In the 1830s and 1840s, Poiseuille, Hagen, and Stokes figured out the basic hydraulic law, that is, how the pressure drop Δp needed to drive a viscous fluid through a channel relates to the volumetric flow rate q. It is now in every fluid mechanics textbook.
💭 But, what happens when a viscous fluid flow reshapes the very channel carrying it?
Pressure changes the geometry ↔ the geometry changes the flow, and so it goes. Simple setups lead to surprisingly rich physics as flow and elasticity compete. Channels that are soft and deformable are common in PDMS-based microfluidic devices, such as lab-on-a-chip and wearable biosensors.
In these compliant systems, fundamental scientific questions arise: What is Poiseuille's law for a compliant microchannel? What is Womersley's law for an oscillatory flow in a deformable conduit? And what if the fluid were non-Newtonian, for example shear thinning (the harder you push it, the easier it flows) or viscoelastic (meaning it flows like a liquid but is also stretchy like an elastic solid) — think saliva, mucus, polymer solutions, protein dispersions?
These questions didn't quite fit any single existing field of mechanics. So, in 2022, we started using the term soft hydraulics, starting with an invited topical review, to give this class of problems a name and an identity.
The phrase soft hydraulics appears to have entered the scientific lexicon through the work of L. Mahadevan (Harvard University, School of Engineering and Applied Sciences), who used it in a talk at a 2005 Gordon Research Conference to describe the physics and physiology of water movement in soft fluid-infiltrated solids (gels, cells, and tissues), as well as at subsequent invited lectures in 2006 and 2007. In Mahadevan's framing, the emphasis is on poroelastic media: fluid permeating through a deformable solid matrix, as in swelling gels or pressurized cells. TMNT-Lab adopted the term and took it in a different direction, namely the engineering setting of internal flow through a compliant conduit, where the fluid pressure deforms the channel walls, which in turn reshape the flow. Same evocative name; a distinct and complementary class of problems.
Soft hydraulics is the science of fluid flow through compliant conduits at the microscale.
TMNT-Lab adopted and extended this framing to the engineering setting: deriving the governing equations for flow-induced deformation in compliant microchannels from first principles — the equations of non-Newtonian fluid mechanics and the theory of elasticity, no fitting parameters, no hand-waving — establishing the dimensionless parameters that control soft hydraulic behavior, and expanding the framework to non-Newtonian fluids, oscillatory flows, and three-dimensional geometries. The field is picking up steam. The 2022 topical review "Soft hydraulics: from Newtonian to complex fluid flows through compliant conduits" in the Journal of Physics: Condensed Matter provided a comprehensive theoretical foundation for this nascent field, as well as a roadmap for the next steps, becoming a standard reference for the field.
TMNT-Lab, directed by Prof. Ivan C. Christov, are pioneers and champions of soft hydraulics as a unified theoretical framework. We derive our laws from the basic equations of fluid mechanics and elasticity and then test them against stringent precision experiments.
... it doesn’t make any difference how beautiful your guess is; it doesn't make any difference how smart you are, who made the guess, or what his name is-if it disagrees with experiment, it's wrong ...
Our body of work was recognized with the 2026 Discovery in Mechanical Engineering Award from Purdue University, awarded for a singular research contribution by ME faculty over the previous five years.
We developed the first theory of flow-induced deformation of compliant microchannels, deriving the nonlinear flow rate–pressure drop relation from the full equations of elasticity and low-Reynolds-number fluid mechanics — no fitting parameters. This work, now highly cited, established the soft hydraulics framework and introduced the key dimensionless compliance parameter that governs the strength of fluid–structure coupling. We subsequently extended the theory beyond plate-like structures to the full elasticity setting, connecting thin- and thick-wall deformation regimes and providing the first rationalization of the original 2006 MIT experiments on microchannel deformation.
Until the 2020s, there was no established quantitative predictive theory for the Δp – q relation when both the fluid is viscoelastic, and the conduit is deformable. We changed that. Starting with the first extension to strongly shear-thinning fluids (in the power-law regime), we expanded soft hydraulics theory to constant-viscosity viscoelastic (Boger) fluids, deriving new predictive hydraulic laws without fitting parameters. One surprise: despite the complexity of this multiphysics problem, a version of the new law fits on a napkin. Our theories were then confirmed by precision experiments in collaboration with Prof. Feng's group at UIUC. Time to update the textbooks.
Physiological flows involving blood or cerebrospinal fluid are pulsatile, not steady. We showed that oscillatory soft hydraulic systems exhibit a new type of flow rectification, or streaming, which Zhang & Rallabandi clarified and termed elastoinertial rectification. A secondary streaming flow emerges from the intricated nonlinear coupling between the flow's inertia and the elastic relaxation of the wall. We recently extended elastoinertial rectification to fully three-dimensional deformable microchannels. This self-induced pumping effect is the internal-flow complement to soft streaming, where an oscillating elastic body rectifies the surrounding external flow, as studied by Prof. Gazzola's group at UIUC. Two perspectives, same underlying physics. Our work sets the stage for understanding soft hydraulics in perivascular spaces and, at the same time, opens the door to direct design guidance for oscillatory soft hydraulic circuits.
Perhaps the most famous experiment in fluid mechanics is Osborne Reynolds' (1883) demonstration that a dye stream in a pipe becomes unstable, and causes mixing, when a dimensionless number (now bearing his name) exceeds roughly 2000. Kumaran's group's experiments in channels barely a few human hairs tall, made of soft materials, have shown instability at Reynolds numbers below 200. Local instability theories were put forward, but none explained all the observations. Using our foundational soft hydraulics theory as the base state, we provided a definitive global instability analysis that rationalizes these unexpected micromixing experiments and shows that wall compliance fundamentally changes the base state, and thus the stability of the flow.
Soft hydraulic systems look benign — just low-Reynolds-number flow in a channel — yet even canonical configurations hide surprising nonlinear physics. The TMNT-Lab has shown that theory can go a long way. At the same time, simulations can provide insight where theory can't. To help experts and non-experts alike explore this interplay, Prof. Christov maintains an open-source solver suite built on the FEniCS finite element platform.
soft-hydraulic-ale-fsi: an arbitrary Lagrangian–Eulerian (ALE) fluid–structure interaction (FSI) code customized for soft hydraulics problems. Specify your channel geometry, wall material behavior and properties, and boundary conditions, then solve the fully coupled flow–deformation problem out of the box. Written for mechanistic clarity, not a 10,000-line black box. It is designed as a sandbox for building intuition: fork it, change parameters, break it, see what happens. That's often how great research questions come up. If you're curious about FSI, soft matter, microfluidics, or continuum mechanics and want to get your hands dirty without investing years developing solvers. Dive in!
Many microfluidic chips are made from PDMS, a polymeric gel. Thus, channels in microfluidic chips are soft and deform under operating pressures. Ignoring this leads to errors in flow rate control and device characterization. Soft hydraulics provides the fitting-parameter-free theory needed for accurate compliant microfluidic system design.
Compliant microchannels aren't a nuisance — they're an opportunity. By letting flows shape the very channels they flow in, it becomes possible to measure more than just viscosity from a pressure drop: a deformable wall reveals rheological features (such as polymer relaxation time) that are simply inaccessible in rigid devices, using only microliters of fluid.
Blood vessels are the original soft hydraulic circuits. Our theoretical framework can improve reduced-order models of cardiovascular flows in the small capillaries, as well as their biomicrofluid analogs. Flows in the brain's perivascular spaces and cerebrospinal fluid circulation present another application of our foundational theories.
New researchers are joining the field all the time. The problems and applications of soft hydraulics are getting richer. Enjoy these several vignettes on notable examples of soft hydraulics applications.
Rosti, Pramanik, Brandt & Mitra (KTH Mechanics / Nordita) applied soft-hydraulics principles to a model porous medium consisting of a lattice of soft elastic pillars immersed in a viscous liquid. Their direct numerical simulations and supporting theory in Soft Matter (2020) showed that wall compliance makes the Darcy flux a steeper-than-linear function of the pressure drop. Treating the gaps between pillars as elastic channels, and using an approach inspired by our soft hydraulics theory, they predict the permeability collapses onto a universal function of Δp/G; Darcy's linear law breaks down exactly where the medium turns soft.
Guyard, Restagno & McGraw (ESPCI Paris / CNRS) showed that compliance in soft microchannels introduces a finite relaxation timescale when pressure changes abruptly. Their time-resolved measurements in Physical Review Letters (2022) revealed that this timescale can vary over a decade depending on channel geometry, which is a critical consideration for transient soft microfluidic device operation. Importantly, their model incorporates our soft hydraulics theory and completely describes the experimental results.
Paludan, Dollet, Marmottant & Jensen (DTU / Université Grenoble Alpes) applied soft-hydraulics principles to intertwined channel geometries in which two microfluidic channels overlap and share an elastomeric membrane wall at the intersection. Their elegant experiments and theory in the Journal of Fluid Mechanics (2024) revealed flow-limitation behaviors and demonstrated applications of soft-hydraulic elements for microfluidic flow rectification and attenuation, functions analogous to electronic diodes and filters.
Ledda, Garg, Østergaard-Clausen, Rudzki, Madary & Pezzulla (University of Cagliari / Århus University) and collaborators demonstrated how elastic snap-through instability in spherical shells embedded in viscous flows can be harnessed to create passive soft-hydraulic valves. Their desktop-scale experiments and mathematical model, reported in Physical Review Letters (2025) was selected as an Editors' Suggestion and featured in a Physics Magazine Focus story, where Prof. Christov was quoted: "Such basic soft-hydraulic principles are invaluable for designing microfluidic systems, as one needs to rapidly develop designs without running lengthy simulations or building many prototypes."