The Solution Assumed Normal: The Chapman–Enskog Ansatz and the Licence Inside Every Transport Coefficient
Year: 2026
Type: preprint
Abstract: In 1916 and 1917, independently, Sydney Chapman and David Enskog made the transport coefficients of a gas computable: expanding the Boltzmann equation in the Knudsen parameter returns the shear viscosity and heat conductivity from molecular parameters, with Maxwell's density-independent viscosity among the returns — derivations, and permanent ones. The expansion runs under a licence stated at its first move: the solution is sought as a normal solution, one in which the distribution function depends on space and time only through the hydrodynamic fields and their gradients. That shape is postulated in both founding papers, not derived. This paper evaluates the ansatz in its own parameter: for a declared hard-sphere gas (argon parameters at 300.000000 K) the first-order returns are \eta_1 = 2.528084\times10^{-5} Pa·s and \kappa_1 = 1.973159\times10^{-2} W/(m·K), and the first neglected (Burnett) order stands at 0.000080 of the retained order at millimetre gradient scales, 0.796014 at a tenth of a micron, and 1.000000 — no shrinkage between orders at all — at gradient lengths of one mean free path, which is where shock interiors and Knudsen layers live and where the solutions the ansatz excludes are the physics. The coefficients given the ansatz are the founding papers' derivations and stand entire. What is counted here is the derivation status of the solution class they are computed in: 1 normal-solution ansatz, 0 derivations in 1916/1917 that the Boltzmann equation's solutions are normal where the method is used. Census headline, unchanged: NOT SOLVED / PARTIAL · producing-count 0.
Author: Aldred, William Alexander
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225386188