Rheofluidics: Dynamics of Complex Fluids

• Rheofluidics is an interdisciplinary field that combines the study of complex fluid deformation and flow with advanced micro- and mesoscale device technologies.
• It leverages specialized setups like extensional rheometry, magnetically and electrically controlled flows, and programmable capillary platforms to quantify soft matter behavior.
• Data-driven operator learning and stochastic hydrodynamics enable real-time simulation and optimization of nonlinear, history-dependent, and field-responsive fluid dynamics.

Rheofluidics is the interdisciplinary field concerned with the manipulation, measurement, and modeling of complex fluid properties within microscale and mesoscale devices. It integrates rheology—the study of the deformation and flow behavior of materials—with microfluidics, mesofluidics, and operator-theoretic data-driven modeling, enabling the exploration of nonlinear, history-dependent, and field-responsive effects in flowing soft matter. Rheofluidic systems are foundational to advancing simulation and control in soft condensed matter, biotechnology, biomedical engineering, active matter physics, and process optimization for non-Newtonian materials.

1. Governing Principles and Rheological Modeling

Rheofluidic flows are governed by the incompressible Navier–Stokes and conservation laws supplemented by constitutive equations, which relate the total stress tensor σ to the local deformation rate tensor D and to internal state variables. For complex fluids, the extra stress τ is nonlinear and history-dependent, posed through models such as Oldroyd-B, Giesekus, thixotropic elasto-viscoplastic (TEVP), and Bingham constitutive relations. Representative equations include:

• Mass conservation:

$\nabla \cdot \mathbf{u} = 0$

• Momentum balance:

$$\rho (\partial_t \mathbf{u} + \mathbf{u} \cdot \nabla \mathbf{u}) = \nabla \cdot \sigma + \mathbf{f}$$

• Stress decomposition:

$\sigma = -p I + \tau$

• Rate-of-strain tensor:

$$D = \tfrac{1}{2}(\nabla \mathbf{u} + \nabla \mathbf{u}^T)$$

Closure for τ accounts for nonlinear elasticity (e.g., Oldroyd-B: $\tau + \lambda D\tau/Dt = 2\eta D$), viscoplasticity (e.g., Bingham), and history effects. In microfluidic contexts, these are further coupled to dynamic boundary conditions and electrokinetic, magnetic, or thermal fields (Saberi et al., 1 Oct 2025).

2. Microfluidic Architectures and Rheofluidic Rheometry

Advances in micro- and mesoscale device design have enabled the precise imposition of controlled flow fields, unlocking new measurement modalities for single particles, drops, gels, and tissues.

2.1 Extensional and Shear Flows

Devices such as the OUBER (Optimized Uniaxial and Biaxial Extensional Rheometer) generate spatially uniform uniaxial or biaxial extensional flows by optimized 3D shaping of microchannel geometries. This allows persistent strain rates for quantitative extensional rheometry, critical for probing coil–stretch transitions, strain hardening, and conformational dynamics in polymers (Haward et al., 2023). Similarly, optimized hyperbolic channels achieve millimeter-scale plateaus of homogeneous strain rate suitable for tracking dilute DNA, actin filaments, or aggregates and extracting relaxation times and bending moduli (Liu et al., 2020).

2.2 Single-Object Rheology

The “Rheofluidics” technique applies temporally modulated extensional stress via microfluidic constrictions to measure frequency-dependent moduli of individual microgels, vesicles, or droplets. The deformation response yields $G^*(\omega)$ for soft objects at frequencies inaccessible to bulk rheometry (Milani et al., 12 Jan 2026).

2.3 Magnetically and Electrically Controlled Flows

Techniques leveraging magnetic wire-based active microrheology extract local viscoelastic parameters (η₀, τ, G) from the dynamics of magnetically actuated probes in microchannels (Chevry et al., 2013). Electroosmotic and ferrohydrodynamic microfluidics superpose field-driven forces with hydrodynamics to generate tunable flow fields, particle migration, and separation according to rheological properties (Zhang et al., 30 Oct 2025, Maiti et al., 2023, Mukherjee et al., 2019, Patel et al., 2019, Hamid et al., 2022).

2.4 Programmable Capillary Platforms

Elasto-capillary drop fluidics utilize magnetic field-mediated deformation of soft elastomer films, sculpting energy landscapes for aqueous droplet transport and fusion within oil layers. Controlled capillary pressure gradients yield digital “valveless” droplet routing and enable elementary operations for reaction engineering (Biswas et al., 2016).

3. Data-Driven Modeling and Operator Learning

Traditional CFD methods for non-Newtonian flows are computationally expensive due to the necessity of evolving tensorial memory fields and resolving coupled nonlinear and history-dependent equations. Recent developments in operator learning have enabled the construction of generative surrogate models that map inputs (geometry, boundary conditions, material parameters) directly to the full space–time resolved fields $(\mathbf{u}(x,t), p(x,t), \tau(x,t))$, bypassing explicit numerical integration.

The RheOFormer model applies transformer-based self-attention to learn flexible operator kernels from synthetic and high-fidelity CFD datasets, achieving millisecond-scale inference of spatio-temporal flow fields for a wide range of rheological models. Key features include:

• Encoding of history effects via latent propagation ($z_{t+1} = z_t + N(z_t)$), enabling memory without explicit storage of full stress histories.
• Cross-attention mapping to arbitrary spatial/temporal points for zero-shot prediction with new geometries and parameters.
• Strong generalization across varying Weissenberg (Wi) or Bingham numbers (Bn), demonstrating extrapolation with $<5\%$ L₂ errors (Saberi et al., 1 Oct 2025).

After training, inference scales as $O(n\,d\log d)$ per time step, independent of the particular geometry or protocol, providing an effective surrogate for process optimization and real-time flow control.

4. Multiphase and Multiscale Rheofluidics

Rheofluidics extends to the multiscale regime, encompassing interactions between particle/molecular mechanics and macroscopic flow, and between mechanical and chemical/biochemical processes.

4.1 Biomimetic Tissue Mechanics

Programmable assembly of giant unilamellar vesicle (GUV) prototissues in microfluidic chips enables multiscale rheological measurements, linking single-vesicle response (eccentricity, inter-vesicle strain) to collective viscoelastic or viscoelastoplastic tissue-scale moduli. By tuning adhesion energy ($W$), the material transitions from viscous to viscoelastic and plastic rheology, characterized by the emergence of significant storage modulus $G$ and relaxation time $\lambda$ (Layachi et al., 6 Nov 2025).

4.2 Hydrodynamic Interactions and Fiber Networks

Immersed fiber networks (e.g., biopolymer hydrogels) exhibit four dynamic regimes in their frequency-dependent storage and loss moduli when coupled to viscous fluids. A unique “fluid-coupled non-affine plateau” emerges, reflecting nonlocal drag and momentum transfer not present in the dry case. This regime, controlled by the dimensionless coupling $C = \omega \zeta / k$ and $M = \omega \eta / k$, can be harnessed to tune the dissipation and stiffness of soft materials (Head et al., 2019).

4.3 Active Fluids and Dissipative Control

In active nematic films, submersed micropatterned structures induce gradients in effective friction, yielding “virtual boundaries” that control flow topology, defect dynamics, and local material concentration without solid walls (Thijssen et al., 2021). Depth-induced friction gradients segment flow, guide topological defects, and selectively deplete active material—enabling programmable active-matter circuits.

5. Impact of Non-Newtonian and Field-Responsive Effects

Non-Newtonian and field-responsive behaviors are central to rheofluidic phenomena. Key findings are:

• Shear-thinning and mixing: Power-law fluids with $n<1$ greatly enhance electrokinetic instability-induced mixing in microchannels, producing larger and more coherent chaotic rolls, and raising mixing efficiencies by up to 40 percentage points compared to Newtonian or shear-thickening fluids (Hamid et al., 2022).
• Electroosmotic flows in non-uniform geometries: For Herschel–Bulkley fluids, nonlinear superposition and parametric control via yield stress, flow index, and cross-sectional shape enable fine-tuning of throughput and plug-layer thickness in lab-on-chip and biomedical devices (Maiti et al., 2023, Mukherjee et al., 2019, Patel et al., 2019).
• Field-tunable viscosity: In ferrohydrodynamic microfluidics, the magnetoviscous effect increases $\eta_{eff}$ by 10–50%, and the magnetophoretic-to-drag ratio $Mp$ can be tuned for size- or phenotype-based sorting of cells and vesicles (Zhang et al., 30 Oct 2025).
• Frequency-dependent interfacial dynamics: At the single-particle level, interfacial and membrane moduli show ω-dependent dispersion, revealing fast surfactant adsorption ($\Gamma(\omega)$), high-frequency gel viscoelasticity, and nonlinear membrane tension in vesicles (Milani et al., 12 Jan 2026).

6. Computational Methods and Stochastic Hydrodynamics

The theoretical modeling of rheofluidic systems increasingly employs stochastic Eulerian–Lagrangian frameworks, in which continuum fluctuating hydrodynamics are solved with microstructure (e.g., polymers, vesicles, gels) coupled via consistent thermal fluctuations and boundary imposition (Lees–Edwards conditions). These approaches allow the direct simulation of shear-thinning, oscillatory rheology, and thixotropic aging in soft materials, maintaining fluctuation–dissipation balance and near-equilibrium statistical mechanics (Atzberger, 2022).

Key results include:

• Emergence of shear-alignment and power-law thinning at high $\dot\gamma$.
• Extraction of $G'(\omega)$, $G''(\omega)$ from oscillatory simulations of model vesicles.
• Simulation of dynamic aging (time-dependent viscosity) in crosslinked gels following shear-induced structural transitions.

7. Outlook and Applications

Rheofluidics provides a unifying architecture for experimental, theoretical, and data-driven investigation of soft and complex fluids under flow. Its applications span:

• High-throughput characterization of polymers, gels, vesicles, and biomaterials at single-object to tissue scales.
• Real-time process control and optimization in additive manufacturing, injection molding, and biofabrication.
• Biomedical platforms for diagnostics, separation, and mechanobiology, leveraging field-responsive or programmable microfluidic devices.
• Fundamental studies of soft matter physics, including active turbulence, extensional flow nonlinearity, viscoelastic instabilities, and non-affine mechanics.

Limitations include the need for broader datasets to encompass extreme elasticity (Wi $\gg$ 10), highly nonlinear or yield-stress-dominated regimes, and scalable extension to three-dimensional and free-surface flows. Ongoing research explores inverse design, multiplexed detection, and machine learning integration for next-generation adaptive rheofluidic systems (Saberi et al., 1 Oct 2025, Zhang et al., 30 Oct 2025, Liu et al., 2020, Biswas et al., 2016).