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.

Flow rate-pressure drop relation for deformable channels via fluidic and elastic reciprocal theorems

“…Finally, while we considered viscous Newtonian fluids, it would be of interest to understand how the rheological response of complex fluids (such as shear thinning and viscoelasticity) influences the flow rate-pressure drop relation in deformable channels. One convenient approach to accomplish this task would be to rely on reciprocal theorems, and use a combination of the present approach and the approach recently established by Boyko and Stone [27,28] for calculating the effect of complex fluid rheology on the flow rate-pressure drop relation for rigid non-uniform channels. These calculations are left for future investigation.…”

“…flow rate or flow rate-pressure drop relation for confined viscous Newtonian and complex fluid flows, such as in rigid channels [25][26][27][28].…”

Peeling of linearly elastic sheets using complex fluids at low Reynolds numbers

“…( 6) important. In this situation, the relaxation of the polymer chain will be comparable to the average transit time in the conduit, giving a non-quasi steady flow profile [58]. We will not discuss this situation in this manuscript.…”

Exact solutions of flow and pressure variation inside a horizontalfilter chamber using Lie symmetry analysis

“…Analysis of unsteady pulsatile pressure-driven flow through an elliptic cylindrical microchannel with the Navier slip condition was examined by Chuchard et al [11] using the Mathieu and modified Mathieu functions. Recently analytical solutions to the pressure-driven flow of the Oldroyd-B fluid inside a variable narrow-shaped channel were obtained by Boyko and Stone [12] using the perturbation expansion method to investigate the relationship between the flow rate and pressure drop. The study found that pressure decreases for narrow contracting channels due to shear stress.…”