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Abstract Non-Newtonian fluid mechanics and computational rheology widely exploit elastic dumbbell models such as Oldroyd-B and FENE-P for a continuum description of viscoelastic fluid flows. However, these constitutive equations fail to accurately capture some characteristics of realistic polymers, such as the steady extension in simple shear and extensional flows, thus questioning the ability of continuum-level modeling to predict the hydrodynamic behavior of viscoelastic fluids in more complex flows. Here, we present seven elastic dumbbell models, which include different microstructurally inspired terms, i.e., (i) the finite polymer extensibility, (ii) the conformation-dependent friction coefficient, and (iii) the conformation-dependent non-affine deformation. We provide the expressions for the steady dumbbell extension in shear and extensional flows and the corresponding viscosities for various elastic dumbbell models incorporating different microscopic features. We show the necessity of including these microscopic features in a constitutive equation to reproduce the experimentally observed polymer extension in shear and extensional flows, highlighting their potential significance in accurately modeling viscoelastic channel flow with mixed kinematics. 

Lubrication theory is adapted to incorporate the large normal stresses that occur for order-one Deborah numbers,$$De$$, the ratio of the relaxation time to the residence time. Comparing with the pressure drop for a Newtonian viscous fluid with a viscosity equal to that of an Oldroyd-B fluid in steady simple shear, we find numerically a reduced pressure drop through a contraction and an increased pressure drop through an expansion, both changing linearly with$$De$$at high$$De$$. For a constriction, there is a smaller pressure drop that plateaus at high$$De$$. For a contraction, much of the change in pressure drop occurs in the stress relaxation in a long exit channel. An asymptotic analysis for high$$De$$, based on the idea that normal stresses are stretched by an accelerating flow in proportion to the square of the velocity, reveals that the large linear changes in pressure drop are due to higher normal stresses pulling the fluid through the narrowest gap. A secondary cause of the reduction is that the elastic shear stresses do not have time to build up to their steady-state equilibrium value while they accelerate through a contraction. We find for a contraction or expansion that the high$$De$$analysis works well for$$De>0.4$$. 

Pressure-driven flows of viscoelastic fluids in narrow non-uniform geometries are common in physiological flows and various industrial applications. For such flows, one of the main interests is understanding the relationship between the flow rate$$q$$and the pressure drop$$\Delta p$$, which, to date, is studied primarily using numerical simulations. We analyse the flow of the Oldroyd-B fluid in slowly varying arbitrarily shaped, contracting channels and present a theoretical framework for calculating the$$q-\Delta p$$relation. We apply lubrication theory and consider the ultra-dilute limit, in which the velocity profile remains parabolic and Newtonian, resulting in a one-way coupling between the velocity and polymer conformation tensor. This one-way coupling enables us to derive closed-form expressions for the conformation tensor and the flow rate–pressure drop relation for arbitrary values of the Deborah number ($$De$$). Furthermore, we provide analytical expressions for the conformation tensor and the$$q-\Delta p$$relation in the high-Deborah-number limit, complementing our previous low-Deborah-number lubrication analysis. We reveal that the pressure drop in the contraction monotonically decreases with$$De$$, having linear scaling at high Deborah numbers, and identify the physical mechanisms governing the pressure drop reduction. We further elucidate the spatial relaxation of elastic stresses and pressure gradient in the exit channel following the contraction and show that the downstream distance required for such relaxation scales linearly with$$De$$. 

We provide a direct derivation of the typical time derivatives used in a continuum description of complex fluid flows, relying on principles of the kinematics of line elements. 

We analyse the pressure-driven flow of the Oldroyd-B fluid in slowly varying arbitrarily shaped, narrow channels and present a theoretical framework for calculating the relationship between the flow rate $$q$$ and pressure drop $$\Delta p$$ . We first identify the characteristic scales and dimensionless parameters governing the flow in the lubrication limit. Employing a perturbation expansion in powers of the Deborah number ( $De$ ), we provide analytical expressions for the velocity, stress and the $$q$$ – $$\Delta p$$ relation in the weakly viscoelastic limit up to $O(De^2)$ . Furthermore, we exploit the reciprocal theorem derived by Boyko $$\&$$ Stone ( Phys. Rev. Fluids , vol. 6, 2021, L081301) to obtain the $$q$$ – $$\Delta p$$ relation at the next order, $O(De^3)$ , using only the velocity and stress fields at the previous orders. We validate our analytical results with two-dimensional numerical simulations in the case of a hyperbolic, symmetric contracting channel and find excellent agreement. While the velocity remains approximately Newtonian in the weakly viscoelastic limit (i.e. the theorem of Tanner and Pipkin), we reveal that the pressure drop strongly depends on the viscoelastic effects and decreases with $De$ . We elucidate the relative importance of different terms in the momentum equation contributing to the pressure drop along the symmetry line and identify that a pressure drop reduction for narrow contracting geometries is primarily due to gradients in the viscoelastic shear stresses. We further show that, although for narrow geometries the viscoelastic axial stresses are negligible along the symmetry line, they are comparable or larger than shear stresses in the rest of the domain. 

Non-Newtonian fluids are characterized by complex rheological behaviour that affects the hydrodynamic features, such as the flow rate–pressure drop relation. While flow rate–pressure drop measurements of such fluids are common in the literature, a comparison of experimental data with theory is rare, even for shear-thinning fluids at low Reynolds number, presumably due to the lack of analytical expressions for the flow rate–pressure drop relation covering the entire range of pressures and flow rates. Such a comparison, however, is of fundamental importance as it may provide insight into the adequacy of the constitutive model that was used and the values of the rheological parameters. In this work, we present a theoretical approach to calculating the flow rate–pressure drop relation of shear-thinning fluids in long, narrow channels that can be used for comparison with experimental measurements. We utilize the Carreau constitutive model and provide a semi-analytical expression for the flow rate–pressure drop relation. In particular, we derive three asymptotic solutions for small, intermediate and large values of the dimensionless pressures or flow rates, which agree with distinct limits previously known and allow us to approximate analytically the entire flow rate–pressure drop curve. We compare our semi-analytical and asymptotic results with the experimental measurements of Pipe et al. ( Rheol. Acta , vol. 47, 2008, pp. 621–642) and find excellent agreement. Our results rationalize the change in the slope of the flow rate–pressure drop data, when reported in log–log coordinates, at high flow rates, which cannot be explained using a simple power-law model.