Flow rate-pressure drop relation for deformable channels via fluidic and elastic reciprocal theorems

Viscous flows through configurations manufactured or naturally assembled from soft materials apply both pressure and shear stress at the solid-liquid interface, leading to deformation of the fluidic conduit's cross section, which, in turn, affects the flow rate-pressure drop relation. Conventionally, calculating this flow rate-pressure drop relation requires solving the complete elastohydrodynamic problem, which couples the fluid flow and elastic deformation. In this Letter, we use the reciprocal theorems for Stokes flow and linear elasticity to derive a closed-form expression for the flow rate-pressure drop relation in deformable microchannels, bypassing the detailed calculation of the solution to the fluid-structure-interaction problem. For small deformations (under a domain perturbation scheme), our theory provides the leading-order effect of the interplay between the fluid stresses and the compliance of the channel on the flow rate-pressure drop relation. Our approach uses solely the fluid flow solution and the elastic deformation due to the corresponding fluid stress distribution in an undeformed channel, eliminating the need to solve the coupled elastohydrodynamic problem. Unlike previous theoretical studies that neglected the presence of lateral sidewalls (and considered shallow geometries of effectively infinite width), our approach allows us to determine the influence of confining sidewalls on the flow rate-pressure drop relation. In particular, for the flow-rate-controlled situation and the Kirchhoff-Love plate-bending theory for the elastic deformation, we show a trade-off between the effect of compliance of the deforming top wall and the drag due to sidewalls on the pressure drop. Whereas compliance decreases the pressure drop, the drag due to sidewalls increases it. Our theoretical framework may provide insight into existing experimental data and pave the way for the design of novel optimized soft microfluidic configurations of different cross-sectional shapes.

Figure 1

Schematic illustration of the geometry consisting of a three-dimensional deformable channel of length $\ell$ with an initially rectangular cross section of width $w$ and height $h_0$. The channel contains a viscous fluid steadily driven by an imposed flow rate $q$, leading to the deformation $u_y(x,z)$ of the top wall of thickness $b$, which, in turn, affects the pressure drop $\mathrm{\Delta }p$ over the axial distance $\ell$. The sidewalls are assumed to be rigid.

Figure 2

Dimensionless pressure drop $\mathrm{\Delta }P/12=\mathrm{\Delta }pw{h}_0^3/12\mu q\ell$ in a deformable channel modeled using the plate-bending theory. (a) Contour plot of the dimensionless pressure drop as a function of $\tilde{\beta }=u_c/24h_0$ and $\delta =h_0/w$. (b) Dimensionless pressure drop as a function of $\delta =h_0/w$ (dotted line) for $\tilde{\beta }=0.2$ and as a function of $\tilde{\beta }=u_c/24h_0$ (solid and dashed curves) for $\delta =0$. The solid line represents the first-order asymptotic solution, and the dashed line represents the solution based on the lubrication theory (also for $\delta =0$), derived by Christov et al. [14].