Kinematic and rheological equivalence of steady shearing and planar extensional flows

The nonlinear canonical rheology of complex fluids is typically characterized in homogeneous flow fields such as simple shear and pure extension. These two flows probe different aspects of the microstructure through different stretch histories, and in general, one cannot be predicted from the other [1, 2]. Although shear flows are routinely employed in a torsional rheometer to obtain material functions such as the shear viscosity and the first normal stress coefficient, steady extensional flows have been more difficult to achieve [3]. Nevertheless, extensional flows are essential for understanding many polymer processing operations [4], as well as filament breakup [5], fracture [6] and jetting phenomena [7]. Furthermore, in flows with mixed kinematics, the extensional component will often dominate at high strain rates [8].

It is commonly assumed that planar elongation probes nonlinear material responses that are independent of those probed by shearing deformations [9]. However, generating true planar extensional flows has posed a major challenge experimentally, especially compared to implementing uniaxial or biaxial extensional deformations. Various devices have been developed for generating uniaxial elongation, though these deformations are typically transient and inhomogeneous. Examples include the use of converging flows [9] or the rotating clamp Meissner apparatus [10] for viscous polymer melts, and filament stretching configurations [5] such as Dripping onto Substrate [11], and jetting devices such as ROJER [7] for less viscous materials. The only bespoke devices able to generate planar extensional flow for dilute polymer solutions are the OSCER microfluidic device [12] and a modified filament stretching device [13]. The difficulties in generating planar extensional flow, especially at sufficiently high extension rates to achieve large microstructural deformations, have led to few published studies.

More broadly, flow fields can be classified kinematically using invariants of frame-invariant tensors such as the strain rate $\dot{\boldsymbol{\gamma}}$ and the Finger strain $\mathbf{B}$. Because of fluid incompressibility, the first invariant of $\dot{\boldsymbol{\gamma}}$ is zero, and the third invariant of $\mathbf{B}$ is always unity. The remaining two invariants for each tensor thus form two separate maps of all physically realizable flows. Such maps were first put forth by Sawyers [14] in terms of the Finger strain tensor, and have been popularized by Chong, Perry and Cantwell [15] for classifying turbulent flow fields in terms of the strain rate tensor. Both representations are featured in the classic textbook by Bird et al. [16] and are summarized in the SI. These maps are bounded by uniaxial and biaxial extensional flows, but interestingly, simple shear flow and planar extension reside on the same invariant lines and are thus kinematically indistinguishable. This raises a more fundamental question about whether a mapping exists between steady shear material functions and the steady planar extensional viscosity. In this Letter, we show that for any materially objective constitutive equation, the steady planar extensional viscosity can be completely determined from the steady shear viscosity and the first normal stress difference. This relationship reveals that the two flows, although ostensibly distinct, are kinematically equivalent. Our key development is to systematically remove the contribution of rotation from the shearing flow, by defining an “effective extension rate” to establish this equivalence. This establishes an accessible route to infer steady planar extensional behavior from steady shear measurements alone, and opens avenues for understanding the physics inherent in a complex fluid subjected to two flows previously thought to be distinct.

In a recent study, Hillebrand et al. [17] have shown that suspensions of elastic discs and solutions of long flexible polymers, described respectively by the Oldroyd-A and Oldroyd-B models, produce identical velocity fields in any planar 2D flow. Our finding builds on this development, since both constitutive models generate identical shear stress and first normal stress differences in shear flow, and we show more broadly that the rheological material functions defined in planar extensional flows are kinematically connected to those in steady shear flow, once the rate of rotation has been accounted for.

By symmetry arguments, the extra stress tensor $\boldsymbol{\sigma}$ in any planar flow (i.e. including simple shear and planar extensional flow) has the following form:

Meaningful material functions are always defined in terms of the system’s response (i.e. the output) divided by a measure of the imposed deformation history (input) [18]. For steady shear, we define 1 as the flow direction and 2 as the velocity gradient direction (Figure 1a). The only non-zero component of the velocity field is $v_1 = \dot{\gamma}x_2$. Hence, the shear viscosity is defined as $\eta(\dot{\gamma}) \equiv\sigma_{12}/\dot{\gamma}$, and rheologists also measure the first normal stress difference $N_1 = \sigma_{11} - \sigma_{22}$ and the second normal stress difference $N_2 = \sigma_{22} - \sigma_{33}$. For steady homogeneous planar extensional flow, we define 1 as the extension direction and 2 as the compression direction (Figure 1b). The velocity field has two non-zero components $v_1 = \dot{\epsilon}x_1$ and $v_2 = -\dot{\epsilon}x_2$. The steady planar extensional viscosity is defined by $\eta_{P1}(\dot{\epsilon}) \equiv(\sigma_{11} - \sigma_{22})/\dot{\epsilon}$. For completeness, we also define a second planar extensional viscosity $\eta_{P2}(\dot{\epsilon}) \equiv(\sigma_{33} - \sigma_{22})/\dot{\epsilon}$. Note that from symmetry arguments based on reversing the flow direction (such that $\dot{\gamma} \to-\dot{\gamma}$ or $\dot{\epsilon} \to-\dot{\epsilon}$), we expect that the normal stress difference $N_1$ in steady shear is an even function of the applied shear rate, $(\sigma_{11} - \sigma_{22}) \sim\dot{\gamma}^{2}$, whereas the corresponding quantity in planar extensional flow is an odd function of the imposed extension rate, i.e. $(\sigma_{11} - \sigma_{22}) \sim\dot{\epsilon}$ [16].

The extra stress tensor for any planar flow can also be expressed in terms of three principal stresses $\sigma_1$, $\sigma_2$ and $\sigma_3$ when rotated into its principal axes:

In planar extensional flow, the stress tensor is already diagonal since $\sigma_{12} = 0$, hence $\sigma_1 = \sigma_{11}$, $\sigma_2 = \sigma_{22}$ and $\sigma_3 = \sigma_{33}$. It is frequently described by rheologists as a ‘shear-free’ flow [16], but is also referred to as ‘pure shear’ by solid mechanicians [19]. In simple shear flow, the eigenvalues of $\boldsymbol{\sigma}$ are [20]:

When shearing deformations are expressed in this coordinate frame, we can define a principal stress difference $\Delta\sigma\equiv(\sigma_1-\sigma_2) = \sqrt{N_1^2 + 4\sigma_{12}^2}$ (see Figure 1c). As the shear flow strength increases, the microstructure becomes increasingly aligned with the 1 direction, and this can be quantified by an orientation angle $\chi= \frac{1}{2}\tan^{-1}(2\sigma_{12}/N_1)$ that decreases from $45^\circ$ towards $0^\circ$ [20].

The question is thus: can we define a composite viscosity function that maps the material responses determined in shear and is consistent with the true planar extensional viscosity $\eta_{P1} = (\sigma_{11} - \sigma_{22})/\dot{\epsilon}$ defined in Figure 1?

To answer this question, we define an effective extension rate $\dot{\epsilon}_{\mathrm{eff}}$ by projecting the imposed deformation rate field onto the evolving principal axes of the resulting stress tensor. For general deformations, the symmetric strain rate tensor is defined as $\dot{\boldsymbol{\gamma}} = \nabla\mathbf{v} + \nabla\mathbf{v}^{T}$, in which $\nabla\mathbf{v}$ is the velocity gradient tensor. In planar extensional flow, this tensor is diagonal and can be expressed compactly as $\dot{\boldsymbol{\gamma}} = 2\dot{\epsilon}\mathbf{e}_1\mathbf{e}_1 - 2\dot{\epsilon}\mathbf{e}_2\mathbf{e}_2$. Therefore, the eigenvectors of the strain rate tensor and the stress tensor (cf. Eq. 2) are always coincident and fixed with time. On the other hand, the strain rate tensor in simple shear is $\dot{\boldsymbol{\gamma}} = \dot{\gamma}\mathbf{e}_1\mathbf{e}_2 + \dot{\gamma}\mathbf{e}_2\mathbf{e}_1$. As we show schematically in Figure 1c, the eigenvectors of the strain rate tensor are always fixed in time (even in a transient shear experiment) and rotated $45^\circ$ anticlockwise from the flow axis $\mathbf{e}_1$. However, the eigenvectors of the stress tensor change their orientation at a rate that is dependent on the evolution of the stress field given by the material’s constitutive response. The full expressions for the time-dependent eigenvectors of the stress tensor are provided in the SI. Since the principal axes of the strain rate tensor and the principal axes of the stress tensor are not coincident, the proper way to define the effective extension rate that is aligned in the direction of the stress is to find the projection of the imposed deformation rate onto the direction of the principal axes of the stress tensor, as we sketch in Figure 1c.

We define the normalized eigenvectors of the stress tensor as $e'_1$, $e'_2$, and $e'_3$. By rotating the strain rate tensor into this coordinate frame and extracting the 11 component of the resultant tensor, the “effective extension rate” can be shown to be (see details in the SI):

This is the main finding of this Letter. The factor of 1/2 arises because we constrain the “flow strength”, measured by the second invariant of the strain rate tensor, to be equivalent for both simple shear flow and planar extension. In simple shear flow, $II_{\dot{\gamma}} = \operatorname{tr}(\dot{\gamma}^2) = 2\dot{\gamma}^2$, while in a general planar extension, $II_{\dot{\gamma}} = 8\dot{\epsilon}^2$. Equating the two flow strengths, for internal consistency we have $\dot{\epsilon}_{\mathrm{eff}} = \dot{\gamma}/2$.

This effective extension rate $\dot{\epsilon}_{\mathrm{eff}}$ now isolates the stretching component of the shear flow projected onto a deforming material element at time $t$ by removing the vorticity inherent to a simple shearing deformation. This then leads to a proper definition of an equivalent composite planar extensional viscosity:

where we have substituted $\eta(\dot{\gamma}) \equiv\sigma_{12}/\dot{\gamma}$ in the last equality. We show that when this new composite function is plotted against the effective extension rate, we can completely reconstruct the a priori unknown steady planar extensional viscosity curve, using only steady shear data.

An expression for the second planar extensional viscosity can also be obtained by dividing the appropriate principal stress difference by the effective extension rate:

However, Petrie [21] has shown that $\eta_{P2}$ becomes vanishingly small compared to $\eta_{P1}$ in planar extension for most rheological models. Furthermore, it is more difficult to measure $N_2$ in a rheometer [22] especially since its magnitude is often much smaller than $N_1$. In view of these limitations, the rest of this Letter focuses on the dominant planar extensional viscosity $\eta_{[P1]}$. Further results on $\eta_{[P2]}$ are provided in the SI.

Inelastic fluids exhibit no normal stress differences, $N_1 = N_2 = 0$. In the Newtonian limit, $\sigma_{12} = \eta_0\dot{\gamma}$ in shear flow, and the planar extensional viscosity $\eta_{[P1]}$ predicted from (7) is four times the shear viscosity $\eta_0$, in agreement with the well-known Trouton ratio $Tr = \eta_{P1}/\eta_0 = 4$ in planar extensional flow. Similarly, due to its inelasticity ($N_1 \to0$), a generalized Newtonian fluid will always have a planar extensional viscosity that is four times the shear viscosity, even if it shear thins. Numerical simulations on a suspension of solid spherical particles, which flows like an inelastic fluid, in planar extension [23] have also shown that the Trouton ratio remains at four at large extension rates.

With the insight of (7), we are able to replicate the steady state planar extensional viscosity curves for a wide range of constitutive models. We first show that this composite material function is valid for a phenomenological nonlinear model based on polymer network formation and destruction, the Phan-Thien Tanner (PTT) model [24]. Specifically, the linear PTT model has similarities to the widely used FENE-P model for polymer solutions when the flows are steady [25]. It has the equation:

where the operator ($\overset{\triangledown}{\cdot}$) indicates the upper convected derivative $\overset{\triangledown}{\boldsymbol{\sigma}} = \frac{D\boldsymbol{\sigma}}{Dt} - (\nabla\mathbf{v})^T\cdot\boldsymbol{\sigma} - \boldsymbol{\sigma}\cdot\nabla\mathbf{v}$, and $\tau$ is the relaxation time of the fluid. The model nonlinearity is controlled by a parameter $\varepsilon$, which is a measure of the enhanced rate of rupture events in the transient entangled network, and the model simplifies to the quasilinear Upper Convected Maxwell (UCM) model when $\varepsilon= 0$. To solve for the stress tensor $\boldsymbol{\sigma}$, we consider the start-up of steady shear $\dot{\gamma}(t)=\dot{\gamma}_0\mathcal{H}(t)$ and planar extensional flow $\dot{\epsilon}(t)=\dot{\epsilon}_0\mathcal{H}(t)$, where $\mathcal{H}(t)$ is the Heaviside step function. We define an accumulated Hencky strain for extensional flow, $\int_0^t \dot{\epsilon}(t')\,dt'$ and an accumulated strain for shear flow $\int_0^t \dot{\gamma}(t')\,dt'=\dot{\gamma}_0t$. As we show in Figure 2(a), the shear stress growth coefficient [18] $\eta^+ = \sigma_{12}(t)/\dot{\gamma}_0$ evolves nonlinearly with shear strain and saturates. In extensional flow the stress difference is larger, but $\eta^+_{P1}$ also saturates at large Hencky strains. Our new composite material function $\eta^+_{[P1]}$ closely follows this material response and, importantly, saturates at exactly the same value. The dimensionless ratio of the extension rate $\dot{\epsilon}_{\mathrm{eff}}(t)/\dot{\gamma}_0$ (Figure 2(b)) decreases from an initial value of $1/2$ and plateaus to a steady value at large strains. The predictions from

the UCM model, which is the universal asymptotic limit at low strain rates, are shown as dotted lines, emphasizing the transition from linear to nonlinear fluid response at high shear rates ($\dot{\gamma}\tau\gg1$) and large strains. The UCM model does not predict a steady extensional viscosity at dimensionless extension rates beyond $\tau\dot{\epsilon}_0 > \frac{1}{2}$.

We represent the steady state responses of this dynamical system over a wide range of deformation rates in fig:2c. We show that the trajectories of the three transient stress growth coefficients each evolve towards the corresponding steady state attractors. Importantly, our new composite material function $\eta_{[P1]}$ (determined from transient shear properties alone) converges to the same value as $\eta_{P1}$ (obtained from planar extension). This is because the effective extension rate begins at $\dot{\gamma}/2$ at low strain rates, and later decreases (in a model-dependent way) because $\dot{\epsilon}_{\mathrm{eff}}/\dot{\gamma} \to\sigma_{12}/N_1$ at high strain rates ((6)). Asymptotic analysis of the linear PTT model at large strain rates [26] gives $\sigma_{12} \sim\eta_0(2\varepsilon\tau^2)^{-1/3}\dot{\gamma}^{1/3}$ and $N_1 \sim\eta_0(\varepsilon\tau/2)^{-1/3}\dot{\gamma}^{2/3}$, hence as shown by the dashed line in fig:2d, $\dot{\epsilon}_{\mathrm{eff}}/\dot{\gamma} \sim(4/\varepsilon)^{-1/3}(\tau\dot{\gamma})^{-1/3}$.

These conclusions are further supported by numerical calculations (fig:3a). The planar extensional viscosity $\eta_{P1}(\dot{\epsilon})$ increases due to the coil-stretch transition [27] at a critical extension rate corresponding to a critical Weissenberg number $Wi_c = \tau\dot{\epsilon}_c = \frac{1}{2}$. The material response softens at higher extension rates, causing $\eta_{P1}$ to saturate smoothly.

Our analysis also applies to microstructurally motivated polymer models. We illustrate this by considering a state-of-the-art continuum model for entangled polymer melts, the Rolie-Poly model [28]. In terms of the polymer configuration tensor $\mathbf{c}$ and the identity tensor $\mathbf{I}$, the Rolie-Poly model is written as:

where the disengagement time is $\tau_d$, and the chain stretch $\lambda= \sqrt{\operatorname{tr}\mathbf{c}/3}$ evolves separately with the Rouse time $\tau_R$. The number of entanglements per polymer chain $Z$ is often used to describe the degree of entanglements in the melt, and controls the ratio $\tau_d/\tau_R$ [29, 30]. Two scalar model parameters $\beta$ and $\delta$ modulate the effectiveness of convective constraint release and the chain sensitivity to stretching [31]. The finite extensibility of the polymer chains is incorporated using the Cohen-Padé approximation to the inverse Langevin function for the spring law [32], making the spring constant $k_s$ a function of the maximum extensibility ($L_{\max}$) of the polymer chain:

The stress is related to the polymer configuration tensor $\mathbf{c}$, the spring nonlinearity $k_s$ and the plateau modulus $G$:

The entangled fluid described by the Rolie-Poly model exhibits shear thinning of the shear viscosity beyond $Wi_d \equiv\dot{\gamma}\tau_d \gtrsim1$ (fig:3b). Once again, our proposed composite material function $\eta_{[P1]}$ accurately replicates the entire planar extensional viscosity curve. At low $Wi_d$, $N_1$ increases quadratically, but when $N_1 \gtrsim\sigma_{12}$, nonlinear elastic effects grow substantially and $\eta_{[P1]}$ evolves non-monotonically. Physically, the initial decrease in $\eta_{P1}$ is due to the deformation-induced orientation of the chains within their tubes, followed by a coil-stretch transition and a sharp increase in the tensile stress difference at a critical Weissenberg number based on the Rouse time $\dot{\epsilon}_c\tau_R \sim O(1)$ (or equivalently, $Wi_c \sim\tau_d/\tau_R \sim3Z$). Finally, $\eta_{P1}$ saturates due to finite extensibility effects and this is also faithfully captured by $\eta_{[P1]}$.

In each of the examples above, simply observing either the flow curve of $\eta(\dot{\gamma})$ or the evolution of $N_1(\dot{\gamma})$ with $\dot{\gamma}$ does not clearly reveal the existence of this stretching phenomenon because of the increasing angle between the eigenvectors of the imposed deformation rate and the resulting stress difference. Evaluating the new composite material function ((7)) in terms of the measurable quantities in steady shear clearly reveals the physics of the coil-stretch transition, at no additional experimental cost. We show in the SI that $\eta_{[P1]}$ also fully captures the planar extensional flow response of other commonly used constitutive models such as the UCM, Giesekus, and exponential PTT models.

Finally, we illustrate the application of this composite planar extensional function experimentally. Due to the experimental difficulties in generating a steady planar extensional flow, data in the literature for $\eta_{P1}$ is sparse. We perform steady shear experiments on a 2 wt% poly-isobutylene (PIB) fluid in a paraffin oil (solvent viscosity $\eta_s = 18$ mPa.s) at 20°C in a strain-controlled rheometer (TA Instruments ARES-G2) using a 25 mm 0.1 rad cone and plate fixture. Semidilute entangled shear-thinning viscoelastic materials are widely used in industry as adhesives, coolants and lubricants. However, they are not viscous enough to characterize in extension using rotating clamp devices [33], and also too viscous for microfluidic experiments [12, 34].

This shear-thinning elastic polymer solution was studied in detail by More et al. [35] and has been characterized as being in the semi-dilute regime, with an overlap concentration $c^{*} = 0.3\ \mathrm{wt}\%$. The data shown in Figure 4 is truncated at the critical shear rate $\dot{\gamma}_{c}$ beyond which elasto-inertial instabilities onset and the flow is no longer viscometric. For illustrative purposes, we show that a single mode Rolie-Poly model is able to describe the measured steady shear viscosity (●) and $N_{1}$ (▲) data. The average molecular weight of the polydisperse dissolved PIB polymer is $M \approx10^{6}\ \mathrm{g/mol}$. The molecular weight between entanglements of the polymer melt $M_{\mathrm{e},0} \approx7000\ \mathrm{g/mol}$ [36], the mass fraction of polymer $\phi= 2\ \mathrm{wt}\%$, and the solvent quality was determined by More et al. [35] from viscoelastic measurements to be $\nu= 0.593$. The material is a ‘barely entangled’ semidilute solution and the modified entanglement number in the solution [30] is $Z_{\mathrm{sol}} = (M/M_{\mathrm{e},0})\phi^{1/(3\nu-1)} \simeq0.7$. Taking the first order truncation of the perturbation expansion by Likhtman and McLeish [29], $\tau_{\mathrm{d}}/\tau_{\mathrm{R}} \sim3Z = 2.1$. This is close to the best fit value $\tau_{\mathrm{d}}/\tau_{\mathrm{R}} = 1.3$, and rationalizes the fit.

The Rolie-Poly model predicts that the steady planar extensional viscosity (—) starts at $4\eta_{0}$ at low extension rates, and begins to increase at deformation rates $\dot{\epsilon} \sim2$–$5\ \mathrm{s}^{-1}$ due to the onset of chain stretching. We have shown in our theoretical analysis above that the ratio of the effective extension rate to the imposed shear rate decreases monotonically as the stresses in the fluid grow. As we show in Figure 4 by the stars, the new composite material function $\eta_{[P1]}$, computed from the steady shear viscosity and $N_{1}$ data and plotted as a function of $\dot{\epsilon}_{\mathrm{eff}}$, matches very well with the predicted $\eta_{P1}$ curve.

viscosity and $N_{1}$ data and plotted as a function of $\dot{\epsilon}_{\mathrm{eff}}$, matches very well with the predicted $\eta_{P1}$ curve. The Trouton ratio $Tr = \eta_{P1}(\dot{\epsilon})/\eta_{0}(\dot{\gamma})$ increases from $Tr = 4$ at low extension rates to $Tr \simeq50$ at $\dot{\epsilon}_{\mathrm{eff}} \simeq7\ \mathrm{s}^{-1}$ (corresponding to the maximum shear rate $\dot{\gamma} \simeq80\ \mathrm{s}^{-1}$).

We have shown in this Letter that the steady planar extensional viscosity can be reconstructed purely from the rheological information contained in the steady shear viscosity and the first normal stress difference. This result follows from the congruity of the kinematics of the two flows (quantified in terms of both $II_{\gamma}$ and $II_{B}$) and is independent of the material constitutive equation. We have illustrated this finding using classical nonlinear continuum models as well as a molecularly-informed reptation model. We have also demonstrated it using experimental data for a representative, weakly entangled, polymer solution which cannot be characterized in planar extension using other techniques. The dynamical response of a viscoelastic material in steady shear is entirely equivalent to that in planar extension when expressed in a suitable reference frame, but the principal stress difference is distributed between two material responses $\sigma_{12}$ and $N_{1}$. This is a step towards understanding planar extensional flows, and prompts a deeper re-evaluation of other previously published viscometric data in steady shear flows for which both $\sigma_{12}$ and $N_{1}$ have been carefully measured.

Our new analysis augments the well-known Cox-Merz rule [37] — which states that for viscoelastic fluids, the nonlinear shear viscosity curve can be approximated by the magnitude of the complex viscosity obtained from linear viscoelasticity measurements as a function of angular frequency. The analogous concept here is that the nonlinear material response in one type of deformation (readily achievable in a torsional rheometer) enables prediction of the corresponding behavior in a very different (shear-free) flow. Moving forward, carefully resolved measurements of $\sigma_{12}(\dot{\gamma})$ and $N_{1}(\dot{\gamma})$ performed in steady shear flow over a range of shear rates can be used to construct steady planar extensional viscosity curves and provide greater insight into physical mechanisms underpinning strain-rate stiffening or softening, such as coil-stretch transitions and chain alignment, without incurring the additional costs and difficulties of performing more complicated planar extensional experiments.

Acknowledgments

We acknowledge Lubrizol Corporation for financial support and for providing the PIB fluid, as well as the Non-Newtonian Fluids group at MIT for the highly enjoyable reading group on flow kinematics in Fall 2025. N.K. also acknowledges conference grant support from the National University of Singapore Overseas Graduate Scholarship (NUS-OGS).