Fast flow of an Oldroyd-B model fluid through a narrow slowly varying contraction

Abstract

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 $De>0.4$ .