Viscoelastic oscillatory flow and streaming in a deformable 3D microchannel

We develop the theory of elastoinertial flow rectification in an oscillatory flow of a viscoelastic Boger fluid in a deformable 3D channel following Zhang & Rallabandi, J. Fluid Mech., 2024, doi:10.1017/jfm.2024.612 and Huang, Pande, Feng & Christov, arXiv:2504.05132. We focus on Boger fluids, which have constant viscosity and a polymeric relaxation time and are well described by the Oldroyd-B constitutive relation. By reducing the Cauchy momentum and the Oldroyd-B constitutive equations under the lubrication approximation, we derive the governing equations for the fluid flow. This type of flow features three kinds of nonlinearity: (i) advective inertia in the momentum equation, (ii) upper-convected terms in the constitutive equation, (iii) geometric (cross-sectional area) changes driven by the pressure. To decouple these nonlinearities and make progress, we employ a double perturbation expansion, assuming weak viscoelasticity (a small Weissenberg number but an arbitrary Deborah number) and weak wall compliance. We are thus able to solve for the primary pressure exactly and obtain a linear boundary-value problem for the cycle-averaged secondary (streaming) pressure. We find that the Deborah number (the ratio of the polymeric relaxation timescale to the timescale of the oscillatory forcing) strongly influences both the primary and streaming pressures. Through a parameter space study, focusing on the elastoviscous and Deborah numbers, we demonstrate that the streaming pressure is highly tunable via the Deborah number, including the possibility of significant enhancement of elastoinertial rectification due to fluid viscoelasticity. Our theoretical results could be used to design novel pulsatile microfluidic systems for sorting cells and particles in viscoelastic biological fluids in lab-on-a-chip and organ-on-a-chip devices.