Mechanical Visualization of FLRW-Type Expansion Kinematics in Hele-Shaw Compression Flow: Tracer Trajectories, Comoving Scalar Modes, and Anisotropic Cells

Technical Summary: Mechanical Visualization of FLRW-Type Expansion Kinematics in Hele-Shaw Compression Flow

Problem Statement
The paper addresses the challenge of creating a macroscopic, bench-scale mechanical system that accurately visualizes the kinematics of metric expansion found in Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology. While fluid systems have previously been used to model relativistic horizons and particle production via effective acoustic metrics, this work seeks to reproduce the specific kinematic features of expansion: the scale factor $a(t)$, the expansion rate (Hubble parameter), comoving separation, and the transport of superposed scalar fields. The core problem is identifying which physical observables in a compressed Hele-Shaw flow (squeeze flow) strictly follow the depth-averaged kinematic laws derived from mass conservation, and distinguishing these from the behavior of physical tracer particles which are subject to transverse velocity gradients.

Methodology
The study employs a combination of analytical derivation, exact solutions to the lubrication equations, and high-fidelity three-dimensional computational fluid dynamics (CFD).

1. Analytical Framework: The system is modeled as an incompressible Newtonian fluid in a horizontal axisymmetric (or elliptical) cell with a time-dependent gap $h(t)$. Under the thin-gap ($\epsilon = h/R \ll 1$) and creeping flow ($Re_{squeeze} \ll 1$) limits, the radial velocity profile is derived as parabolic across the gap.
2. Tracer Dynamics: The trajectories of physical tracers are analyzed by solving the coupled radial and vertical equations of motion. This involves determining the height-dependent expansion rate and the stability of the mid-plane.
3. Scalar Transport: The transport of a passive scalar (e.g., temperature or dye) is analyzed in comoving coordinates. The study derives the reduction of the advection-diffusion equation to pure diffusion with a metric-suppressed diffusivity, allowing for the calculation of wavelength stretching and amplitude damping.
4. Anisotropic Analogue: The geometry is extended to elliptical cells to model Bianchi Type-I anisotropic expansion. The exact solution for the pressure and velocity fields is derived and verified against CFD simulations.
5. Numerical Verification: Three-dimensional CFD simulations (using OpenFOAM) are performed to verify the anisotropic scaling laws, test the limits of the creeping flow approximation, and quantify finite-Reynolds number corrections.

Key Contributions and Results

• Distinction Between Depth-Averaged Fields and Tracers:
  The paper establishes that the exact comoving scaling law $r(t) = a(t)r_0$ with $a(t) = \sqrt{h_0/h(t)}$ applies strictly to the depth-averaged velocity field, not to physical tracer particles.

  • Tracer Behavior: Physical tracers experience a height-dependent expansion rate $H(\zeta) = 6\zeta(1-\zeta)\bar{H}$. Tracers at different heights recede at different rates, spanning a factor of 3.3 in radius over a compression to $a=7.4$.
  • Mid-Plane Attractor: The mid-plane ($\zeta = 0.5$) is an asymptotically stable attractor. Off-plane displacements decay as $a^{-1}$. Tracers on the mid-plane obey a modified law: $r(t) = a(t)^{3/2}r_0$.
  • Measurement Implication: Standard particle tracking without depth resolution yields erroneous scale factors. Only light-sheet velocimetry (µPIV) focused on the mid-plane recovers a clean power-law correspondence (exponent 3/4 relative to $h_0/h$).

• Physical Realization of the Depth-Averaged Law:
  The depth-averaged FLRW correspondence is physically realized only by observables that mix across the gap faster than the cell expands. This requires the condition $\Pi = \bar{H}h^2/\kappa \ll 1$, where $\kappa$ is diffusivity.

  • Particle tracers fail this condition by five orders of magnitude.
  • Molecular dyes in millimeter gaps fail by three orders of magnitude.
  • Thermal scalars in thin gaps (e.g., $h=100\,\mu\text{m}$) satisfy the condition, making them the correct physical analogue for the depth-averaged field.

• Comoving Scalar Modes and Damping:
  For well-mixed scalars, the transport equation in comoving coordinates reduces to pure diffusion with a time-dependent diffusivity $\kappa/a^2(t)$.

  • Wavelength Stretching: The wavelength stretches as $\lambda(t) = a(t)\lambda_0$.
  • Silk-Type Damping: The amplitude decays according to $A(t) = A_0 \exp\left(-\kappa k_0^2 \int a^{-2} dt\right)$.
  • Freeze-Out Criterion: A relic amplitude survives only if the scale factor grows asymptotically faster than $t^{1/2}$. The radiation-like case ($a \propto t^{1/2}$) is marginal, resulting in complete erasure.

• Anisotropic (Bianchi Type-I) Analogue:
  In an elliptical cell with semi-axes $a_e$ and $b_e$, the flow generates an exact homogeneous anisotropic Hubble law:
  $$\frac{H_x}{H_y} = \frac{b_e^2}{a_e^2}$$
  This ratio is a genuine prediction of the lubrication solution, independent of mass conservation alone.

  • CFD Verification: Three-dimensional simulations confirm this ratio to within 0.1% in the creeping-flow limit.
  • Finite-Reynolds Effects: The ratio acquires a correction of approximately $+15\%$ per unit squeeze Reynolds number ($Re_{squeeze}$), quantified by an empirical fit. The anisotropy is "frozen" by the boundary geometry, unlike in dynamical Bianchi-I cosmologies where shear evolves.

• Design Window:
  The paper defines a narrow but non-empty design space where all constraints (thin-gap, creeping flow, cross-gap mixing, and Taylor-Aris dispersion limits) are satisfied simultaneously. A viable configuration is identified as a thermal scalar in a $100\,\mu\text{m}$ gap with a $50\,\text{mm}$ radius cell.

Significance and Scope
The paper claims to provide a quantitatively accurate mechanical visualization of FLRW-type expansion kinematics, provided the correspondence is applied to the correct observables (depth-averaged fields or well-mixed scalars, not ballistic tracers).

The significance lies in the ability to "program" an expansion history $h(t)$ mechanically and observe the resulting kinematic effects (recession, wavelength stretching, damping, and anisotropy) directly on a bench top. The system serves as a kinematic analogue, reproducing scale factors, expansion rates, and comoving transport.

Explicit Limitations:
The author explicitly states what the system does not model:

• It is purely kinematic; there is no effective metric, wave equation, or causal horizons.
• It does not reproduce the Einstein or Friedmann equations (expansion is prescribed, not determined by stress-energy).
• It does not model true Silk damping (which involves acoustic oscillations), only the diffusive erasure of modes.
• The anisotropy is geometrically frozen, not dynamically evolving.
• No experimental realization has yet been performed; the design window remains a theoretical proposal.

In summary, the work bridges fluid mechanics and cosmological kinematics by rigorously defining the conditions under which a squeezed viscous flow mimics the expansion of the universe, offering a precise framework for future educational or experimental demonstrations.