Modelfluid Representation in Fluid Modeling

• Modelfluid representation is a fluid modeling strategy that replaces inessential details with structured, continuous descriptors tailored for optimization and simulation.
• It is applied across domains such as distillation, porous-media upscaling, and fluid-mechanics visualization, each emphasizing computational efficiency and physical interpretability.
• The approach reduces computational cost and enhances fidelity by mapping fluid properties into reusable surrogate models and effective constitutive relations.

Searching arXiv for papers explicitly using or closely related to “modelfluid representation,” plus adjacent fluid-representation frameworks. Modelfluid representation denotes a family of task-specific fluid encodings in which the object being manipulated is not the raw fluid identity or a fully resolved field, but a structured surrogate chosen to preserve the information needed by a downstream calculation. In chemical-process research, it is defined most explicitly as a continuous, physically interpretable vector of vapor-liquid-equilibrium descriptors that can be mapped back into standard thermodynamic correlations for distillation and flowsheet optimization (Bubel et al., 2 Sep 2025, Bubel et al., 8 Sep 2025). In multiscale porous-media simulation, the same idea appears as an effective fluid representation that replaces unresolved pore geometry by constitutive laws for capillarity, viscous drag, residual saturation, and anisotropic resistivity (Yang et al., 25 Dec 2025). Related work represents fluids by printable coherent-structure geometries, boundary-oriented topological objects, spectral mode amplitudes, finite cochains, or compositional energy-based subsystems (Taira et al., 2017, Zhang et al., 2019, Floerchinger et al., 2018, Wilson, 2011, Lohmayer et al., 2024). Taken together, these works suggest a common modeling move: replace inessential detail by a representation matched to the target question.

1. Terminological scope

The literature does not use the phrase to denote a single fixed formalism. Instead, it appears in several domain-specific senses that differ in what is represented, what is preserved, and what is discarded.

Domain	Representation object	Primary purpose
Distillation and flowsheet optimization	Continuous VLE-derived feature vector	Reusable optimization and surrogate modeling (Bubel et al., 2 Sep 2025, Bubel et al., 8 Sep 2025)
Under-resolved porous media	Local constitutive “effective fluid representation”	Replace missing pore geometry by source terms (Yang et al., 25 Dec 2025)
Fluid-mechanics visualization	3D-printed isosurface artifact	Materialize coherent structures (Taira et al., 2017)
Geometric/algebraic fluid formalisms	Boundary curves, cochains, modes, ports	Topology, discretization, or compositional structure (Zhang et al., 2019, Wilson, 2011, Floerchinger et al., 2018, Lohmayer et al., 2024)

In the chemical-engineering usage, the central problem is reusability across chemical systems. Prior surrogates were described as “system-specific,” because they encoded component identity rather than thermodynamic behavior; the modelfluid replaces this with continuous descriptors of boiling behavior, nonideality, and infinite-dilution volatility (Bubel et al., 2 Sep 2025, Bubel et al., 8 Sep 2025). In the porous-media usage, the issue is unresolved geometry: subgrid pore space is not meshed directly, but represented by local permeability-, capillary-, and resistivity-based laws (Yang et al., 25 Dec 2025).

A broader related literature studies fluid representation without always using the same label. These works encode fluids as oriented Jordan curves and Boolean algebras, finite-dimensional cochain complexes, symmetry-adapted mode expansions, or energy-based subsystems coupled through ports (Zhang et al., 2019, Wilson, 2011, Floerchinger et al., 2018, Lohmayer et al., 2024). This suggests that “modelfluid representation” is best understood as a representational strategy rather than a single canonical data structure.

2. Thermodynamic feature-space modelfluids for distillation

The most explicit and technically developed meaning of modelfluid representation appears in distillation and flowsheet optimization. The representation is designed to satisfy four requirements: R1 availability, R2 integrability, R3 interpretability, and R4 continuity. Its defining move is to encode a fluid system by physically meaningful VLE descriptors rather than by discrete component labels, one-hot vectors, SMILES strings, or system-specific parameter sets (Bubel et al., 2 Sep 2025).

For a binary system, the feature vector consists of pure-component saturated vapor temperatures at the system pressure, pure-component vaporization enthalpies, activity coefficients at infinite dilution, derivatives of vapor mole fraction with respect to liquid mole fraction at infinite dilution, and $\ln p$. The infinite-dilution slope is connected directly to Henry’s law through

$$K_i^H = \lim_{x_i\to 0}\frac{y_i p}{x_i\vert_j} = \frac{\partial y_i}{\partial x_i}\Big\vert_j\, p,$$

and the associated relative volatility is

$$\alpha_{ij} = \frac{\frac{\partial y_i}{\partial x_i}\vert_j} {\frac{\partial y_j}{\partial x_j}\vert_i}.$$

These choices make the representation explicitly tied to separation difficulty, azeotropy, and energy demand (Bubel et al., 2 Sep 2025).

The feature layer is not used as a standalone thermodynamic model. Instead, it is mapped deterministically into standard correlations: a three-suffix Margules activity-coefficient model, a simplified two-parameter Antoine vapor-pressure model, and a composition-weighted enthalpy-of-vaporization model. The equilibrium relation remains extended Raoult’s law,

$y_i\, p = x_i\, \gamma_i\, p_i^{sat}(T),$

so the representation is optimization-friendly while remaining compatible with MESH-based distillation simulation (Bubel et al., 2 Sep 2025).

The reusable-surrogate formulation extends the same logic to homogeneous ternary vapor-liquid mixtures. Its ternary modelfluid is built from saturated vapor temperatures $T_i$, vaporization enthalpies $h_i$, activity coefficients at infinite dilution $\gamma^\infty$, and infinite-dilution slope features. The raw ternary description contains 19 features, reduced to 16 independent features because unique vapor-pressure models must hold across binary subsystems (Bubel et al., 8 Sep 2025). This feature vector is concatenated with the column-specification variables $s$, $RR$, $N_S^{BF}$, and $$K_i^H = \lim_{x_i\to 0}\frac{y_i p}{x_i\vert_j} = \frac{\partial y_i}{\partial x_i}\Big\vert_j\, p,$$0, with pressure already included in the modelfluid component (Bubel et al., 8 Sep 2025).

The representation enables large-scale dataset generation. The reported pipeline starts from pure-component data from DIPPR 801, predicts missing infinite-dilution activity coefficients using the Matrix Completion method, constructs the modelfluid vector, maps it into thermodynamic parameters, and runs a rigorous stage-to-stage distillation simulator based on MESH equations. The resulting dataset contains 167,221 unique ternary mixtures, 1,972,085 feature/target vectors for training and validation, 420 unique ternary mixtures in the test set, and 13,494 test feature/target vectors (Bubel et al., 8 Sep 2025).

The surrogate itself is a feed-forward ANN with one input layer, four hidden layers, and one output layer, using ReLU activations except in the final layer. The reported layer sizes are

$$K_i^H = \lim_{x_i\to 0}\frac{y_i p}{x_i\vert_j} = \frac{\partial y_i}{\partial x_i}\Big\vert_j\, p,$$1

Its targets are reboiler heat duty and stream compositions, with the last component recovered from the summation constraint (Bubel et al., 8 Sep 2025).

Validation is explicitly out-of-distribution: training uses ML-predicted activity coefficients, whereas testing uses Aspen Plus NRTL parameters. The reported RMSE values are $$K_i^H = \lim_{x_i\to 0}\frac{y_i p}{x_i\vert_j} = \frac{\partial y_i}{\partial x_i}\Big\vert_j\, p,$$2 mol for $$K_i^H = \lim_{x_i\to 0}\frac{y_i p}{x_i\vert_j} = \frac{\partial y_i}{\partial x_i}\Big\vert_j\, p,$$3, $$K_i^H = \lim_{x_i\to 0}\frac{y_i p}{x_i\vert_j} = \frac{\partial y_i}{\partial x_i}\Big\vert_j\, p,$$4 mol for $$K_i^H = \lim_{x_i\to 0}\frac{y_i p}{x_i\vert_j} = \frac{\partial y_i}{\partial x_i}\Big\vert_j\, p,$$5, $$K_i^H = \lim_{x_i\to 0}\frac{y_i p}{x_i\vert_j} = \frac{\partial y_i}{\partial x_i}\Big\vert_j\, p,$$6 mol for $$K_i^H = \lim_{x_i\to 0}\frac{y_i p}{x_i\vert_j} = \frac{\partial y_i}{\partial x_i}\Big\vert_j\, p,$$7, $$K_i^H = \lim_{x_i\to 0}\frac{y_i p}{x_i\vert_j} = \frac{\partial y_i}{\partial x_i}\Big\vert_j\, p,$$8 mol for $$K_i^H = \lim_{x_i\to 0}\frac{y_i p}{x_i\vert_j} = \frac{\partial y_i}{\partial x_i}\Big\vert_j\, p,$$9, and $$\alpha_{ij} = \frac{\frac{\partial y_i}{\partial x_i}\vert_j} {\frac{\partial y_j}{\partial x_j}\vert_i}.$$0 kW for reboiler duty. Uncertainty is quantified by conformal prediction, with calibrated intervals such as $$\alpha_{ij} = \frac{\frac{\partial y_i}{\partial x_i}\vert_j} {\frac{\partial y_j}{\partial x_j}\vert_i}.$$1 mol, $$\alpha_{ij} = \frac{\frac{\partial y_i}{\partial x_i}\vert_j} {\frac{\partial y_j}{\partial x_j}\vert_i}.$$2 mol, and $$\alpha_{ij} = \frac{\frac{\partial y_i}{\partial x_i}\vert_j} {\frac{\partial y_j}{\partial x_j}\vert_i}.$$3 kW (Bubel et al., 8 Sep 2025).

A major practical use is entrainer selection for azeotropic or extractive distillation. In the flowsheet-optimization formulation, the modelfluid makes continuous optimization over hypothetical entrainer properties possible, after which real candidates are ranked by proximity in feature space. In the reusable-surrogate formulation, one surrogate is reused across three columns in an entrainer-distillation flowsheet. The reported ranking results include correct identification of the top 10 candidates, a mean absolute ranking error of 3.73 for the surrogate-based ranking, compared with 4.84 for the earlier approximate method, and a reduction of optimization runtime from hours to minutes (Bubel et al., 2 Sep 2025, Bubel et al., 8 Sep 2025).

3. Effective constitutive modelfluids in unresolved porous media

In multiscale porous-media simulation, modelfluid representation has a different meaning. The problem is not chemical reusability but geometric under-resolution: some pores are fully resolved, while others are smaller than the grid and cannot be represented explicitly. The unresolved zones are therefore treated as an effective fluid representation, namely a local constitutive model whose action enters the two-component solver through source terms (Yang et al., 25 Dec 2025).

The baseline formulation couples a pseudo-potential lattice Boltzmann multiphase solver to subgrid constitutive laws for absolute permeability, capillary pressure, and relative permeability. In under-resolved regions, the source term contains a capillary contribution active near the interface through a switching function and a viscous resistance analogous to a Brinkman term. This earlier model already reproduced global permeability, capillary pressure, and relative permeability curves reasonably well compared with a fully resolved simulation at up to 10× finer resolution (Yang et al., 25 Dec 2025).

The enhanced formulation adds three capabilities. First, it introduces controllable surface tension in a Shan–Chen pseudo-potential LB model, implemented as a continuous-surface-force-type correction that is applied only in under-resolved porous-media regions. Validation shows that the surface tension can be reduced by about two orders of magnitude while keeping interface thickness and spurious currents essentially unchanged (Yang et al., 25 Dec 2025).

Second, it adds a local constitutive relation for residual fluid components smaller than a cell. Residual water and oil amounts are determined from intersections of the local $$\alpha_{ij} = \frac{\frac{\partial y_i}{\partial x_i}\vert_j} {\frac{\partial y_j}{\partial x_j}\vert_i}.$$4-$$\alpha_{ij} = \frac{\frac{\partial y_i}{\partial x_i}\vert_j} {\frac{\partial y_j}{\partial x_j}\vert_i}.$$5 curve with reference pressures

$$\alpha_{ij} = \frac{\frac{\partial y_i}{\partial x_i}\vert_j} {\frac{\partial y_j}{\partial x_j}\vert_i}.$$6

and the local capillary curve is transferred across porosities using a Leverett-type scaling,

$$\alpha_{ij} = \frac{\frac{\partial y_i}{\partial x_i}\vert_j} {\frac{\partial y_j}{\partial x_j}\vert_i}.$$7

This lets the model retain irreducible oil or water that diffusive-interface solvers miss when residual blobs are cell-sized or sub-cell-sized (Yang et al., 25 Dec 2025).

Third, it replaces scalar resistance by a tensorial resistivity model for heterogeneous unresolved structures such as fiber bundles. The principal axes are extracted from the eigenvectors of the Hessian of the local porosity field, and resistivity is aligned with these directions. This permits directional flow restriction rather than isotropic damping, which is essential in zigzag or strongly anisotropic media (Yang et al., 25 Dec 2025).

The decisive point is representational: a fully resolved case simulates every pore and interface directly, whereas the modelfluid substitutes the unresolved microstructure by locally applied constitutive relations obtained from representative subdomains. Benchmark cases show improved transient interface dynamics, better capture of residual phases, and correct directional effects. In one reported test, the approach yields about a three-order-of-magnitude reduction in computational cost while substantially improving fidelity relative to the earlier multiscale formulation (Yang et al., 25 Dec 2025).

4. Geometric and materialized representations of fluid structure

A distinct usage represents fluid structure geometrically or even physically. In fluid-mechanics visualization, a coherent structure can be turned into a 3D-printable object: start with a scalar field $$\alpha_{ij} = \frac{\frac{\partial y_i}{\partial x_i}\vert_j} {\frac{\partial y_j}{\partial x_j}\vert_i}.$$8, choose an isovalue, extract an isosurface with MATLAB’s isosurface, export it with stlwrite, and fabricate it as an STL model (Taira et al., 2017). The printed object is therefore a selected, reduced, and materialized representation of a flow feature, not a simulation of the full field.

The reported examples are a $$\alpha_{ij} = \frac{\frac{\partial y_i}{\partial x_i}\vert_j} {\frac{\partial y_j}{\partial x_j}\vert_i}.$$9-criterion isosurface for vortical structures in the wake of a pitching rectangular flat-plate wing at $y_i\, p = x_i\, \gamma_i\, p_i^{sat}(T),$0, and the positive and negative isosurfaces of a dominant global stability mode in a compressible cavity flow at $y_i\, p = x_i\, \gamma_i\, p_i^{sat}(T),$1, $y_i\, p = x_i\, \gamma_i\, p_i^{sat}(T),$2, $y_i\, p = x_i\, \gamma_i\, p_i^{sat}(T),$3. The paper emphasizes educational and research value, but also notes limitations: printer resolution restricts fine structures, multi-component objects may require separate printing and assembly, and the result depends on the chosen scalar and isovalue because it represents one level set rather than the entire flow field (Taira et al., 2017).

A more formal geometric representation appears in two-dimensional multiphase topology modeling. There, a fluid phase is a Yin set: a regular open semianalytic subset of $y_i\, p = x_i\, \gamma_i\, p_i^{sat}(T),$4 with bounded boundary. Its boundary can be decomposed uniquely into oriented Jordan curves, and the corresponding Jordan space is in bijection with the fluid modeling space. Boolean operations on fluids are then carried out by operating on curve segments rather than dense regions, and topological data such as connected components and holes are readable in $y_i\, p = x_i\, \gamma_i\, p_i^{sat}(T),$5 time from the number and orientation of curves (Zhang et al., 2019).

Other geometric-interface representations replace sharp boundaries by richer constitutive or mixture objects. A homogeneous two-fluid model for violent free-surface compressible flow replaces the air–water interface by a thin three-dimensional interpenetrating mixture zone with local volume fractions $y_i\, p = x_i\, \gamma_i\, p_i^{sat}(T),$6 and $y_i\, p = x_i\, \gamma_i\, p_i^{sat}(T),$7, while both phases share the same velocity, pressure, and temperature. This yields a conservative, hyperbolic model that handles wave breaking and topological changes implicitly (0802.3013). In topology optimization for Stokes flow, an anisotropic tensor-mixture model represents pure fluid, pure solid, no-slip, and free-slip within one differentiable constitutive law on a background grid, using directional tensors $y_i\, p = x_i\, \gamma_i\, p_i^{sat}(T),$8, $y_i\, p = x_i\, \gamma_i\, p_i^{sat}(T),$9, and $T_i$0 to encode normal and tangential behavior near interfaces (Li et al., 2022).

These approaches differ in purpose, but all shift the representational burden from pointwise fluid state to a geometric object: an isosurface, an oriented boundary complex, a mixture zone, or an anisotropic constitutive tensor.

5. Algebraic, spectral, and compositional state representations

Another large class of related work represents fluids in algebraic or modal state spaces. One line rewrites incompressible Navier–Stokes in terms of differential forms and reproduces the same structure on finite-dimensional cochain complexes. In that setting, $T_i$1 is replaced by the coboundary $T_i$2, the wedge product by a cup product, and incompressibility by projection onto $T_i$3. Two concrete models are given: a purely combinatorial toy model and a Whitney model on triangulated manifolds. For co-closed solutions, the finite models obey the exact energy identity

$T_i$4

and the Whitney model approximates the smooth fluid vector field weakly under mesh refinement (Wilson, 2011).

A spectral representation is developed for relativistic heavy-ion fluid dynamics in FluiduM. The fireball is decomposed into a smooth background plus fluctuations, and the fluctuations are expanded in basis functions adapted to azimuthal rotations and longitudinal boosts. The azimuthal dependence is written as $T_i$5, radial structure is expanded in a Bessel-type basis, and the resulting mode amplitudes evolve in proper time. Numerically, this is solved with a pseudo-spectral method, and comparison with the analytic Gubser solution is used as a high-accuracy benchmark (Floerchinger et al., 2018).

A compositional representation is provided by Exergetic Port-Hamiltonian Systems. There, the fluid is built from storage components, reversible interconnection components, and irreversible Onsager-type components coupled by ports. The ideal fluid model is represented by kinetic-energy and internal-energy subsystems; Navier–Stokes–Fourier is obtained by adding thermal conduction, bulk viscosity, and shear viscosity; and the resulting NSF fluid is then reused as a subsystem inside electro-magneto-hydrodynamics and MHD models. The formal structure guarantees conservation of energy, non-negative entropy production, and Onsager reciprocal relations (Lohmayer et al., 2024).

These representations have a common technical character: the fluid state is not a raw field on a mesh alone, but an element of a structured algebra, basis, or network whose formal properties are chosen to preserve incompressibility, energy balance, symmetry classification, or thermodynamic consistency.

6. Adaptive and stochastic multi-model representations

Some representations are dynamic: the model itself changes during the simulation, or the state is represented by a coupled stochastic object rather than a deterministic field. In adaptive coupling of 3D and 2D fluid flow, the central idea is model adaptivity: instead of changing only resolution, the governing equations switch locally between 3D incompressible Navier–Stokes and a 2D thin-film model. The decision is particle-wise, based on user-prescribed resolution and a local principal component analysis of neighboring particles. When a 3D particle is converted to 2D, nearby interior particles are absorbed and their data are mapped onto a wall particle; when a 2D particle is promoted to 3D, new bulk particles are generated along the normal direction to match the local film height. The framework includes two-way coupling through ghost particles and conservative mass transfer, and the reported industrial example achieves about a 3.2× speedup at the finest reported resolution while maintaining comparable results in the region of interest (Suchde, 2024).

A conceptually related multi-level strategy appears in a vortex-based simulation that can change its level of description during runtime. Fine-grained vortex particles are clustered by Delaunay triangulation, minimal spanning trees, convex hulls, and ellipse fitting; accepted structures are then reified as higher-level vortex entities managed by multiplicity automata and eco-resolution-like behaviors such as growth, shrinkage, fusion, and disintegration. Solids are handled through virtual particles. The representation is explicitly designed so that the active grain size can shift as coherent structures emerge or decay (0712.2643).

Stochastic fluid-fluid models provide a different kind of nonstandard representation. The state is a three-dimensional Markov process $T_i$6 consisting of a finite-state phase process and two nonnegative fluid levels. The main contribution is to translate an operator-analytic framework into a matrix-analytic one, with evolution operators written as matrix exponentials such as $T_i$7 and $T_i$8. This makes distributions at level-dependent stopping times, first-return probabilities, and stationary quantities computationally accessible in the same style as classical stochastic fluid queues (Bean et al., 2020).

These adaptive and stochastic formulations extend the representational idea beyond static encodings. The represented object may now be a locally switchable model family, a hierarchy of dynamically reified entities, or a matrix-analytic process with fluid levels and phase states.

7. Shared principles, misconceptions, and limits

Across the cited literature, modelfluid representation is consistently task-driven. The distillation papers make this explicit by requiring availability, integrability, interpretability, and continuity; the porous-media paper emphasizes local constitutive closure for unresolved regions; the EPHS formulation emphasizes conservation laws and entropy production; and the geometric papers emphasize exact topology or inspectable structure (Bubel et al., 2 Sep 2025, Yang et al., 25 Dec 2025, Lohmayer et al., 2024, Zhang et al., 2019).

A recurrent misconception is to equate representation with complete physical replication. The chemical-process modelfluid is not a molecular identity encoding but a continuous feature layer mapped into Margules and Antoine parameters (Bubel et al., 2 Sep 2025). The porous-media modelfluid does not simulate each subgrid pore but replaces missing geometry by constitutive relations (Yang et al., 25 Dec 2025). The 3D-printed object is not the whole flow field but one extracted isosurface of a chosen scalar (Taira et al., 2017). The adaptive 3D/2D framework does not retain a single governing PDE everywhere, but explicitly changes model and discretization in space and time (Suchde, 2024).

The chief advantages follow from that reduction. Continuous thermodynamic descriptors enable gradient-based optimization and reusable surrogates across broad chemical spaces (Bubel et al., 2 Sep 2025, Bubel et al., 8 Sep 2025). Effective constitutive closures retain unresolved capillary, viscous, trapping, and anisotropic effects at coarse resolution (Yang et al., 25 Dec 2025). Boundary-based and algebraic formalisms make topology, energy balance, or symmetry channels explicit (Zhang et al., 2019, Wilson, 2011, Floerchinger et al., 2018). Compositional energy-based models make subsystem reuse and replacement straightforward while preserving thermodynamic structure (Lohmayer et al., 2024).

The limitations are equally representation-specific. Distillation modelfluids are tailored mainly to homogeneous VLE systems and rely on simplified Margules, Antoine, and enthalpy models (Bubel et al., 2 Sep 2025). Porous-media closures depend on representative constitutive libraries and remain sensitive to unresolved-scale assumptions (Yang et al., 25 Dec 2025). Printed or boundary-based representations depend on threshold choice, geometric regularity, or fabrication limits (Taira et al., 2017, Zhang et al., 2019). Adaptive multi-model schemes introduce heuristic switching criteria and data-transfer complexity (Suchde, 2024).

In this sense, modelfluid representation is best characterized not by one universal encoding, but by a recurring methodological principle: choose a fluid representation whose variables, structure, and constraints are aligned with the dominant physics and the intended computation. The literature shows this principle operating in thermodynamic optimization, porous-media upscaling, topology-aware geometry, spectral hydrodynamics, finite-model discretization, compositional multiphysics, and adaptive multi-model flow simulation (Bubel et al., 8 Sep 2025, Yang et al., 25 Dec 2025, Floerchinger et al., 2018, Wilson, 2011, Lohmayer et al., 2024).