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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$$De>0.4$$.
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This content will become publicly available on April 25, 2026
Motion of a deformable droplet in a rectangular, straight channel
The motion and deformation of a neutrally buoyant drop in a rectangular channel experiencing a pressure-driven flow at a low Reynolds number has been investigated both experimentally and numerically. A moving-frame boundary-integral algorithm was used to simulate the drop dynamics, with a focus on steady-state drop velocity and deformation. Results are presented for drops of varying undeformed diameters relative to channel height ($$D/H$$), drop-to-bulk viscosity ratio ($$\lambda$$), capillary number ($$Ca$$, ratio of deforming viscous forces to shape-preserving interfacial tension) and initial position in the channel in a parameter space larger than considered previously. The general trend shows that the drop steady-state velocity decreases with increasing drop diameter and viscosity ratio but increases with increasing$$Ca$$. An opposite trend is seen for drops with small viscosity ratio, however, where the steady-state velocity increases with increasing$$D/H$$and can exceed the maximum background flow velocity. Experimental results verify theoretical predictions. A deformable drop with a size comparable to the channel height when placed off centre migrates towards the centreline and attains a steady state there. In general, a drop with a low viscosity ratio and high capillary number experiences faster cross-stream migration. With increasing aspect ratio, there is a competition between the effect of reduced wall interactions and lower maximum channel centreline velocity at fixed average velocity, with the former helping drops attain higher steady-state velocities at low aspect ratios, but the latter takes over at aspect ratios above approximately 1.5.
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- Award ID(s):
- 2301910
- PAR ID:
- 10595302
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 1009
- ISSN:
- 0022-1120
- Page Range / eLocation ID:
- A61
- Subject(s) / Keyword(s):
- drops boundary integral methods microfluidics
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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