Dimensional Consistency of Fractional Time Derivatives in Fluid Flow, Heat Transfer, and Mass Transport Models

Abstract

Fractional transport models are widely used to represent memory and anomalous relaxation, but replacing an ordinary time derivative by a fractional operator changes the temporal dimension of the governing balance. This paper develops a unified audit framework for fluid momentum, heat transfer, and mass transport that separates two dimensionally consistent conventions: generalized fractional transport coefficients and classical coefficients accompanied by an explicit memory time. The contribution is not the already established dimensional property of fractional derivatives; rather, it is the transport-wide synthesis of coupled balance laws, characteristic-time scaling, coefficient-convention equivalence, parameter identifiability, and reproducible verification tests. A dimensionally completed Atangana--Baleanu--Caputo operator is distinguished explicitly from its standard mathematical form and is nondimensionalized with both the reference-time and kernel-decay scales retained. A quantitative one-dimensional Caputo diffusion benchmark demonstrates unit-system invariance, agreement with the Mittag--Leffler solution, equivalence of the two coefficient conventions, and convergence toward the classical limit. The analysis further shows that transient data often identify a combined diffusivity--memory parameter more robustly than its constituent parameters separately. These results provide a practical verification protocol for fractional CFD, inverse modeling, and physics-informed computation, where unit ambiguity can otherwise be absorbed into fitted coefficients or residual weights.