Radical scaling: beyond our feet and fingers

Abstract

Scaling laws arise and are eulogized across disciplines from natural to social sciences for providing pithy, quantitative, ‘scale-free’, and ‘universal’ power law relationships between two variables. On a log–log plot, the power laws display as straight lines, with a slope set by the exponent of the scaling law. In practice, a scaling relationship works only for a limited range, bookended by crossovers to other scaling laws. Leading with Taylor's oft-cited scaling law for the blast radius of an explosion against time, and by collating an unprecedented amount of datasets for laser-induced, chemical and nuclear explosions, we show distinct kinematics arise at the early and late stages. We illustrate that picking objective scales for the two axes using the transitions between regimes leads to the collapse of the data for the two regimes and their crossover, but the third regime is typically not mapped to the master curve. The objective scales permit us to abandon the arbitrarily chosen anthropocentric units of measurement, like feet for length and heart-beat for time, but the decimal system with ten digits (fingers) is still part of the picture. We show a remarkable collapse of all three regimes onto a common master curve occurs if we replace the base 10 by a dimensionless radix that combines the scales from the two crossovers. We also illustrate this approach of radical scaling for capillarity-driven pinching, coalescence and spreading of drops and bubbles, expecting such generalizations will be made for datasets across many disciplines.