# Extensional rheometry of mobile fluids. Part II: Comparison between the
uniaxial, planar and biaxial extensional rheology of dilute polymer solutions
using numerically-optimized stagnation point microfluidic devices

Simon J. Haward<EMAIL_ADDRESS>Okinawa Institute of Science and
Technology, Onna, Okinawa 904-0495, Japan.  Stylianos Varchanis  Okinawa
Institute of Science and Technology, Onna, Okinawa 904-0495, Japan.  Gareth H.
McKinley  Hatsopoulos Microfluids Laboratory, Department of Mechanical
Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue,
Cambridge, MA 02139, USA  Manuel A. Alves  Departamento de Engenharia Química,
ALiCE, CEFT, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto
Frias, 4200-465 Porto, Portugal.  Amy Q. Shen  Okinawa Institute of Science
and Technology, Onna, Okinawa 904-0495, Japan.

(7 April 2022)

###### Abstract

In Part I of this paper [Haward et al. submitted (2023)], we presented a new
three-dimensional microfluidic device (the optimized uniaxial and biaxial
extensional rheometer, OUBER) for generating near-homogeneous uniaxial and
biaxial elongational flows. In this Part II of the paper, we employ the OUBER
device to examine the uniaxial and biaxial extensional rheology of some model
dilute polymer solutions. We also compare the results with measurements made
under planar extension in the optimized-shape cross-slot extensional rheometer
[or OSCER, Haward et al. Phys. Rev. Lett. (2012)]. In each case (uniaxial,
planar and biaxial extension), we use micro-particle image velocimetry to
measure the extension rate as a function of the imposed flow rate, and we
measure the excess pressure drop across each device in order to estimate the
tensile stress difference generated in the fluid. We present a new analysis,
based on solving the macroscopic power balance for flow through each device,
to refine the estimate of the tensile stress difference obtained from the
measured pressure drop. Based on this analysis, we find that for our most
dilute polymer sample, which is “ultradilute”, the extensional viscosity is
well described by the finitely extensible non-linear elastic dumbbell model.
In this limit, the biaxial extensional viscosity at high Weissenberg numbers
(Wi) is half that of the uniaxial and planar extensional viscosities. At
higher polymer concentrations, the experimental measurements deviate from the
model predictions, which is attributed to the onset of intermolecular
interactions as polymers unravel in the extensional flows. Of practical
significance (and fundamental interest), elastic instability occurs at a
significantly lower Wi in uniaxial extensional flow than in either biaxial or
planar extensional flow, limiting the utility of this flow type for
extensional viscosity measurement.

††preprint: AIP/123-QED

##  I Introduction

For an incompressible Newtonian fluid of shear viscosity $\eta$, it is well
known that the uniaxial extensional viscosity is $\eta_{E}=3\eta$, the planar
extensional viscosity is $\eta_{P}=4\eta$, and the biaxial extensional
viscosity is $\eta_{B}=6\eta$, where the coefficients, 3, 4, and 6, are
commonly referred to as the respective Trouton ratio Tr. Trouton (1906);
Petrie (2006) By contrast, for viscoelastic fluids such as polymer solutions
and melts, these limiting values of the extensional viscosity are only
approached at small rates of strain. At higher rates of strain, such that the
dimensionless Weissenberg number $\text{Wi}\gtrsim 0.5$, the unraveling and
orientation of polymer chains De Gennes (1974); Hinch (1974); Keller and Odell
(1985); Larson and Magda (1989); Perkins, Smith, and Chu (1997) results in an
increased elastic tensile stress difference $\Delta\sigma$ in the fluid and
hence a non-linear increase in the extensional viscosity and apparent Trouton
ratio $\text{Tr}_{app}$.

Understanding how the extensional viscosity and $\text{Tr}_{app}$ for
viscoelastic fluids at $\text{Wi}>0.5$ depends on the imposed mode of
extension has interested a number of researchers over many years. Stevenson,
Chung, and Jenkins (1975); Meissner  _et al._ (1982); Petrie (1984); Demarmels
and Meissner (1985); Jones, Walters, and Williams (1987); Khan and Larson
(1987); Khan, Prud’homme, and Larson (1987); Petrie (1990); Isaki  _et al._
(1991); Wagner  _et al._ (1998); Nishioka  _et al._ (2000); Kwan and Shaqfeh
(2001); Hachmann and Meissner (2003); Stadler  _et al._ (2007); Shogin (2021)
A large number of studies have involved constitutive modeling, while most of
the experimental work for the validation of those models has involved the
extensional flow of polymer melts. For such highly elastic fluids, various
instrumentation has been developed based on, e.g., the stretching of filaments
or sheets of material held in rotary clamps,Meissner, Raible, and Stephenson
(1981); Meissner (1987); Hachmann and Meissner (2003) or by lubricated
squeezing,Chatraei, Macosko, and Winter (1981); Khan, Prud’homme, and Larson
(1987); Nishioka  _et al._ (2000) and the high elastic stresses resulting from
the imposed deformation are quite readily measurable. By contrast, for less
viscous, more mobile, viscoelastic fluids such as polymeric solutions, which
can not be fixed in clamps and which generate relatively weak elastic
stresses, the development of extensional rheometers is far more challenging.
James and Walters (1993); Macosko (1994); Haward (2016) In this case,
experimental comparisons between the response of viscoelastic fluids under
different modes of extension are extremely rare. Jones, Walters, and Williams
(1987)

Extensional flows are potential flows characterized by diagonal rate-of-strain
tensors D, and come in three fundamental types. Uniaxial extension has one
positive extensional axis and is compressional along the remaining directions,
e.g.:

$\text{\bf{D}}_{U}=\begin{pmatrix}-\dot{\varepsilon}/2&0&0\\\
0&-\dot{\varepsilon}/2&0\\\ 0&0&\dot{\varepsilon}\end{pmatrix}.$ (1)

Planar extension has one neutral direction with equal and opposite extension
and compression along two perpendicular directions, e.g.:

$\text{\bf{D}}_{P}=\begin{pmatrix}\dot{\varepsilon}&0&0\\\
0&-\dot{\varepsilon}&0\\\ 0&0&0\end{pmatrix}.$ (2)

Finally, biaxial extension is the kinematic reverse of uniaxial extension,
having one compressional axis and with extension along the remaining
directions, thus:

$\text{\bf{D}}_{B}=\begin{pmatrix}\dot{\varepsilon}_{B}&0&0\\\
0&\dot{\varepsilon}_{B}&0\\\ 0&0&-2\dot{\varepsilon}_{B}\end{pmatrix}.$ (3)

Note that here we use similar definitions for uniaxial and biaxial extension
as those used by Meissner and coworkers, Meissner  _et al._ (1982); Demarmels
and Meissner (1985) and adopted by Petrie, Petrie (1984, 1990, 2006) in the
sense that we always consider the relevant strain rate metric on which
material functions will be defined as that along the stretching direction(s).
In accordance with Society of Rheology notation, Dealy (1984, 1995) we place
the subscript “$B$” on $\dot{\varepsilon}$ for biaxial extension. This serves
to distinguish these expressions from alternative definitions, such as those
suggested by Stevenson et al, Stevenson, Chung, and Jenkins (1975) and by
Bird, Bird, Armstrong, and Hassager (1987) for which biaxial extension is
considered equivalent to uniaxial compression and is thus described by Eq. 1
with a reversed sign, resulting in a strain rate of $\dot{\varepsilon}/2$ in
the two orthogonal stretching directions.

The extra stresses that arise in viscoelastic polymer solutions for
$\text{Wi}=\lambda\dot{\varepsilon}>0.5$ (or
$\text{Wi}=\lambda\dot{\varepsilon}_{B}>0.5$ in biaxial flow) result from the
entropic elasticity of the polymer chains (with characteristic relaxation time
$\lambda$), causing them to resist deformation and stretching. The
hydrodynamically-forced stretching causes optical anisotropy in the fluid,
often visible in experiments as flow-induced birefringence. Fuller (1995); te
Nijenhuis  _et al._ (2007) Birefringence is thus an optical signature of the
elastic stress in the fluid; indeed the two may be directly proportional in
cases for which the stress-optical rule is obeyed. Fuller (1995) In hyperbolic
stagnation point extensional flows (such as those described by Eqs. 1 to 3),
due to the long residence time (or equivalently the large accumulated strain)
available for polymers to unravel, the birefringence is predominantly aligned
along the axes of positive extension rate. Here, the fluid has passed through
(or near) the hyperbolic point at the coordinate origin, where the residence
time in the straining flow (and hence the strain) is theoretically infinite.
The localization of the birefringence about the stretching axes gives rise to
the common descriptor of “birefringent strand”. Crowley, Frank, and Mackley
(1976); Keller and Odell (1985); Harlen, Rallison, and Chilcott (1990);
Harlen, Hinch, and Rallison (1992); Remmelgas, Singh, and Leal (1999);
Becherer, Morozov, and van Saarloos (2008); Becherer, van Saarloos, and
Morozov (2009); Haward  _et al._ (2012, 2019) It is the growth of the elastic
stress along the stretching axes only that leads us to consider the relevant
strain rate for determination of the extensional viscosity as also being that
directed along the same axes.

Figure 1: The apparent Trouton ratio $\text{Tr}_{app}$ predicted in the three
fundamental modes of extensional deformation by various common constitutive
models used to describe dilute polymer solutions: (a) the Oldroyd-B model, (b)
the Giesekus model (mobility factor $\alpha=0.01$), (c) the FENE-P model
(extensibility $L=10$), and (c) the linear PTT (l-PTT) model (PTT
extensibility parameter $\text{\char 15\relax}=0.01$). In all cases, the
solvent viscosity is set to $\eta_{s}=0$.

To obtain a flow curve of extensional viscosity as a function of the strain
rate, the extensional kinematics applied to the fluid should be both
homogeneous in space and constant in time (i.e., _persistent_). Petrie (2006)
Thus, at each imposed extension rate, the polymer chains have sufficient time
to achieve an equilibrium degree of stretching in the flow, and for the
elastic stresses to reach a steady state as the strain
$\varepsilon=\dot{\varepsilon}t\rightarrow\infty$ (or
$\varepsilon_{B}=\dot{\varepsilon}_{B}t\rightarrow\infty$) over a residence
time, $t\gg\lambda$. If the steady-state diagonal stress tensor resulting from
the homogeneous extensional deformation is:

$\boldsymbol{\upsigma}=\begin{pmatrix}\sigma_{xx}&0&0\\\ 0&\sigma_{yy}&0\\\
0&0&\sigma_{zz}\end{pmatrix},$ (4)

then the uniaxial extensional viscosity will be
$\eta_{E}(\dot{\varepsilon})~{}=~{}(\sigma_{zz}~{}-~{}\sigma_{xx})/\dot{\varepsilon}$,
the planar extensional viscosity will be
$\eta_{P}(\dot{\varepsilon})~{}=~{}(\sigma_{xx}~{}-~{}\sigma_{yy})/\dot{\varepsilon}$,
and the biaxial extensional viscosity will be
$\eta_{B}(\dot{\varepsilon}_{B})~{}=~{}(\sigma_{xx}~{}-~{}\sigma_{zz})/\dot{\varepsilon}_{B}$.
The apparent Trouton ratio can be defined as
$\text{Tr}_{app}=\eta_{E}/\eta_{0}$, $\eta_{P}/\eta_{0}$, or
$\eta_{B}/\eta_{0}$ for uniaxial, planar, or biaxial extension (respectively),
where $\eta_{0}$ is the steady shear viscosity of the fluid at zero shear
rate.

Petrie (1990) obtained asymptotic results for the uniaxial and planar
extensional viscosities given by various viscoelastic constitutive models
commonly used to describe polymeric solutions. Petrie (1990) For all models
examined, including the Phan-Thien and Tanner (PTT), the Giesekus, and the
finitely extensible non-linear elastic dumbbell with Peterlin closure (FENE-P)
model, the two extensional viscosities were equal at high Wi (apart from the
relatively small differences due to the different contribution of the solvent
in each flow type). In fact, all viscoelastic constitutive models predict that
at low deformation rates in uniaxial, planar, and biaxial elongation
$\text{Tr}_{app}$ approaches the Newtonian limit of 3, 4, or 6 (respectively)
as $\text{Wi}\rightarrow 0$, and that in all three flows, as the Weissenberg
number exceeds 0.5, $\text{Tr}_{app}$ undergoes an abrupt increase (see Fig.
1). For the infinitely extensible Oldroyd-B model (Fig. 1(a)),
$\text{Tr}_{app}\rightarrow\infty$ for $\text{Wi}\geq 0.5$, in all cases.
Models with a bounded elasticity show that under uniaxial and planar
elongation, $\text{Tr}_{app}$ approaches the same limiting plateau value as
$\text{Wi}\rightarrow\infty$ (Fig. 1(b-d)). However, there is a disagreement
between the predictions of different models in terms of the high Wi limit of
$\text{Tr}_{app}$ in biaxial elongation. Some models, such as the Giesekus
model, predict that $\text{Tr}_{app}$ will tend to the same high Wi plateau in
biaxial extension as it does in uniaxial and planar extension (Fig. 1(b)).
However, the FENE-P and PTT models predict that in biaxial extension the
limiting value of $\text{Tr}_{app}$ at high Wi will be one-half of that for
uniaxial and planar extension (Fig. 1(c,d)). Note that, using the alternative
definition of the deformation rate tensor formulation for biaxial extension
outlined by Bird Bird, Armstrong, and Hassager (1987) results in a doubling of
the Weissenberg number and a halving of the apparent Trouton ratio compared to
the formulation of Meissner et al, Meissner  _et al._ (1982) as indicated by
the dotted blue lines in Fig. 1. Shogin (2021)

Due to the great difficulty associated with experimental extensional rheometry
of low viscosity, mobile viscoelastic fluids such as dilute polymer solutions,
James and Walters (1993); Macosko (1994); Haward (2016) these theoretical
predictions are largely untested experimentally. Using a “spin-line” rheometer
and a converging channel rheometer to generate uniaxial and planar extension,
respectively, Jones et al (1987) found a “satisfactory” (meaning order-of-
magnitude) correspondence between $\eta_{E}$ and $\eta_{P}$ for a variety of
polymer solutions. Williams and Williams (1985); Jones, Walters, and Williams
(1987) However, the two measurement methods employed differed greatly in terms
of how the deformation was applied, its spatial homogeneity, the range of
deformation rates probed, and how the tensile stress was estimated. Williams
and Williams (1985); Jones, Walters, and Williams (1987) The authors
themselves expressed apparent surprise at the agreement they obtained given
the inherent problems with making such measurements, and remarking that they
were not “comparing like with like” since one method was planar and the other
uniaxial. To this day, a systematic experimental comparison of the uniaxial,
planar and biaxial extensional responses of dilute polymer solutions using
comparable measurement methods is still missing from the literature.

In the present work, we employ numerically optimized stagnation point
microfluidic devices to make measurements of the uniaxial, planar and biaxial
extensional viscosities of a variety of model solutions formulated from dilute
concentrations of linear polymers. For planar extensional viscosity
measurements we utilize the two-dimensional (2D) Optimized-Shape Cross-slot
Extensional Rheometer (OSCER, Fig. 2(a)), Alves (2008) which over the last
decade has proven useful for characterizing the extensional rheology and flow
behavior of a variety of viscoelastic fluids. Haward  _et al._ (2012, 2013);
Haward, McKinley, and Shen (2016) For uniaxial and biaxial extensional
viscosity measurements we utilize the three-dimensional (3D) Optimized
Uniaxial and Biaxial Extensional Rheometer (OUBER, Fig. 2(b)) presented in
Part I of this paper. Haward  _et al._ (2023) The devices are designed to
provide close approximations to the respective deformation rate tensors (given
in Eqs. 1-3) over multiple characteristic device lengthscales in each spatial
direction. Both devices allow the strain rate to be controlled by simply
varying the volumetric flow rate. They also both generate stagnation points at
the center of the flow field, such that strain can accumulate indefinitely at
the set strain rate (a requirement for measuring the extensional viscosity).
Under all three modes of elongation, we employ micro-particle image
velocimetry ($\upmu$-PIV) to confirm and quantify the extensional strain
rates, coupled with pressure drop measurements designed to enable estimation
of the elastic tensile stress difference. The comparable (microfluidic) size
scales of the two devices allow similar extension rates to be obtained in each
mode of extension while always keeping inertia negligible.

Figure 2: Schematic illustrations of numerically optimized stagnation point
elongational flow devices. (a) The optimized shape cross-slot extensional
rheometer (OSCER) geometry consisting of two pairs of opposed planar inlet
(outlet) channels (height $2H$, width $2W$) oriented along the $y$ ($x$) axes
and joined by a numerically-determined profile designed to generate an optimal
approximation to planar elongational flow. (b) The optimized uniaxial and
biaxial extensional rheometer (OUBER), consisting of two pairs of opposed
planar inlet (outlet) channels (height $2H$, width $2W$) oriented along the
$x$ and $y$ axes and connected via a numerically-determined profile to a pair
of opposing outlet (inlet) channels of circular cross-section oriented along
the $z$ axis. Depending on the choice of imposed flow direction, the geometry
can produce an optimal approximation to either uniaxial or biaxial
elongational flow. For the OUBER geometry, along with the standard $(x,y,z)$
coordinate system, a $45^{\circ}$-rotated coordinate system
$(x^{\prime},y^{\prime},z)$ is employed (see main text for details). The
coordinate origin is located at the center of each device

We remark that experimental extensional viscosity measurements are always an
approximation, and that extensional “rheometers” must always be considered
“indexers” to some extent. In this work, for the first time we have assembled
a pair of highly comparable and sophisticated indexers that permit a fair
comparison between the extensional rheology of viscoelastic fluids under each
of the three fundamental modes of extension. For the most dilute polymer
solution that we test (which can be considered “ultradilute” Clasen  _et al._
(2006)), our results at high $\text{Wi}>0.5$ indicate that
$\eta_{E}\approx\eta_{P}\approx 2\eta_{B}$, in agreement with the prediction
of the FENE-P constitutive model. Of some interest, we observe that these
elongational flows lose stability at different Wi in each of the three flows
(lowest in uniaxial and highest in biaxial extension), which we discuss in
terms of the region occupied by the elastic “birefringent strand” that forms
along the stretching axis (or over the stretching plane). These observations
have important implications for viscoelastic constitutive modeling as well as
for experimental extensional rheometry.

##  II Experimental Methods

###  II.1 Microfluidic geometries

#### Planar extensional flow OSCER device

The OSCER device, shown schematically in Fig. 2(a), has been described in
detail in several prior works. Haward  _et al._ (2012, 2013); Haward,
McKinley, and Shen (2016) Briefly, the channel is cut in stainless steel by
wire-electrical discharge machining and sealed about the $z$ direction with
soda glass viewing windows. The channel has a uniform half-height $H=1$ mm and
a characteristic half-width $W=0.1$ mm upstream and downstream of the
optimized region. The channel shape is optimized over a region spanning
$\lvert x\rvert,\lvert y\rvert\leq 15W$, and generates a close approximation
to pure planar elongation over a large portion of that region.Haward  _et al._
(2012); Haward, McKinley, and Shen (2016) The high aspect ratio of the device
($H/W=10$) gives a good approximation to a two-dimensional (2D) flow ensuring
that the flow field is also uniform through most of the channel height.

#### Uni- and biaxial extensional flow OUBER device

The fabrication of an OUBER device (Fig. 2(b)), Haward  _et al._ (2023) is
achieved by the technique of selective laser-induced etching (SLE) in fused
silica glass, Gottmann, Hermans, and Ortmann (2012); Meineke  _et al._ (2016);
Burshtein  _et al._ (2019) and is described in detail in Part I of this paper.
Haward  _et al._ (2023)

The circular cross-section channels aligned along $z$ have a radius $R=0.4$
mm, while the four planar channels aligned along $x$ and $y$ each have half-
width $W=0.64$ mm and half-height $H=0.16$ mm. The channel shape is optimized
to provide almost uniform velocity gradients over a region spanning $\lvert
x\rvert,\lvert y\rvert,\lvert z\rvert\leq 5R$, and (depending on how the flow
is imposed) generates a close approximation to either pure uniaxial or pure
biaxial elongation over a large portion of that region. Haward  _et al._
(2023)

Note that, as depicted in Fig. 2, it is natural to align the $x$ and $y$ axes
with adjacent planar inlet/outlet channels. However, obtaining an experimental
view inside of the OUBER device along either of those two directions is
problematic with our current design. As described in Part I of the paper,
optical access to the stagnation point region inside the device is only
possible by viewing at $45^{\circ}$ to the $x$-axis. Haward  _et al._ (2023)
Therefore, in our experimental setup we consider a coordinate system described
by $(x^{\prime},y^{\prime},z)$, where $x^{\prime}=\frac{1}{\sqrt{2}}(x+y)$,
and $y^{\prime}=\frac{1}{\sqrt{2}}(y-x)$ (also shown in Fig. 2).

Figure 3: Rheological response of the various test fluids employed. (a) Shear
viscosity $\eta$ as a function of the applied shear rate $\dot{\gamma}$ of the
Newtonian solvent (89.6% glycerol in water, dashed line) and poly(acrylamide)
(PAA) solutions at various polymer concentrations measured in steady shear
using a stress-controlled TA Instruments DHR3 rotational rheometer with 40 mm
diameter 1∘ cone-and-plate fixture. (b) Decay of the filament diameter $D$ as
a function of time for the polymer solutions during capillary thinning in a
CaBER device, used to obtain the extensional relaxation times $\lambda$ of the
samples.

###  II.2 Test fluids

Due to the surface curvature of the three-dimensional (3D) OUBER device, see
Fig. 2(b), clear imaging inside of the device (e.g., for performing flow
velocimetry, as described below) requires that the channel be filled with a
fluid of similar refractive index $RI$ as the fused silica glass
($RI=1.4584$). Malitson (1965) A mixture of 89.6 wt% glycerol and 10.4 wt%
water, with $RI=1.4582$ at $25^{\circ}$C (measured using an Anton-Paar Abbemat
MW refractometer operating at 589 nm) is found to be a sufficiently close
match. The 89.6 : 10.4 wt% glycerol : water mixture (with density $\rho=1231$
kg m-3 and viscosity $\eta_{s}=0.143$ Pa s, Fig. 3(a)) is used as both a
Newtonian reference fluid and also as a solvent for viscolelastic polymeric
test solutions.

The polymer sample used is a nonionic poly(acrylamide) (PAA) of molecular
weight $M\approx 5$ MDa obtained from Sigma-Aldrich. Polymer solutions are
prepared at four different concentrations $c=50,~{}100,~{}200$, and 400 parts-
per-million (ppm) by first dissolving the required mass of dry polymer powder
in the aqueous component of the solvent, before adding the mass of glycerol
necessary to achieve the final desired composition. To avoid mechanical
degradation of the polymer during the solution preparation, mechanical
stirring is not used. Rather the fluids are mixed by gentle agitation on a
roller-mixer (Ika, Japan). Typically, 24 h is required for complete
dissolution of the polymer powder into the water, and a further 24 h for
complete mixing with the glycerol. Subsequent to preparation, the fluids are
stored at $5^{\circ}$C in unlit conditions, and are discarded if not used
within one month.

The concentration regime and equilibrium conformation of the PAA in solution
can be estimated based on the number of backbone bonds $n=2M/m\approx 140,000$
(where $m=71$ Da is the monomer molecular weight), the average length per bond
$l=0.154$ nm, and the characteristic ratio $C_{\infty}=6.9$. von Scholtan
(1954); Winston and Winston (1980); Kulicke, Kniewske, and Klein (1982) From
this information, it is possible to estimate the contour length
$L_{c}=nl\approx 21.6~{}\upmu$m and the equilibrium mean-square end-to-end
length $\langle r_{0}^{2}\rangle=C_{\infty}nl^{2}\approx
23,000~{}\text{nm}^{2}$. The radius of gyration is then
$R_{g}=\frac{1}{\sqrt{6}}\langle r_{0}^{2}\rangle^{1/2}\approx 62$ nm, which
can be used to estimate the overlap concentration
$c^{*}=M/N_{A}(2R_{g})^{3}\approx 4400$ ppm (where $N_{A}$ is Avogadro’s
number). Graessley (1980) It can also be estimated that to achieve full
stretch of the polymer chain, the end-to-end separation needs to be increased
from its equilibrium value by an extensibility factor (or stretch ratio) of
$L=L_{c}/\langle r_{0}^{2}\rangle^{1/2}\approx 143$. We note that these
molecular parameters are estimated based on the value of $C_{\infty}$ reported
for PAA in water at $25^{\circ}$C. Kulicke, Kniewske, and Klein (1982)
However, recent molecular dynamics simulations indicate that the PAA chain
adopts a roughly similar conformation in 90 wt% aqueous glycerol as it does in
pure water. Hopkins  _et al._ (2020) Therefore, we have some confidence that
our test fluids should be safely in the dilute solution regime with
$0.011\lesssim c/c^{*}\lesssim 0.088$, and that the PAA chains should be
highly extensible.

The steady shear rheology of the polymeric test fluids is measured using a
stress-controlled DHR3 rotational rheometer (TA Instruments Inc.) fitted with
a 40 mm diameter 1∘ angle cone-and-plate geometry (see Fig. 3(a)). Over the
range of accessible shear rates, the fluids each have a near-constant
viscosity, close to that of the solvent. For this reason, we take the
viscosity $\eta$ of each fluid as being the average of the respective data
shown in Fig. 3(a). The relaxation times $\lambda$ of the fluids are assessed
by means of capillary thinning measurements using a CaBER device (Thermo-
Haake). Anna and McKinley (2001) The device is fitted with 6 mm diameter
plates with the initial separation set to 1 mm and the final separation to 6
mm. Curves of the filament diameter at the midpoint between the plates $D$ as
a function of time are shown in Fig. 3(b). The value of $\lambda$ is extracted
from the time constant of the exponential decay of the filament diameter
observed within the elasto-capillary thinning regime. Anna and McKinley (2001)
The values of $\eta$ and $\lambda$ obtained for each polymeric fluid are given
in Table 1.

###  II.3 Flow control and dimensionless groups

The test fluids are driven through the microfluidic OSCER and OUBER devices
using 29:1 gear ratio neMESYS low pressure syringe pumps (Cetoni, GmbH) to
control the volumetric flow rate through each individual channel. For planar
extensional flow in the OSCER device, two pumps are used to impose equal
volumetric flow rates $Q$ into each of the two inlet channels, while two pumps
withdraw fluid at equal and opposite rates from each of the two outlet
channels. For uniaxial (biaxial) extensional flow in the OUBER device, two
pumps are used to impose equal volumetric flow rates $Q$ through each of the
two circular cross-section outlet (inlet) channels, while four pumps impose
equal volumetric flow rates $Q/2$ through each of the four planar inlet
(outlet) channels. The pumps are fitted with Hamilton Gastight syringes of
appropriate volumes such that the specified “pulsation free” dosing rate is
always exceeded. Connections between the syringes and the microfluidic devices
are made using flexible Tygon tubing.

Table 1: Values of viscosity $\eta$, solvent-to-total viscosity ratio $\beta$, and relaxation time $\lambda$ obtained from rheological characterization of the PAA solutions.  PAA concentration [ppm] | $c/c^{*}$ |  $\eta$ [Pa s] | $\beta=\eta_{s}/\eta$ |  $\lambda$ [s]
---|---|---|---|---
50 | 0.011 | 0.146 | 0.98 | 0.22
100 | 0.022 | 0.151 | 0.95 | 0.38
200 | 0.044 | 0.155 | 0.92 | 0.54
400 | 0.088 | 0.181 | 0.79 | 1.03

For an imposed volumetric flow rate $Q$ in each channel of the OSCER device,
the average flow velocity is $U=Q/4WH$ and the expected (or nominal) extension
rate based on a Newtonian flow field prediction is given by
$\dot{\varepsilon}_{nom}=0.1U/W$. Haward  _et al._ (2012, 2013); Haward,
McKinley, and Shen (2016)

For the OUBER device, we consider the characteristic average flow velocity $U$
as being that in the two channels of circular cross-section, so that for an
imposed volumetric flow rate $Q$ in each of those channels, $U=Q/\uppi R^{2}$.
The expected nominal extension rates are $\dot{\varepsilon}_{nom}=0.4U/R$ for
uniaxial extension, and $\dot{\varepsilon}_{B,nom}=0.2U/R$ for biaxial
extension. Haward  _et al._ (2023)

The Reynolds number Re describes the relative strength of inertial to viscous
forces in the flow experiments. In the OSCER device, we define $\text{Re}=\rho
UD_{h}/\eta$, where $D_{h}=2WH/(W+H)$ is the hydraulic diameter of the
rectangular channels. The maximum Reynolds number reached in experiments using
the OSCER device is $\text{Re}\approx 0.1$. In the case of the OUBER device,
we define $\text{Re}=2\rho UR/\eta$, and the maximum values reached are
$\text{Re}\approx 0.2$ (uniaxial extension), and $\text{Re}\approx 0.7$
(biaxial extension). Since $\text{Re}<1$ in all experiments, inertial effects
in the flow are considered negligible.

The Weissenberg number describes the relative strength of elastic to viscous
forces in the flow and can be quantified by the product of the extension rate
and the relaxation time $\lambda$. However, since we only have _a priori_
knowledge of the nominal extension rate, it is convenient to first also define
a nominal Weissenberg number as
$\text{Wi}_{nom}=\lambda\dot{\varepsilon}_{nom}$ in uniaxial and planar
elongation, and $\text{Wi}_{nom}=\lambda\dot{\varepsilon}_{B,nom}$ in biaxial
elongation.

Typically in elongational flows, it is found that polymer stretching for
$\text{Wi}_{nom}\gtrsim 0.5$ will modify the flow field compared to the
Newtonian case, resulting in a reduction of the true extension rate below its
nominal value. Mackley (1978); Dunlap and Leal (1987); Remmelgas, Singh, and
Leal (1999); Haward  _et al._ (2012, 2013); Haward, McKinley, and Shen (2016)
In the present work, micro-particle image velocimetry ($\upmu$-PIV)
experiments (described in Sec. II.4) will be used to directly measure the true
extension rate (or velocity gradient) along the stretching axis
$\dot{\varepsilon}$ (or $\dot{\varepsilon}_{B}$), allowing the true
Weissenberg number to be evaluated as $\text{Wi}=\lambda\dot{\varepsilon}$ in
uniaxial and planar elongation, and $\text{Wi}=\lambda\dot{\varepsilon}_{B}$
in biaxial elongation.

###  II.4 Microparticle image velocimetry

Quantitative measurement of the flow field in each extensional flow
configuration is achieved using microparticle image velocimetry ($\upmu$-PIV,
TSI Inc., MN). Wereley and Meinhart (2005, 2010) For this purpose, the test
fluids are seeded with a low concentration ($c_{p}\approx 0.02$ wt%) of
$3.2~{}\upmu$m diameter red fluorescent tracer particles (Fluor-Max, Thermo
Scientific) with excitation (emission) wavelength 542 nm (612 nm). The plane
of interest within the geometry (i.e., the $xy$ midplane in the OSCER
geometry, and the $y^{\prime}=0$ plane in the OUBER geometry) is brought into
focus on an inverted microscope (Nikon Eclipse Ti) with a $4\times$
magnification, $\text{NA}=0.13$ numerical aperture Nikon PlanFluor objective
lens. Under these conditions, the measurement depth over which microparticles
contribute to the determination of the velocity field is $\delta_{m}\approx
180~{}\upmu$m. Meinhart, Wereley, and Gray (2000) Excitation with a dual-
pulsed Nd:YLF laser with a wavelength of 527 nm induces the emission of
particle fluorescence, which is detected by a high speed camera (Phantom MIRO,
Vision Research). The camera is operated in frame-straddling mode and is
synchronized with the laser in order to acquire pairs of particle images
corresponding to pairs of laser pulses separated by a small time $\Delta t$.
The value of $\Delta t$ is varied inversely to the imposed flow rate and set
so that the average displacement of particles between the two images in each
pair is always $\approx 4$ pixels. In this work we are only concerned with
steady flows, so at each flow rate tested 50 image pairs are acquired and are
processed using an ensemble average cross-correlation PIV algorithm (TSI
Insight 4G) in order to reduce noise. A recursive Nyquist criterion is
employed with a final interrogation area of $16\times 16$ pixels to enhance
the spatial resolution and obtain two components of the velocity vector
$\bf{u}$ spaced on a square grid of $26.6~{}\upmu$m $\times 26.6~{}\upmu$m. In
the OSCER device, the obtained components of $\bf{u}$ are $u$ and $v$ (the $x$
and $y$ component, respectively). In the OUBER device, the obtained components
of $\bf{u}$ are $u^{\prime}$ and $w$ (the $x^{\prime}$ and $z$ component,
respectively). Subsequent to data acquisition, the software Tecplot Focus
(Tecplot Inc., WA) is used for generation of velocity contour plots and
streamline traces and for extraction of velocity profiles.

###  II.5 Pressure drop measurements and extensional rheometry

Pressure drop measurements are made using a 35.5 kPa wet-wet differential
pressure sensor (Omega Engineering Inc.) connected across one inlet and one
outlet of each device. Pressure taps are taken by installing T-junction
connectors in the upstream and downstream tubing connecting between the
fluidic device and the syringes driving the flow. At each imposed flow rate in
each extensional flow configuration (uniaxial, planar and biaxial), two
independent measurements of the pressure drop are made. The first pressure
drop measurement (labeled $\Delta P_{tot}$) is made with flow imposed in all
the channels of the device (i.e., with the device in normal operation, as
described in Sec II.3) and provides an estimate of the total stress. This
combines stresses due to the shear induced by the walls of the channel and
connecting tubing, as well as any extra stress due to the elongational
kinematics in the flow. A second measurement (labeled $\Delta P_{sh}$) is made
with half of the inlet channels and half of the outlet channels disabled and
allows estimation of the shear stresses only. It is important to state that in
the OUBER device, two adjacent (not opposing) planar channels are disabled
during the measurement of $\Delta P_{sh}$ in order to avoid the formation of a
‘T-channel-like’ flow configuration which would retain a stagnation point in
the center of the device.

Figure 4: Normalized velocity fields with superimposed streamlines for
creeping flow ($\text{Re}<0.05$) of the Newtonian solvent in (a) uniaxial, (b)
planar, and (c) biaxial extension. Parts (d), (e) and (f) show the respective
normalized velocity profiles measured along the flow axes (data points), which
compare favorably against the target numerical profiles (lines). The nominal
elongation rate in each case is indicated within the respective plot.

From the raw pressure drop measurements, we obtain an excess pressure drop
$\Delta P_{ex}=\Delta P_{tot}-\Delta P_{sh}$, which we assume arises
predominantly due to the extensional kinematics present in the flow field
during the measurement of $\Delta P_{tot}$. Of course, this differential
measurement is not able to quantify each individual component of the diagonal
stress tensor in order to precisely evaluate the principal stress difference
$\Delta\sigma$ required to compute the extensional viscosity (see Sec. I). In
previous works involving planar extensional flows in the OSCER device, and
also in the standard cross-slot geometry, it has simply been assumed that
$\Delta P_{ex}\approx\Delta\sigma$, thus the planar extensional viscosity has
been computed as $\eta_{P}\approx\Delta P_{ex}/\dot{\varepsilon}$. Haward,
Sharma, and Odell (2011); Haward and McKinley (2012); Haward  _et al._ (2012,
2013) Some support for this assumption has been shown using birefringent
polymer solutions, for which direct proportionality has been shown between
$\Delta P_{ex}$ and the birefringence $\Delta n$ measured at the stagnation
point. The constants of proportionality approximately matched with the known
stress-optical coefficients $C$, of the respective fluids, suggesting that
$\Delta P_{ex}\approx\Delta n/C=\Delta\sigma$. Sharma  _et al._ (2015); Fuller
(1995)

In this work, we attempt to more properly relate $\Delta P_{ex}$ to
$\Delta\sigma$ by considering the macroscopic power balance for flow through
each of our geometries, thus enabling a more accurate estimation of the
extensional viscosity to be obtained from the experimental pressure drop
measurements (see details in Sec. 9).

##  III Results

###  III.1 Newtonian flow field characterization

Figure 5: Normalized velocity fields with superimposed streamlines for the
flow of the 100 ppm poly(acrylamide) solution in (a,d,g) uniaxial, (b,e,h)
planar, and (c,f,i) biaxial extension at the nominal extension rates and
nominal Weissenberg numbers indicated.

For completeness, we commence the presentation of the experimental results by
showing the Newtonian flow field in each extensional flow configuration (Fig.
4). Normalized fields of the velocity magnitude measured by $\upmu$-PIV, with
streamlines superimposed to indicate the direction of flow, are shown in Fig.
4(a), (b) and (c) for uniaxial, planar and biaxial extension, respectively, at
Reynolds numbers $\text{Re}<0.05$. In each case, the velocity field is
symmetric about the flow axes, with a stagnation point at the coordinate
origin, as expected. Velocity profiles extracted along the flow axes are shown
in Fig. 4(d), (e) and (f) (below the respective velocity field), in comparison
with the numerical predictions for Newtonian creeping flow (available in Refs.
43 and 46, for the OSCER and the OUBER, respectively). It is clear that over
the measurable ranges of the accessible flow axes in each device
configuration, the experimental velocity profiles agree very well with the
respective numerical predictions, giving confidence that the microfluidic
devices and the experimental setup are performing satisfactorily. It should be
noted that over the $z=0$ plane in the OUBER device, the 2D $\upmu$-PIV
measurement provides the $u^{\prime}(x^{\prime})$ velocity profile, which
agrees with the $u(x)$ profile for $\lvert x^{\prime}\rvert\lesssim R$ and
$\lvert x\rvert\lesssim R$, as shown in Part I. Haward  _et al._ (2023) In
other words, the velocity field has good axisymmetry over a radial distance of
$r=\sqrt{x^{2}+y^{2}}\approx R$ about the $z$-axis.

###  III.2 Polymer solution flow field characterization

We proceed to examine the flow field in the case of the viscoelastic PAA-based
test solutions. Fig. 5 shows normalized velocity magnitude fields obtained in
each of the three extensional flow configurations for the 100 ppm PAA solution
over a range of imposed nominal extension rates. At lower nominal rates, such
that $\text{Wi}_{nom}$ is only slightly above unity, the flow field in each
case (uniaxial, planar, and biaxial, shown in Fig. 5(a), (b), and (c),
respectively) appears to be rather similar to that observed for the flow of
Newtonian fluid (shown in Fig. 4(a), (b), and (c), respectively). However,
close inspection of the velocity magnitude contours for uniaxial extension of
the polymer solution (Fig. 5(a)) reveals a local minimum in the velocity along
the stretching axis (i.e., along the $x^{\prime}=0$ centerline). In contrast,
for Newtonian flow, the Poiseuille-like velocity profile is maximal along the
center of the outlet channels. This local minimum in velocity along the
stretching direction is not evident in either the planar (Fig. 5(b)) or
biaxial (Fig. 5(c)) flows at these rather low imposed values of
$\text{Wi}_{nom}$. For the uniaxial flow, increasing the nominal Weissenberg
number to $\text{Wi}_{nom}=2.5$ (Fig. 5(d)), results in the centerline minimum
of the velocity profile across the outlet channels becoming more pronounced.
For the planar extensional flow at $\text{Wi}_{nom}=6.3$ (Fig. 5(e)), and for
biaxial extension at a similar $\text{Wi}_{nom}=5.0$ (Fig. 5(f)), still no
obvious difference from Newtonian flow can be discerned (see Fig. 4(b,c) for
comparison). At sufficiently high $\text{Wi}_{nom}$, uniaxial and planar
extensional flows of the 100 ppm PAA solution exhibit elastic instabilities
manifested as an asymmetry of the flow. For uniaxial extension, this occurs
for a critical nominal Weissenberg number $\text{Wi}_{nom,c}\approx 5$ as a
distinct distortion of the streamlines close to the stagnation point, and
develops with increasing $\text{Wi}_{nom}$ into the strong flow asymmetry
shown in Fig. 5(g) for $\text{Wi}_{nom}=10.1$. For planar extension, a
somewhat higher critical value $\text{Wi}_{nom,c}\approx 13$ is necessary
before the flow exhibits instability. Fig. 5(h) shows a strongly asymmetric
flow state observed in planar extension for $\text{Wi}_{nom}=25.4$. In
contrast, for biaxial extension, no obvious sign of instabilty is observed
even at the highest achievable nominal Weissenberg number
$\text{Wi}_{nom}=20.2$ (Fig. 5(i)); indeed, the kinetics appear to remain
essentially Newtonian-like.

Figure 6: Illustration of the flow modification along the stretching direction
resulting from the flow of poly(acrylamide) solutions in (a,d) uniaxial, (b,e)
planar, and (c,f) biaxial extension. Flow velocity profiles are measured
across a device outlet 1 mm downstream of the stagnation point. (a,b,c) show
normalized profiles of the streamwise flow velocity for the 100 ppm PAA
solution at various nominal extension rates and compared against the result
for the Newtonian solvent. (d,e,f) show normalized profiles of the streamwise
flow velocity for all the tested polymer solutions at the highest nominal
extension rate (indicated in the plot) for which all are deemed to be steady
and symmetric, again compared against the result for the Newtonian solvent.

The flow asymmetry observed in the OSCER device (Fig. 5(h)) has been observed
previously in various experiments involving planar stagnation point
extensional flows. Gardner  _et al._ (1982); Arratia  _et al._ (2006); Poole,
Alves, and Oliveira (2007); Rocha  _et al._ (2009); Haward and McKinley (2012,
2013); Haward, McKinley, and Shen (2016) It is considered to be a purely
elastic phenomenon driven by elastic tensile stress on the strongly curving
streamlines that pass through the birefringent strand in the vicinity of the
stagnation point, a mechanism consistent with the well-known elastic
instability criterion introduced by McKinley and coworkers. Pakdel and
McKinley (1996); McKinley, Pakdel, and Öztekin (1996); Öztekin, Alakus, and
McKinley (1997); Haward, McKinley, and Shen (2016) The asymmetric flow state
observed under uniaxial extension in the OUBER device (Fig. 5(g)) appears to
be similar in form to that observed in the OSCER (Fig. 5(h)), but it is
unclear exactly how this asymmetry is oriented in the 3D ‘axisymmetric’ flow
field of the OUBER. A detailed investigation is beyond the scope of the
present work and will require careful visualization in 3D, possibly using
microtomographic flow velocimetry. Haward  _et al._ (2023) A cursory
investigation indicates that the asymmetry in the OUBER device (Fig. 5(g)) is
steady in time and not rotating around the stretching axis. Most likely, it
selects a favored orientation due to the presence of the four planar inlet
channels which break the perfect axisymmetry of the flow away from the
$z$-axis, similar to the instability reported by Afonso et al. Afonso, Alves,
and Pinho (2010) in numerical simulations of viscoelastic flow in a 6-arm
cross-slot.

In this work, which is focused on extensional rheometry, we wish to avoid
elastic instabilities. Each viscoelastic test fluid is driven to the point of
instability only in order to determine the limiting values of
$\text{Wi}_{nom}$ up to which the extensional flow field generated around the
stagnation point remains stable and symmetric. For experimental determination
of the extensional viscosity under each extensional flow configuration (Sec.
9), the range of extension rates is restricted to
$\text{Wi}_{nom}<0.5\text{Wi}_{nom,c}$ in order to ensure the measurement is
made while the flow is stable and symmetric.

Figure 7: Quantification of extension rates determined from velocity fields
measured with poly(acrylamide) solutions in (a,d) uniaxial, (b,e) planar, and
(c,f) biaxial extension. (a,b,c) show normalized streamwise velocity profiles
along the stretching axes measured for the 100 ppm PAA solution at various
nominal extension rates and compared against the result for the Newtonian
solvent. (d,e,f) show the measured extension rates as a function of the
average flow velocity for all the tested polymer solutions and compared
against the result for the Newtonian solvent. The extension rate in each case
is determined by averaging the velocity gradient on the relevant axis over the
spatial domain indicated in the respective plot. Solid lines are fits to the
experimental data points of the form described in the main text.

The modification of the Newtonian flow field by the presence of polymer (i.e.,
the development of local minima in the velocity profiles, mentioned above) is
rendered more apparent by extracting velocity profiles across the channel
outlets. Fig. 6 shows normalized profiles of the streamwise flow velocity
taken across a channel outlet 1 mm downstream of the stagnation point. Such
profiles for the 100 ppm PAA solution (for which example velocity fields at
different extension rates are provided in Fig. 5) are shown in comparison with
the Newtonian case for uniaxial, planar, and biaxial extension in Fig. 6(a),
(b), and (c), respectively. During uniaxial extension in the OUBER device
(Fig. 6(a)), the velocity profile obtained for the polymer solution agrees
well with that of the Newtonian fluid at low $\dot{\varepsilon}_{nom}$, but an
increasingly pronounced local minimum develops at $x^{\prime}=0$ as the flow
rate through the device is increased. At high $\dot{\varepsilon}_{nom}$, the
profile becomes asymmetric about $x^{\prime}=0$ as the symmetric base flow
becomes unstable and the elastic asymmetry (illustrated in Fig. 5(g))
develops. For planar extension in the OSCER device (Fig. 6(b)), the velocity
profile across the channel outlet for the 100 ppm PAA solution again agrees
well with that of the Newtonian fluid at low $\dot{\varepsilon}_{nom}$. In
this case, with increasing $\dot{\varepsilon}_{nom}$, there is a progressive
modification to the velocity profile, with some flattening of the central peak
at $y=0$, but the profile remains almost parabolic. For the 100 ppm PAA
solution in biaxial extension in the OUBER device, no significant difference
is noticable compared to the Newtonian flow profile, even up to the highest
values of $\dot{\varepsilon}_{B,nom}$ examined (Fig. 6(c)).

To illustrate the effects of increasing the polymer concentration, Fig. 6(d),
(e), and (f) shows the velocity profiles across the channel outlets for all of
the tested fluids in uniaxial, planar, and biaxial extension (respectively).
In each case the profiles are shown for a fixed value of the nominal extension
rate (the highest for which all the flows are considered stable and
symmetric). In general, the degree of flow modification caused by
viscoelasticity becomes increasingly severe with increasing polymer
concentration. For uniaxial extension at
$\dot{\varepsilon}_{nom}=1.66~{}\text{s}^{-1}$ (Fig. 6(d)), at lower polymer
concentrations of 50 and 100 ppm, the velocity profile is flattened compared
with the Newtonian case, but an increasing local minimum in the centerline
flow velocity develops as the PAA concentration is raised to 200 ppm and
above. In planar extension at $\dot{\varepsilon}_{nom}=4.16~{}\text{s}^{-1}$
(Fig. 6(e)), and low polymer concentrations of 50 and 100 ppm of PAA, the
profiles are essentially Newtonian-like. For 200 ppm of polymer, the profile
becomes flattened compared with the Newtonian case, and for 400 ppm a local
minimum around $y=0$ becomes evident. In the case of biaxial extension at
$\dot{\varepsilon}_{B,nom}=13.3~{}\text{s}^{-1}$, flow modification by the
polymer is evident at 200 and 400 ppm of PAA, where the flow velocity is
reduced about the centerline and the profiles become flattened again (Fig.
6(f)). However, in biaxial extension, the flow profiles always remain
essentially parabolic.

Velocity profiles with a local minimum in the streamwise velocity along the
stretching axis have been reported a number of times in the literature
studying stagnation point flows of viscoelastic fluids (e.g., Refs. 75; 66;
59; 35; 76; 40). The reduction in flow velocity on the axis (relative to a
Newtonian fluid) is associated with the localized stretching of polymers that
pass near the stagnation point and are subsequently advected downstream along
the outlet centerline. For polymer solutions that exhibit measurable flow-
induced birefringence, this stretching results in the appearance of a
characteristic ‘birefringent strand’ localized along the stretching axis
(e.g., Refs. 5; 35; 36; 37; 38; 39; 40), and indicative of high extensional
stress. Fuller (1995) Within the strand, the fluid behaves elastically and
exhibits a much higher extensional viscosity than the fluid flowing outside
the strand, where the polymer is relatively unstretched, and the fluid remains
Newtonian-like. The elastic strand thus acts as an internal stress boundary
layer in the flow, driving velocity perturbations that resist the stretching,
and thus giving rise to the modified flow profile observed. Harlen, Rallison,
and Chilcott (1990)

The modification to the Newtonian flow field by the stretching of the polymer
along the extensional axis reduces the true extension rate along the
stretching axis, as assessed in Fig. 7. Normalized profiles of the streamwise
velocity component along the stretching axis are shown for the 100 ppm PAA
solution over a range of nominal extension rates under uniaxial, planar, and
biaxial extension in Fig. 7(a), (b), and (c), respectively. Here we only
consider flows that are deemed stable and symmetric. Under uniaxial extension
(Fig. 7(a)), the velocity profile for the 100 ppm PAA solution agrees well
with the Newtonian profile for $\dot{\varepsilon}_{nom}\leq
0.83~{}\text{s}^{-1}$, but increasingly deviates from the Newtonian profile as
the imposed flow rate is increased beyond $\dot{\varepsilon}_{nom}\approx
0.83~{}\text{s}^{-1}$. Under planar extension (Fig. 7(b)), as
$\dot{\varepsilon}_{nom}$ is increased, only a slight deviation from the
Newtonian profile is evident even at the highest nominal extension rates
tested (up to 16.7 s-1), while for biaxial extension (Fig. 7(c)), the profiles
for the polymer solution remain Newtonian-like for $\dot{\varepsilon}_{B,nom}$
up to 53.1 s-1 (the highest imposed value).

In Fig. 7(d), (e), and (f) we plot the measured extension rate (determined
from velocity profiles such as those shown in Fig. 7(a), (b), and (c)) as a
function of the imposed flow rate or nominal extension rate in uniaxial,
planar, and biaxial extension, respectively. Here data is shown for all the
tested polymer solutions and is compared against the Newtonian result (shown
by the dashed grey lines). For uniaxial extension (Fig. 7(d)), we report
$\dot{\varepsilon}=\partial w/\partial z$, which is evaluated along the
$z$-axis for $\lvert z\rvert\leq 4R$ (i.e., within the range over which the
flow field is optimized Haward  _et al._ (2023)). Similarly, in planar
extension (Fig. 7(e)), we report $\dot{\varepsilon}=\partial u/\partial x$,
evaluated along the $x$-axis for $\lvert x\rvert\leq 12W$. In biaxial
extension, (Fig. 7(f)), we report $\dot{\varepsilon}_{B}=\partial
u^{\prime}/\partial x^{\prime}$, which is evaluated along the
$x^{\prime}$-axis for $\lvert x^{\prime}\rvert\leq R$ (the range over which
axisymmetry applies such that $u^{\prime}(x^{\prime})\equiv u(x)$, see Fig.
4(f)). As shown in Fig. 7(d), (e), and (f), for all three extensional flow
configurations and all polymer concentrations, the polymer solutions follow
the Newtonian trend for lower imposed nominal extension rates, but
progressively deviate below the Newtonian trend at higher flow rates. The
large polymeric stresses induced by the extensional stretching always retard
the evolution of the velocity profile downstream of the stagnation point. For
each polymeric fluid in each extensional flow configuration, the experimental
data are well-described by a curve of the form
$\dot{\varepsilon}=\dot{\varepsilon}_{nom}-A{\dot{\varepsilon}_{nom}}^{B}$
(where $A$ and $B$ are fitting constants), as shown by the respective solid
lines. The fitted curves allow calculation of the true strain rate for
arbitrary imposed flow conditions with the given fluid.

In general, from Figs. 6 and 7 it is evident that the deviation from
Newtonian-like behavior becomes more severe with increasing polymer
concentration and increasing extension rate. Also, the greatest effects are
observed in uniaxial extension while biaxial extension causes the mildest
modification of the flow field. The effect of planar extension appears to be
intermediate between uniaxial and biaxial. A similar general trend is also
apparent from the onset of instability, which for a given polymer solution
occurs at the lowest nominal Weissenberg number in uniaxial extension,
followed by planar extension, and finally biaxial extension. This may have
implications for the utility of the different flows for extensional rheometry,
as will be discussed further below.

Figure 8: Pressure drop measurements made with the Newtonian solvent and PAA
solutions in uniaxial (left column), planar (middle column), and biaxial
(right column) extensional flow. (a), (b) and (c) show representative raw
measurements of the total pressure drop ($\Delta P_{tot}$) and the pressure
due to shear ($\Delta P_{sh}$) versus time as the imposed flow rate $U$ is
increased in steps. Inserts schematically indicate the flow configurations
used for measurement of $\Delta P_{tot}$ and $\Delta P_{sh}$ in each case. The
respective steady state plateau value of the pressure drop at each increment
in $U$ is presented as a function of $U$ in (d), (e) and (f), where the dashed
lines represent linear fits to $\Delta P_{sh}$ passing through the origin, for
low $U<2.5~{}\text{mm~{}s}^{-1}$. (g), (h), and (i) show the excess pressure
drop ($\Delta P_{ex}=\Delta P_{tot}-\Delta P_{sh}$) for all of the tested
fluids as a function of the nominal strain rate in uniaxial, planar, and
biaxial extension, respectively. Dashed grey lines are linear fits through the
data for the Newtonian fluid, with constants of proportionality $\approx 1.3$
Pa s, $\approx 0.9$ Pa s, and $\approx 2.0$ Pa s in parts (g), (h), and (i),
respectively. Error bars on $\Delta P_{ex}$ for the polymer solutions
represent the standard deviation over at least five repeated measurements.

###  III.3 Pressure drop

In Fig. 8(a), (b), and (c), we illustrate raw measurements of the pressure
drop in uniaxial, planar, and biaxial extension (respectively) using a few of
the polymeric fluids and also the Newtonian solvent. As described in Sec. II.5
(and illustrated schematically by the respective inserts to Fig. 8(a), (b),
and (c)), for each fluid and each flow configuration, the pressure drop is
measured once with the device in full operation mode to obtain the total
pressure drop $\Delta P_{tot}$, and once with half of the inlet channels and
half of the outlet channels disabled in order to quantify the contribution of
shear $\Delta P_{sh}$. The measurements are made by programming the syringe
pumps to increment the average flow velocity through the device $U$ in a
stepwise fashion, with sufficient time at each step for the pressure to rise
and stabilize to a steady plateau value. Subsequently, the average plateau
pressure drop is measured at each step in flow rate in order to obtain curves
such as those shown in Fig. 8(d), (e), and (f), which result from the raw
pressure traces shown in Fig. 8(a), (b), and (c), respectively. Note that in
each flow configuration, for the Newtonian fluid $\Delta P_{tot}\approx\Delta
P_{sh}$, with $\Delta P_{sh}\propto U$ (as indicated by the dashed grey
lines). For lower concentration polymer solutions, $\Delta P_{sh}\propto U$
(as indicated by the dashed black and red lines in Fig. 8(d) and (e),
repsectively), although at higher polymer concentrations, $\Delta P_{sh}$ may
increase superlinearly at higher imposed flow rates (as shown by the deviation
of the experimental data points from the dashed blue line in Fig. 8(f)). Most
notably, for the polymer solutions at low average flow velocities $\Delta
P_{tot}\approx\Delta P_{sh}$ (as we also observe for the Newtonian fluid), but
beyond a certain value of $U$, the two curves diverge and $\Delta P_{tot}$
rises clearly above $\Delta P_{sh}$, leading to a significant and clearly
measurable excess pressure drop $\Delta P_{ex}=\Delta P_{tot}-\Delta P_{sh}$.

In Fig. 8(g), (h), and (i), we present $\Delta P_{ex}$ as a function of the
nominal extension rate for all of the tested fluids under uniaxial, planar,
and biaxial extension (respectively). Here, error bars represent the standard
deviation over a minimum of five repeated measurements. Scatter and
uncertainty in the data for the Newtonian solvent fluid is significant, so for
better clarity the data in each plot is represented by a linear fit (dashed
grey line). In general, at low nominal deformation rates the data from the
polymer solutions follow a roughly linear trend (similar to the Newtonian
fluid), but depart from that trend as the extension rate increases, turning
upwards and tending towards eventual plateau values. Note that, in several
cases, the measured pressure difference ($\Delta P_{tot}$ and/or $\Delta
P_{sh}$) exhibits fluctuations at higher extension rates (even though the flow
field may be deemed steady and symmetric, see Sec. III.2). For this reason,
the maximum extension rates at which data is curtailed in Fig. 7(d), (e), and
(f) and in Fig. 8(g), (h), and (i) (respectively) may not always precisely
match, and in some cases there is an increase in the reported error bars in
$\Delta P_{ex}$ at the highest rates tested.

###  III.4 Extensional rheometry

With the required experimental data, i.e., true measured extension rates, and
excess pressure drops in hand (see Figs. 7 and 8, respectively), we turn to
simple theory in order to understand how to obtain a robust estimate of the
extensional viscosity from our measurements.

Considering for simplicity the 2D flow through the OSCER geometry, a
macroscopic power balance leads to the following approximate expression (see
Appendix A for details):

$2\Delta
P_{ex}Q\approx(\sigma_{xx}-\sigma_{yy})\dot{\varepsilon}\mathcal{V}_{P},$ (5)

which allows estimation of the extensional viscosity:

$\eta_{P}\approx 2\Delta P_{ex}Q/\dot{\varepsilon}^{2}\mathcal{V}_{P},$ (6)

where $\mathcal{V}_{P}$ is an appropriate volume of fluid within the device
over which $\sigma_{xx}-\sigma_{yy}$ can be considered ‘homogeneous’ for
averaging purposes.

For a Newtonian fluid, or for a viscoelastic fluid flowing at $\text{Wi}\ll
0.5$, $\mathcal{V}_{P}$ might be expected to roughly equate with the volume of
the optimized region of the OSCER geometry, $V_{OSC}=480W^{2}H~{}(=4.8\times
10^{-9}~{}\text{m}^{3}$ for the specific device being used here with $H=1$ mm
and $W=0.1$ mm, Sec. II.1).

Since for a Newtonian fluid in planar extension the Trouton ratio is known to
be $\text{Tr}=(\sigma_{xx}-\sigma_{yy})/\dot{\varepsilon}\eta=4$, Eq. 5 can be
rewritten and rearranged to give:

$\mathcal{V}_{P,Newt}=2\Delta P_{ex}Q/4\dot{\varepsilon}^{2}\eta.$ (7)

We know that for Newtonian flow in the OSCER device,
$\dot{\varepsilon}\approx\dot{\varepsilon}_{nom}=0.1U/W=0.1Q/4W^{2}H$.
Furthermore, from the fit to the Newtonian data shown in Fig. 8(h), we know
that $\Delta P_{ex}\approx 0.9\dot{\varepsilon}$. Hence, Eq. 7 can be
evaluated to give a unique value $\mathcal{V}_{P,Newt}\approx 126W^{2}H$ (or
$\approx 0.25V_{OSC}$). We should indeed anticipate that
$\mathcal{V}_{P,Newt}<V_{OSC}$ since the extensional kinematics are not
entirely homogeneous over the whole OSCER geometry due to the shear induced at
the channel walls. Haward  _et al._ (2012); Haward, McKinley, and Shen (2016)
In fact, a volume of magnitude $0.25V_{OSC}$ corresponds well to the volume of
the OSCER geometry over which the local extension rate is within 10% of
$\dot{\varepsilon}_{nom}$, i.e., the region where the extensional kinematics
are almost homogeneous.

As discussed in Sec. III.2 and elsewhere, for a viscoelastic fluid in an
extensional flow at $\text{Wi}>0.5$, a localized elastic ‘birefringent strand’
develops along the stretching axis within which $\Delta\sigma$ becomes
dominant. Harlen, Rallison, and Chilcott (1990); Harlen, Hinch, and Rallison
(1992); Becherer, Morozov, and van Saarloos (2008); Becherer, van Saarloos,
and Morozov (2009); Haward  _et al._ (2012) Accordingly, we expect that for
$\text{Wi}>0.5$, the relevant volume $\mathcal{V}_{P}$ to use in Eqs. 5 or 6,
would be that of the birefringent strand.

In principle, in certain cases, it may be possible to directly measure the
dimensions of the birefringent elastic strand in order to determine its volume
experimentally as a function of the Weissenberg number. However, this is not
always practical or even possible; for instance in the present case, the
fluids being used are too weakly birefringent to make the required optical
measurements. For this reason, we seek a simple and pragmatic approach to
estimate the volume of the elastic strand $\mathcal{V}_{P,strand}$, which may
be used more generally in Eqs. 5 and 6 when $\text{Wi}\geq 0.5$.

For the FENE-P model in planar extension, an approximate scaling relation for
the dimensionless half-width $w_{strand}^{*}$ of the sheet-like birefringent
strand in terms of Wi and the polymer extensibility $L$ has been presented by
Becherer et al. Becherer, Morozov, and van Saarloos (2008) The scaling has
been shown to adequately describe measurements of the birefringent strands
that develop in the OSCER device for $\text{Wi}\geq 0.5$. Haward  _et al._
(2012) In the asymptotic limit of high Wi, the dimensionless strand half-width
scales as $w_{strand}^{*}\sim 1/L$. Crowley, Frank, and Mackley (1976);
Rallison and Hinch (1988); Harlen, Hinch, and Rallison (1992); Renardy (2006);
Becherer, Morozov, and van Saarloos (2008) A dimensional strand half-width can
be computed as $w_{strand}\approx l_{opt}/L$, where $l_{opt}=15W$ is the
lengthscale over which the flow field in the OSCER device is optimized and the
flow is purely extensional.

Accordingly, we approximate the asymptotic volume of the sheet-like
birefringent strand as $\mathcal{V}_{P,strand}\approx 1800W^{2}H/L$ (strand
length $l_{strand}=30W$, width $2w_{strand}=30W/L$, height $h_{strand}=2H$),
which for $L=143$ (Sec. II.2) yields
$\mathcal{V}_{P,strand}\approx\mathcal{V}_{P,Newt}/10$. We propose the
following simple piecewise approximation to the volume $\mathcal{V}_{P}$ as a
function of Wi for planar extensional flow of dilute solutions of flexible
polymers in the OSCER device:

$\mathcal{V}_{P}=\begin{cases}\mathcal{V}_{P,Newt}\approx
126W^{2}H&\text{:}\quad\text{Wi}<0.5\\\ \mathcal{V}_{P,strand}\approx
1800W^{2}H/L&\text{:}\quad\text{Wi}\geq 0.5\\\ \end{cases}.$ (8)

By following similar arguments, we arrive at the following equations to
evaluate $\eta_{E}$ in the case of uniaxial extensional flow in the OUBER
device:

$2\Delta
P_{ex}Q\approx(\sigma_{zz}-\sigma_{xx})\dot{\varepsilon}\mathcal{V}_{E},$ (9)

i.e.:

$\eta_{E}\approx 2\Delta P_{ex}Q/\dot{\varepsilon}^{2}\mathcal{V}_{E},$ (10)

where

$\mathcal{V}_{E}=\begin{cases}\mathcal{V}_{E,Newt}\approx
47R^{3}&\text{:}\quad\text{Wi}<0.5\\\ \mathcal{V}_{E,strand}\approx 250\uppi
R^{3}/L&\text{:}\quad\text{Wi}\geq 0.5\\\ \end{cases}.$ (11)

Here, $\mathcal{V}_{E,Newt}\approx 0.31V_{OUB}$, where $V_{OUB}=154R^{3}$ is
the volume of the optimized region of the OUBER device. For the particular
OUBER device being used in this study, with $R=0.4$ mm (Sec. II.1),
$V_{OUB}\approx 9.86\times 10^{-9}~{}\text{m}^{3}$. For uniaxial extension,
Harlen et al (1992) Harlen, Hinch, and Rallison (1992) have shown for the
FENE-CR model that the asymptotic radius of the birefringent strand at high Wi
scales as $1/\sqrt{L}$, which is consistent with experimental measurements
made in classical opposed-jets apparatus, Müller, Odell, and Keller (1988);
Cathey and Fuller (1990) as well as in a 6-arm cross-slot device. Haward  _et
al._ (2019) In Eq. 11, $\mathcal{V}_{E,strand}$ represents the volume of a
columnar birefringent strand of asymptotic diameter $10R/\sqrt{L}$ and length
$10R$.

Figure 9: Extensional viscosity as a function of the extensional strain rate
for the Newtonian solvent and the dilute PAA solutions, determined from excess
pressure drop measurements made in (a) uniaxial, (b) planar, and (c) biaxial
elongational flow.

For biaxial extensional flow in the OUBER device, we obtain:

$\Delta
P_{ex}Q\approx(\sigma_{xx}-\sigma_{zz})\dot{\varepsilon}_{B}\mathcal{V}_{B},$
(12)

i.e.:

$\eta_{B}\approx\Delta P_{ex}Q/\dot{\varepsilon}_{B}^{2}\mathcal{V}_{B},$ (13)

where

$\mathcal{V}_{B}=\begin{cases}\mathcal{V}_{B,Newt}\approx
37.5R^{3}&\text{:}\quad\text{Wi}<0.5\\\ \mathcal{V}_{B,strand}\approx 250\uppi
R^{3}/L&\text{:}\quad\text{Wi}\geq 0.5\\\ \end{cases},$ (14)

and in this case $\mathcal{V}_{B,Newt}\approx 0.24V_{OUB}$. For biaxial
stagnation point extension, the thickness of the disk-like birefringent
“strand” region (e.g., Refs. 81; 82) that forms over the $z=0$ plane has not
been well characterized in the literature. Additionally, since there is a
disagreement between constitutive models regarding the response of polymeric
solutions to biaxial extension, we do not wish to rely on the prediction of a
specific (e.g., FENE-type) model to describe the dimension of the resulting
birefringent region. Limited experimental data obtained from a dilute solution
of near-monodisperse atactic polystyrene in a 6-arm cross-slot device over a
range of Wi in both uniaxial and biaxial elongation is available in Ref. 41.
Assuming that the asymptotic strand radius in uniaxial extension scales as
$1/\sqrt{L}$ (as established above), the data available in Ref. 41 indicates
the asymptotic thickness of the birefringent region in biaxial extension to
scale as $\sim 1/L$, Haward  _et al._ (2019) similar to the result for planar
elongation. Becherer, Morozov, and van Saarloos (2008); Haward  _et al._
(2012) Accordingly, in Eq. 14, $\mathcal{V}_{B,strand}$ represents the volume
of a birefringent disk of asymptotic thickness $10R/L$ and diameter $10R$.

Note that the computation of $\mathcal{V}_{P,Newt}$, $\mathcal{V}_{E,Newt}$,
and $\mathcal{V}_{B,Newt}$ using the excess pressure drop measured for the
Newtonian fluid serves as a Newtonian calibration of the respective flow,
ensuring the correct value of Tr will be obtained for the Newtonian fluid when
those volumes are used to compute the extensional viscosity from Eqs. 6, 10,
and 13, respectively.

The volumes computed for the Newtonian fluid and for the birefringent strand
regions (given in Eqs. 8, 11, and 14), should be valid for
$\text{Wi}\rightarrow 0$ and $\text{Wi}\rightarrow\infty$, respectively. The
step change in $\mathcal{V}_{P}$, $\mathcal{V}_{E}$, and $\mathcal{V}_{B}$ at
$\text{Wi}=0.5$ is clearly unphysical, however at present the functional form
that the volume should take across this transition between Newtonian-like and
viscoelastic behavior is unclear. The question over this is further
complicated if we are to consider a Wi-dependent strand volume, which vanishes
for $\text{Wi}\leq 0.5$, Becherer, Morozov, and van Saarloos (2008) suggesting
a possibly nonmonotonic variation of the volume with Wi. Despite this
shortcoming, we consider the formulation described above to be an advance on
earlier estimates of the extensional viscosity from pressure drop
measurements. For instance, in the cross-slot and OSCER devices, the rather
coarse approximation $\eta_{P}\approx\Delta P_{ex}/\dot{\varepsilon}$ was
generally used (e.g., Ref. 40), although some prior attempts have also been
made to account for the dimensions of the birefringent strand. Haward  _et
al._ (2010b, a) Given the experimentally-established linear relation between
$Q$ and $\dot{\varepsilon}$ (at least for the Newtonian fluid) it can be seen
that our new approximation to the planar extensional viscosity measured in the
OSCER device (Eq. 6) can be written $\eta_{P}\approx(\Delta
P_{ex}/\dot{\varepsilon})\times F$, where $F=8W^{2}H/0.1\mathcal{V}_{P}$ is a
dimensionless correction factor essentially consisting of a ratio of geometric
parameters. An analogy can be drawn with the determination of the shear
viscosity from the experimentally-measured pressure drop along a pipe or
channel of arbitrary cross-section, where the pressure drop must be scaled by
the ratio of the hydraulic diameter to the length of the conduit. Walters
(1975)

The extensional viscosities $\eta_{E}(\dot{\varepsilon})$,
$\eta_{P}(\dot{\varepsilon})$, and $\eta_{B}(\dot{\varepsilon}_{B})$, computed
as described above, are shown for each of the experimental test fluids in Fig.
9(a), (b), and (c), respectively. In each plot, the Newtonian result (dashed
gray line) is computed using the respective fit to the excess pressure drop
data shown in Fig. 8(g,h,i), resulting in a constant value for the extensional
viscosity equal to $3\eta_{s}$, $4\eta_{s}$, and $6\eta_{s}$ in uniaxial,
planar and biaxial extension, respectively. At low extension rates, the
results obtained for the dilute polymer solutions generally approach a
constant value close to (or slightly higher than) that of the Newtonian
solvent, as expected given the slightly higher shear viscosities of the
polymer solutions (Table 1). With increasing extension rate, each of the
polymeric fluids undergo a gradual increase in the extensional viscosity,
before an abrupt jump takes place at a specific extension rate (corresponding
to $\text{Wi}=0.5$) that reduces with the polymer concentration (due to the
increasing relaxation time, see Table 1). Subsequently, for further increasing
extension rate, there is a general trend for the extensional viscosity to
gradually increase towards an apparent plateau.

Figure 10: Apparent Trouton ratio $\text{Tr}_{app}$ as a function of the
Weissenberg number Wi for (a) 50 ppm PAA, (b) 100 ppm PAA, (c) 200 ppm PAA,
and (d) 400 ppm PAA in uniaxial, planar and biaxial elongational flow. Data
points are experimentally determined from pressure loss measurements. Lines
are computed from the FENE-P model with the solvent-to-total viscosity ratio
$\beta$ matched to the respective fluid (given in Table 1), and the
extensibility parameter (or stretch ratio) $L=143$ (Sec. II.2).

Similarities and differences between the responses of the fluids to the
different modes of extensional flow are made more obvious by viewing their
apparent Trouton ratio as a function of the Weissenberg number in Fig. 10.
Here, we also plot the response predicted by the FENE-P model under
homogeneous uniaxial, planar and biaxial elongation conditions, where the
model parameters $\beta$ and $L^{2}$ are matched to those of the fluids
(solvent-to-total viscosity ratio $\beta$ given in Table 1, and extensibility
$L=143$, as computed in Sec. II.2). In general, within experimental
uncertainty, for all polymer solutions at low Wi, $\text{Tr}_{app}$ approaches
the expected (i.e., Newtonian) limiting value. Also, consistent with the model
prediction, in all cases $\text{Wi}=0.5$ marks the point of an abrupt increase
in $\text{Tr}_{app}$. For the most dilute 50 ppm PAA solution (Fig. 10(a)),
for $\text{Wi}>0.5$ the experimental data obtained from uniaxial, planar, and
biaxial extension closely follow the respective FENE-P prediction towards the
high-Wi plateau, where
$\text{Tr}_{app}(\text{uniaxial})=\text{Tr}_{app}(\text{planar})=2\times\text{Tr}_{app}(\text{biaxial})$.
As the PAA concentration is increased through Fig. 10(b), (c), and (d), the
agreement with the FENE-P model prediction becomes less convincing, with an
increasingly gradual approach of the experimental data towards the eventual
plateau in the extensional viscosty and a less distinct difference between the
response in biaxial extension from that in uniaxial and planar extension.
These changes with polymer concentration may be because the polymer solutions
(although dilute with $c/c^{*}\leq 0.1$) can not all be considered
“ultradilute”, and at higher polymer concentrations intermolecular
interactions may play an increasingly important role when the molecules become
stretched by the flow. Dunlap and Leal (1987); Harrison, Remmelgas, and Leal
(1998); Clasen  _et al._ (2006); Stoltz, de Pablo, and Graham (2006)
Experimental data from uniaxial extension Clasen  _et al._ (2006) and
molecular dynamics simulations in planar extension Stoltz, de Pablo, and
Graham (2006) suggest that the ultradilute limit, for which interchain
interactions are negligible even at high polymer extensions, is approached as
the polymer concentration is decreased towards $c/c^{*}\approx 0.01$, similar
to the concentration regime of our 50 ppm PAA solution. Notably, at the higher
polymer concentrations tested (Fig. 10(c,d)), we are unable to see a
convincing high-Weissenberg number plateau in $\text{Tr}_{app}$ for uniaxial
extensional flow. In these cases, the onset of elastic flow instability
curtails the measurement before a plateau is reached. In fact, the flow
modification is so severe in these cases (see Fig. 7(d)) that
$\dot{\varepsilon}$ almost ceases to increase with the imposed flow velocity.
Since the excess pressure drop across the device continues to increase with
the imposed flow velocity (Fig. 8(g)), this causes an apparent upturn in
$\eta_{E}$ and $\text{Tr}_{app}$ to an asymptote as $\text{Wi}\rightarrow 1$,
before the flow field breaks symmetry.

##  IV Discussion and Conclusions

In this work we have used the new OUBER device (developed in Part I of this
paper Haward  _et al._ (2023)), and also the pre-existing OSCER device, Haward
_et al._ (2012) to perform the first experimental comparison of the
extensional rheology of dilute mobile polymer solutions in planar, uniaxial
and biaxial extensional flow. In each case the extensional viscosity is
assessed using common methods: micro-particle image velocimetry is used to
quantify the relevant extensional strain rate along the stretching axis (or
axes) and excess pressure drop measurements are used to estimate the
respective tensile stress difference as a function of the extension rate. The
estimate of the tensile stress difference is based on a new analysis of the
macroscopic power balance for each extensional flow configuration. In each
case, the Reynolds number of the flow is maintained sufficiently low for
inertial contributions to the pressure drop to be ignored.

Several differences are observed between the responses of the polymer
solutions in the various extensional flow configurations. Specifically, for a
given nominal extension rate, the flow field is most severely modified
(compared to that of a Newtonian fluid) in uniaxial extension. By contrast, in
biaxial extensional flow of the polymer solutions the kinematics remain
essentially Newtonian-like even at much higher nominal extension rates. Planar
extensional deformations of the polymer solutions have an intermediate effect,
showing more significant flow modification than in biaxial extension, but
being less severe than in uniaxial extension. Stability constraints follow a
similar trend: for a given polymer solution, uniaxial flow destabilizes and
becomes asymmetric at the lowest extension rate, while biaxial flow remains
stable to much higher extension rates, with planar flow being intermediate.

Our estimates of the extensional viscosities and apparent Trouton ratios of
the polymer solutions, based on our new analysis method, are broadly
consistent with the predictions of the FENE-P constitutive model. Within
experimental error, the data approach the expected limiting values at low
extension rates or Weissenberg numbers, and all of the polymeric test
solutions exhibit an increase in the extensional viscosity (or
$\text{Tr}_{app}$) at $\text{Wi}=0.5$. For $\text{Wi}>0.5$, the extensional
viscosities of the polymeric fluids generally approach towards high-Wi plateau
values. For our most dilute 50 ppm polymer solution (for which $c/c^{*}\approx
0.01$ and which can be considered “ultradilute”), the high-Wi plateau values
of the extensional viscosity agree very well with the prediction of the FENE-P
model, for which $\eta_{E}=\eta_{P}=2\eta_{B}$. However, this agreement
progressively deteriorates with increasing polymer concentration. This is
likely because, although the polymer chains are dilute and non-interacting
under quiescent conditions (with $c/c^{*}\leq 0.1$), interchain interactions
become increasingly important at higher concentrations as the molecules
unravel in the extensional flow.

From a practical point of view an important consideration is the early onset
of instability in the uniaxial extensional flow. This can cause difficulty in
reaching the high-Wi plateau of the extensional viscosity, and therefore
limits the utility of the device for measurement of $\eta_{E}$. The greater
relative stability of planar and biaxial extensional flows allow measurements
to be made to much higher extension rates (or larger limiting Weissenberg
numbers), and for plateau values of $\eta_{P}$ and $\eta_{B}$ to be found more
convincingly. On the other hand, it will be of fundamental interest to
understand the three-dimensional form of the symmetry-breaking flow
instability that occurs in uniaxial extension, which is not readily
ascertained from the 2D flow velocimetry performed in the present work (see
Fig. 5(g)). It will also be important to better understand the physical reason
for why uniaxial extension is the most prone to the onset of elastic
instability. We speculate at present that this is related to the greater
thickness of the birefringent strand (radius $\sim 1/\sqrt{L}$ in uniaxial
extension, but half-width $\sim 1/L$ in planar and biaxial extension). We
think this is likely to be the reason why the flow modification along the
stretching axis is most severe in the case of uniaxial extension, and that
this probably contributes to the onset of instability, too.

We reiterate that our estimates of the extensional viscosities
$\eta_{E},~{}\eta_{P},$ and $\eta_{B}$ are just that (i.e., _estimates_), as
will necessarily always be the case since generating a spatially homogeneous
extensional flow throughout the whole of the rheometric device is practically
impossible. However, we have for the first time designed and fabricated
microfluidic devices that generate reasonably homogeneous approximations to
uniaxial, planar and biaxial extension over spatial regions much larger than
the characteristic lengthscale of the geometry, and which also permit
comparable assessments to be made of the tensile stress difference as a
function of the imposed extension rate, all at low levels of fluid inertia. We
believe that our new approach to estimating the tensile stress difference from
the excess pressure drop, based on an approximate solution to the macroscopic
power balance (see Sec. III.4) represents a significant advance on prior
analyses in similar such devices. Nevertheless, there remains significant
scope for further improvement. Specifically, at the transition between
Newtonian-like and viscoelastic behavior at $\text{Wi}=0.5$, the abrupt step
down in the volume used to compute the extensional viscosity in Eqs. 6, 10,
and 13 is unphysical. Clearly this transition should be smooth, but at present
it is unclear how it should be described mathematically. Numerical simulations
may provide insight to this computational rheology problem, although it is
possible that polydispersity of the polymer molecular weight also contributes
to the form of this transition region, which may be confounding. Furthermore,
the estimation of the volume of the birefringent strand for $\text{Wi}>0.5$
should strictly depend on Wi, which further complicates the analysis. An
analytical solution (based on the FENE-P model) for the width of the
birefringent strand as a function of Wi is available for planar extension,
Becherer, Morozov, and van Saarloos (2008) but the corresponding elastic
boundary layer analysis needs to be solved (and confirmed experimentally) for
uniaxial and biaxial extension. In our ongoing work, we intend to focus our
research efforts towards addressing these issues.

###### Acknowledgements.

S.J.H, S.V., and A.Q.S. gratefully acknowledge the support of the Okinawa
Institute of Science and Technology Graduate University (OIST) with subsidy
funding from the Cabinet Office, Government of Japan, along with funding from
the Japan Society for the Promotion of Science (JSPS, Grant Nos. 21K03884 and
22K14184). MAA acknowledges the support by LA/P/0045/2020 (ALiCE),
UIDB/00532/2020 and UIDP/00532/2020 (CEFT), funded by national funds through
FCT/MCTES (PIDDAC). We are indebted to Prof. R. J. Poole for insightful
discussions.

## Conflict of Interest Statement

The authors have no conflicts to disclose.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  Appendix A Origin of $\Delta P_{ex}$ in the OSCER device

We first consider 2D flow in the OSCER device with flow injected at volumetric
rate $Q$ through two opposing inlet channels and withdrawn at volumetric rate
$Q$ through two opposing outlet channels (i.e., normal operation mode) to
generate the stagnation point extensional flow field. Assuming that the flow
is fully developed at the inflow and outflow boundaries, the total pressure
drop $\Delta P_{tot}$ measured across an inlet and an outlet of the device
must obey the following macroscopic power balance equation:

$\Delta P_{tot}Q=\int\limits_{V/2}\biggl{(}\sigma_{xx}\frac{\partial
u}{\partial x}+\sigma_{xy}\Bigl{(}\frac{\partial u}{\partial y}+\frac{\partial
v}{\partial x}\Bigr{)}+\sigma_{yy}\frac{\partial v}{\partial y}\biggr{)}dV,$
(15)

which by application of the continuity equation can be written as:

$\begin{split}\Delta
P_{tot}Q&=\int\limits_{V/2}(\sigma_{xx}-\sigma_{yy})\frac{\partial u}{\partial
x}dV+\int\limits_{V/2}\sigma_{xy}\Bigl{(}\frac{\partial u}{\partial
y}+\frac{\partial v}{\partial x}\Bigr{)}dV.\end{split}$ (16)

We split the total volume $V$ into two parts: $V_{OSC}$, the extensional flow-
dominated volume of the optimized region of the OSCER device, and $V-V_{OSC}$,
which comprises the remaining regions between the upstream and downstream
pressure taps (i.e., shear-dominated regions in the inlets, outlets, and
connecting tubing). Thus Eq. 16 can be rewritten:

$\begin{split}\Delta
P_{tot}Q&=\int\limits_{V_{OSC}/2}(\sigma_{xx}-\sigma_{yy})\frac{\partial
u}{\partial
x}dV+\int\limits_{(V-V_{OSC})/2}(\sigma_{xx}-\sigma_{yy})\frac{\partial
u}{\partial x}dV\\\ &+\int\limits_{V_{OSC}/2}\sigma_{xy}\Bigl{(}\frac{\partial
u}{\partial y}+\frac{\partial v}{\partial
x}\Bigr{)}dV+\int\limits_{(V-V_{OSC})/2}\sigma_{xy}\Bigl{(}\frac{\partial
u}{\partial y}+\frac{\partial v}{\partial x}\Bigr{)}dV.\end{split}$ (17)

Since the flow is shear-dominated in the volume $V-V_{OSC}$, and extension-
dominated in the volume $V_{OSC}$ the second and third terms in Eq. 17 can be
neglected, leaving:

$\Delta
P_{tot}Q\approx\int\limits_{V_{OSC}/2}(\sigma_{xx}-\sigma_{yy})\frac{\partial
u}{\partial x}dV+\int\limits_{(V-V_{OSC})/2}\sigma_{xy}\Bigl{(}\frac{\partial
u}{\partial y}+\frac{\partial v}{\partial x}\Bigr{)}dV.$ (18)

Similar to the derivation of Eq. 17, the pressure drop measured when one inlet
and one outlet of the OSCER device is disabled ($\Delta P_{sh}$) must obey the
following:

$\begin{split}\Delta
P_{sh}Q&=\int\limits_{V_{OSC}}(\sigma_{xx}-\sigma_{yy})\frac{\partial
u}{\partial
x}dV+\int\limits_{(V-V_{OSC})/2}(\sigma_{xx}-\sigma_{yy})\frac{\partial
u}{\partial x}dV\\\ &+\int\limits_{V_{OSC}}\sigma_{xy}\Bigl{(}\frac{\partial
u}{\partial y}+\frac{\partial v}{\partial
x}\Bigr{)}dV+\int\limits_{(V-V_{OSC})/2}\sigma_{xy}\Bigl{(}\frac{\partial
u}{\partial y}+\frac{\partial v}{\partial x}\Bigr{)}dV.\end{split}$ (19)

Since, in this case, the extensional component in the flow is absent in the
entire volume, the first and second terms in Eq. 19 can be neglected. We also
neglect the third term since the shear rate in $V_{OSC}$ is much smaller than
that in the inlet and outlet regions ($V-V_{OSC}$). Thus:

$\Delta
P_{sh}Q\approx\int\limits_{(V-V_{OSC})/2}\sigma_{xy}\Bigl{(}\frac{\partial
u}{\partial y}+\frac{\partial v}{\partial x}\Bigr{)}dV.$ (20)

Subtracting Eq. 20 from Eq. 18 to solve for the excess pressure drop $\Delta
P_{ex}=\Delta P_{tot}-\Delta P_{sh}$, we obtain:

$\Delta
P_{ex}Q\approx\int\limits_{V_{OSC}/2}(\sigma_{xx}-\sigma_{yy})\frac{\partial
u}{\partial x}dV.$ (21)

The integrated volume $V_{OSC}/2$ can be further subdivided if the tensile
stress difference $\sigma_{xx}-\sigma_{yy}$ becomes dominant in a specific
region, such as within the birefringent strand when $\text{Wi}>0.5$. Haward
_et al._ (2012) Thus:

$\Delta
P_{ex}Q\approx\int\limits_{\mathcal{V}_{P}/2}(\sigma_{xx}-\sigma_{yy})\frac{\partial
u}{\partial
x}dV+\int\limits_{(V_{OSC}-\mathcal{V}_{P})/2}(\sigma_{xx}-\sigma_{yy})\frac{\partial
u}{\partial x}dV,$ (22)

where $\mathcal{V}_{P}$ represents the relevant volume of fluid over which the
tensile stress difference should be averaged. Accordingly, the second term in
Eq. 22 can be dropped. Assuming homogeneous extension within the volume
$\mathcal{V}_{P}$ (e.g., the strand, Becherer, Morozov, and van Saarloos
(2008)) we obtain:

$2\Delta
P_{ex}Q\approx(\sigma_{xx}-\sigma_{yy})\dot{\varepsilon}\mathcal{V}_{P}.$ (23)

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