© 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
Theoretical and numerical studies of die swell flow
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
Alaa H. Al-Muslimawi
232 Korea-Australia Rheology J., 28(3), 2016
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
Theoretical and numerical studies of die swell flow
Korea-Australia Rheology J., 28(3), 2016 233
, (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.
Alaa H. Al-Muslimawi
234 Korea-Australia Rheology J., 28(3), 2016
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
Theoretical and numerical studies of die swell flow
Korea-Australia Rheology J., 28(3), 2016 235
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.
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