Pressure-driven flow of the viscoelastic Oldroyd-B fluid in narrow non-uniform geometries: analytical results and comparison with simulations

Abstract

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.