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Experiments have shown that flow in compliant microchannels can become unstable at a much lower Reynolds number than the corresponding flow in a rigid conduit. Therefore, it has been suggested that the wall's elastic compliance can be exploited towards new modalities of microscale mixing. While previous studies mainly focused on the local instability induced by the fluid–structure interactions (FSIs) in the system, we derive a one-dimensional (1-D) model to study the FSI's effect on the global instability. The proposed 1-D FSI model is tailored to long, shallow rectangular microchannels with a deformable top wall, similar to the experiments. Going beyond the usual lubrication flows analysed in these geometries, we include finite fluid inertia and couple the reduced flow equations to a novel reduced 1-D wall deformation equation. Although a quantitative comparison with previous experiments is difficult, the behaviours of the proposed model show, qualitatively, agreement with the experimental observations, and capture several key effects. Specifically, we find the critical conditions under which the inflated base state of the 1-D FSI model is linearly unstable to infinitesimal perturbations. The critical Reynolds numbers predicted are in agreement with experimental observations. The unstable modes are highly oscillatory, with frequencies close to the naturalmore »frequency of the wall, suggesting that the observed instabilities are resonance phenomena. Furthermore, during the start-up from an undeformed initial state, self-sustained oscillations can be triggered by FSI. Our modelling framework can be applied to other microfluidic systems with similar geometric scale separation under different operating conditions.« less

We study the dynamics of a ferrofluid thin film confined in a Hele-Shaw cell, and subjected to a tilted non-uniform magnetic field. It is shown that the interface between the ferrofluid and an inviscid outer fluid (air) supports travelling waves, governed by a novel modified Kuramoto–Sivashinsky-type equation derived under the long-wave approximation. The balance between energy production and dissipation in this long-wave equation allows for the existence of dissipative solitons. These permanent travelling waves’ propagation velocity and profile shape are shown to be tunable via the external magnetic field. A multiple-scale analysis is performed to obtain the correction to the linear prediction of the propagation velocity, and to reveal how the nonlinearity arrests the linear instability. The travelling periodic interfacial waves discovered are identified as fixed points in an energy phase plane. It is shown that transitions between states (wave profiles) occur. These transitions are explained via the spectral stability of the travelling waves. Interestingly, multi-periodic waves, which are a non-integrable analogue of the double cnoidal wave, are also found to propagate under the model long-wave equation. These multi-periodic solutions are investigated numerically, and they are found to be long-lived transients, but ultimately abruptly transition to one of the stablemore »periodic states identified.« less

Abstract Microfluidic devices manufactured from soft polymeric materials have emerged as a paradigm for cheap, disposable and easy-to-prototype fluidic platforms for integrating chemical and biological assays and analyses. The interplay between the flow forces and the inherently compliant conduits of such microfluidic devices requires careful consideration. While mechanical compliance was initially a side-effect of the manufacturing process and materials used, compliance has now become a paradigm, enabling new approaches to microrheological measurements, new modalities of micromixing, and improved sieving of micro- and nano-particles, to name a few applications. This topical review provides an introduction to the physics of these systems. Specifically, the goal of this review is to summarize the recent progress towards a mechanistic understanding of the interaction between non-Newtonian (complex) fluid flows and their deformable confining boundaries. In this context, key experimental results and relevant applications are also explored, hand-in-hand with the fundamental principles for their physics-based modeling. The key topics covered include shear-dependent viscosity of non-Newtonian fluids, hydrodynamic pressure gradients during flow, the elastic response (deformation and bulging) of soft conduits due to flow within, the effect of cross-sectional conduit geometry on the resulting fluid–structure interaction, and key dimensionless groups describing the coupled physics. Open problems andmore »future directions in this nascent field of soft hydraulics, at the intersection of non-Newtonian fluid mechanics, soft matter physics, and microfluidics, are noted.« less

A flow vessel with an elastic wall can deform significantly due to viscous fluid flow within it, even at vanishing Reynolds number (no fluid inertia). Deformation leads to an enhancement of throughput due to the change in cross‐sectional area. The latter gives rise to a non‐constant pressure gradient in the flow‐wise direction and, hence, to a nonlinear flow rate–pressure drop relation (unlike the Hagen–Poiseuille law for a rigid tube). Many biofluids are non‐Newtonian, and are well approximated by generalized Newtonian (say, power‐law) rheological models. Consequently, we analyze the problem of steady low Reynolds number flow of a generalized Newtonian fluid through a slender elastic tube by coupling fluid lubrication theory to a structural problem posed in terms of Donnell shell theory. A perturbative approach (in the slenderness parameter) yields analytical solutions for both the flow and the deformation. Using matched asymptotics, we obtain a uniformly valid solution for the tube's radial displacement, which features both a boundary layer and a corner layer caused by localized bending near the clamped ends. In doing so, we obtain a “generalized Hagen–Poiseuille law” for soft microtubes. We benchmark the mathematical predictions against three‐dimensional two‐way coupled direct numerical simulations (DNS) of flow and deformation performedmore »using the commercial computational engineering platform by ANSYS. The simulations show good agreement and establish the range of validity of the theory. Finally, we discuss the implications of the theory on the problem of the flow‐induced deformation of a blood vessel, which is featured in some textbooks.« less