Fluid: Theory, Simulation, and Applications

• Fluid is defined as any material system that flows under any applied shear stress, characterized by continuous deformation and a rate-dependent response.
• Continuum models, such as the Navier–Stokes equations and phase-field approaches, capture key dynamics including interface motion and complex rheology.
• Recent advances integrate machine learning and statistical mechanics to enhance simulation accuracy and predict fluid-structure interactions in diverse applications.

A fluid is any material system that cannot sustain a finite shear force in static equilibrium, undergoing continuous deformation (“flow”) under arbitrarily small applied shear stress. This phenomenological definition encompasses classical liquids, dense and ideal gases, plasmas, and even highly viscous or plastic solids that flow on sufficiently long timescales. At the theoretical level, a fluid is characterized by a constitutive response in which shear stress depends on the instantaneous rate of deformation, not the magnitude of deformation itself. The mathematical and computational modeling of fluids, their interfaces, phase transitions, and their interactions with other materials is foundational to numerous fields, including continuum mechanics, statistical physics, computational science, geosciences, and engineering design (Mryglod et al., 2022).

1. Fundamental Properties and Microscopic Structure

A defining feature of all fluids is their inability to support static shear stresses—they flow in response to infinitesimal tangential forces, storing negligible elastic energy except in transient microstructural arrangements. Shear stress $\tau_{xy}$ in Newtonian fluids is linearly proportional to the local strain rate $\partial v_x/\partial y$, with proportionality constant $\eta$ (shear viscosity), i.e., $\tau_{xy} = \eta\,\partial v_x/\partial y$. For non-Newtonian fluids, this dependence is nonlinear or history-dependent, reflecting memory effects and microscopic structure (Mryglod et al., 2022); (Bui et al., 27 Nov 2025).

At the microscopic scale, fluids are distinguished by their lack of infinite-range positional order. Liquids and amorphous solids possess pronounced short-range order, manifested in shells of nearest-neighbor coordination captured by the pair correlation function $g(r)$. The absence or decay of long-range oscillations in $g(r)$, together with broad peaks in the structure factor $S(k)$, distinguish fluid phases from crystalline solids (Mryglod et al., 2022). This local structure strongly influences dynamics and transport—phenomena such as the cage effect (transient particle localization), nonmonotonic velocity autocorrelation, and viscoelastic response are direct consequences of sample-spanning but impermanent microenvironments.

2. Governing Equations of Fluid Dynamics

The continuum mechanical description of fluids is specified by the conservation laws for mass, momentum, and, where appropriate, energy:

• Continuity equation (mass conservation):

$$\frac{\partial\rho}{\partial t} + \nabla \cdot (\rho\,\mathbf{v}) = 0.$$

• Navier–Stokes equation (momentum balance):

$$\rho\left(\frac{\partial\mathbf{v}}{\partial t} + \mathbf{v}\cdot\nabla\mathbf{v}\right) = -\nabla p + \nabla\cdot\boldsymbol{\sigma} + \mathbf{f}.$$

Here, $p$ is pressure, $\boldsymbol{\sigma}$ is the viscous (deviatoric) stress tensor, and $\mathbf{f}$ external body forces (Mryglod et al., 2022).

The Newtonian viscous stress tensor for an isotropic incompressible fluid is:

$$\sigma_{ij} = \eta (\partial_i v_j + \partial_j v_i).$$

Extensions to compressible, viscoelastic, or multiphase systems introduce additional terms, such as bulk viscosity, relaxation terms, or interfacial stresses (surface tension) (Mahady et al., 2014).

For multiphase, interface-rich, or complex fluid systems, phase-field, volume-of-fluid, and other Eulerian level-set or indicator-function formulations generalize the Newtonian framework, enabling the modeling of interface motion, topological changes, and surface interactions (1803.02354); (Prajapati et al., 28 May 2025).

3. Fluid-Fluid and Fluid-Solid Interfaces

At interfaces between fluid phases (fluid-fluid) or between fluid and solid phases (fluid-solid), additional physical principles apply:

• Fluid-fluid demixing: In binary or multicomponent fluid mixtures, phase separation (demixing) may occur, typically governed by excluded-volume interactions and differences in microscopic affinity. Fundamental measure density functional theory (FMT-DFT) and Ornstein–Zernike integral equations provide quantitative tools for elucidating bulk phase coexistence, interface structure, the asymptotic decay of pair correlations, and associated Fisher–Widom crossovers between oscillatory and monotonic correlation decay (Hopkins et al., 2010).
• Fluid-solid demixing and crystallization: In high-dimensional systems, entropy-driven separation into coexisting fluid and solid phases preempts fluid-fluid demixing, as shown by spinodal and bifurcation analysis in mixtures of parallel hard hypercubes (Lafuente et al., 2010).
• Wetting and contact-line dynamics: Fluid/solid surface interactions (e.g., van der Waals, electrostatic) generate microscopic body forces that set equilibrium contact angles (Young’s law) and regularize moving contact lines. Explicit inclusion of these forces in direct Navier–Stokes solvers with VOF interface tracking captures the correct static and dynamic behavior of contact lines for arbitrary equilibrium angles, obviating the need for geometric angle imposition or slip models (Mahady et al., 2014).

4. Complex Rheology and Multiphase Behavior

While the canonical fluid is Newtonian, most geophysical, biological, and industrially significant fluids are non-Newtonian. These materials exhibit yield stresses, shear-thinning or thickening, viscoelasticity, or plasticity (Bui et al., 27 Nov 2025); (Xian et al., 2023). Examples include:

• Herschel–Bulkley–Papanastasiou fluids: These exhibit a finite yield stress $\tau_0$, power-law dependence of viscosity, and require regularization for numerical simulation. Non-Newtonianity introduces key effects in flow through porous and fractured media: partitioning of flow into yielded and unyielded ("rigid") zones, nonlinear pressure–flow relationships (e.g., pre-Darcy, Forchheimer), multimodal velocity distributions, and pronounced scale- and geometry-dependent inertia (Bui et al., 27 Nov 2025).
• Complex materials in simulation: Cutting-edge simulators employ the Material Point Method (MPM) or Eulerian one-continuum VOF formalisms to handle fluids, plastics, and hyperelastic solids within a unified computational framework. Interface-capturing approaches (geometric VOF/PLIC, phase field, high-order WENO advection) maintain sharp interfaces, robust mass conservation, and accommodate topological changes without remeshing (Xian et al., 2023); (Prajapati et al., 28 May 2025); (1803.02354).

5. Statistical Mechanical and Theoretical Approaches

Beyond continuum models, fluid dynamics is tightly linked to non-equilibrium statistical mechanics. Approaches such as the non-equilibrium statistical operator (NSO) and generalized collective modes (GCM) frameworks enable systematic derivation of transport coefficients, prediction of emergent propagating modes (sound, "fast sound," shear waves), and explanation of anomalies in ionic melts, supercritical fluids, and glassy states (Mryglod et al., 2022). Sum rules, mode-coupling, and memory-function analyses explicitly connect microscopic time correlation functions to macroscopic hydrodynamics and generalized transport behavior.

Table: Examples of Fluid Phenomena and Their Statistical/Theoretical Significance

Phenomenon	Theoretical Framework	Key Microscopic Feature
Cage effect	GCM/NSO	Local transient "cage" order
Shear wave onset in liquids	GCM, viscoelasticity	Finite local shear modulus
Fast sound in mixtures	GCM, mode coupling	Disparate mass/charge species
Nonlinear transport	Memory function, GCM	Non-Markovian relaxation

The systematic extension of the set of slow variables or moments in such frameworks ensures sum-rule consistency and accurate prediction of hydrodynamic–kinetic crossovers (Mryglod et al., 2022).

6. Machine Learning, Numerical, and Algorithmic Advances

Recent advances in data-driven modeling, especially the integration of deep convolutional neural networks (CNNs) and LLMs, have demonstrated that high-fidelity, differentiable, and generalizable fluid simulators can be constructed without reliance on classical simulation data. Notably:

• Physics-constrained neural solvers: CNNs trained by penalizing PDE residuals on random domain ensembles can produce stable, accurate, and fully differentiable incompressible flow solvers, generalizing to out-of-distribution geometries and achieving real-time performance. These architectures exploit physical invariance, divergence-free velocity fields via Helmholtz decomposition, and U-Net update steps (Wandel et al., 2020).
• LLM-based spatiotemporal frameworks: FLUID-LLM combines pre-trained transformer models with patch-based spatiotemporal encoding and local GNN decoding to accurately autoregress unsteady CFD on standard benchmarks, outperforming mesh GNN baselines even at long prediction horizons. The global attention and pretraining bias intrinsic to LLMs facilitate few-shot generalization and context-based reasoning about fluid behavior (Zhu et al., 2024).

Fluid manipulation tasks in robotics, synthetic biology, and materials processing mandate simulation of coupled fluid–solid and multicomponent interactions in differentiable environments (e.g., FluidLab), further driving demand for robust, unifying, and GPU-accelerated frameworks (Xian et al., 2023); (Prajapati et al., 28 May 2025).

7. Fluid-Structure and Contact Interactions

The interaction of fluids with deformable or moving solid boundaries—fluid–structure interaction (FSI)—is fundamental in biomechanics, engineering, and natural systems. Accurate FSI modeling requires:

• Monolithic, sharp-interface Eulerian or ALE methods: These formulations enforce no-slip, slip, or frictionless contact conditions, treat interface conditions via weak (Nitsche) or penalty-based methods, and maintain consistency as the interface moves or undergoes topological changes (Sauer et al., 2017); (Ager et al., 2019).
• Diffuse-interface and phase-field approaches: Fully Eulerian models employing phase fields, capturing both viscous and elastic stresses within unified energy-driven frameworks, naturally enable simulation of large deformations, contact, adhesion, and topological transitions. Matched asymptotic analysis confirms convergence to classical sharp-interface limits (1803.02354); (Prajapati et al., 28 May 2025).
• Benchmarks covering a wide Re regime: Verified implementations correctly replicate key physical benchmarks, from laminar deformation and reversibility to turbulent interactions in compliant-walled channels, robustly capturing flow, pressure, and interface dynamics across multiple orders of magnitude in Reynolds number (Prajapati et al., 28 May 2025); (Sauer et al., 2017).

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In sum, “fluid” embeds a hierarchy of theoretical, computational, and empirical constructs, connecting fundamental symmetry-breaking at the material level to intricate multiscale phenomena spanning flow, phase transition, and interacting domains. The intersection of rigorous statistical mechanics, advanced simulation methodologies, and emerging machine learning paradigms continues to expand the boundaries of fluid research, with implications for both foundational physics and practical applications (Mryglod et al., 2022); (Hopkins et al., 2010); (Wandel et al., 2020); (Zhu et al., 2024); (Prajapati et al., 28 May 2025); (Bui et al., 27 Nov 2025).