Spatially-Resolved Fluidity Model

• The spatially-resolved fluidity model defines a position-dependent parameter that quantifies material flow by linking local shear, microstructure, and defect contributions.
• It incorporates nonlocal effects via PDEs and kinetic closures to capture multiscale physics in systems like granular flows, glasses, membranes, and porous media.
• The model informs experimental and computational design by providing predictive constitutive relations that couple microscale properties with macroscopic transport phenomena.

A spatially-resolved fluidity model introduces a local, typically position-dependent, field quantifying a material's ability to flow or deform, with spatial variations arising from microstructure, defects, composition, or external forcing. Such models generalize classic constitutive laws to include nonlocal and inhomogeneous phenomena, and are central to the description of granular materials, glasses, membranes, thin films, and multiscale porous flows. Spatial fluidity models can admit operational, microscopic, or phenomenological definitions, but universally facilitate the bridging of scales—linking atomistic or mesoscopic structure to macroscopic flow behavior via partial differential equations (PDEs), kinetic models, or numerical closure relations.

1. Definitions of Spatially-Resolved Fluidity Fields

The precise definition of fluidity is system-dependent. For dense granular flows, the "granular fluidity" field $g(\mathbf{x})$ admits both an operational and a microscopic form (Zhang et al., 2016):

• Operational (macroscopic):

$$g(\mathbf{x}) = \frac{\dot\gamma(\mathbf{x})}{\mu(\mathbf{x})} = \frac{\text{local shear rate}}{\text{local stress ratio}}$$

• Microscopic (kinematic):

$$g(\mathbf{x}) = \frac{\delta v(\mathbf{x})}{d} F(\Phi(\mathbf{x}))$$

where $\delta v$ is the root-mean-square velocity fluctuation, $d$ is particle diameter, $\Phi$ is the local packing fraction, and $F(\Phi)$ vanishes at random close packing.

In amorphous thin films with radiation-enhanced fluidity, the local inverse viscosity is a linear function of defect concentrations (Evans et al., 12 Oct 2025):

$$\frac{1}{\eta(\mathbf{x},t)} = \frac{1}{\eta^\ast} + \frac{C_I(\mathbf{x},t)}{\eta_I} + \frac{C_V(\mathbf{x},t)}{\eta_V}$$

where $C_I$ and $C_V$ are interstitial and vacancy concentrations.

For glassy films, the local fluidity $f(z)$ at depth $z$ is the inverse of the local relaxation time,

$f(z) = \tau_\alpha(z)^{-1},$

derived via dynamical overlap functions and Maxwell’s relation (Zhai et al., 2024).

Lipid bilayers present two competing definitions for the lateral fluidity:

• Local fluidity: $\mu(z) \equiv 0$ everywhere in the monolayer thickness $z$,
• Global fluidity: $\int \mu(z)\,dz = 0$, allowing finite, sign-varying local shear modulus (Pinigin, 2024).

2. Governing Equations and Constitutive Relations

Spatially-resolved fluidity models are encoded either as explicit dynamic fields (satisfying PDEs) or through local constitutive laws closed by auxiliary fields:

• Granular flows: Fluidity satisfies a nonlocal, diffusive-type PDE (Zhang et al., 2016):

$$t_0\,\dot g = A^2 d^2\nabla^2g - \Delta\mu\frac{\mu_s-\mu}{\mu_2-\mu}g - b\sqrt{\frac{\rho_s d^2}{P}\mu}g^2$$

subject to boundary conditions (Dirichlet for $g$ at rough walls, Neumann at free surfaces).

• Porous media: A Lattice Boltzmann approach introduces effective body-forces:

$$F^{\mathrm{ms}}_y(\mathbf{x}) = -P_c(S_w(\mathbf{x}))\, n_y\, H(\mathrm{At},|\nabla\mathrm{At}|) - \mu_a\,K_{\rm abs}(\phi(\mathbf{x}))\,K_{r,a}(S_a(\mathbf{x}),\phi(\mathbf{x}))\,u_y(\mathbf{x})$$

with locally determined permeability and capillarity from a constitutive library (Yang et al., 25 Dec 2025).

• Lipid vesicles: Surface fluidity and viscosity $\eta(c)$, depending on phase composition $c(\mathbf{x})$, enter a Navier–Stokes–Cahn–Hilliard system posed on the curved membrane with surface differential operators (Wang et al., 2021).
• Thin films under irradiation: Reaction–diffusion–recombination equations for defects are tightly coupled to the lubrication-Stokes equations for the flow, with viscosity $\eta(\mathbf{x},t)$ slaved to the evolving defect densities (Evans et al., 12 Oct 2025).

3. Physical Origins and Microscopic Justification

The emergence and spatial variation of fluidity derive from a range of microscale mechanisms:

• Granular materials: Micro-kinetic theory relates $g$ to velocity fluctuations and packing fraction via dense-gas analogies, while Eyring-type models connect $g$ to activated rearrangement rates (Zhang et al., 2016).
• Polymeric and glassy films: Facilitation emerges through void-mediated hopping and relaxation, with elevated surface mobility propagating into the bulk as a dynamic “front” (Zhai et al., 2024).
• Lipid bilayer mechanics: Distinguishing between local and global fluidity is crucial for accurate prediction of stress distributions, as only global fluidity is compatible with nonzero local shear stress in curved geometries (Pinigin, 2024).
• Irradiated films: Ion flux, energy deposition, and recombination define nonuniform defect fields, resulting in depth-dependent fluidity that must be resolved to capture short-wavelength instabilities (Evans et al., 12 Oct 2025).

4. Numerical Implementation and Boundary Conditions

Spatially-resolved fluidity models are computationally demanding due to multiscale coupling:

• Granular flows: Finite-difference or finite-element schemes march the $g$-PDE in pseudo-time, updating stress and velocity until a steady state is reached (Zhang et al., 2016).
• Porous media: Constitutive closure is achieved by coarse-graining pore-scale data and storing lookup tables for permeability, capillary pressure, and relative permeability; multiscale forcing is incorporated at each LB step (Yang et al., 25 Dec 2025).
• Lipid vesicles: Trace/cut-FEM methods discretize surface PDEs, with phase-dependent viscosity smoothly interpolated across interfaces (Wang et al., 2021).
• Glassy films: Simulation relies on lattice models with Metropolis dynamics and depth-resolved overlap analysis; continuum flow is solved using lubrication theory (Zhai et al., 2024).
• Thin films: Standard boundary conditions (no-slip at solid boundaries, free-surface stress balance) are imposed for the flow, along with no-flux for defect fields at the substrate (Evans et al., 12 Oct 2025).

5. Key Results, Validation, and Nonlocal Effects

Spatial fluidity resolution enables quantitative agreements with experiments and reproduces nonlocal or emergent dynamics:

• Granular flows: The PDE, kinetic, and operational definitions of $g$ collapse across 20 different flow geometries. Nonlocal phenomena such as creeping flow below yield, stopping-angle dependence, and stress propagation from boundaries are captured (Zhang et al., 2016).
• Multiphase porous flow: The enhanced LB model recovers breakthrough curves, capillary fingering, and residual saturation patterns matching high-resolution simulations at a fraction of the cost. Directional transport in anisotropic fiber bundles is accurately modeled using tensorial resistivity (Yang et al., 25 Dec 2025).
• Polymers and glasses: The surface mobility gradient and transition to surface-dominated flow as temperature decreases are reproduced without additional fitting parameters. The penetration depth of fluidity depends on temperature, with a clear regime transition from bulk to surface control (Zhai et al., 2024).
• Irradiated films: Spatial variation in fluidity alters the surface morphological stability and relaxation rates by up to 10–20% in regimes with low defect mobility or high nonuniformity, in contrast to constant-fluidity approximations (Evans et al., 12 Oct 2025).
• Lipid membranes: MD simulations confirm that local stress anisotropy and finite lateral shear modulus exist in curved bilayers, favoring the global over the local fluidity hypothesis. This finding impacts the calculation of bending and Gaussian curvature moduli (Pinigin, 2024).

6. Theoretical and Practical Significance

Spatially-resolved fluidity models constitute a unifying framework to address complex flow and mechanical phenomena not accessible to local or constant-property theories. They:

• Enable predictive modeling of size-dependent, nonlocal, and interface-driven transport phenomena, crucial for understanding rheology in small systems, thin films, and confined geometries.
• Bridge scales from microstructure (defect kinetics, packing, phase composition) to macroscopic flow.
• Provide quantitative guidance for experimental design and interpretation in soft condensed matter, granular physics, porous media, thin-film nanofabrication, and biological membranes.

The table below summarizes characteristic ingredients of spatially-resolved fluidity models in several systems:

System	Fluidity Field	Microscale Origin / Closure
Granular flows	$g = \dot\gamma/\mu$; $g = (\delta v/d) F(\Phi)$	Kinetic theory, DEM, Eyring-like models
Glassy films	$f(z) = \tau_\alpha(z)^{-1}$	Void/hopping facilitation, overlap metrics
Lipid bilayers (global)	$\int \mu(z)\,dz = 0$, with $\mu(z) \neq 0$	Lateral monolayer elasticity, MD
Porous media	Cell-wise $K_{abs}$, $K_r$, $P_c$, tensorial resistivity from local porosity	Pore-scale simulation, closure library
Irradiated thin films	$\eta^{-1} = \eta^{*-1} + C_I/\eta_I + C_V/\eta_V$	Point-defect kinetics + hydrodynamics

7. Limitations and Outstanding Questions

While spatially-resolved fluidity models yield improved quantitative and qualitative predictions, several limitations remain:

• Molecular parameterization of fluidity fields is frequently phenomenological or empirically fitted; ab initio computation of mobility or viscosity landscapes is system-dependent and computationally intensive.
• Closure relations (e.g., $F(\Phi)$ in granular flows) or empirical exponents (penetration depth in glassy films) may lack universality across materials or regimes.
• Coupling of spatially-varying fluidity with anisotropic, inhomogeneous, and dynamically evolving microstructure (e.g., in multiphase or living systems) is an open challenge.
• The interplay of stochasticity, thermal fluctuations, and spatial disorder is often neglected or only partially captured in continuum closures.

Despite these open issues, spatially-resolved fluidity models form a foundational tool in modern rheology, microfluidics, membrane biophysics, and mesoscopic materials science, enabling description, simulation, and engineering of transport and stability in complex fluids and soft materials (Zhang et al., 2016, Zhai et al., 2024, Pinigin, 2024, Yang et al., 25 Dec 2025, Evans et al., 12 Oct 2025, Wang et al., 2021).