We present new theoretical developments on the incompressible viscoelastic flow of an Oldroyd-B fluid in a hyperbolic contracting symmetric channel. We show uniformity of the effect of viscoelasticity in hyperbolic planar and cylindrical axisymmetric geometries by revealing a modified Deborah number according to which the lubrication solutions for the pressure drop and the Trouton ratio are simplified. We also develop an expression for the Trouton ratio in terms of the fluid velocity at the midplane only, valid beyond the lubrication limit. Using the Newtonian velocity as a first approximation, we show the enhancement of the Trouton ratio with increasing the fluid elasticity.

Viscoelastic flow in narrow and confined geometries is a huge class of interesting flows with many practical applications.^{1–3} However, the theoretical study of these flows is a formidable task due to the highly non-linear character of the relevant governing equations. A theory that has been developed and widely used due to its simplicity and efficiency is the lubrication theory.^{4,5} Application of the lubrication theory to study viscoelastic lubricants flows in confined geometries with solid walls was performed for the first time by Tichy^{6} who derived the lubrication equations for the UCM model, followed by others such as Zhang *et al.*,^{7} Li,^{8} Gamaniel *et al.*,^{9} and Ahmed and Biancofiore^{10,11} who utilized more realistic constitutive models than the UCM model.

Contraction flows in hyperbolic geometries have attracted a lot of attention because of the almost constant extensional rate developed along the midplane for planar geometries or along the axis of symmetry for axisymmetric ones. These flows have been used to measure the elongational viscosity of the fluid, $ \eta e l *$, by relating the pressure drop along the tube, $ \Delta \Pi *$, to the volumetric flow rate, $ Q *$ (see, e.g., Refs. 12–17) although it should be noted that the measurement of the extensional viscosity^{15,18–22} or its theoretical prediction, as for instance presented first by Cogswell,^{23,24} improved by Binding,^{25,26} and advanced by Lubansky *et al.*^{13} and James,^{27} are both challenging because a steady and spatially uniform (i.e., homogeneous) flow cannot be achieved easily;^{28,29} see also the experimental work in Ref. 30, the simulation works in Refs. 16 and 31–33, and the optimization work in Ref. 34. Other experimental works with highly elastic polymer solutions in axisymmetric circular geometries include the papers of Rothstein and McKinley^{35,36} and, more recently, of James and Roos^{37} (for fluids with constant shear viscosity) and James and Tripathy^{38} (for fluids with strongly shear thinning, power-law type, viscosity). Based on some authors, however, it is difficult to estimate $ \eta e l *$ directly from the $ \Delta \Pi * \u2212 Q *$ experimental data due to the fact that the flow in this geometry is not purely extensional.^{16,27,39}

Theoretical analysis on viscoelastic flows in contracting channels with hyperbolic geometry has been carried out by Perez-Salas *et al.*,^{40} Boyko and Stone,^{41} and Housiadas and Beris.^{42} In the latter, we extended substantially the results presented in Ref. 41 for the viscoelastic flow in a symmetric channel with variable cross section. In particular, we derived and presented eight-order asymptotic solutions of the lubrication equations in terms of the Deborah number (defined as the ratio of the relaxation time of the polymer to a characteristic residence time) using the UCM/Oldroyd-B, Phan-Thien & Tanner (PTT), Giesekus, and Finite-Extensibility Non-Linear Elastic dumbbell with the Peterlin approximation (FENE-P) models. We also exploited the Padé diagonal approximants to improve the accuracy of the perturbation solutions and to demonstrate their convergence. The results showed a decrease in the average pressure drop, required to maintain a constant flow rate through the channel, with increasing the fluid elasticity and/or the polymer viscosity ratio.

More recently, we developed an analysis of the viscoelastic flow in axisymmetric (pipe) hyperbolic contraction flow.^{43} There, in addition to developing high order perturbation approximations in terms of the Deborah number for the solution of the lubrication equations, we made two significant advances. First, a general solution (independent of the lubrication approximation) for the Trouton ratio of the fluid was developed that only involved the centerline velocity. Moreover, as the centerline velocity was found to be insensitive to either higher order corrections to lubrication approximations or viscoelasticity, accurate approximations were obtained just employing the Newtonian solution for the centerline velocity. This allowed us to easily obtain solutions for large values of the Deborah number too. The solution showed a smooth increase in the Trouton ratio with increasing elasticity, well beyond the critical value corresponding to infinite Trouton ratio in homogeneous uniaxial extensional flow, a well-known feature of the Oldroyd-B fluid model due to its ideal, elastic Hookean spring-originating, connection.^{44} The form of the exact solution, involving an exponential of a term weighted by the inverse of the Deborah number, also explains the loss of convergence of the perturbation solution at a finite Deborah number, whereas the actual solution exists well beyond that. Second, we were able to show that the solution of the lubrication equations for the Trouton ratio as well as for the average pressure drop reduced by the corresponding Newtonian solution can be described as perturbation series only involving a modified Deborah number and the polymer viscosity ratio. Thus, a very efficient parametric representation of the results was achieved.

In this Letter, we present new results for the viscoelastic flow in a hyperbolic converging channel, which parallel those developed for the axisymmetric cylindrical geometry,^{43} thereby demonstrating their universality and general applicability to hyperbolic converging confined flows. Specifically, we show that for the planar geometry, it is also possible to obtain an expression of the Trouton ratio of the fluid, with an explicit dependence only on the midplane velocity; this expression goes beyond the lubrication approximation/low-Deborah number asymptotic analysis developed in previous works.^{41,42} The form of the exact solution explains the loss of convergence of the perturbation solution at a finite Deborah number as previously presented for the axisymmetric geometry,^{43} whereas the actual solution exists well beyond that. Furthermore, by approximating the velocity based on the Newtonian lubrication solution we develop an easy extension of the results to high Deborah numbers showing that there is not any finite Deborah number limitation (which is predicted for homogeneous extensional flows). This is also confirmed from the formula in series form for the reduced pressure drop, i.e., the ratio of the pressure drop along the channel divided by the corresponding Newtonian solution, which is shown to depend only on a modified Deborah number and the polymer viscosity ratio. The new results are in full similarity to those presented for a long axisymmetric and hyperbolic pipe,^{43} showing a generality of the viscoelastic behavior in hyperbolic confined geometries. The results also reaffirm that for Deborah numbers less than the limiting Deborah number predicted for homogeneous extensional flow, the hyperbolic confined geometries can be used to obtain directly results in agreement with those of unbounded (homogenous) steady extensional flows.

The geometry and the flow configuration are shown in Fig. 1. This consists of a varying main region along with an entrance and an exit region. Note, however, that for the analysis performed here, the exit region is not taken into account and does not affect the results. The distance between the walls of the channel in the entrance region is $ 2 h 0 *$, that in the exit region is $ 2 h f *$, and the length of the varying region of the channel is $ \u2113 *$; a star superscript throughout the paper denotes a dimensional quantity. We assume a constant volumetric flow rate per unit length in the neutral direction, $ Q * = \u222b \u2212 h 0 * h 0 * U * ( y * , 0 ) d y * = constant$, and we define the aspect ratio of the varying region of the channel $ \epsilon \u2261 h 0 * / \u2113 *$, the characteristic velocity $ u c * \u2261 Q * / h 0 *$, and the contraction ratio of the channel $ \Lambda \u2261 h 0 * / h f *$.

An incompressible and isothermal viscoelastic fluid, namely, a polymeric material dissolved into a Newtonian solvent, enters the channel. The solvent has constant dynamic viscosity $ \eta s *$, and the polymeric material has zero shear-rate viscosity $ \eta p *$ and longest relaxation time $ \lambda *$. We also assume that the fluid is characterized by negligible shear thinning behavior, i.e., we consider an elastic, Boger-type fluid. We use a Cartesian coordinate system $ ( x * , y * , z * )$ to describe the flow field, where *z ^{*}* is the main flow direction,

*y*is the transverse direction between the two walls, and

^{*}*x*is the neutral direction, normal to the $ ( y * , z * )$ plane. We consider a channel that is symmetric with respect to the midplane, and thus, the rigid walls of the channel are described by the same shape function $ H * = H * ( z * ) > 0$ for $ z * \u2208 [ 0 , \u2113 * ]$, i.e., $ y * = H * ( z * )$ for the upper wall and $ y * = \u2212 H * ( z * )$ for the lower one.

^{*}*P*where $ e y$ and $ e z$ are the unit vectors in the main flow and transverse directions, respectively. For incompressible, creeping steady flow and in the absence of any external forces and torques, the mass and momentum balances that govern the flow in the channel are

^{*},^{6,10,11,41,42}Specifically, the

*z*-coordinate is scaled by $ \u2113 *$, while the

^{*}*y*-coordinate and the shape function

^{*}*H*by $ h 0 *$. The main velocity component $ U *$ is scaled by $ u c *$, and the vertical velocity component $ V *$ by $ \epsilon \u2009 u c * .$ The characteristic scale for the pressure difference $ P * \u2212 P ref *$ is $ ( \eta s * + \eta p * ) \u2009 u c * \u2009 \u2113 * / h 0 * 2$, and the extra-stress components due to viscoelasticity $ \tau y z *$, $ \tau z z *$, and $ \tau y y *$ are scaled by $ \eta p * u c * / h 0 *$, $ \eta p * u c * \u2113 * / h 0 * 2$, and $ \eta p * u c * / \u2113 *$, respectively. Thus, the dimensionless balance equations in scalar form are

^{*}The Trouton ratio of the fluid (i.e., the dimensionless elongational viscosity of the fluid) is investigated starting from the symmetry conditions and the continuity equation, both evaluated at the midplane. Therefore, the continuity equation implies $ \u2202 V / \u2202 y | y = 0 = \u2212 \u2202 U / \u2202 z | y = 0$ and the rate-of-deformation tensor is given by $ \gamma \u0307 | y = 0 = 2 \epsilon \u2202 U / \u2202 z | y = 0 ( e z e z \u2212 e y e y )$. Thus, the flow at the midplane is a pure planar extensional flow where the quantity $ \u2202 U / \u2202 z | y = 0$ is a dimensionless rate of extension (or strain-rate). Note, however, that the flow has both extensional and shear characteristics since it is purely extensional along the midplane, and shear dominated along the wall.^{13,25,26}

*N*denotes the Newtonian solution. It is important to emphasize that the viscoelastic contribution due to the $ z z$-component of the extra-stress tensor requires special attention because even at the limit of a vanishing small aspect ratio of the channel, the ratio $ \tau z z / \epsilon 2$ is always $ O ( 1 )$ and must be taken into account in Eq. (9).

^{42,43,45}evaluated at the midplane. Equation (11) is significant as it reveals the reason for the difficulties experienced in the regular perturbation analysis in terms of $ D e$ in approximating $ \phi = \phi ( z )$. This was also found for the axisymmetric geometry too

^{43}and, as mentioned there, makes the case for the necessity of a special treatment, for example, such as the one appearing in Ref. 46, for a perturbation approach to be successful. Alternatively, a dual (two-point) perturbation analysis, post-processed with an acceleration of convergence technique, may be used.

^{47}

^{41,42,48,49}and $ u N \u2032 ( z ) = \u2212 3 H \u2032 ( z ) / ( 4 H 2 ( z ) )$ $ = 3 ( \Lambda \u2212 1 ) / 4$. Interestingly, $ u N \u2032 ( z )$ is constant for any $ z \u2208 [ 0 , 1 ]$ and depends only on the contraction ratio. For $ \Lambda \u2260 1$, the Trouton ratio is merely the ratio of $ N 1$ to the constant elongational rate $ u \u2032 ( z )$, which for the Newtonian case gives the well-established result for planar extension $ T r N \u2261 N 1 , N / u N \u2032 ( z ) = 4$.

^{28,44,50}For the viscoelastic case, one can also use the Newtonian velocity to find an approximation of the Trouton ratio, as commonly done in the literature. For the hyperbolic channel studied here, Eq. (12) yields $ T r = T r ( z , \eta , \Lambda , D e m )$,

^{44,50}(see below). In contrast to the Trouton ratio for a Newtonian fluid, Eq. (13) reveals the dependence of $ T r$ on $z$. Evaluating Eq. (13) at the exit of the pipe, we find $ T r ( 1 , \eta , \Lambda , D e m )$, where in Eq. (13) the quantity $ 1 + ( \Lambda \u2212 1 ) z$ is simply replaced with $\Lambda $. We also comment on the peculiar behavior of $ C z z ( 0 , z )$ as $ D e m \u2192 1 / 2.$ In this case, $ C z z ( 0 , z )$ is defined only as a limit that is given by

^{42,43}for the solution of Eqs. (3)–(8). Since all the details are described in Refs. 42 and 43, here we provide only an outline. We invoke the lubrication theory, valid for $ 0 < \epsilon \u226a 1$, and we apply a regular perturbation scheme in terms of $ \epsilon 2$, i.e., the solution for each dependent flow variable $X$ is approximated as $ X ( y , z ) \u2248 \u2211 j \u2265 0 \epsilon 2 j X ( 2 j ) ( y , z )$ for $ \epsilon 2 \u2192 0$. Substituting in Eqs. (3)–(8) and neglecting the terms of $ \epsilon 2$ or higher, we derive the final simplified equations at the classic lubrication limit with unknowns $ U ( 0 ) , V ( 0 ) , P ( 0 ) , \tau z z ( 0 ) , \tau y z ( 0 )$, and $ \tau y y ( 0 )$. These equations are solved asymptotically, using a regular perturbation scheme in terms of the Deborah number, i.e., assuming that $ X ( 0 ) ( y , z ) \u2248 \u2211 k \u2265 0 D e k X ( 0 ) ( k ) ( y , z )$. Similarly, for the total first normal stress difference, we start from Eq. (9) and follow the lubrication analysis. Keeping all terms up to

*O*(

*1*), we get

^{42}we assume that $ \tau z z ( 2 ) \u2248 \u2211 k = 0 8 D e k \tau z z ( 2 ) ( k )$, substitute $ U ( 0 )$ and $ \tau z z ( 2 )$ in Eq. (19) and follow the standard perturbation procedure to find $ \tau z z ( 2 ) ( k )$, $ k = 0 , 1 , 2 , \u2026 , 8$, analytically.

To increase the accuracy of our eight-order formula for the Trouton ratio, we apply the non-linear rearrangement of the terms that appear in Eq. (21) based on the diagonal [M/M] Padé approximant^{51} where M = 1, 2, 3, and 4, and we derive new, transformed, analytical formulas. We remind the reader that the diagonal [M/M] Padé approximant of a function $ f = f ( \delta )$ agrees with the corresponding Taylor series of $f$ about the point $ \delta = 0$ up to $ O ( \delta 2 M )$. The successive approximants for M = 1,2,3, and 4 can be used to check the convergence of the approximants (for more details and a systematic study of various techniques that accelerate the convergence of series in viscoelasticity, see Refs. 47 and 52).

^{44,50}(dotted red line),

^{43}has revealed that the Trouton ratio is insensitive to small changes from the Newtonian velocity profile caused due to the fluid viscoelasticity, or due to the lubrication approximation.

*O*(1) and

*O*(

*ε*

^{2}), solves the resulting equations for the constitutive model at the midplane exactly, and use them in the definition of the first normal stress difference, then deduces the exact solution for the Trouton given by Eq. (12) in which the full velocity profile $ u ( z )$ is replaced with $ u ( 0 ) ( z )$, i.e., Eq. (23). This fully justifies that in our first approach, we have approximated the velocity with the Newtonian solution valid at the classic lubrication limit.

^{41,42}We remind the reader that the leading-order pressure at the classic lubrication limit, $ P ( 0 )$, is independent of the transverse coordinate $y$ [see Eq. (5), neglecting the higher-order terms with respect to $ \epsilon 2$]. It turns out that $ \Delta \Pi $ reduced by the corresponding Newtonian value, i.e., the ratio $ \Delta \Pi / \Delta \Pi N$, can be given in series form in terms of $ D e m$ and $\eta $ as follows:

The results for $ \Delta \Pi / \Delta \Pi N$, for $ \eta = 4 / 10$ up to second-, fourth-, sixth, and eight-order asymptotic solution as a function of $ D e m$ are presented in Fig. 3(a). One can observe that convergence is achieved gradually, as more terms are included in the series. Based on the results, we can safely claim that the radius of convergence of the series is at least 0.3; it also seems that the radius of convergence approaches $ 1 2$. Furthermore, the accelerated (transformed) solutions, which are depicted in Fig. 3(b), clearly show the convergence of the perturbation results when at least five terms in the series are taken into account for the construction of the Padé [M/M] diagonal approximant (i.e., for M = 2). Indeed, the curves with M = 2, 3, and 4, constructed using the first five, seven, and nine terms in Eq. (25), respectively, are practically indistinguishable in the window $ 0 \u2264 D e m \u2264 1$. The convergent results show a decrease in $ \Delta \Pi / \Delta \Pi N$ with increasing the modified Deborah number as previously predicted theoretically in Refs. 41 and 42, and numerically by many researchers (see, for instance, in Refs. 53–55).

^{42,43}also showed that for the Oldroyd-B model $ \Delta \Pi $ is decomposed into two contributions. A pure viscous contribution due to the tangential shear stress at the walls, $ \gamma \u0307 w$, plus an additional viscoelastic contribution that results exclusively from the $ z z$-component of the polymer extra-stress tensor, $\tau $ [see Eq. (83) in Ref. 42]

*η*= 4/10 and

*η*= 1 based on the Padé [M/M] diagonal approximants for M = 2 and M = 4. The convergence for both $ \gamma \u0307 w / \Delta \Pi N$ and $ \tau / \Delta \Pi N$ is clear at the lower polymer viscosity ratio, since the differences between M = 2 and M = 4 cannot be seen. Very small differences can be seen at the maximum polymer viscosity ratio, for large values for the modified Deborah number only. Note that in all cases, the results for M = 3 (not shown in Fig. 4 though) are indistinguishable from those for M = 4. We also see that with increasing $ D e m$, an increase in the magnitude of the positive viscous contribution at the wall, $ \gamma \u0307 w$, is predicted, as well as an increase in the magnitude of the negative viscoelastic contribution, $\tau $. However, the negative viscoelastic contribution overwhelms the positive viscous contribution leading to a decrease in the average pressure drop.

In this Letter, we clarified two important new aspects for the viscoelastic flow of an Oldroyd-B fluid in a contracting and symmetric hyperbolic channel. First, we presented the theoretical framework for the evaluation of the Trouton ratio for the Oldroyd-B model, which is beyond the lubrication expansion and/or the low-Deborah number analysis. Our analysis also avoids the extraction of the Trouton ratio from the pressure-drop/flow rate data commonly used in the literature. With the aid of the symmetries of the flow, we showed that the pure extensional character of the flow at the midplane allows for the evaluation of the Trouton ratio revealing a finite increase with fluid viscoelasticity and/or polymer viscosity ratio. Note that this analysis can be extended to more realistic constitutive models if one exploits the lubrication approximation; this issue is under current investigation.

Second, by relying on our previous analysis^{42} which was based on the lubrication approximation/low-Deborah number analysis, we revealed that the reduced average pressure drop, Eq. (25), its individual contributions due to viscous and elastic forces, Eqs. (28) and (29), respectively, and the Trouton ratio, Eq. (21), can be recast in terms of an effective (or modified) Deborah number and the polymer viscosity ratio only, i.e., they are independent of the contraction ratio. This theoretical issue, which we confirmed numerically as well, is reported for the first time in the literature and is valid for the cylindrical axisymmetry hyperbolic pipe too.^{43} On the contrary, this feature holds true only for the UCM/Oldroyd-B models, namely, the contraction ratio affects the contribution that results from the additional non-linear terms of the FENE-P/Giesekus/PTT models studied before in Ref. 42. The results showed that with increasing the fluid elasticity, the positive viscous contribution due to shearing at the walls increases, but the negative elastic contribution decreases much faster, and thus, a net decrease in the average pressure drop is predicted.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Kostas D. Housiadas:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). **Antony N. Beris:** Conceptualization (equal); Formal analysis (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

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

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