitrary Lagrangian Eulerian (ALE) based finite element

method, J. Non-Newton. Fluid Mech. 156, 21-28.

Matallah, H., P. Townsend, and M.F. Webster, 1998, Recovery

and stress-splitting schemes for viscoelastic flows, J. Non-

Newton. Fluid Mech. 75, 139-166.

Mitsoulis, E., S.S. Abdali, and N.C. Markatos, 1993, Flow sim-

ulation of Herschel-Bulkley fluids through extrusion dies, Can.

J. Chem. Eng. 71, 147-160.

Ngamaramvaranggul, N. and M.F. Webster, 2001, Viscoelastic

simulations of stick-slip and die swell flows, Int. J. Numer.

Methods Fluids 36, 539-595.

Oishi, C.M., F.P. Martins, M.F. Tomé, J.A. Cuminato, and S.

McKee, 2011, Numerical solution of the extended Pom–Pom

model for viscoelastic free surface flows, J. Non- Newton.

Fluid Mech. 166, 165-179.

Papanastasiou, T.C., 1987, Flows of materials with yield, J.

Rheol. 31, 385-404.

Sizaire, R. and V. Legat, 1997, Finite element simulation of a fil-

ament stretching extensional rheometer, J. Non-Newton. Fluid

Mech. 71, 89-107.

Szabo, P., J.M. Rallison, and E.J. Hinch, 1997, Start-up of flow

of a FENE-fluid through a 4:1:4 constriction in a tube, J. Non-


Tanner, R.I., 1970, A theory of die-swell, J. Polym. Sci. Pt. B-

Polym. Phys. 8, 2067-2078.

Tanner, R.I., 2005, A theory of die-swell revisited, J. Non-New-

ton. Fluid Mech. 129, 85-87.

Tomé, M.F., L. Grossi, A. Castelo, J.A. Cuminato, S. McKee, and

K. Walters, 2007, Die-swell, splashing drop and a numerical

technique for solving the Oldroyd-B model for axisymmetric

free surface flows, J. Non-Newton. Fluid Mech. 141, 148-166.

Wapperom, P. and M.F. Webster, 1998, A second order hybrid

finite-element/volume method for viscoelastic flows, J. Non-


Webster, M.F., H.R. Tamaddon-Jahromi, and M. Aboubacar,

2005, Time-dependent algorithm for viscoelastic flow: Finite

element/volume schemes, Numer. Meth. Part. Differ. Equ. 21,

272-296.

Theoretical and numerical studies of die swell flo · 2016. 8. 24. · Theoretical and numerical studies of die swell flow Korea-Australia Rheology J., 28(3), 2016 231 incompressible

Documents