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 Tichy6 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 Biancofiore10,11 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, η e l *, by relating the pressure drop along the tube, Δ Π *, to the volumetric flow rate, Q * (see, e.g., Refs. 12–17) although it should be noted that the measurement of the extensional viscosity15,18–22 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 McKinley35,36 and, more recently, of James and Roos37 (for fluids with constant shear viscosity) and James and Tripathy38 (for fluids with strongly shear thinning, power-law type, viscosity). Based on some authors, however, it is difficult to estimate η e l * directly from the Δ Π * 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 *; 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 * = h 0 * h 0 * U * ( y * , 0 ) d y * = constant, and we define the aspect ratio of the varying region of the channel ε h 0 * / *, the characteristic velocity u c * Q * / h 0 *, and the contraction ratio of the channel Λ h 0 * / h f *.

FIG. 1.

Cartesian coordinate system (y*,z*) for a symmetric hyperbolic channel with aspect ratio ε = 1/4 and contraction ratio ∧ =  h 0 * / h f * = 4.

FIG. 1.

Cartesian coordinate system (y*,z*) for a symmetric hyperbolic channel with aspect ratio ε = 1/4 and contraction ratio ∧ =  h 0 * / h f * = 4.

Close modal

An incompressible and isothermal viscoelastic fluid, namely, a polymeric material dissolved into a Newtonian solvent, enters the channel. The solvent has constant dynamic viscosity η s *, and the polymeric material has zero shear-rate viscosity η p * and longest relaxation time λ *. 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 * [ 0 , * ], i.e., y * = H * ( z * ) for the upper wall and y * = H * ( z * ) for the lower one.

Using the characteristic timescale h 0 * / u c * (equivalently, the characteristic inverse shear-rate in the entrance region), and the average residence time of the fluid in the varying region of the channel * / u c *, we define the Weissenberg number W i λ * u c * / h 0 * and the Deborah number, D e λ * u c * / *; from the definitions, we see that D e = ε W i. Since we are interested in long channels, the aspect ratio is small ( 0 < ε 1), and for the theoretical analysis that follows, we assume that W i D e (or, in asymptotic notation, we consider W i = O ( 1 / ε ) and D e = O ( 1 ) as ε 0). The velocity vector in the flow domain is denoted by u * = V * ( y * , z * ) e y + U * ( y * , z * ) e z, and the total pressure is denoted by 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
* u * = 0 ,
(1a)
* ( P * I + η s * γ ̇ * + τ * ) = 0
(1b)
where I is the unit tensor, γ ̇ * = * u * + ( * u * ) T is the rate-of-deformation tensor, τ * is the viscoelastic extra-stress tensor, and * e y ( / y * ) + e z ( / z * ) is the gradient operator. In order to determine τ *, we utilized the Oldroyd-B constitutive model
τ * + λ * ( u * * τ * τ * * v * ( * v * ) T τ * ) = η p * γ ̇ * .
(2)
Equations (1a), 1(b), and (2) are solved with the usual no-slip and no-penetration boundary conditions along the channel walls, i.e., U * = V * = 0 at y = ± H * ( z * ), 0 z * *. Alternatively, one can apply no-slip and no-penetration conditions at one of the two walls, and symmetry conditions at the midplane ( y * = 0), i.e., V * ( 0 , z * ) = U * / y * | y * = 0 = τ y z * ( 0 , z * ) = 0. The integral constraint of mass at any distance from the inlet is also utilized Q * = constant , 0 z * *, as well as a reference pressure, P ref *, at the top wall of the exit plane, i.e., P ref * = P * ( H * ( * ) , * ).
Dimensionless variables are introduced based on the lubrication theory using the transformation X = X * / X c * where X c * is the relevant characteristic scale for X *.6,10,11,41,42 Specifically, the z*-coordinate is scaled by *, 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 ε u c * . The characteristic scale for the pressure difference P * P ref * is ( η s * + η p * ) u c * * / h 0 * 2, and the extra-stress components due to viscoelasticity τ y z *, τ z z *, and τ y y * are scaled by η p * u c * / h 0 *, η p * u c * * / h 0 * 2, and η p * u c * / *, respectively. Thus, the dimensionless balance equations in scalar form are
U z + V y = 0 ,
(3)
0 = P z + ( 1 η ) ( 2 U y 2 + ε 2 2 U z 2 ) + η ( τ y z y + τ z z z ) ,
(4)
0 = P y + ( 1 η ) ε 2 ( 2 V y 2 + ε 2 2 V z 2 ) + η ε 2 ( τ y y y + τ y z z ) ,
(5)
where in Eqs. (4) and (5), the polymer viscosity ratio η η p * / ( η s * + η p * ) appears. The Oldroyd-B model for a planar incompressible flow is given by the following equations:
τ z z + D e ( U τ z z z + V τ z z y 2 τ z z U z 2 τ y z U y ) = 2 ε 2 U z ,
(6)
τ y y + D e ( U τ y y z + V τ y y y 2 τ y z V z 2 τ y y V y ) = 2 V y ,
(7)
τ y z + D e ( U τ y z z + V τ y z y τ z z V z τ y y U y ) = U y + ε 2 V z .
(8)
The dimensionless boundary, symmetry, and integral conditions, imposed for 0 z 1, are V = U = 0 at y = ± H ( z ), V = U / y = τ y z = 0 at y = 0 and H ( z ) H ( z ) U ( y , z ) d y = 1. As far as the shape function is concerned, we focus on the hyperbolic converging channel described by the function, H ( z ) = 1 / ( ( Λ 1 ) z + 1 ), 0 z 1, Λ > 1; notice that H is a continuous and piecewise differentiable smooth function with H ( 0 ) = 1 , H ( 1 ) = 1 / Λ.

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 V / y | y = 0 = U / z | y = 0 and the rate-of-deformation tensor is given by γ ̇ | y = 0 = 2 ε U / z | y = 0 ( e z e z e y e y ). Thus, the flow at the midplane is a pure planar extensional flow where the quantity U / 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

When the rate-of-deformation tensor is given in diagonal form as for γ ̇ | y = 0, the first normal stress difference, scaled by η 0 * u c * / *, is N 1 : = ( T z z T y y ) / ε 2 where T z z = P + 2 ( 1 η ) ε 2 U z + η τ z z and T y y = P 2 ( 1 η ) ε 2 U z + η ε 2 τ y y are the diagonal components of the total stress tensor. Substituting T z z and T y y into the definition for N 1, we find
N 1 = 4 ( 1 η ) U z + η ( τ z z ε 2 τ y y ) at   y = 0 .
(9)
For a Newtonian fluid, i.e., for D e = 0, Eqs. (6) and (7) give τ z z , N = 2 ε 2 U / z | y = 0 and τ y y , N = 2 U / z | y = 0. In this case, Eq. (9) reduces to N 1 , N = 4 U / z | y = 0; hereafter, the subscript 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 τ z z / ε 2 is always O ( 1 ) and must be taken into account in Eq. (9).
For the viscoelastic case, one needs to solve Eqs. (3)–(8) to find U , τ z z and τ y y at y = 0. We proceed with two approaches. According to our first approach, we start by considering the symmetry conditions at the midplane and solving Eqs. (6) and (7) analytically. Indeed, denoting with u ( z ) U ( y = 0 , z ) and u ( z ) U / z | y = 0, for simplicity, the exact analytical solution of Eqs. (6) and (7) is
τ z z ( 0 , z ) = ε 2 D e ( C z z ( 0 , z ) 1 ) , C z z ( 0 , z ) = u 2 ( z ) φ ( z ) { C z z ( 0 , 0 ) u 2 ( 0 ) + 1 D e 0 z d x φ ( x ) u 3 ( x ) }
(10a)
and
τ y y ( 0 , z ) = C y y ( 0 , z ) 1 D e , C y y ( 0 , z ) = φ ( z ) u 2 ( z ) { u 2 ( 0 ) C y y ( 0 , 0 ) + 1 D e 0 z u ( x ) φ ( x ) d x } ,
(10b)
where
φ ( z ) = exp ( 1 D e 0 z d s u ( s ) ) .
(11)
Notice that the intermediate variables C z z ( 0 , z ) and C y y ( 0 , z ), used for the derivation of the solution for τ z z ( 0 , z ) and τ y y ( 0 , z ), respectively, are merely the components for the original conformation tensor for the polymer,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 φ = φ ( z ). This was also found for the axisymmetric geometry too43 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 
Finally, by plugging Eqs. (10a), (10b) and (11) in Eq. (9), we find
N 1 = 4 ( 1 η ) u ( z ) + η D e 2 φ ( z ) ( u 2 ( z ) 0 z d x φ ( x ) u 3 ( x ) 1 u 2 ( z ) 0 z u ( x ) φ ( x ) d x ) + η D e φ ( z ) ( u 2 ( z ) u 2 ( 0 ) C z z ( 0 , 0 ) u 2 ( 0 ) u 2 ( z ) C y y ( 0 , 0 ) ) .
(12)
Equation (12) is the general formula for the evaluation of the dimensionless first normal stress difference based on the flow field at the midplane. We do emphasize that Eq. (12) is exact, namely, no approximations at all have been made its derivation. If we assume that the polymer molecules are unstretched at the inlet of the varying region of the channel, then C z z ( 0 , 0 ) = C y y ( 0 , 0 ) = 1, and Eq. (12) is simplified accordingly. Note that the initial conditions for the polymer molecules at the center of the inlet plane of the varying region of the channel are in full agreement with the exact analytical solution for the fully developed unidirectional laminar flow of an Oldroyd-B fluid in a straight channel, i.e., with the solution of Eqs. (3)–(8) at the entrance region for which H ( z ) = 1 (as clearly reported in Refs. 41 and 42 for the planar geometry, and in Ref. 43 for the axisymmetric cylindrical one). We also emphasize that these initial conditions are in full agreement with the Newtonian solution at the classic lubrication limit, i.e., for a vanishing small Deborah number, as previously reported in Refs. 41–43 too.
The evaluation of N 1 requires the midplane velocity u = u ( z ). Obviously, the solution of the full governing equations is needed, albeit approximation(s) of u can be utilized too. For instance, for a Newtonian fluid ( D e = 0) and creeping conditions, the velocity at the axis of symmetry for a long channel is u N ( z ) U N ( y = 0 , z ) = 3 / ( 4 H ( z ) )41,42,48,49 and u N ( z ) = 3 H ( z ) / ( 4 H 2 ( z ) ) = 3 ( Λ 1 ) / 4. Interestingly, u N ( z ) is constant for any z [ 0 , 1 ] and depends only on the contraction ratio. For Λ 1, the Trouton ratio is merely the ratio of N 1 to the constant elongational rate u ( z ), which for the Newtonian case gives the well-established result for planar extension T r N N 1 , N / u N ( 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 , η , Λ , D e m ),
T r = 4 ( 1 η ) + η D e m ( 1 2 D e m ( 1 + ( Λ 1 ) z ) 2 1 D e m 1 2 D e m C z z ( 0 , z ) 1 + 2 D e m ( 1 + ( Λ 1 ) z ) 2 1 D e m 1 + 2 D e m C y y ( 0 , z ) ) .
(13)
The first term in parenthesis on the right-hand side of Eq. (13) is C z z ( 0 , z ), while the second is C y y ( 0 , z ); for completeness, we also report that φ ( z ) = ( 1 + ( Λ 1 ) z ) 1 / D e m. In the latter and in Eq. (13), we have used an effective (or modified) Deborah number, which we define as the original Deborah number times the constant strain-rate at the midplane
D e m : = u N ( z ) D e = 3 ( Λ 1 ) 4 D e .
(14)
Note that the effective, or modified, Deborah number given in Eq. (14) parallels the definition of the Deborah number in pure steady homogenous extension44,50 (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 , η , Λ , D e m ), where in Eq. (13) the quantity 1 + ( Λ 1 ) z is simply replaced with Λ. We also comment on the peculiar behavior of C z z ( 0 , z ) as D e m 1 / 2. In this case, C z z ( 0 , z ) is defined only as a limit that is given by
lim D e m 1 / 2 C z z ( 0 , z ) = 1 + 2 ln ( 1 + ( Λ 1 ) z ) z 1 1 + 2 ln ( Λ ) .
(15)
We note however that C z z ( 0 , z ) is continuous and differentiable for any D e m > 0 (including D e m = 1 / 2). From the solution for C z z ( 0 , z ) and Eq. (15), we find the corresponding limit of T r
lim D e m 1 / 2 T r = 4 + η ( 4 ln ( 1 + z ( Λ 1 ) ) 3 1 ( 1 + z ( Λ 1 ) ) 4 ) z 1 4 + η ( 4 ln ( Λ ) 3 1 Λ 4 ) .
(16)
We proceed with the second approach for the evaluation of the Trouton ratio, using the solution procedure previously followed and described by Housiadas and Beris42,43 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 < ε 1, and we apply a regular perturbation scheme in terms of ε 2, i.e., the solution for each dependent flow variable X is approximated as X ( y , z ) j 0 ε 2 j X ( 2 j ) ( y , z ) for ε 2 0. Substituting in Eqs. (3)–(8) and neglecting the terms of ε 2 or higher, we derive the final simplified equations at the classic lubrication limit with unknowns U ( 0 ) , V ( 0 ) , P ( 0 ) , τ z z ( 0 ) , τ y z ( 0 ), and τ 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 ) k 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
N 1 4 ( 1 η ) U ( 0 ) z + η ( τ z z ( 0 ) ε 2 + τ z z ( 2 ) τ y y ( 0 ) ) + O ( ε 2 ) at   y = 0 .
(17)
The equation that determines τ z z ( 0 ) is deduced from Eq. (6)
τ z z ( 0 ) + D e ( U ( 0 ) τ z z ( 0 ) z 2 τ z z ( 0 ) U ( 0 ) z ) = 0 at   y = 0 .
(18)
Assuming that the polymer molecules are unstretched at the middle of the inlet cross section ( z = 0) the solution of Eq. (18) is τ z z ( 0 ) = 0 everywhere. Then, from Eq. (6) at O ( ε 2 ), we find
τ z z ( 2 ) + D e ( U ( 0 ) τ z z ( 2 ) z 2 τ z z ( 2 ) U ( 0 ) z ) = 2 U ( 0 ) z at   y = 0 .
(19)
Equation (19) reveals that only the fluid velocity along at the midplane, U ( 0 ) ( y = 0 , z ), influences τ z z ( 2 ). Note that U ( 0 ) ( y = 0 , z ) is fully determined by solving the O ( ε 0 ) equations, namely, the simplified governing equations at the classic lubrication limit. Since U ( 0 ) k = 0 8 D e k U ( 0 ) ( k ) is available,42 we assume that τ z z ( 2 ) k = 0 8 D e k τ z z ( 2 ) ( k ), substitute U ( 0 ) and τ z z ( 2 ) in Eq. (19) and follow the standard perturbation procedure to find τ z z ( 2 ) ( k ), k = 0 , 1 , 2 , , 8, analytically.
Finally, the equation that determines τ y y ( 0 ) is deduced from Eq. (7)
τ y y ( 0 ) + D e ( U ( 0 ) τ y y ( 0 ) z + 2 τ y y ( 0 ) U ( 0 ) z ) = 2 U ( 0 ) z at   y = 0 .
(20)
Similarly, we assume τ y y ( 0 ) k = 0 8 D e k τ y y ( 0 ) ( k ) and find the solution for τ y y ( 0 ) ( k ), k = 0 , 1 , 2 , , 8, analytically. Substituting all known solutions in Eq. (17), we determine the first normal stress difference in series form as N 1 k = 0 8 D e k ( 4 ( 1 η ) U ( 0 ) ( k ) + η ( τ z z ( 2 ) ( k ) τ y y ( 0 ) ( k ) ) ). Last, by forming the ratio T r N 1 / U ( 0 ) and simplifying the result, we find
T r 4 + η ( 33 35 ( 4 D e m ) 2 + 2 η 30 ( 4 D e m ) 3 + ( 181 924 + 719 η 32340 5 η 2 231 ) ( 4 D e m ) 4 + O ( ( 4 D e m ) 5 ) ) ,
(21)
where the modified Deborah number D e m has been used. Note that if we form the ratio N 1 / U ( 0 ) where U ( 0 ) k = 0 8 U ( 0 ) ( k ) D e k, we arrive at an equation for the Trouton ratio with similar form but slightly different coefficients than Eq. (21). An unexpecting but interesting feature of Eq. (21) is that this lubrication/low-Deborah number approximation for the Trouton ratio depends only on two dimensionless numbers (the modified Deborah number and the polymer viscosity ratio, D e m and η, respectively). Thus, the contraction ratio does not affect the results for T r. It does affect the exact solution at high values for D e m though; see also Fig. 2.
FIG. 2.

Trouton ratio as a function of Dem = 3 (∧-1)De/4 for η = 4/10. In Fig. 2(a), the perturbation solutions (black dashed lines) are calculated to second, fourth, sixth, and eighth order in Dem, where the arrow shows in the direction of increasing the order of perturbation. The accelerated solution (dashed red line with circles) is constructed based on the perturbation solutions up to O(De8). The solutions shown in Fig. 2(b) for ∧ = 3, 6, 9, and 12 are based on the Newtonian velocity at the exit of the channel.

FIG. 2.

Trouton ratio as a function of Dem = 3 (∧-1)De/4 for η = 4/10. In Fig. 2(a), the perturbation solutions (black dashed lines) are calculated to second, fourth, sixth, and eighth order in Dem, where the arrow shows in the direction of increasing the order of perturbation. The accelerated solution (dashed red line with circles) is constructed based on the perturbation solutions up to O(De8). The solutions shown in Fig. 2(b) for ∧ = 3, 6, 9, and 12 are based on the Newtonian velocity at the exit of the channel.

Close modal

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 approximant51 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 ( δ ) agrees with the corresponding Taylor series of f about the point δ = 0 up to O ( δ 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).

In Fig. 2, we present the Trouton ratio as predicted by the lubrication approximation/low-Deboarh number analysis, Eq. (21), by gradually increasing the number of terms that are included in the series (shown with dashed lines), the corresponding Padé approximants with M = 2, 3, and 4 (show with solid lines), along with the solution for the homogenous case44,50 (dotted red line),
T r h = 4 ( 1 + η ( 2 D e m ) 2 1 ( 2 D e m ) 2 ) .
(22)
Equation (22) is derived assuming that there are not any effects from the boundaries and that the flow has reached at steady state, namely, a pure steady planar elongation is imposed. First, we observe that the Padé approximants are indistinguishable, as well as that they diverge near D e m 1 / 2. Second, we see that the homogenous solution is practically the same as the transformed/accelerated solution, i.e., with the Padé [M/M] approximant for M = 4; note that the differences of the approximants for M = 2 and 3 with that for M = 4 are negligible [not shown in Figs. 2(a) and 2(b) because the curves are indistinguishable]. Third, we see that the perturbation solutions converge slowly and only for D e m < 1 / 2. This information in conjunction with the convergence of the successive Padé approximants implies that the radius of convergence of the perturbation series for the Trouton ratio, Eq. (21), is less than one half. Moreover, in Fig. 2(b), we compare the accelerated solution for M = 4 up to O ( D e m 8 ) with the exact solution based on the Newtonian velocity u N U ( 0 ) ( 0 ) ( y = 0 , z ) for different values of the contraction ratio Λ; recall that Λ does not appear in Eq. (21). It is seen that for D e m > 0.3 approximately, the accelerated solution cannot follow closely the exact solution as it should be expected due to its divergence near D e m = 1 / 2. The exact solution based on the Newtonian velocity profile, derived for a long channel, shows a smooth enhancement of the Trouton ration, although at high enough values of D e m a slight decrease appears. We also see that as the contraction ratio increases, the exact solution for the Trouton ratio (based on the Newtonian velocity profile) approaches the steady homogenous solution for a wider range of the modified Deborah number (always in the window 0 < D e m < 1 / 2 though). Worth mentioning is also that our recent work for the hyperbolic axisymmetric cylindrical pipe43 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.
A final comment for the two procedures to calculate the Trouton ratio is worth making here. Equations (19) and (20) have precisely the same form as the original equations (6) and (7) at y = 0, respectively, with the only difference that in Eqs. (19) and (20), the leading order velocity profile at the classic lubrication limit appears, u ( 0 ) ( z ), instead of u ( z ). Therefore, one can solve Eqs. (19) and (20) exactly, as it was done for Eqs. (6) and (7), substitute in Eq. (16) and using the result τ z z ( 0 ) = 0 to derive the formula for N 1 at the classic lubrication limit
N 1 = 4 ( 1 η ) u 0 ( z ) + η D e 2 φ ( 0 ) ( z ) ( u ( 0 ) 2 ( z ) 0 z d x φ ( 0 ) ( x ) u ( 0 ) 3 ( x ) 1 u ( 0 ) 2 ( z ) 0 z u ( 0 ) ( x ) φ ( 0 ) ( x ) d x ) + η D e φ ( 0 ) ( z ) ( u ( 0 ) 2 ( z ) u ( 0 ) 2 ( 0 ) u ( 0 ) 2 ( 0 ) u ( 0 ) 2 ( z ) ) ,
(23)
where φ ( 0 ) ( z ) = exp ( 1 D e 0 z d s u ( 0 ) ( s ) ). In other words, if one uses the lubrication equations at O(1) and O(ε2), 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.
We now show a further result concerning the asymptotic series solution with respect to the Deborah number of the lubrication equations for the evaluation of the average pressure drop along the channel, Δ Π. The expression for Δ Π is
Δ Π = 0 1 P ( 0 ) ( z ) d z 0 1 P ( 0 ) ( 0 ) ( z ) d z Δ Π N k = 1 8 D e k 0 1 P ( 0 ) ( k ) ( z ) d z ,
(24)
where Δ Π N = 3 ( 1 + Λ ) ( 1 + Λ 2 ) / 8 is the solution for a Newtonian fluid.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 ε 2]. It turns out that Δ Π reduced by the corresponding Newtonian value, i.e., the ratio Δ Π / Δ Π N, can be given in series form in terms of D e m and η as follows:
Δ Π Δ Π N 1 η ( 4 D e m ) + 24 35 η ( 4 D e m ) 2 + η 4 ( η 11 ) 105 ( 4 D e m ) 3 + η ( 8 33 1117 13 475 η + 8 385 η 2 ) ( 4 D e m ) 4 + η k = 5 8 δ k ( η ) ( 4 D e m ) k ,
(25)
where
δ 5 ( η ) = 136 1001 + 2788 25 025 η 11 175 η 2 + 64 5005 η 3 ,
(26a)
δ 6 ( η ) = 32 429 187 072 1 576 575 η + 95 752 848 925 η 2 6016 121 275 η 3 + 128 15 015 η 4 ,
(26b)
δ 7 ( η ) = 1472 36 465 + 13 072 119 119 η 113 608 728 875 η 2 + 2 315 736 20 845 825 η 3 120 136 2 977 975 η 4 + 512 85 085 η 5 ,
(26c)
δ 8 ( η ) = 384 17 765 702 784 7 621 185 η + 1 025 055 424 5 601 570 975 η 2 257 618 664 437 1 372 384 888 875 η 3 + 847 399 528 7 842 199 365 η 4 188 139 968 5 601 570 975 η 5 + 1024 230 945 η 6 .
(26d)
As it was identified above for the Trouton ratio, the remarkable feature of Eq. (25) is that Δ Π / Δ Π N depends only on D e m and η, i.e., the formula is independent of Λ. The authors in Refs. 41 and 42 did not notice and did not report this feature. We also note that this theoretical feature of the formula for the pressure drop has been confirmed numerically too, using the codes developed in Ref. 42. It has also been confirmed for the cylindrical axisymmetric geometry in Ref. 43.

The results for Δ Π / Δ Π N, for η = 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 D e m 1. The convergent results show a decrease in Δ Π / Δ Π 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).

FIG. 3.

Reduced pressure drop vs Dem = 3 (∧-1)De/4 for a viscoelastic fluid with η = 0.4 (a) Perturbation solutions: the arrow shows at the direction of an increasing order of perturbation; solid (black) line: second order solution; dashed (red) line: fourth order solution; dotted (blue) line: sixth order solution; dot-dashed (magenta) line: eighth order solution (b) accelerated solutions: solid (black) line: up to second order; dashed (red) line: up to fourth order; dotted (blue) line: up to sixth order; dot-dashed (magenta) line: up to eighth order.

FIG. 3.

Reduced pressure drop vs Dem = 3 (∧-1)De/4 for a viscoelastic fluid with η = 0.4 (a) Perturbation solutions: the arrow shows at the direction of an increasing order of perturbation; solid (black) line: second order solution; dashed (red) line: fourth order solution; dotted (blue) line: sixth order solution; dot-dashed (magenta) line: eighth order solution (b) accelerated solutions: solid (black) line: up to second order; dashed (red) line: up to fourth order; dotted (blue) line: up to sixth order; dot-dashed (magenta) line: up to eighth order.

Close modal
Housiadas and Beris42,43 also showed that for the Oldroyd-B model Δ Π is decomposed into two contributions. A pure viscous contribution due to the tangential shear stress at the walls, γ ̇ w, plus an additional viscoelastic contribution that results exclusively from the z z-component of the polymer extra-stress tensor, τ [see Eq. (83) in Ref. 42]
Δ Π = 0 1 γ ̇ + ( z ) H ( z ) d z γ ̇ w + η ( I ( 0 ) Λ I ( 1 ) 0 1 H ( z ) I ( z ) H 2 ( z ) d z ¯ ) τ ,
(27)
where γ ̇ + ( z ) : = U ( 0 ) / y | y = H ( z ) < 0 and I ( z ) : = 0 H ( z ) τ z z ( 0 ) ( y , z ) d y > 0. Note that for the hyperbolic channel, H ( z ) / H 2 ( z ) = 1 Λ and thus, the last integral on the right-hand-side of Eq. (27) can be simplified further. Also note that the polymer contribution in Eq. (27) is given in a slightly different form than that given in Ref. 42 in which the integral 0 1 τ z z ( 0 ) ( y , z ) d y was used instead of I = I ( z ) as defined above. From the perturbation solution, we find the reduced contributions γ ̇ w / Δ Π N and τ / Δ Π N which are given below up to fourth order in D e m
γ ̇ w Δ Π N 1 + 3 35 η ( 4 D e m ) 2 + 4 105 ( η 2 ) η ( 4 D e m ) 3 + ( 4 77 809 13 475 η + 8 385 η 2 ) η ( 4 D e m ) 4 ,
(28)
and
τ Δ Π N 4 η D e m ( 1 + 3 5 ( 4 D e m ) 12 35 ( 4 D e m ) 2 + 4 ( 1 21 η 175 ) ( 4 D e m ) 3 ) .
(29)
In Fig. 4, we present γ ̇ w / Δ Π N and τ / Δ Π N as functions of D e m for η = 4/10 and η = 1 based on the Padé [M/M] diagonal approximants for M = 2 and M = 4. The convergence for both γ ̇ w / Δ Π N and τ / Δ Π 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, γ ̇ w, is predicted, as well as an increase in the magnitude of the negative viscoelastic contribution, τ. However, the negative viscoelastic contribution overwhelms the positive viscous contribution leading to a decrease in the average pressure drop.
FIG. 4.

Decomposition of the average pressure drop, ΔΠ, in viscous and viscoelastic contributions normalized by ΔΠN, (a) wall shear stress contribution, (b) viscoelastic contribution solid (black) lines with circles and solid (red) lines are acceleration formulas up to O(De4) and O(De8), respectively.

FIG. 4.

Decomposition of the average pressure drop, ΔΠ, in viscous and viscoelastic contributions normalized by ΔΠN, (a) wall shear stress contribution, (b) viscoelastic contribution solid (black) lines with circles and solid (red) lines are acceleration formulas up to O(De4) and O(De8), respectively.

Close modal

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 analysis42 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.

The authors have no conflicts to disclose.

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).

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

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