Oscillatory flows in three-dimensional deformable microchannels

TL;DR

Oscillatory viscous flow in deformable microchannels exhibits two-way coupling between pressure-driven flow and wall deformation. The authors develop a lubrication-based 3D theory for slender, shallow channels with a deformable top wall, introducing the dimensionless groups $Wo$ and $γ$ and applying thin-plate elasticity. They solve for the primary oscillatory pressure distribution $P_0(Z,T)$ and predict a cycle-averaged streaming pressure $⟨P_1⟩(Z)$ arising from elastoinertial rectification, validating these predictions against a PDMS-based experimental platform. The results demonstrate strong flow–structure coupling beyond one-way models and provide a practical framework for soft hydraulics in lab-on-a-chip and organ-on-a-chip devices.

Abstract

Deformable microchannels emulate a key characteristic of soft biological systems and flexible engineering devices: the flow-induced deformation of the conduit due to slow viscous flow within. Elucidating the two-way coupling between oscillatory flow and deformation of a three-dimensional (3D) rectangular channel is crucial for designing lab- and organ-on-a-chip microsystems and eventually understanding flow-structure instabilities that can enhance mixing and transport. To this end, we determine the axial variations of the primary flow, pressure, and deformation for Newtonian fluids in the canonical geometry of a slender (long) and shallow (wide) 3D rectangular channel with a deformable top wall under the assumption of weak compliance and without restriction on the oscillation frequency (\textit{i.e.}, on the Womersley number). Unlike rigid conduits, the pressure distribution is not linear with the axial coordinate. To validate this prediction, we design a PDMS-based experimental platform with a speaker-based flow-generation apparatus and a pressure acquisition system with multiple ports along the axial length of the channel. The experimental measurements show good agreement with the predicted pressure profiles across a wide range of the key dimensionless quantities: the Womersley number, the compliance number, and the elastoviscous number. Finally, we explore how the nonlinear flow-deformation coupling leads to self-induced streaming (rectification of the oscillatory flow). Following Zhang and Rallabandi (\textit{J.\ Fluid Mech.}, vol.~996, 2024, A16), we develop a theory for the cycle-averaged pressure based on the primary problem's solution, and we validate the predictions for the axial distribution of the streaming pressure against the experimental measurements.

Figures (7)

• Figure 1: Schematic of the 3D deformable shallow and slender rectangular microchannel geometry of initial (undeformed) height $h_0$, axial length $\ell$, and transverse width $w$, such that $\ell \gg w \gg h_0$. The top wall (darker color) is an elastic plate structure of thickness $b$ that can deform from $y=h_0$ to $y=h(x,z,t)$, where $h(x,z,t)-h_0=u_y(x,z,t)$ is the vertical displacement of the fluid--solid interface. The top wall is clamped (no displacement) along the planes $x=\pm w/2$ (and $0\le z\le \ell$), while taking the outlet pressure as gauge, $p|_{z=\ell}=0$, ensures no deformation along the plane $z=\ell$ (and $-w/2 \le x \le +w/2$). An oscillatory inlet pressure, $p|_{z=0} = p_\mathrm{in}(t)$ of amplitude $p_0$ and angular frequency $\omega$, drives the flow.
• Figure 2: Experimental system with oscillatory flow in a 3D deformable rectangular microchannel. (a) Setup schematic. The entire interior space of the system is completely filled with the fluid prior to the experiments. To initiate the flow, an analog sinusoidal signal generated by the function generator is transmitted into the speaker, enabling its diaphragm to vibrate. The deformable membrane of the liquid chamber (linked to the speaker diaphragm via a rigid, 3D-printed connector shown in dark blue) transmits these vibrations, causing the oscillation of the fluid within both the chamber and the following channel. (b) Microchannel configuration. The channel features five pressure ports connecting to the data acquisition system (pressure transducer & PC). The five ports (each of width $w_p$) are evenly spaced with an axis-to-axis interval $\ell_p$ in the flow direction. The microchannel section between the first port and the outlet is covered with a deformable PDMS film of length $\ell$ at the top, and the section of the channel ahead of the first port is covered by a rigid glass slide, with its front edge precisely aligned to the center of the first port.
• Figure 3: (a) Dependence of the reduced complex "wavenumber" $\kappa/\sqrt{(1+\mathscr{T})\gamma} = \sqrt{\mathrm{i}/\mathfrak{f}({\text{Wo}})}$ on ${\text{Wo}}$ and its asymptotic behaviors (dashed curves labeled with arrows). (b) Shape of the primary pressure amplitude's axial distribution, $\Real[P_{0,a}(Z)]$ from (\ref{['eq:Pa0_soln']}a), for ${\text{Wo}} = 1$ (solid) and ${\text{Wo}} = 3$ (dashed) and across a range of $\gamma$ values.
• Figure 4: Pressure distribution and evolution in a rigid channel ($\gamma=0$) with ${\text{Wo}}=2.5$. (a) Experimental time series of the evolution of the pressure over time at the different axial positions of the ports, recall figure \ref{['expstp']}(b). (b) Comparison between the evolution of the dimensionless axial pressure distribution from the experiments (solid symbols) and the rigid-channel theory (solid lines) over half a cycle.
• Figure 5: Experimental measurements of the evolution of the pressure over time at different axial positions (pressure port locations) for the deformable channel for smaller and larger compliance numbers (left column versus right column) and smaller and larger Womersley numbers (top row versus bottom row). Specifically, (a) ${\text{Wo}}=0.537$ ($\gamma=0.109$), (b) ${\text{Wo}}=0.537$ ($\gamma=0.913$), (c) ${\text{Wo}}=2.15$ ($\gamma=1.745$), and (d) ${\text{Wo}}=2.15$ ($\gamma=14.6$), respectively.