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Soft-walled microchannels arise in many applications, ranging from organ-on-a-chip platforms to soft-robotic actuators. However, despite extensive research on their static and dynamic response, the potential failure of these devices has not been addressed. To this end, we explore fluid–structure interaction in microchannels whose compliant top wall is governed by a nonlocal mechanical theory capable of simulating both deformation and material failure. We develop a one-dimensional model by coupling viscous flow under the lubrication approximation to a state-based peridynamic formulation of an Euler–Bernoulli beam. The peridynamic formulation enables the wall to be modeled as a genuinely nonlocal beam, and the integral form of its equation of motion remains valid whether the deformation field is smooth or contains discontinuities. Through the proposed computational model, we explore the steady and time-dependent behaviors of this fluid–peridynamic structure interaction. We rationalize the wave and damping dynamics observed in the simulations through a dispersion (linearized) analysis of the coupled system, finding that, with increasing nonlocal influence, wave propagation exhibits a clear departure from classical behavior, characterized by a gradual suppression of the phase velocity. The main contribution of our study is to outline the potential failure scenarios of the microchannel’s soft wall under the hydrodynamic load of the flow. Specifically, we find a dividing curve in the space spanned by the dimensionless Strouhal number (quantifying unsteady inertia of the beam) and the compliance number (quantifying the strength of the fluid–structure coupling) separating scenarios of potential failure during transient conditions from potential failure at the steady load. 

Oscillatory flows between deformable boundaries frequently occur in various processes within engineered and biological systems. The fluid–structure interaction problem, and specifically the two-way pressure-deformation coupling that arises due to the flow in a compliant channel, is central to applications. A complete understanding of oscillatory flow in a compliant 2D channel (a canonical configuration) has not been achieved; specifically, developing the elastoinertial theory and benchmarking its predictions against detailed simulation data remains a knowledge gap. The goal of this work is to address these knowledge gaps. 

We demonstrate the use of the Lorentz reciprocal theorem in obtaining corrections to the steady flow rate due to flow oscillations in rigid channels. Starting from the unsteady Stokes equations, we derive the suitable reciprocity relation, assuming all quantities can be expressed as time-harmonic phasors. The auxiliary problem is the steady Hagen–Poiseuille flow solution, from which the reciprocal theorem allows us to calculate the first-order correction in the Womersley number to the steady flow rate in a straight rigid channel. We also consider nonuniform channels, specifically with variable height in the flow-wise direction, in which case the flow rate correction provides the leading-order effect of the interplay between the oscillations of the fluid flow and the given shape of the channel. 

We propose an analytical approach to solving nonlocal generalizations of the Euler–Bernoulli beam. Specifically, we consider a version of the governing equation recently derived under the theory of peridynamics. We focus on the clamped–clamped case, employing the natural eigenfunctions of the fourth derivative subject to these boundary conditions. Static solutions under different loading conditions are obtained as series in these eigenfunctions. To demonstrate the utility of our proposed approach, we contrast the series solution in terms of fourth-order eigenfunctions to the previously obtained Fourier sine series solution. Our findings reveal that the series in fourth-order eigenfunctions achieve a given error tolerance (with respect to a reference solution) with ten times fewer terms than the sine series. The high level of accuracy of the fourth-order eigenfunction expansion is due to the fact that its expansion coefficients decay rapidly with the number of terms in the series, one order faster than the Fourier series in our examples. 

We study the steady-flow-induced deformation between an incompressible non-Newtonian fluid and a three-dimensional deformable channel. Specifically, we provide a comprehensive experimental-theoretical framework for such flows of constant-viscosity viscoelastic (i.e., Boger) fluids, which allows us to quantify the influence of the fluid’s viscoelasticity on the flow rate–pressure drop (𝑞–Δ⁢𝑝) relation. For a flow-rate-controlled regime and weakly viscoelastic flow, we find excellent agreement between the theoretical prediction and experimental results, specifically providing an experimental demonstration of how both wall compliance and fluid viscoelasticity decrease the pressure drop. To definitively demonstrate the coupled effects of wall compliance and fluid viscoelasticity (change of cross-sectional area and curving of streamlines), we perform experiments in an equivalent rigid channel, showing that both the Boger fluid and a viscosity-matched Newtonian fluid have the same pressure drop. Thus, the experimentally verified theory for the 𝑞–Δ⁢𝑝 relation of weakly viscoelastic flows of Boger fluids in compliant channels can now be used for the design of microfluidics and soft hydraulic systems operating with complex fluids. 

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 (3-D) rectangular channel is crucial for designing lab-on-a-chip 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) 3-D rectangular channel with a deformable top wall under the assumption of weak compliance and without restriction on the oscillation frequency (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 polydimethylsiloxane-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 (J. Fluid Mech., vol. 996, 2024, p. 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. 

A linear sixth-order partial differential equation (PDE) of “parabolic” type describes the dynamics of thin liquid films beneath surfaces with elastic bending resistance when deflections from the equilibrium film height are small. On a finite domain, the associated sixth-order eigenvalue problem is self-adjoint for the boundary conditions corresponding to a thin film in a closed trough, and the eigenfunctions form a complete orthonormal set. Using these eigenfunctions, we derive the Green’s function for the governing sixth-order PDE on a finite interval and compare it to the known infinite-line solution. Further, we propose a Galerkin spectral method based on the constructed sixth-order eigenfunctions and their derivative expansions. The system of ordinary differential equations for the time-dependent expansion coefficients is solved by standard numerical methods. The numerical approach is applied to versions of the governing PDE with a second-order spatial derivative (in addition to the sixth-order one), which arises from gravity acting on the film. In the absence of gravity, we demonstrate the self-similar intermediate asymptotics of initially localized disturbances on the film surface, at least until the disturbances “feel” the finite boundaries, and show that the derived Green’s function is an attractor for such solutions. In the presence of gravity, we use the proposed Galerkin numerical method to demonstrate that self-similar behavior persists, albeit for shortened intervals of time, even for large values of the gravity-to-bending ratio. 

We develop a theory of fluid--structure interaction (FSI) between an oscillatory Newtonian fluid flow and a compliant conduit. We consider the canonical geometries of a 2D channel with a deformable top wall and an axisymmetric deformable tube. Focusing on the hydrodynamics, we employ a linear relationship between wall displacement and hydrodynamic pressure, which has been shown to be suitable for a leading-order-in-slenderness theory. The slenderness assumption also allows the use of lubrication theory, and the flow rate is related to the pressure gradient (and the tube/wall deformation) via the classical solutions for oscillatory flow in a channel and in a tube (attributed to Womersley). Then, by two-way coupling the oscillatory flow and the wall deformation via the continuity equation, a one-dimensional nonlinear partial differential equation (PDE) governing the instantaneous pressure distribution along the conduit is obtained, without \textit{a priori} assumptions on the magnitude of the oscillation frequency (\textit{i.e.}, at arbitrary Womersley number). We find that the cycle-averaged pressure (for harmonic pressure-controlled conditions) deviates from the expected steady pressure distribution, suggesting the presence of a streaming flow. An analytical perturbative solution for a weakly deformable conduit is obtained to rationalize how FSI induces such streaming. In the case of a compliant tube, the results obtained from the proposed reduced-order PDE and its perturbative solutions are validated against three-dimensional, two-way-coupled direct numerical simulations. We find good agreement between theory and simulations for a range of dimensionless parameters characterizing the oscillatory flow and the FSI, demonstrating the validity of the proposed theory of oscillatory flows in compliant conduits at arbitrary Womersley number.