Nonlinear 2D Reynolds Model

• The nonlinear 2D Reynolds model is a reduced-order approach capturing essential hydrodynamic interactions in viscous flows via quadratic nonlinearities and complex boundary effects.
• It utilizes techniques like asymptotic expansion, conformal mapping, and mode reduction to derive simplified equations that accurately reflect phenomena such as turbulence, thin-film lubrication, and MHD dynamics.
• The models bridge rigorous mathematical theory with practical applications, enabling insights into flow instabilities and nonlinear phenomena in diverse regimes including low-Re swimming and non-Newtonian lubrication.

The nonlinear two-dimensional Reynolds model is a class of reduced-order hydrodynamic models that capture essential nonlinear effects in two-dimensional (2D) viscous flows, either by retaining nonlinear interactions derived from the Navier–Stokes equations or by integrating the impact of physical or geometric complexities such as boundary deformation, pressure- and shear-dependence of viscosity, non-Newtonian rheology, surface tension, or turbulence. These models arise in diverse contexts, including low-Reynolds number swimming, turbulent shear flows, thin-film lubrication, MHD, transitional turbulence, and data-driven reduction of high-dimensional dynamics, and feature rigorous mathematical derivation to account for the dominant nonlinear mechanisms specific to the regime and physical problem under consideration.

1. Mathematical Structure and Nonlinear Coupling

A central trait of nonlinear 2D Reynolds models is the explicit inclusion of nonlinear terms that derive from the physical setting, such as the full quadratic nonlinearity of the Navier–Stokes equations, or additional nonlinearities arising from viscosity laws, elastic stresses, or coupling with deformable interfaces.

• Complex Variable Formulation in 2D Low-Reynolds Flows: For creeping flows interacting with deformable interfaces, the streamfunction $\psi(x,y)$ can be expressed in terms of complex analytic functions (“Goursat functions”):

$$\psi(x, y) = \mathrm{Im}\left[\overline{z}\, f(z) + g(z)\right],$$

where the disturbance flow generated by singularities (e.g., stresslet, dipole, quadrupole) interacts nonlinearly with a deforming boundary, and the resultant system is closed via highly nonlinear free-boundary conditions (Crowdy et al., 2010). Only certain deformed interface shapes admit steady, translating solutions, due to this nonlinear hydrodynamic coupling.

• Nonlinear Power Series and Reynolds Number Expansion: For in-plane flows at arbitrary (but not large) Re, direct expansion in powers of Re:

$u^N(t) = \sum_{j=0}^N R^j u_j(t)$

constructs approximate solutions with recursively included nonlinear interactions up to order $N$ (Morosi et al., 2013). The remainder scales as $O(R^{N+1})$, and a posteriori estimates provide global existence criteria in terms of a critical Reynolds number $R_*(N)$.

• Band-limited Nonlinear Models for Wall-bounded Turbulence: The restricted nonlinear (RNL) model for wall-bounded turbulence separates the flow into a streamwise-constant mean and streamwise-varying perturbations, restricting nonlinear self-interaction to the mean and reducing the perturbation dynamics to forced linear problems. Band-limited variants select a small set of kinematic modes, dramatically reducing computational cost yet capturing key nonlinear feedbacks and mean profile features (Bretheim et al., 2014).
• Nonlinear Thin-Film and Lubrication Models: For thin-film flows with pressure- or shear-dependent viscosity, asymptotic expansion, scaling, and homogenization yield two-dimensional Reynolds-type equations where nonlinearity enters through mobility coefficients (e.g., Carreau or power-law rheology) and new terms coupling shear and cross-film pressure variation. For example, for power-law fluids (Anguiano et al., 2017) or Carreau-law fluids (Anguiano et al., 6 Aug 2025),

$$\text{div}_{x'}\left( \mathcal{U}(\nabla_{x'}\tilde{p} - f') \right) = 0,$$

where $\mathcal{U}$ encapsulates nonlinear permeability and roughness effects, determined by cell problems at the micro-scale.

• MHD and Rotating Systems: In MHD, asymptotic expansions account for Hartmann layer and core inertial effects, and the resulting quasi-2D equations include “Reynolds stress” and nonlinear wall-friction corrections due to vertical recirculation (Pothérat et al., 2020). For rapidly rotating flows, 3D perturbations are stabilized by rotation, and the long-time dynamics become strictly 2D regardless of Reynolds number (Gallet, 2015).

2. Physical Regimes and Application Domains

The nonlinear 2D Reynolds framework is applied across diverse subfields:

Regime	Nonlinearity Type	Physical Problem/Effect
Low-Re, deformable interface	Hydrodynamic, free-boundary	Microswimmer near capillary interface, steady locomotion (Crowdy et al., 2010)
Moderate-Re, spectral expansion	Quadratic, Re-expansion	Existence and error bounds for 2D/3D Navier–Stokes (Morosi et al., 2013)
Wall-bounded, turbulent shear flow	Mode-restricted nonlinearity	RNL, band-limited RNL, logarithmic mean profiles (Bretheim et al., 2014)
Thin-film, non-Newtonian lubrication	Non-Newtonian, roughness coupling	Effective Reynolds equations for power-law/Carreau fluids (Anguiano et al., 2017, Anguiano et al., 6 Aug 2025)
Quasi-2D MHD	Layer-wise, inertial correction	Hartmann wall damping, core “barrel” effect (Pothérat et al., 2020)

The nonlinearities adapt to the leading-order physical interaction: stress–free surface deformation, band-limited shear transfer, pressure/sheer dependence, or external force (e.g., magnetic field or rotation).

3. Mathematical and Computational Techniques

Derivations of nonlinear two-dimensional Reynolds models employ:

• Asymptotic and Homogenization Methods: Thin-film and lubrication limits use domain rescaling and rigorous two-scale homogenization/unfolding to handle oscillatory boundaries and extract effective coefficients, ensuring convergence via a priori estimates and monotonicity arguments (Anguiano et al., 2017, Anguiano et al., 6 Aug 2025).
• Conformal Mapping and Complex Analysis: For 2D Stokes flows beneath deformable interfaces, conformal maps parameterize the evolving free boundary as part of solving the nonlinear free-boundary value problem (Crowdy et al., 2010).
• Mode Reduction and Projection: Galerkin truncations or vertical mode expansions (e.g., minimal wall-normal modes in transitional turbulence (Benavides et al., 2023)) are used to derive reduced PDE systems retaining the essential nonlinear couplings of the trend and perturbation fields.
• Data-driven Reduction (Autoencoders, Symmetry Reduction): In turbulent Kolmogorov flow, deep autoencoders with symmetry reduction and implicit rank minimization distill the high-dimensional turbulent attractor to a low-dimensional nonlinear manifold, enabling quantitative estimates of the attractor dimension and identifying dynamically relevant structures (Cleary et al., 27 May 2025).
• Numerical Continuation and Relaxed Iteration: To solve nonlinear BVPs at high Re, iterative relaxation and adaptive step strategies are required to capture non-unique stationary solutions and bifurcations (Lomasov et al., 29 Oct 2024).

4. Emergent Nonlinear Phenomena and Physical Insights

Nonlinear two-dimensional Reynolds models reveal several physically significant phenomena:

• Nonlinear Self-Selection: In swimming under deformable interfaces (Crowdy et al., 2010), only certain surface shapes, self-consistently selected through the nonlinear problem, admit steady translation—an outcome unattainable for a flat surface.
• Limited Attractor Dimension: In 2D Kolmogorov turbulence, despite the formally infinite-dimensional state space, the effective dimension of the chaotic attractor scales only as $Re^{1/3}$, much weaker than rigorous global bounds, indicating strong nonlinear constraints and the efficiency of reduced manifolds (Cleary et al., 27 May 2025).
• Negative Reynolds Stress in EIT: For two-dimensional elasto-inertial turbulence, the Reynolds stress becomes negative under sufficiently strong elasticity, a signature of non-inertial, elasticity-driven turbulence not observed in Newtonian IT (Cheng et al., 8 Feb 2025).
• Pattern Selection in Transitional Flows: Nonlinear reduced models with accurate Reynolds stress closures can capture the onset and orientation of turbulent-laminar patterns, predicting critical wavenumbers and tilt angles consistent with direct simulations (Benavides et al., 2023).
• Non-monotonic Scaling and Multiple Solutions: In 2D or quasi-2D MHD and high-Re stationary flows, nonlinear effects induce multiple attractors, vortex spreading, and enhanced dissipation, all captured via nonlinear Reynolds-type models (Pothérat et al., 2020, Lomasov et al., 29 Oct 2024).

5. Limitations, Validity, and Broader Impact

While these nonlinear 2D Reynolds models capture dominant order nonlinear effects rigorously under the scaling regimes for which they are derived, several points delimit their validity:

• For high Reynolds numbers in three dimensions or cases with strong three-dimensional instabilities, models based on strictly two-dimensional or “quasi-two-dimensional” closures (even when including advanced nonlinear corrections) may lose quantitative accuracy or predictive capability unless additional physics (such as fully three-dimensional coherence or instability mechanisms) are included.
• In lubrication and thin-film settings, breakdown of model assumptions (e.g., loss of ellipticity when nonlinear stress terms are too large (Gustafsson et al., 2014)) can occur, indicating the need for careful checking of parameter regimes.
• The performance of band-limited and reduced-order models depends on correct empirical or analytical identification of the “dominant” modes or features; universality is not guaranteed, especially far from the tested parameter range.
• Data-driven reductions relying on symmetry reduction and learned embeddings presume exotic nonlinear dynamics are encoded faithfully in the latent variables; the “dynamical irrelevance” of physically exact but feature-distant periodic orbits (Cleary et al., 27 May 2025) raises nuanced questions about the interpretation of traditional dynamical systems objects.

Despite these caveats, nonlinear two-dimensional Reynolds models remain an indispensable class of tools across fluid dynamics, offering both analytical tractability and deep physical insight, and playing a pivotal role in connecting rigorous mathematical theory, computation, and experimental phenomenology across regimes where essential nonlinearities shape the effective dynamics.