Thirteen numbers · three applications each · not peer reviewed
Thirteen dimensionless numbers, three applications each, taken to domains where they are not usually applied. If a group governs flow in a pipe, it may govern flow in an aneurysm; the volume tests whether that transfer holds.
Thrombosis in an aneurysm sac begins where the local Reynolds number passes a threshold, not where the artery is widest, so a small sac with fast flow clots before a large slow one.
Fish school structure changes at a critical Reynolds number set by body length and swimming speed, so the same species forms different formations at different speeds.
Fine sediment resuspension occurs at a threshold in particle Reynolds number rather than in bed shear stress, so cohesive and non-cohesive beds resuspend differently at the same shear.
Penetration of a drug into a solid tumour is set by the interstitial Peclet number, so a drug with high potency but poor diffusion reaches less tissue than a weaker one that moves well.
Nutrient capture by roots is set by the Peclet number at the root surface, so transpiration rate changes uptake efficiency non-monotonically.
The shape and length of a groundwater contaminant plume is set by the Peclet number, so the same mass produces a compact plume in one aquifer and a long dilute one in another.
Wound closure rate is set by the ratio of cell proliferation rate to nutrient transport rate, so a well-perfused wound heals faster even with the same cells.
The fraction of an electrode that participates in a fast discharge is set by the ratio of reaction rate to ion transport rate, so thick electrodes at high rate use only their surface.
Hot spots in a catalytic bed appear above a critical ratio of reaction rate to heat removal, so a modest reaction in a poorly conducting bed can run away.
How much an organ's temperature rises during fever is set by its Biot number, so a large well-perfused organ heats less than a small poorly perfused one.
Cooking time for a solid food is set by its Biot number, so two pieces of equal mass cook at different rates depending on shape and surface transfer.
Whether a failing cell ignites its neighbour is set by the neighbour's Biot number, so a well-cooled cell resists propagation even at close spacing.
The insulating performance of blubber is set by the Prandtl number of the tissue, so the same thickness insulates differently depending on blood flow through it.
The transition between stable and convective boundary layers is set by the Prandtl number of the surface layer rather than by temperature gradient alone.
Fouling rate in a heat exchanger is set by the Prandtl number at the wall, so a fluid with high thermal diffusivity fouls differently at the same velocity.
Leaf transpiration is set by a Sherwood-like number for the leaf boundary layer, so wind changes water loss only until the boundary layer thins enough that stomata take over.
Solute clearance in dialysis is set by the Sherwood number of the membrane module, so a larger membrane with poor flow distribution clears less than a smaller well-distributed one.
Coral calcification rate is set by the Sherwood number at the tissue surface, so flow conditions change calcification independently of saturation state.
The walk-to-run transition occurs at a critical Froude number rather than at a critical speed, so leg length sets the speed at which people switch.
Tsunami run-up height is set by the Froude number of the wave at the shore, so two waves of equal height produce different inundation depending on their speed.
Shoreline damage from vessel wakes is set by the Froude number rather than by vessel size, so a small fast craft can do more than a large slow one.
Gas production from nanoporous shale is set by the local Mach number in the pore network, so the same pressure drawdown produces different flow regimes in different pore sizes.
The size distribution of droplets produced by a cough is set by the local Mach number in the airway, so a harder cough produces smaller, farther-travelling droplets.
Volcanic plume height is set by the Mach number at the vent, so an eruption with a choked vent produces a different plume than one with the same mass flux but lower exit velocity.
Gas separation in nanoporous membranes is set by the Knudsen number, so a membrane designed for one pressure regime can invert its selectivity in another.
Where inhaled nanoparticles deposit in the lung is set by the Knudsen number, so the same mass of particles deposits differently depending on their size relative to the airway.
Vacuum drying of food is set by the Knudsen number in the pore network, so reducing pressure below a threshold stops improving drying rate.
Copepod encounter rates with prey are set by the Stokes number of the prey, so a prey that follows the flow is harder to capture than one that crosses streamlines.
Particle capture in a fibrous filter is set by the Stokes number of the particle, so filters have a minimum efficiency at an intermediate particle size.
Dust accumulation on solar panels is set by the Stokes number of the particles, so wind removes some size classes and deposits others.
The effectiveness of sweating is set by the Lewis number of the air at the skin, so humid and dry air of equal temperature cool very differently.
The drying of a coating is set by its Lewis number, so two solvents with identical boiling points dry at different rates because their thermal and mass diffusivities differ.
Spread rate through a porous fuel bed is set by the Lewis number of the gas mixture, so fuel moisture changes spread differently than temperature alone predicts.
Whether a spray droplet adheres to a leaf or bounces off is set by its Weber number, so finer sprays are not necessarily better for retention.
Foam stability in fermentation is set by the Weber number of the sparged bubbles, so changing aeration rate changes foaming without changing the broth chemistry.
Stable drop-on-demand printing requires the Weber number to lie within a window, so increasing drive amplitude beyond it produces satellites rather than better drops.
The propulsive efficiency of swimming and flying animals peaks at a narrow Strouhal number, so robots tuned to that number should be more efficient at any size.
Vocal fold oscillation frequency is set by the Strouhal number of the airflow rather than by fold tension alone, so the same tension produces different pitch at different pressures.
Flow-induced vibration of heat exchanger tubes becomes severe when the shedding Strouhal number matches the tube's natural frequency, so failures cluster at particular velocities.
A field-by-field catalogue tells you what each discipline thinks. A mechanism-by-mechanism catalogue tells you what kind of argument a discipline is making, and lets you check whether the same argument holds somewhere else. Each family here is a distinct causal shape; the eleven instances test whether it generalises.
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