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arxiv: 2605.30659 · v1 · pith:UUUYHSGR · submitted 2026-05-28 · physics.flu-dyn

Neural-Network-based Viscosity Closure for Non-Newtonian Multiphase Flows

Suresh Murugaiyan , Claire L. Nelson , Dhruv Gamdha , Austin Cunniff , Cheng-Hau Yang , Abraham Wiletsky , Kaitlyn W. Dilley , Patrick Babb
show 6 more authors
Andrew Rhode Christopher M. Bates Angela A. Pitenis Michael L. Chabinyc Adarsh Krishnamurthy Baskar Ganapathysubramanian
This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-06-29 05:06 UTCgrok-4.3pith:UUUYHSGRrecord.jsonopen to challenge →

classification physics.flu-dyn
keywords neural networkviscosity closurenon-Newtonian rheologyCahn-Hilliard-Navier-Stokesadditive manufacturingmultiphase flowONNX runtimeshear-thinning
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The pith

A neural network trained on rheometer data serves as the viscosity closure inside a Cahn-Hilliard-Navier-Stokes solver and reproduces experimental droplet rise velocities.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a workflow where experimental viscosity-versus-shear-rate curves are learned by a neural network, exported in ONNX format, and called directly by an existing finite-element multiphase solver at every time step. This removes the need to pick a functional form such as Carreau-Yasuda or power-law and to re-code it inside the solver for each new material. The same adaptive octree mesh and CHNS discretization are used for both Newtonian validation cases from the literature and for two silicone inks whose rise dynamics were recorded on high-speed video. Simulated terminal velocities lie inside the experimental scatter, and the computed steady shapes match the observed ones. The approach therefore supplies a practical route for bringing measured non-Newtonian rheology into interface-resolved simulations without solver modification.

Core claim

A neural network trained on experimental rheometry data and exported via ONNX produces simulated rise velocities that fall within the experimentally measured spread and a steady-state droplet shape that agrees with observations for the characterized silicone inks when used as the viscosity closure inside a Cahn-Hilliard-Navier-Stokes finite element solver.

What carries the argument

The ONNX-exported neural network queried at runtime by the CHNS solver to supply local viscosity from the instantaneous shear rate, with Lipschitz regularization applied during training to keep the viscosity field smooth.

If this is right

  • A single solver binary can handle Newtonian and non-Newtonian materials by swapping only the closure call.
  • New ink formulations require only rheometer data collection and network retraining, not new code inside the flow solver.
  • The adaptive mesh refinement already concentrates degrees of freedom where shear rates and interfaces are steepest.
  • The separation of rheology characterization from solver development allows experimental groups to supply closures without finite-element expertise.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same ONNX workflow could be applied to other local constitutive relations such as viscoelastic stress tensors if suitable experimental data exist.
  • Because the network is queried pointwise, the method extends immediately to other finite-element or finite-volume codes that already support external function calls.
  • If shear-rate history or temperature dependence becomes important, the network input vector can be enlarged without changing the solver interface.

Load-bearing premise

Viscosity measured as a function of shear rate alone on a rheometer is sufficient to determine the local viscosity field inside the dynamic, interface-containing flow solved by the CHNS equations.

What would settle it

A simulation of one of the characterized silicone inks in which the predicted rise velocity lies outside the experimental measurement spread while the network still reproduces the original rheometer curve to high accuracy.

Figures

Figures reproduced from arXiv: 2605.30659 by Abraham Wiletsky, Adarsh Krishnamurthy, Andrew Rhode, Angela A. Pitenis, Austin Cunniff, Baskar Ganapathysubramanian, Cheng-Hau Yang, Christopher M. Bates, Claire L. Nelson, Dhruv Gamdha, Kaitlyn W. Dilley, Michael L. Chabinyc, Patrick Babb, Suresh Murugaiyan.

Figure 1
Figure 1. Figure 1: Flow Chart for the Solution Strategy The Cahn–Hilliard equation is treated as a nonlinear system and solved with the Newton-Raphson method. For the Navier–Stokes system, a projection-based semi-implicit scheme [15] is used, with the convection term ∂j(uiuj) linearized [35] using the advecting velocity from the previous time step. The same block-iterative philosophy ap￾plies within the Navier–Stokes solve: … view at source ↗
Figure 2
Figure 2. Figure 2: Comparison between regular and Lipschitz regularized neural networks in predicting a noisy function. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Neural network architecture learning rate of 10−3 , full-batch updates, and 50,000 epochs. The loss function is the Lipschitz-regularized mean￾squared error in Eq. 9, with regularization weight α = 10−4 selected by manual tuning. The benchmark networks (Section 5.1) are trained on shear-rate/viscosity pairs generated from the analytical Carreau–Yasuda formula with the parameters in [PITH_FULL_IMAGE:figure… view at source ↗
Figure 4
Figure 4. Figure 4: Schematic for rising droplet experimental setup. 10−1 100 101 102 1.5 2 2.5 Shear rate [1/s] Viscosity [Pa·s] (a) Material A (1.0 wt% fumed silica) 10−1 100 101 102 1 1.1 1.2 1.3 Shear rate [1/s] Viscosity [Pa·s] (b) Material B (0.5 wt% fumed silica) [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Steady-shear viscosity as a function of shear rate for the two silicone ink formulations used in the droplet rise experiments. 4.2. Bubble Ink Characterization Shear rate vs. viscosity plots were generated using an AR-G2 Rheometer from TA Instruments using 40 mm parallel plates at a gap height of 0.5 mm. Samples were pre-sheared at 100 s−1 for 60 seconds, followed by an equilibration time of 60 minutes to … view at source ↗
Figure 6
Figure 6. Figure 6: shows the problem setup. The simulations in this section are two-dimensional, matching the setup of Pang and Lu [26]. The computational domain has length l = 4.0 and height h = 8.0, with a circular bubble of radius ro = 0.5 placed initially at (xo, yo) = (2.0, 2.0). With no characteristic velocity available, the reference velocity is chosen so that the Froude number is unity, ur = p gr lr . The Peclet numb… view at source ↗
Figure 7
Figure 7. Figure 7: Normalized rise velocity versus non-dimensional time for Case 5 (Re = 3, We = 5, n = 0.4). Velocities are normalized by the corresponding n = 0.2 terminal velocity, matching the convention of [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 10
Figure 10. Figure 10: Both simulated trajectories rise monotonically, reach a peak, and settle onto a quasi-steady plateau. The [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of the bubble shape between experimental, analytical, and our neural network-based approach. The shapes vary with the governing dimensionless parameters in the expected way. Across Cases 1–3 (all at We = 20, Figures 8a–8c), reducing the power-law index n from 0.8 to 0.4 produces progressively greater deformation and a more pronounced skirted shape: lower n means stronger shear-thinning, which lo… view at source ↗
Figure 9
Figure 9. Figure 9: Synthetic non-monotonic constitutive law. Shear-thinning–to–thickening crossovers of this kind are reported for associative polymer solutions and concentrated particulate suspensions [40]. The flow setup matches Case 4 of Section 5.1: Re = 3, We = 5, Fr = 1, Cn = 0.01, Pe = 3333.33, with the same domain, initial bubble position, and boundary conditions. Only the constitutive law changes. We run two simulat… view at source ↗
Figure 10
Figure 10. Figure 10: Synthetic non-monotonic case. (a) Bubble shape at quasi-steady time. (b) Rise velocity versus non-dimensional time. In both panels: solid blue is the solver with the analytical constitutive law as closure; red is the solver with the neural-network closure trained on data from the analytical law. 10−3 10−2 10−1 100 200 300 400 500 Non-dimensional shear rate ˙γ ∗ Non-dimensional viscosity η ∗ NN prediction … view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of the trained Lipschitz-regularized neural network closure with the averaged rheometer measurements for the two silicone ink formulations. Open circles: experimental rheometer data. Solid blue line: prediction from the trained neural network used as the viscosity closure in the droplet rise simulations. Both axes are non-dimensional, scaled by the reference quantities defined in Section 5.3. T… view at source ↗
Figure 12
Figure 12. Figure 12: Overlay of simulated and experimental droplet contours for the four cases. Solid red line: experimental contour. Dashed blue line: simulated contour. (a) t = 0 (b) t = 2.5 (c) t = 5 (d) t = 10 [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Case 1 (Material A, 20G needle): phase-field evolution. (a) t = 0 (b) t = 2.5 (c) t = 5 (d) t = 10 [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Case 2 (Material A, 25G needle): phase-field evolution. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Case 3 (Material B, 20G needle): phase-field evolution. (a) t = 0 (b) t = 2.5 (c) t = 5 (d) t = 10 [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Case 4 (Material B, 25G needle): phase-field evolution. The Hausdorff distances span 0.044–0.079 and the Chamfer distances span 0.040–0.062, with the smallest values for Case 1 and the largest for Case 2. Each contour is normalized by the average of its horizontal and vertical extents before comparison, so these values are fractions of the average droplet size: simulated and experimental contour points ar… view at source ↗
Figure 17
Figure 17. Figure 17: Computed and experimentally measured rise velocities for the four droplet cases (Cases 1–2: Material A; Cases 3–4: Material B). The violin plots show the distribution of measured rise velocities over the duration of each droplet’s rise, and the curve shows the simulated rise velocity from the neural-constitutive CHNS solver. target. In the droplet cases (Section 5.3), the network is trained on experimenta… view at source ↗
read the original abstract

Materials used in polymer-based additive manufacturing processes, such as Digital Light Processing (DLP) and direct ink writing (DIW), typically exhibit non-Newtonian rheology. Carreau--Yasuda and power-law models describe basic shear-thinning and shear-thickening behavior well, but applying them to a new material requires choosing a functional form, deriving it, and re-implementing it inside the flow solver. We present a deployment workflow in which a neural network trained on experimental rheometry data serves as the viscosity closure inside a Cahn--Hilliard--Navier--Stokes (CHNS) finite element solver. Lipschitz regularization during training produces smooth viscosity predictions, and the trained network is exported in the Open Neural Network Exchange (ONNX) format and queried by the solver at runtime via the ONNX runtime, without solver modification or network reimplementation. The framework is built on a parallel octree-based adaptive mesh refinement infrastructure that concentrates resolution at the fluid interface. We validate the CHNS solver against benchmark shear-thinning bubble-rise cases from the literature, reproducing reported bubble shapes across varying power-law indices and Weber numbers. We characterized two silicone ink formulations, recorded their rise dynamics in perfluorodecalin on high-speed video, and used the resulting data to test the full workflow. Simulated rise velocities fall within the experimentally measured spread, and the simulated steady-state droplet shape agrees with the observed one. This work contributes to a growing body of literature on integrating neural constitutive closures into multiphysics simulations, and demonstrates a practical path for deploying experimentally trained rheological surrogates inside finite element solvers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces a workflow for using a neural network trained on experimental rheometry data as a viscosity closure (via ONNX export) inside a parallel adaptive-mesh Cahn-Hilliard-Navier-Stokes solver for non-Newtonian multiphase flows. It validates the solver on literature shear-thinning bubble-rise benchmarks and then applies the full pipeline to two characterized silicone inks, reporting that simulated rise velocities lie within the experimental spread and that steady-state shapes match high-speed video observations.

Significance. If the central claim holds, the work supplies a reproducible, solver-agnostic route for embedding experimentally trained rheological surrogates into multiphysics codes without manual reimplementation of constitutive models. The combination of Lipschitz-regularized training, ONNX deployment, and octree AMR infrastructure is a concrete engineering contribution to data-driven non-Newtonian simulation.

major comments (2)
  1. [Validation against experimental data] Validation section (experimental droplet-rise comparison): the reported agreement with rise velocity and shape tests the integrated workflow but does not isolate whether the single-variable NN closure η(γ̇) remains adequate once the flow becomes unsteady and spatially inhomogeneous near a moving interface. No discussion is provided of possible thixotropy, viscoelastic normal stresses, or interfacial rheology in the silicone inks that would violate the steady-homogeneous rheometer assumption.
  2. [Neural-network viscosity closure and CHNS coupling] Workflow description: the claim that the NN can be queried at runtime with the instantaneous local shear rate computed from the velocity gradient inside the CHNS solver rests on the untested premise that rheometer-derived η(γ̇) suffices everywhere, including in regions of high interface curvature and transient shear. No auxiliary test (e.g., comparison against a known viscoelastic or thixotropic model) is shown to bound the error introduced by this assumption.
minor comments (1)
  1. [Training procedure] The abstract states that Lipschitz regularization produces smooth viscosity predictions, yet the manuscript does not quantify the resulting Lipschitz constant or demonstrate that the ONNX-evaluated field remains sufficiently smooth for the finite-element discretization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment below and indicate where revisions will be made to the manuscript.

read point-by-point responses

  1. Referee: [Validation against experimental data] Validation section (experimental droplet-rise comparison): the reported agreement with rise velocity and shape tests the integrated workflow but does not isolate whether the single-variable NN closure η(γ̇) remains adequate once the flow becomes unsteady and spatially inhomogeneous near a moving interface. No discussion is provided of possible thixotropy, viscoelastic normal stresses, or interfacial rheology in the silicone inks that would violate the steady-homogeneous rheometer assumption.

    Authors: We agree that the experimental comparisons validate the end-to-end workflow rather than isolating the performance of the NN closure under unsteady or spatially varying conditions. The inks were characterized exclusively with steady shear rheometry, and no separate measurements of thixotropy, viscoelasticity, or interfacial effects were conducted. The observed agreement in rise velocity (within experimental spread) and steady shape provides indirect support for the closure in these cases, but does not bound errors from the steady-homogeneous assumption. We will add an explicit limitations paragraph in the discussion section acknowledging these assumptions and their potential violation for other materials. revision: partial

  2. Referee: [Neural-network viscosity closure and CHNS coupling] Workflow description: the claim that the NN can be queried at runtime with the instantaneous local shear rate computed from the velocity gradient inside the CHNS solver rests on the untested premise that rheometer-derived η(γ̇) suffices everywhere, including in regions of high interface curvature and transient shear. No auxiliary test (e.g., comparison against a known viscoelastic or thixotropic model) is shown to bound the error introduced by this assumption.

    Authors: The paper's primary contribution is the solver-agnostic deployment workflow (Lipschitz-regularized training, ONNX export, runtime querying) rather than exhaustive validation of the single-variable closure against all possible rheological complexities. Benchmark cases use known power-law and Carreau-Yasuda models, while the experimental cases use the measured inks directly. No auxiliary comparison to viscoelastic or thixotropic constitutive models was performed, as this lies outside the stated scope of demonstrating practical integration of experimental rheometry data. We will revise the methods and conclusions sections to state the modeling assumption more explicitly and list the absence of such auxiliary tests as a limitation and avenue for future work. revision: partial

Circularity Check

0 steps flagged

No circularity: NN closure trained on independent rheometry, validated on separate video and literature benchmarks

full rationale

The paper trains an NN on experimental rheometry data (shear-rate vs viscosity curves) and deploys it via ONNX inside a CHNS solver. Validation uses separate high-speed video of droplet rise (rise velocity and shape) plus literature benchmark cases for shear-thinning bubbles. No equation or step reduces the reported agreement to quantities already fitted inside the training set. The workflow is self-contained against external data; the central claim does not reduce by construction to its inputs. Minor self-citations on the solver infrastructure or prior NN work are not load-bearing for the reported validation results.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the neural network serving as a faithful surrogate for shear-rate-dependent viscosity; the main added value is the integration workflow rather than new physical axioms. The NN weights are fitted parameters; the assumption that rheometer data alone suffices is a domain assumption.

free parameters (1)
  • Neural network weights and biases
    Fitted during supervised training on the experimental rheometry curves; these parameters directly determine the viscosity prediction used by the solver.
axioms (2)
  • domain assumption Viscosity depends only on the local shear rate (no dependence on pressure, temperature, or history).
    This is the modeling choice that allows the network to be trained solely on rheometer data of viscosity versus shear rate.
  • domain assumption The underlying CHNS finite-element solver correctly transports the interface and computes the flow field once supplied with an accurate local viscosity.
    Invoked when the authors interpret agreement between simulation and experiment as validation of the NN closure.

pith-pipeline@v0.9.1-grok · 5896 in / 1725 out tokens · 43539 ms · 2026-06-29T05:06:11.258495+00:00 · methodology

discussion (0)

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Reference graph

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