FlexiFlow: A Multi-Domain Concept

• FlexiFlow is a multifaceted concept defined across domains such as passive microfluidic rectification, elasticity-driven streaming, high-order simulations, flexible electronics, and molecular generative modeling.
• In fluid dynamics and soft robotics, FlexiFlow leverages fluid–structure interactions and elastic corrections to achieve tunable, passive flow rectification and steady streaming flows.
• It underpins innovative applications by combining analytic modeling, experimental validation, and computational frameworks to drive advancements in sustainable, adaptive, and molecular design technologies.

FlexiFlow denotes several distinct concepts in contemporary research, spanning passive microfluidic rectification with soft structures, elasticity-driven fluid streaming, high-order numerical simulation in computational fluid dynamics, lifetime-aware electronic design, and generative modeling in molecular science. Each usage of “FlexiFlow” is domain-specific, supported by primary literature, and unambiguously defined within its field.

1. FlexiFlow in Passive Rectifying Microfluidics: Bio-Inspired Soft Leaflets

The original FlexiFlow concept, developed by Brandenbourger et al. (Brandenbourger et al., 2019), refers to a passive microfluidic or soft robotics element that uses bio-inspired, asymmetric soft leaflets embedded in channels. These soft-leaflet valves increase the hydraulic resistance for forward flows and decrease it for backward flows, enabling strong tunable flow rectification solely via geometric and elastic design—without external actuation or materials beyond elastomers. The core principle is a coupled fluid–structure interaction (FSI), where leaflet deflection is governed by the hydrodynamic torque, balanced against the bending of a hinged elastic plate.

The FSI model admits a nonlinear pressure–flow relation: $\Delta p = R Q (1 + \varepsilon Q)$ where $R$ is the linear resistance and $\varepsilon$ quantifies nonlinear directional asymmetry. The asymmetry ratio

$$A(Q) = \frac{|\Delta p(-Q)|}{\Delta p(Q)} = \frac{1 - \varepsilon Q}{1 + \varepsilon Q}$$

measures rectification. Maximizing $|\varepsilon|$ at fixed channel and leaflet geometry yields design maps for optimal rest angle $\theta_0^{\mathrm{opt}}$ and aspect ratio $h/a$. Closed-form analytic expressions for $R$ and $\varepsilon$, validated against finite-element simulation and experiment, give a complete recipe for robust microfluidic check-valve design. Applications include microfluidic logic, soft-robotic diodes, and lymphatic-mimetic biomedical devices (Brandenbourger et al., 2019).

2. FlexiFlow as Elasticity-Driven Streaming in Low-Reynolds-Number Flows

In a distinct but related domain, FlexiFlow describes an elasticity-driven enhancement of viscous streaming—steady, time-averaged flows generated by oscillatory actuation—in microfluidic environments (Bhosale et al., 2022). When the boundary is compliant rather than rigid (e.g., soft cylinders), the rectified flow contains an additional contribution independent of inertia, even in the Stokes limit. This is formalized via a two-phase fluid–solid problem, with solution structure: $$\langle \psi_1 \rangle(r, \theta) = \sin 2\theta [\Theta(r) + \Lambda(r)]$$ where $\Theta$ is the classical rigid-body contribution and $\Lambda$ is the elastic correction, scaling as $O(\epsilon^3 \omega a / G)$. Streaming velocities in the near-field region thus have independent tunability via the body compliance (Cauchy number), equivalent to increasing actuation frequency in a rigid setting.

The mechanism affords tunable steady flows at lower actuation power, relevant for wearable microfluidics, soft robots, and biological systems whose active appendages operate at low Womersley numbers. Arrays of soft posts exploiting FlexiFlow can achieve programmable flow rectification, particle manipulation, and efficient passive mixing (Bhosale et al., 2022).

3. FlexiFlow (FLEXI) Framework: High-Order DG Simulation of Hyperbolic-Parabolic Conservation Laws

The FLEXI framework (referred to as “FlexiFlow” in codebase and ancillary literature) is an open-source, high-order discontinuous Galerkin spectral element (DGSEM) solver for hyperbolic-parabolic conservation laws such as the compressible Navier–Stokes equations (Krais et al., 2019). The system features modular mesh generation (HOPR), simulation (FLEXI/FlexiFlow), and in situ post-processing (POSTI), supporting curvilinear high-order meshes, hybrid DG/finite-volume switch for shock-capturing, and explicit Runge-Kutta time integration.

The main numerical algorithm discretizes the strong conservation law

$$\frac{dU}{dt} + \nabla \cdot \big(F^c(U) - F^v(U, \nabla U)\big) = 0$$

on each mesh element using a local polynomial basis and numerical fluxes at interfaces. Nonlinear stability is ensured by split-form DG with summation-by-parts (SBP) operators and entropy/kinetic energy-preserving fluxes. The framework attains excellent parallel scalability ($>70\%$ on $\sim50$k cores), supports complex physical models (e.g., LES, RANS, wall modeling), and is fully reproducible from code and data artifacts.

Representative applications include large-eddy simulation of airfoils at $Re=10^6$, shock–vortex interactions, and aeroacoustic cavity noise, all featuring mesh curving and advanced boundary conditions. Ongoing work targets multiphase extensions, real-gas EOS, and GPU acceleration (Krais et al., 2019).

4. FlexiFlow for Lifetime-Aware Design of Item-Level Intelligence (ILI)

FlexiFlow designates a holistic, lifetime-aware design and optimization framework for integrating electronics in disposable, flexible, item-level intelligence (ILI) applications such as food packaging and medical patches (Prakash et al., 9 Sep 2025). The framework targets trillion-unit scales, accounting for the unique operational profile—kHz clock rates, $10^3$–$10^4$ gates, lifetimes from days to years, and severe cost/power/yield constraints—of flexible electronics.

A central aspect is total carbon footprint modeling, balancing embodied carbon (manufacturing) and operational carbon (active power $\times$ lifetime $\times$ carbon intensity). This is formalized as: $$C_{\text{Total}} = C_{\text{Embodied}} + C_{\text{Operational}}$$ with explicit formulas for each term in terms of die area, wafer-level data, execution counts, and grid carbon intensity.

FlexiFlow integrates three pillars:

• FlexiBench: Eleven benchmarks spanning sustainability (e.g., spoilage detection, pollution monitoring) with explicit lifetime, memory, and instruction profiles.
• FlexiBits: Bit-serial RISC-V cores (1/4/8 bits; SERV, QERV, HERV) synthesized in Pragmatic FlexIC PDK, trade off die area versus energy-per-operation.
• Carbon-Aware Selection: Analytical design-space exploration identifies the architecture minimizing $C_{\text{Total}}$ per workload and deployment scenario. The result is a non-monotonic mapping, e.g., 1-bit SERV optimal for days-to-week, 8-bit HERV optimal for multi-year.

Tape-out validation demonstrates the viability of open-source, non-silicon EDA flows, while use-case studies (e.g., nation-scale spoilage mitigation) quantify the system-scale carbon break-evens.

5. FlexiFlow in Generative Molecular Modeling: Flexible Ensemble Generation

FlexiFlow designates a decomposable, E(3)-equivariant, permutation-invariant flow-matching architecture for simultaneous generation of molecular graphs and their full ensemble of low-energy 3D conformations (Tedoldi et al., 21 Nov 2025). Existing 3D molecular generative models typically sample a single conformation per molecule, insufficient for characterizing bioactivity and binding thermodynamics.

In FlexiFlow, the molecule is defined as $\mathcal G = (\mathcal V, \mathcal E, \mathcal S)$, where $\mathcal S$ is the set of conformers. The model is trained via flow-matching under conditional independence: $$p_t(x, \mathcal S) = \prod_{y \in \mathcal S} p_t(x, y)$$ and optimized via conditional flow matching, with network layers preserving both Euclidean equivariance and permutation symmetry.

The architecture leverages graph-attention modules, scalar-invariant message passing, and coordinate updates to ensure generative quality and diversity. Empirical results on QM9 and GEOM Drugs datasets show that FlexiFlow achieves state-of-the-art atom/molecule stability, validity, uniqueness, and coverage of low-energy conformation space, with dramatic speedup in ensemble generation (sampling 100 conformers in $<1$s vs minutes for CREST). The protein-conditionable form enables ligand generation with multiple conformers per target, producing physically realistic docking candidates. Ablation confirms improved conformation flexibility translates to better energetics and docking outcomes (Tedoldi et al., 21 Nov 2025).

6. Distinctions and Commonalities Across FlexiFlow Usages

While FlexiFlow labels frameworks in microfluidics, computational mechanics, physical electronics, and molecular machine learning, several unifying themes emerge:

• Physical flexibility and compliance are core in microfluidic/FSI applications (Brandenbourger et al., 2019, Bhosale et al., 2022).
• Flexible architectural or optimization schemes are central in the lifetime-aware electronics and molecular ensemble generation frameworks (Prakash et al., 9 Sep 2025, Tedoldi et al., 21 Nov 2025).
• Analytic-empirical co-design—combining closed-form modeling, simulation, and experimental or numerical validation—underpins all domains.

However, the term is not interchangeable between domains; each usage is rigorously defined within its original context.

7. Impact and Future Directions

FlexiFlow, as independently developed across scientific domains, catalyzes advances in sustainable electronics, programmable microfluidics, high-fidelity fluid simulation, and AI-driven molecular design. In microfluidics and soft robotics, it enables fully passive, geometry-tunable rectification for next-generation lab-on-chip devices (Brandenbourger et al., 2019, Bhosale et al., 2022). In fluid mechanics, it provides a testbed for high-order simulation and turbulence modeling at scale (Krais et al., 2019). In sustainable engineering, it presents a blueprint for carbon-optimal electronic systems at trillion-unit deployment, with formal tools for benchmarking and co-design (Prakash et al., 9 Sep 2025). In drug discovery, it delivers a state-of-the-art generative model for joint sampling over chemical graphs and low-energy conformer ensembles, with demonstrated alignment to gold-standard methods (Tedoldi et al., 21 Nov 2025).

Future research will likely address integration of these FlexiFlow concepts into multifunctional, feedback-coupled systems—such as smart disposable sensors with microfluidic actuation or adaptive materials with embedded intelligence—capitalizing on their tunable, physically expressive, and compositionally modular design principles.