Swelling-induced flow evolution in shallow compliant microchannels

Fluid transport through compliant microchannels is widely observed in both natural and engineered systems, including the transport of nutrients through xylem, mechano-chemically sensitive valves, hydrogel-based drug delivery systems and filter-paper-based diagnostic kits. In such flows, the two-way coupled interaction between fluid stresses and compliance of the bounding walls gives rise to a rich flow dynamics. These soft solid walls can absorb fluid and swell, and the resulting coupling between swelling kinetics and wall compliance further complicates the flow evolution, which can be observed in water flow through hydrogel microchannels or toluene flow through polydimethylsiloxane microchannels. Here, we develop a mathematical framework to investigate the spatio-temporal evolution of a Newtonian fluid flowing through a shallow rectangular microchannel bounded by soft, swelling walls. Using lubrication approximations, we derive a nonlinear diffusion equation for fluid pressure and channel height, which reveals a non-monotonic flow dynamics arising from the competition between swelling-induced collapse and fluid-pressure-induced restraint. A simplified analytical solution of the nonlinear diffusion equation, validated through fully coupled numerical simulations, identifies distinct regimes of flow evolution, and captures the transition from swelling-dominated to fluid-pressure-dominated behaviour, governed by the interplay between the swelling ratio and wall compliance. These results elucidate non-intuitive, yet fundamental flow responses in swelling, compliant channels, and highlight their implications for passive flow generation and flow control via mechano-chemical interactions.