Respiratory Mechanics and Fluid-Dynamics Questions
Clinical observations motivate questions about geometry, surface tension and distributed time constants. They do not establish a correspondence between respiratory failure and a Navier–Stokes singularity.
Incomplete conjecture
The earlier surface-tension threshold argument does not establish a conditional regularity theorem. Vortex stretching and boundary estimates remain uncontrolled. The inference from H¹ to L∞ in three dimensions was false. The proposed shrinking closed material sphere violates incompressible volume conservation (a collapsing alveolus empties through an open airway, which that closed model does not describe). The clinical observations that motivated the conjecture are listed on the outline page, apart from the mathematics.
Read the conjecture, its gaps and the analytical sanity checks →Static spherical equilibrium
With outward normal and traction jump (T_in − T_out)n = −γκn, this unforced stationary sphere is an equilibrium for any positive constant surface tension. Large pressure-scale ratios therefore do not force every solution to blow up. This does not refute an existential conjecture about other data. Reducing γ to a smaller positive constant does not remove the 1/r divergence along a prescribed path toward zero radius.
Synthetic teaching instruments
The interactive demo prescribes and floors compliance, then draws RC-shaped waveforms. Its recovery schedule is an arbitrary input, not surfactant pharmacology. Scalar flow derivatives measure temporal changes, not spatial vorticity or enstrophy.
Explore descriptive compliance trajectories →The separate two-compartment calculation shows how equal total compliance and initial excess volume can yield different passive-emptying trajectories. That exact linear-model result does not supply a clinical-to-PDE bridge. Shared Air remains an exploratory demonstration with placeholder coupling parameters.
Explore Shared Air →Evidence tiers remain distinct: motivation, incomplete conjecture, analytical identities, closed-form teaching models and numerical implementation checks. No patient-specific prediction, treatment recommendation or validated clinical tool is provided. Use synthetic data only.