1 Introduction

It is a conundrum of non-Newtonian fluid mechanics and rheology that common continuum constitutive models successfully describe certain features of the flows of polymer solutions but fail to capture others. One example of evidence for this puzzle includes imaging of the extension of fluorescently labeled single-molecule DNA in steady shear and extensional flows [1,2,3,4,5]. The commonly used continuum elastic dumbbell models such as Oldroyd-B and FENE-P (a finitely extensible nonlinear elastic model with the Peterlin approximation [6,7,8]) are insufficient to accurately predict the extension in these flows [9], even though, in many scientific and engineering applications, these continuum models form the basis for large-scale flow simulations of complex fluids [10,11,12,13,14,15].

The failure in predicting the polymer extension is perhaps even more surprising given the ability of these continuum models to reasonably describe the viscometric properties, such as a shear viscosity and a first-normal-stress-difference, of a wide range of dilute polymer solutions [7, 16]. Furthermore, there are examples of the ability of the continuum models to predict experimentally observed characteristics of some viscoelastic fluid flows with mixed kinematics involving a combination of shear and extension in different geometries. For instance, various continuum models, including the upper-convected Maxwell, FENE-P, and FENE-CR (a FENE model introduced by Chilcott and Rallison [17]), were utilized in simulations [18, 19] to predict purely elastic flow asymmetries in the cross-slot geometry, consistent with experimental observations [20, 21]. Varchanis et al. [22] and Haward et al. [23] also obtained qualitative agreement between continuum-level simulations and experimental results for the asymmetric flow of viscoelastic polymer solutions around confined cylinders (see also [24]). In addition, considerable attention in the fluid mechanics and rheology communities was devoted to elucidating the experimentally observed flow instabilities and elastic turbulence using Oldroyd-B and FENE-P models. We refer the reader to the recent review papers on the topic [10, 13,14,15, 25].

Besides the polymer extension, Oldroyd-B and FENE-P models also fail to predict a conformation hysteresis in the coil-stretch transition observed in extensional flow experiments [3, 26]; this phenomenon was first hypothesized independently by de Gennes and Hinch in 1974 by introducing a conformation-dependent friction coefficient or relaxation time [27,28,29]. However, for a simple shear flow, this model predicts a non-physical full extension as well as an increase rather than a decrease of the viscosity at high shear rates, as shown in Sects. 3 and 4.

In contrast to the Oldroyd-B and FENE-P models, Brownian dynamics simulations can capture the polymer extension in shear flow and the conformation hysteresis in the coil-stretch transition in extensional flow [3, 4, 30,31,32]. However, these simulations are computationally expensive and difficult to perform in complex geometries [10, 33]. Therefore, despite the limitations of continuum modeling, most non-Newtonian fluid mechanics studies still exploit the constitutive equations, such as Oldroyd-B, FENE-CR, and FENE-P, due to their simplicity. Over the years, several modifications have been introduced into these elastic dumbbell models to incorporate additional microscopic features of realistic polymer chains [9, 27,28,29, 34,35,36,37,38].

Here, we provide a theoretical framework for calculating the dumbbell extension in shear and extensional flows and the corresponding viscosities for different elastic dumbbell models that include various microstructurally inspired terms. Specifically, we elucidate the influence of (i) the finite polymer extensibility, (ii) the conformation-dependent drag as the microstructure is elongated, and (iii) the inefficiency of rotation, allowing non-affine deformation, on the dumbbell extension in simple flows. We qualitatively compare the predictions of different models with the available experimental results and show the importance of including all three microstructurally inspired terms in a constitutive equation to reproduce the experimentally observed polymer extension in shear and extensional flows at high shear and extension rates. While some of the theoretical results summarized in this work are available in the literature, we are not otherwise aware of the style of presentation and systematic comparison between the models and connection to the experiments we provide here. In particular, this work has a similar spirit to the papers of Fuller and Leal [35, 39] and Phan-Thien et al. [9], but presents additional elastic dumbbell models and a more modern discussion on single-polymer experiments performed since the important contributions of these authors. We believe the three microstructurally inspired terms are essential to properly model viscoelastic channel flows with mixed kinematics and, in particular, can help to resolve the well-known discrepancy between the experimental measurements and the Oldroyd-B model predictions for the steady-state flow rate-pressure drop relation of viscoelastic fluids in contracting channel flows [14, 40,41,42,43].

2 Elastic dumbbell modeling

In an elastic dumbbell model, first introduced by Kuhn [44] in 1934, the response of a polymer chain to flow is described only by the end-to-end vector \({\textbf {r}}\) joining the endpoints of the chain. The statistics of \({\textbf {r}}\) are obtained by using the Fokker–Planck or diffusion equation for the probability density function. The continuum-level elastic dumbbell models are then derived from the Fokker–Planck equation using closures that usually involve some form of pre-averaging, resulting in an evolution equation for the conformation tensor \({\textbf {A}}\equiv \left\langle {\textbf {rr}}\right\rangle \), representing the ensemble average of the second moment of \({\textbf {r}}\) [8, 45]. To the best of our knowledge, the most general representation of the continuum-level elastic dumbbell model accounting for the polymer’s finite extensibility, conformation-dependent drag, and conformation-dependent non-affine deformation was proposed by Phan-Thien, Manero, and Leal [9]. We refer to this model as the FENE-PTML model, which is given by

$$\begin{aligned} \frac{D{\textbf {A}}}{Dt}- & {} \underset{\mathrm {Stretching\,in\,flow}}{\underbrace{\left[ (\varvec{\nabla }{} {\textbf {u}})^{\textrm{T}}\cdot {\textbf {A}}+{\textbf {A}}\cdot (\varvec{\nabla }{} {\textbf {u}})\right] }}+\underset{\mathrm {Inefficiency\,of\,rotation\,in\,straining\,flow}}{\underbrace{\mathcal {E}(\textrm{tr}({\textbf {A}}))\left( {\textbf {E}}\cdot {\textbf {A}}+{\textbf {A}}\cdot {\textbf {E}}\right) }}\nonumber \\= & {} -\frac{1}{\underset{\mathrm {Conformation\text{- }dependent\,drag}}{\lambda \underbrace{\mathcal {Z}(\textrm{tr}({\textbf {A}}))}}}\left[ \overset{\mathrm {Finite\,extensibility\,of\,dumbbell}}{\overbrace{\mathcal {F}(\textrm{tr}({\textbf {A}}))}{} {\textbf {A}}}-\mathcal {G}(\textrm{tr}({\textbf {A}})){\textbf {I}}\right] , \end{aligned}$$
(1)

where we have labeled the different physical contributions. Here, t is time, \(D[...]/Dt=\partial [...]/\partial t+{\textbf {u}}\cdot \varvec{\nabla }[...]\) is the material derivative, \(\lambda \) is the longest relaxation time of the polymers, \({\textbf {E}}=\frac{1}{2}(\varvec{\nabla }{} {\textbf {u}}+(\varvec{\nabla }{} {\textbf {u}})^{\textrm{T}})\) is the rate-of-strain tensor, and \(\mathcal {F}\), \(\mathcal {G}\), \(\mathcal {Z}\), and \(\mathcal {E}\) are the positively defined non-dimensional functions of the trace of \({\textbf {A}}\), i.e., \(\textrm{tr}({\textbf {A}})\), which is the square of the dumbbell end-to-end distance (extension) scaled with its equilibrium value.

We note that the first two terms on the left-hand side of Eq. (1) are known as the upper-convected or contravariant time derivative [7, 45],

$$\begin{aligned} \overset{\nabla }{{\textbf {A}}}\equiv \frac{D{\textbf {A}}}{Dt}-(\varvec{\nabla }{} {\textbf {u}})^{\textrm{T}}\varvec{\cdot }{} {\textbf {A}}-{\textbf {A}}\varvec{\cdot }(\varvec{\nabla }{} {\textbf {u}})=\frac{\partial {\textbf {A}}}{\partial t}+{\textbf {u}}\varvec{\cdot }\varvec{\nabla }{} {\textbf {A}}-(\varvec{\nabla }{} {\textbf {u}})^{\textrm{T}}\varvec{\cdot }{} {\textbf {A}}-{\textbf {A}}\varvec{\cdot }(\varvec{\nabla }{} {\textbf {u}}). \end{aligned}$$
(2)

Furthermore, the left-hand side of Eq. (1) can be also expressed as

$$\begin{aligned} \frac{D{\textbf {A}}}{Dt}-(\varvec{\nabla }{} {\textbf {u}})^{\textrm{T}}\varvec{\cdot }{} {\textbf {A}}-{\textbf {A}}\varvec{\cdot }(\varvec{\nabla }{} {\textbf {u}})+\mathcal {E}\left( {\textbf {E}}\varvec{\cdot }{} {\textbf {A}}+{\textbf {A}}\varvec{\cdot }{} {\textbf {E}}\right){} & {} = \frac{D{\textbf {A}}}{Dt}+\varvec{\Omega }\varvec{\cdot }{} {\textbf {A}}-{\textbf {A}}\varvec{\cdot }\varvec{\Omega }\nonumber \\{} & {} \quad -\left( 1-\mathcal {E}\right) \left( {\textbf {E}}\varvec{\cdot }{} {\textbf {A}}+{\textbf {A}}\varvec{\cdot }{} {\textbf {E}}\right) ,\nonumber \\ \end{aligned}$$
(3)

where \(\varvec{\Omega }=\frac{1}{2}(\varvec{\nabla }{} {\textbf {u}}-(\varvec{\nabla }{} {\textbf {u}})^{\textrm{T}})\) is the vorticity tensor and \(D{\textbf {A}}/Dt+\varvec{\Omega }\varvec{\cdot }{} {\textbf {A}}-{\textbf {A}}\varvec{\cdot }\varvec{\Omega }\) is the (Jaumann) co-rotational derivative [46, 47], which represents the rate of change when advected with the flow and rotating with the vorticity. When \(\mathcal {E}\) is a positive constant, Eq. (3) corresponds to the Gordon–Schowalter convected time derivative \(\overset{\square }{{\textbf {A}}}\) (see, e.g., [12, 45, 48, 49]).

Table 1 A summary of different microstructurally inspired terms incorporated in the common elastic dumbbell models. The only model that accounts for (i) the finite polymer extensibility, (ii) conformation-dependent drag, and (iii) conformation-dependent non-affine deformation is the so-called FENE-PTML model [9]
Full size table

A wide class of different constitutive equations can be represented in the form of Eq. (1) [10, 15], including non-dumbbell models such as the Phan-Thien–Tanner model, derived from network theory [50, 51]. The elastic dumbbell models that can be represented by Eq. (1) include the Oldroyd-B model, the Johnson–Segalman/Gordon–Schowalter model [48, 52], the FENE-P [6, 8] and FENE-CR [17] models, and the FENE-P-CD and the FENE-CD models used by Leal and co-workers [35, 37,38,39] that are modifications of the FENE-P and FENE-CR models, respectively, to account for the conformation dependence of the bead friction coefficient, as first proposed by de Gennes [27] and Hinch [28].

In Table 1, we summarize the different microstructurally inspired terms and their explicit expressions that are incorporated in common elastic dumbbell models mentioned above. This table shows that the only constitutive equation that accounts for (i) the finite polymer extensibility, (ii) conformation-dependent drag, and (iii) conformation-dependent non-affine deformation is the so-called FENE-PTML model.

The function \(\mathcal {F}\) in Eq. (1) accounts for the finite extensibility of polymers represented by elastic dumbbells and is modeled using the Warner spring function [54], where L is the extensibility parameter defined as the polymer contour length normalized with the equilibrium coil size. Typically, models of ideal polymers have L proportional to \(\sqrt{N}\), where N is the number of statistical Kuhn segments that comprise the polymer chain.

Next, the function \(\mathcal {Z}\) in Eq. (1) incorporates the dependence of the friction coefficient (drag) on the conformation of the dumbbell. Importantly, the Oldroyd-B, Johnson–Segalman/Gordon–Schowalter, FENE-P, and FENE-CR models do not account for this conformation dependence and have a constant friction, i.e., \(\mathcal {Z}=1\) (\(\kappa =0\)). For \(\kappa =1\), \(\mathcal {Z}(\textrm{tr}({\textbf {A}}))\) takes the form proposed in [27] and [28], and for \(\kappa =0.02\) leads to a friction coefficient similar to the one proposed in [55]. Thus, we assume that \(0\le \kappa \le 1\). Note that \(\mathcal {Z}=1\) in equilibrium (when \({\textbf {A}}={\textbf {I}}\)).

When considering a dumbbell as a deformable particle of finite aspect ratio, the function \(\mathcal {\mathcal {E}}\) accounts for the inefficiency of rotation in extensional flow (e.g., Jeffery rotation [56]), allowing non-affine deformation by adding a slip between the velocity gradient experienced by the dumbbell and the gradient imposed by the continuum deformation. This effect is incorporated only in the Johnson–Segalman/Gordon–Schowalter and FENE-PTML models, while the latter model also accounts for the conformation dependence of this feature, i.e., \(\mathcal {E}(\textrm{tr}({\textbf {A}}))\). Note that \(\varepsilon _{0}\) (see Table 1) is a dimensionless positive parameter and \(\mathcal {E}=\varepsilon _{0}\) when \({\textbf {A}}={\textbf {I}}\).

We note that for \(L\gg 1\), the function \(\mathcal {G}\) is defined as \(\mathcal {G}(\textrm{tr}({\textbf {A}}))=1/(1-(3/L^{2}))\approx 1\) for all models except for FENE-CR and FENE-CD models, in which \(\mathcal {G}(\textrm{tr}({\textbf {A}}))=\mathcal {F}(\textrm{tr}({\textbf {A}}))=1/[1-(\textrm{tr}({\textbf {A}})/L^{2})]\). It is known that \(\mathcal {G}\ne \mathcal {F}\) results in a shear-thinning effect observed in a simple shear flow for the FENE-P model above a certain threshold value of shear rate, as shown in Fig. 2(a) below (see also [6, 57]).

The elastic dumbbell model aims to describe the behavior of viscoelastic dilute polymer solutions for which the deviatoric stress tensor \(\varvec{\tau }=\varvec{\tau }_{s}+\varvec{\tau }_{p}\) is the sum of the solvent (\(\varvec{\tau }_{s}\)) and the polymer (\(\varvec{\tau }_{p}\)) contributions. The solvent is usually assumed to behave as a Newtonian fluid with a constant viscosity \(\eta _{s}\), so that \(\varvec{\tau }_{s}=2\eta _{s}{} {\textbf {E}}\). The response of polymer chains to fluid flow gives rise to the stress distribution, \(\varvec{\tau }_{p}\), which is related to the deformation of the microstructure (conformation tensor) \({\textbf {A}}\) through the constitutive equation

$$\begin{aligned} \varvec{\tau }_{p}=\frac{\eta _{p}}{\lambda }[\mathcal {F}(\textrm{tr}({\textbf {A}})){\textbf {A}}-\mathcal {G}(\textrm{tr}({\textbf {A}})){\textbf {I}}] \quad \text {for Oldroyd-B and FENE models}, \end{aligned}$$
(4)

where \(\eta _{p}\) is the polymer contribution to the shear viscosity at zero shear rate [9]. For the Johnson–Segalman/Gordon–Schowalter model, the relation between \(\varvec{\tau }_{p} \) and \({\textbf {A}}\) is slightly different and is given by \(\varvec{\tau }_{p}=\eta _{p}/((1-\varepsilon _0)\lambda )({\textbf {A}}-{\textbf {I}})\) [58].

In the following sections, we provide expressions for the extension of the conformation tensor, as measured by \(\textrm{tr}({\textbf {A}})\), in simple shear and planar extensional flows, along with the corresponding shear and extensional viscosities for different constitutive equations. Then, we discuss the ability of these continuum-level dumbbell-like models to reproduce the experimental results of dilute polymer solutions, highlighting the importance of the inclusion of the microstructurally inspired terms summarized in Table 1.

3 Dumbbell extension in shear and extensional flow

In this section, we provide the closed-form (implicit) expressions for the steady-state dumbbell extension as a function of the Weissenberg number for simple shear and planar extensional flows.

3.1 Simple shear flow

For a simple shear flow \({\textbf {u}}=(u_{x},u_{y},u_{z}) =(\dot{\gamma }y,0,0)\), where \(\dot{\gamma }\) is the constant shear rate, Eq. (1), understood in a Lagrangian sense following a material point, yields for the components of \({\textbf {A}}\)

$$\begin{aligned}{} & {} \hbox {{ xx}-component:}\qquad \textrm{Wi}\left( \mathcal {E}-2\right) A_{xy}=-\frac{1}{\mathcal {Z}}(\mathcal {F}A_{xx}-\mathcal {G}), \end{aligned}$$
(5a)
$$\begin{aligned}{} & {} \hbox {{ yy}-component:} \qquad \textrm{Wi} ~\mathcal {E}A_{xy}=-\frac{1}{\mathcal {Z}}(\mathcal {F}A_{yy}-\mathcal {G}), \end{aligned}$$
(5b)
$$\begin{aligned}{} & {} \hbox {{ xy}-component:} \qquad \textrm{Wi}\left[ -A_{yy}+\frac{\mathcal {E}}{2}(A_{xx}+A_{yy})\right] =-\frac{1}{\mathcal {Z}}\mathcal {F}A_{xy}, \end{aligned}$$
(5c)
$$\begin{aligned}{} & {} \hbox {{ zz}-component:} \qquad A_{zz}=\frac{\mathcal {G}}{\mathcal {F}},\end{aligned}$$
(5d)
$$\begin{aligned}{} & {} \hbox {{ xz}- and { yz}-components:}\qquad A_{xz}= A_{yz}=0, \end{aligned}$$
(5e)

where \(\textrm{Wi}=\lambda \dot{\gamma }\) is the Weissenberg number. Recall, as summarized in Table 1, that \(\mathcal {E}\), \(\mathcal {F}\), \(\mathcal {G}\), and \(\mathcal {Z}\) are functions of \(\textrm{tr}({\textbf {A}})=A_{xx} + A_{yy} + A_{zz}\). The solution of Eqs. (5) is

$$\begin{aligned} A_{xx}= & {} \frac{\mathcal {G}\left( \mathcal {F}^{2}+(2-\mathcal {E})\mathcal {Z}^{2}\textrm{Wi}^{2}\right) }{\mathcal {F}\left( \mathcal {F}^{2}+\mathcal {E}(2-\mathcal {E})\mathcal {Z}^{2}\textrm{Wi}^{2}\right) }, \end{aligned}$$
(6a)
$$\begin{aligned} A_{yy}= & {} \frac{\mathcal {G}\left( \mathcal {F}^{2}+\mathcal {E}\mathcal {Z}^{2}\textrm{Wi}^{2}\right) }{\mathcal {F}\left( \mathcal {F}^{2}+\mathcal {E}(2-\mathcal {E})\mathcal {Z}^{2}\textrm{Wi}^{2}\right) }, \end{aligned}$$
(6b)
$$\begin{aligned} A_{xy}= & {} \frac{\mathcal{G}\mathcal{Z}(1-\mathcal {E})\textrm{Wi}}{\mathcal {F}^{2}+\mathcal {E}(2-\mathcal {E})\mathcal {Z}^{2}\textrm{Wi}^{2}}. \end{aligned}$$
(6c)

Using Eqs. (5e)−(6b), we obtain an equation for the square of the dumbbell extension, \(\textrm{tr}({\textbf {A}})=A_{xx}+A_{yy}+A_{zz}\), for a shear flow,

$$\begin{aligned} \textrm{tr}({\textbf {A}})=\frac{\mathcal {G}\left( 3\mathcal {F}+(2+\mathcal {E}(2-\mathcal {E}))\mathcal {Z}^{2}\textrm{Wi}^{2}\right) }{\mathcal {F}\left( \mathcal {F}^{2}+\mathcal {E}(2-\mathcal {E})\mathcal {Z}^{2}\textrm{Wi}^{2}\right) }, \end{aligned}$$
(7)

which can be rearranged as

$$\begin{aligned} \textrm{Wi}=\frac{\mathcal {F}}{\mathcal {Z}}\frac{\sqrt{\mathcal {F}\textrm{tr}({\textbf {A}})-3\mathcal {G}}}{\sqrt{\mathcal {G}(2+(2-\mathcal {E})\mathcal {E})-\mathcal{F}\mathcal{E}(2-\mathcal {E})\textrm{tr}({\textbf {A}})}}. \end{aligned}$$
(8)

Rather than expressing the dumbbell extension as a function of \(\textrm{Wi}\), we find it convenient for analysis and illustration (see Fig. 1) to express \(\textrm{Wi}\) in terms of \(\sqrt{\textrm{tr}({\textbf {A}})}\) as in Eq. (8). With this result, we summarize in Table 2 the explicit or implicit expressions for the steady-state extension as a function of \(\textrm{Wi}\) in a shear flow for the considered constitutive models.

Table 2 A summary of the explicit or implicit expressions for the steady-state extension, as measured by \(\textrm{tr}({\textbf {A}})\), as a function of \(\textrm{Wi}\) in a simple shear flow for each of the considered constitutive models
Full size table

3.2 Planar extensional flow

For a planar extension flow \({\textbf {u}}=(u_{x},u_{y},u_{z})=(\dot{\varepsilon }x,-\dot{\varepsilon }y,0)\), where \(\dot{\varepsilon }\) is the constant extensional rate, Eq. (1) yields

$$\begin{aligned}{} & {} \hbox {{ xx}-component:}\qquad 2\textrm{Wi}\left( \mathcal {E}-1\right) A_{xx}=-\frac{\mathcal {F}}{\mathcal {Z}}(A_{xx}-1), \end{aligned}$$
(9a)
$$\begin{aligned}{} & {} \hbox {{ yy}-component:}\qquad 2\textrm{Wi}\left( \mathcal {E}-1\right) A_{yy}=-\frac{\mathcal {F}}{\mathcal {Z}}(A_{yy}-1), \end{aligned}$$
(9b)
$$\begin{aligned}{} & {} \hbox {{ zz}-component:}\qquad A_{zz}=\frac{\mathcal {G}}{\mathcal {F}},\end{aligned}$$
(9c)
$$\begin{aligned}{} & {} \hbox {{ xy}-, { xz}-, and { yz}-components:}\qquad A_{xy}=A_{xz}=A_{yz}=0, \end{aligned}$$
(9d)

where \(\textrm{Wi}=\lambda \dot{\varepsilon }\) is the Weissenberg number. From Eqs. (9a)−(9b), it follows that \(A_{xx}\) and \(A_{yy}\) can be expressed as

$$\begin{aligned} A_{xx}=\frac{\mathcal {G}}{\mathcal {F}-2\textrm{Wi}(1-\mathcal {E})\mathcal {Z}},\qquad A_{yy}=\frac{\mathcal {G}}{\mathcal {F}+2\textrm{Wi}(1-\mathcal {E})\mathcal {Z}}. \end{aligned}$$
(10)

Combining Eqs. (9d) and (10) provides an equation for the square of the dumbbell extension for a planar extensional flow

$$\begin{aligned} \textrm{tr}({\textbf {A}})=\frac{2\mathcal{F}\mathcal{G}}{\mathcal {F}^{2}-4\mathcal {Z}^{2}(\mathcal {E}-1)^{2}\textrm{Wi}^{2}}+\frac{\mathcal {G}}{\mathcal {F}}, \end{aligned}$$
(11)

which can be rearranged as

$$\begin{aligned} \textrm{Wi}=\frac{\mathcal {F}}{2\mathcal {Z}}\frac{\sqrt{\mathcal {F}\textrm{tr}({\textbf {A}})-3\mathcal {G}}}{\sqrt{(1-\mathcal {E})^{2}(\mathcal {F}\textrm{tr}({\textbf {A}})-\mathcal {G}})}, \end{aligned}$$
(12)

representing an implicit equation for the dumbbell extension, \(\sqrt{\textrm{tr}({\textbf {A}})}\), as a function of \(\textrm{Wi}\) in a planar extensional flow. We summarize in Table 3 the explicit or implicit expressions for the steady-state extension as a function of \(\textrm{Wi}\) in a planar extensional flow for the considered constitutive models.

Fig. 1
figure 1

Steady fractional extension in (a) shear flow and (b) planar extensional flow for different elastic dumbbell models. All calculations were performed using \(\kappa =1\), \(\varepsilon _0=0.1\), and \(L^2=1000\) (see Table 1)

Full size image

3.3 Fractional extension for different continuum elastic dumbbell models

In this section, we compare and contrast the different models. We present in Fig. 1(a,b) the steady fractional extension \(\sqrt{\textrm{tr}({\textbf {A}})}/L\) as a function of \(\textrm{Wi}\) in (a) shear flow and (b) planar extensional flow for different elastic dumbbell models. When the Weissenberg number is low, all models behave similarly, but significant differences arise when viscoelastic effects become apparent. It is evident from Fig. 1(a) that the Oldroyd-B model (cyan dotted line) allows a non-physical infinite dumbbell extension in shear flow. Furthermore, as expected, the FENE-CR (black solid line) shows behavior similar to the FENE-P model (gray dots), whereas the FENE-CD (black dashed line) exhibits behavior similar to the FENE-P-CD model (red solid line). More importantly, all FENE models, except the FENE-PTML model, predict a full dumbbell extension in shear flow at high Weissenberg numbers, similar to extensional flows. However, a full dumbbell extension in steady shear flow is unrealistic [9], and experiments with a single polymer show only a partial extension [2]. We observe that only the Johnson–Segalman/Gordon–Schowalter (JS/GS; gray dashed line) and FENE-PTML (purple triangles) models, which account for the inefficiency of rotation and allow non-affine deformation, can reproduce a partial dumbbell extension at high Weissenberg numbers, consistent with experimental observations.

Table 3 A summary of the implicit or explicit expressions for the steady-state extension, as measured by \(\textrm{tr}({\textbf {A}})\), as a function of \(\textrm{Wi}\) in a planar extensional flow for each of the considered constitutive models
Full size table

For a planar extensional flow, it follows from Fig. 1(b) that both Oldroyd-B and Johnson–Segalman/Gordon–Schowalter models diverge at \(\textrm{Wi}=1/2\) and \(\textrm{Wi}=1/2(1-\varepsilon _0)\), respectively. The FENE-CR (black solid line) and FENE-P (gray dots) models predict a rapid but continuous transition to a nearly full extension at \(\textrm{Wi}\approx 1/2\) without hysteresis. These models assume a constant friction, i.e., \(\mathcal {Z}=1\), and thus cannot reproduce the coil-stretch transition. On the other hand, we observe that the FENE-CD (black dashed line), FENE-P-CD (red solid line), and FENE-PTML (purple solid line) models, which account for the conformation-dependent drag, reproduce the S-shaped extension curve and the hysteresis in the coil-stretch transition, as was first proposed by de Gennes [27] and Hinch [28], and then confirmed experimentally by Schroeder et al. [3]. As pointed out by Fuller and Leal [35], the details of the coil-stretch transition depend on the value of L. For sufficiently small values of L (when \(\kappa =1\), \(L^2\) should be smaller than O(30)), the hysteresis loop disappears, resulting in a rapid but continuous transition to a nearly full extension [35, 38], similar to predictions of the FENE-CR and FENE-P models. Experiments also show that the onset of hysteresis strongly depends on the contour length of the polymer chain, i.e., L. For example, while single-polymer experiments using \(\lambda \)-DNA with a contour length of 21 \(\mu \)m (\(N\approx 150\) Kuhn segments, \(L^{2}=3(N-1)\approx 450\) (see, e.g., [8, p.76])) show a nearly full extension at high extension rates without hysteresis [1], experiments using the E. coli DNA fragments with contour length of 1.3 mm (\(N\approx 9250\) Kuhn segments, \(L^{2}=3(N-1)\approx 27700\) [8, p.76]) clearly demonstrate the hysteresis and the coil-stretch transition [3, 4].

We note that while the Johnson–Segalman/Gordon–Schowalter model (gray dashed line) can reproduce a partial dumbbell extension at high Weissenberg numbers for shear flow, consistent with experimental observations, it diverges at high Weissenberg numbers for extensional flow. Thus, only the FENE-PTML model (purple triangles and dashed line) which accounts for (i) the finite polymer extensibility, (ii) conformation-dependent drag, and (iii) conformation-dependent non-affine deformation can reproduce a steady-state extension at high Weissenberg numbers for both shear and extensional flows, consistent with experimental results. We conclude that the inclusion of all three microstructurally inspired terms in a constitutive equation is required to reproduce the experimentally observed polymer extension in shear and extensional flows at non-small Weissenberg numbers. Furthermore, we believe these three terms can help accurately model viscoelastic channel flows with mixed kinematics, i.e., a combination of shear and extensional components.

4 Shear and extensional viscosities

In this section, we provide the expressions for the shear and extensional viscosities for different elastic dumbbell models using the relations for the variation of the conformation tensor \({\textbf {A}}\) with \(\textrm{Wi}\) obtained in Sec. 3.

4.1 Shear viscosity

For a simple shear flow, the shear viscosity \(\eta _{\textrm{shear}}\) is defined as

$$\begin{aligned} \eta _{\textrm{shear}}=\frac{\tau _{xy}}{\dot{\gamma }}, \end{aligned}$$
(13)

where \(\tau _{xy}=\eta _{s}\dot{\gamma }+\tau _{p,xy}\) and \(\dot{\gamma }\) is the constant shear rate. Using the latter relation together with Eq. (4) yields

$$\begin{aligned} \eta _{\textrm{shear}}=\frac{\tau _{xy}}{\dot{\gamma }}=\eta _{s}+\frac{\tau _{p,xy}}{\dot{\gamma }}=\eta _{s}+\eta _{p}\frac{\mathcal {F}(\textrm{tr}({\textbf {A}}))}{\textrm{Wi}}A_{xy}. \end{aligned}$$
(14)

Substituting Eq. (6c) into Eq. (14), we obtain

$$\begin{aligned} \eta _{\textrm{shear}}=\eta _{s}+\eta _{p}\frac{\mathcal {F}\mathcal{G}\mathcal{Z}(1-\mathcal {E})}{\mathcal {F}^{2}+\mathcal {E}(2-\mathcal {E})\mathcal {Z}^{2}\textrm{Wi}^{2}}, \end{aligned}$$
(15)

where \(\mathcal {E}\), \(\mathcal {F}\), \(\mathcal {G}\), and \(\mathcal {Z}\) are functions of \(\textrm{tr}({\textbf {A}})\). Using Eq. (8), Eq. (15) can be expressed as

$$\begin{aligned} \eta _{\textrm{shear}}=\eta _{s}+\eta _{p}\frac{\mathcal {Z}(\mathcal {G}(2+(2-\mathcal {E})\mathcal {E})-\mathcal{F}\mathcal{E}(2-\mathcal {E})\textrm{tr}({\textbf {A}}))}{2\mathcal {F}(1-\mathcal {E})}. \end{aligned}$$
(16)

Equations (15) and (16) can be rearranged to yield the polymer contribution to the shear viscosity scaled by its contribution at zero shear rate,

$$\begin{aligned} \frac{\eta _{\textrm{shear}}-\eta _{s}}{\eta _{\textrm{0}}-\eta _{s}}=\frac{\mathcal {F}\mathcal{G}\mathcal{Z}(1-\mathcal {E})}{\mathcal {F}^{2}+\mathcal {E}(2-\mathcal {E})\mathcal {Z}^{2}\textrm{Wi}^{2}}=\frac{\mathcal {Z}(\mathcal {G}(2+(2-\mathcal {E})\mathcal {E})-\mathcal{F}\mathcal{E}(2-\mathcal {E})\textrm{tr}({\textbf {A}}))}{2\mathcal {F}(1-\mathcal {E})}, \end{aligned}$$
(17)

where \(\eta _{\textrm{0}}\) is the zero-shear-rate viscosity, \(\eta _{\textrm{0}}=\eta _{s}+\eta _{p}\).

We note that Eq. (17) does not hold for the Johnson–Segalman/Gordon–Schowalter model since this model has a slightly different relation between the polymer stress tensor \(\varvec{\tau }_{p}\) and the conformation tensor \({\textbf {A}}\), i.e., \(\varvec{\tau }_{p}=\eta _{p}/((1-\varepsilon _0)\lambda )({\textbf {A}}-{\textbf {I}})\). Using the latter expression and following similar steps, we find that the scaled polymer contribution to the shear viscosity for the Johnson–Segalman/Gordon–Schowalter model is [45, 48]

$$\begin{aligned} \frac{\eta _{\textrm{shear}}-\eta _{s}}{\eta _{\textrm{0}}-\eta _{s}}=\frac{1}{1+\varepsilon _0(2-\varepsilon _0)\textrm{Wi}^{2}} . \end{aligned}$$
(18)

While in this work we discuss only the scaled polymer contribution to the shear viscosity, it is well known that the Johnson–Segalman/Gordon–Schowalter model predicts a non-monotonic shear stress versus shear rate curve, \(\tau _{xy}(\dot{\gamma })\), showing shear banding unless the ratio of solvent to polymer zero-shear-rate viscosity is sufficiently large [59,60,61]. Specifically, for \(\eta _s/\eta _p<1/8\), the curve \(\tau _{xy}(\dot{\gamma })\) exhibits a maximum and a minimum, as shown in Fig. 1 of Renardy [59] and Espanol et al. [60].

Table 4 presents a summary of the expressions for the scaled polymer contribution to the shear viscosity, \((\eta _{\textrm{shear}}-\eta _{s})/(\eta _{\textrm{0}}-\eta _{s})\), in a general case and in the high-\(\textrm{Wi}\) asymptotic limit for each of the considered constitutive models.

Table 4 A summary of the expressions for the scaled polymer contribution to the shear viscosity, \((\eta _{\textrm{shear}}-\eta _{s})/(\eta _{0}-\eta _{s})\), in a general case and in the high-\(\textrm{Wi}\) asymptotic limit for each of the considered constitutive models
Full size table

4.2 Extensional viscosity

Our presentation here follows the basic steps of the previous subsection. For a two-dimensional extensional flow, the extensional viscosity \(\eta _{\textrm{ext}}\) is defined as

$$\begin{aligned} \eta _{\textrm{ext}}=\frac{\tau _{xx}-\tau _{yy}}{\dot{\varepsilon }}, \end{aligned}$$
(19)

where \(\dot{\varepsilon }\) is the constant extension rate. Using the relation \(\tau _{xx}-\tau _{yy}=4\eta _{s}\dot{\varepsilon }+\tau _{p,xx}-\tau _{p,yy}\) and Eq. (4), Eq. (19) takes the form

$$\begin{aligned} \eta _{\textrm{ext}}=\frac{\tau _{xx}-\tau _{yy}}{\dot{\varepsilon }}=4\eta _{s}+\frac{\tau _{p,xx}-\tau _{p,yy}}{\dot{\varepsilon }}=4\eta _{s}+\eta _{p}\frac{\mathcal {F}(\textrm{tr}({\textbf {A}}))}{\textrm{Wi}}(A_{xx}-A_{yy}). \end{aligned}$$
(20)

Using Eqs. (10) and (11), the difference \(A_{xx}-A_{yy}\) can be expressed as

$$\begin{aligned} A_{xx}-A_{yy}= & {} \frac{4\textrm{Wi}\mathcal {F}(\textrm{tr}({\textbf {A}}))\mathcal {Z}(\textrm{tr}({\textbf {A}}))(1-\mathcal {E}(\textrm{tr}({\textbf {A}})))}{\mathcal {F}(\textrm{tr}({\textbf {A}}))^{2}-4\mathcal {Z}(\textrm{tr}({\textbf {A}}))^{2}(\mathcal {E}(\textrm{tr}({\textbf {A}}))-1)^{2}\textrm{Wi}^{2}}\nonumber \\= & {} \frac{2\textrm{Wi}\mathcal {Z}(\textrm{tr}({\textbf {A}}))(1-\mathcal {E}(\textrm{tr}({\textbf {A}})))}{\mathcal {F}(\textrm{tr}({\textbf {A}}))}(\textrm{tr}({\textbf {A}})-1). \end{aligned}$$
(21)

Substituting Eq. (21) into Eq. (20), we obtain

$$\begin{aligned} \eta _{\textrm{ext}}=4\eta _{s}+2\eta _{p}\mathcal {Z}(\textrm{tr}({\textbf {A}}))\left[ 1-\mathcal {E}(\textrm{tr}({\textbf {A}}))\right] \left[ \textrm{tr}({\textbf {A}})-\frac{\mathcal {G}(\textrm{tr}({\textbf {A}}))}{\mathcal {F}(\textrm{tr}({\textbf {A}}))}\right] . \end{aligned}$$
(22)

Equation (22) can be rearranged to yield the polymer contribution to the extensional viscosity scaled by its contribution at zero strain rate,

$$\begin{aligned} \frac{\eta _{\textrm{ext}}-4\eta _{s}}{4(\eta _{0}-\eta _{s})}=\frac{1}{2}\mathcal {Z}(\textrm{tr}({\textbf {A}}))\left[ 1-\mathcal {E}(\textrm{tr}({\textbf {A}}))\right] \left[ \textrm{tr}({\textbf {A}})-\frac{\mathcal {G}(\textrm{tr}({\textbf {A}}))}{\mathcal {F}(\textrm{tr}({\textbf {A}}))}\right] . \end{aligned}$$
(23)

Similar to the case of shear viscosity, Eq. (23) does not apply to the Johnson–Segalman/Gordon–Schowalter model, for which the scaled polymer contribution to the extensional viscosity is given as

$$\begin{aligned} \frac{\eta _{\textrm{ext}}-4\eta _{s}}{4(\eta _{0}-\eta _{s})}=\frac{1}{1-4(\varepsilon _0-1)^{2}\textrm{Wi}^{2}}. \end{aligned}$$
(24)

Table 5 presents a summary of the expressions for the scaled polymer contribution to the extensional viscosity, \((\eta _{\textrm{ext}}-4\eta _{s})/(4\eta _{0}-4\eta _{s})\), in a general case and in the high-\(\textrm{Wi}\) asymptotic limit for each of the considered constitutive models.

Table 5 A summary of the expressions for the scaled polymer contribution to the extensional viscosity, \((\eta _{\textrm{ext}}-4\eta _{s})/(4\eta _{0}-4\eta _{s})\), in a general case and in the high-\(\textrm{Wi}\) asymptotic limit for each of the considered constitutive models
Full size table

4.3 Shear and extensional viscosities for different continuum elastic dumbbell models

First, we present the scaled polymer contribution to the shear viscosity, \((\eta _{\textrm{shear}}-\eta _{s})/(\eta _{\textrm{0}}-\eta _{s})\), as a function of \(\textrm{Wi}\) for different elastic dumbbell models, as displayed in Fig. 2(a). As expected, both Oldroyd-B (cyan dotted line) and FENE-CR (black solid line) models predict a constant shear viscosity, whereas the FENE-P model (gray dots) indicates a shear-thinning behavior at high Weissenberg numbers and scales as \(\textrm{Wi}^{-2/3}\) (see Table 4), consistent with previous studies (see, e.g., [6, 57]). The Johnson–Segalman/Gordon–Schowalter model (gray dashed line) also predicts shear thinning but with a different scaling of \(\textrm{Wi}^{-2}\) for \(\textrm{Wi}\gg 1\) [45]. In contrast, the FENE-P-CD (red crosses) and FENE-CD (black circles) models, which account for the conformation-dependent drag but no strain inefficiency in rotation, exhibit unrealistic shear thickening at moderate and high shear rates, respectively, and thus contradict experimental observations [9]. Notably, when accounting for the inefficiency of rotation and allowing for non-affine deformation, the FENE-PTML model (purple triangles) resolves this unrealistic increase in viscosity and predicts a shear-thinning behavior with scaling of \(\textrm{Wi}^{-2}\) at high Weissenberg numbers, similar to the Johnson–Segalman/Gordon–Schowalter model.

Fig. 2
figure 2

(a) Scaled polymer contribution to the shear viscosity and (b) scaled polymer contribution to the extensional viscosity for different elastic dumbbell models. All calculations were performed using \(\kappa =1\), \(\varepsilon _0=0.1\), and \(L^2=1000\)

Full size image

Next, we present the scaled polymer contribution to the extensional viscosity, \((\eta _{\textrm{ext}}-4\eta _{s})/(4\eta _{0}-4\eta _{s})\), as a function of \(\textrm{Wi}\) for different elastic dumbbell models, as shown in Fig. 2(b). For all models, we observe that the scaled extensional viscosity exhibits qualitatively similar behavior to the fractional extension in a planar extensional flow, shown in Fig. 1(b). In particular, both Oldroyd-B and Johnson–Segalman/Gordon–Schowalter models diverge at the same values of the Weissenberg number. The FENE-CR (black solid line) and FENE-P (gray dots) models predict a rapid but continuous increase in the extensional viscosity at \(\textrm{Wi}\approx 1/2\) to the asymptotic value of \(L^2/2=500\) (see Table 5). Similar to the case of dumbbell extension, the FENE-CD (black dashed line), FENE-P-CD (red solid line), and FENE-PTML (purple solid line) models exhibit hysteresis in the extensional viscosity. Furthermore, models with conformation-dependent drag have a much larger polymer stress, scaling as \(L^3\), compared to the FENE-CR and FENE-P models with a constant friction coefficient, where polymer stress scales as \(L^2\) [38]. As a result, the extensional viscosity for these models also scales as \(L^3\), approaching the asymptotic value of \(L^3/\sqrt{12}\approx 9129\) for \(\textrm{Wi}\gg 1\) and \(\kappa =1\), represented in Fig. 2(b) by a light gray dashed-dot line.

We note that measuring the steady extensional viscosity is challenging compared to the steady shear viscosity [62]. As pointed out by Petrie [63], the extensional viscosity is defined for steady, spatially uniform flow, but in practical applications, flow is rarely steady and spatially uniform. Indeed, most studies to date have reported the transient extensional viscosity as a function of time t or strain \(\varepsilon =\dot{\varepsilon }t\) [64,65,66,67,68]. Thus, performing the theory/experiments comparison for a steady extensional viscosity is difficult due to the lack of reliable experimental data.

5 Concluding remarks

Elastic dumbbell models such as Oldroyd-B and FENE-P are commonly used in the fluid mechanics and rheology communities for a continuum description of viscoelastic fluid flows. However, these models fail to accurately predict some characteristics of realistic flows of polymer solutions even in simple cases, e.g., the steady extension in shear and extensional flows. Here, we considered seven different elastic dumbbell models that include various microstructurally inspired terms. We showed that only the FENE-PTML model which accounts for (i) the finite polymer extensibility, (ii) conformation-dependent drag, and (iii) conformation-dependent non-affine deformation can reproduce a steady-state polymer extension at high Weissenberg numbers for both shear and extensional flows, consistent with experimental observations.

In addition to the elastic dumbbell models considered in this work, many other FENE-type models have been proposed over the years. We believe that some readers may wish to extend our approach to other FENE-type models available in the literature, such as the FENE-L [69, 70], FENE-QE [71], FENE-YDL [72], and FENE-D [73] models.

In this work, we have focused on the steady-state conformation extension and the corresponding shear and extensional viscosities. Therefore, as a future research direction, it would be interesting to analyze and compare the response of different elastic dumbbell models in unsteady shear and extensional flows. Furthermore, while in this work we examined the viscometric properties and steady-state conformation extension with respect to existing experimental measurements, a reader may wish to compare the predictions of different models for other flow characteristics, such as the configuration dissipation function and energy dissipation [74]. Another interesting extension of the present work, which considered simple shear and extensional flows, is to use the FENE-PTML model that incorporates three microstructurally inspired terms discussed herein for studying more complex flows with mixed kinematics, e.g., viscoelastic channel flows. These flows are dominated by shear flow with a small extensional component, and we thus believe that including all three microstructurally inspired terms in a constitutive equation may be required for accurate modeling of such flows.