Construction of Dimensionless Groups by Entropic Similarity

Since the early 20th century, dimensional analysis and similarity arguments have provided a critical tool for the analysis of scientific, engineering, and thermodynamic systems. Traditionally, the resulting dimensionless groups are categorized into those defined by (i) geometric similarity, involving ratios of length scales; (ii) kinematic similarity, involving ratios of velocities or accelerations, and (iii) dynamic similarity, involving ratios of forces. This study considers an additional category based on entropic similarity, with three variants defined by the following: (i) ratios of global or local entropy production terms Π entrop = σ ˙ 1 / σ ˙ 2 or Π ^ entrop = σ ˙ ^ 1 / σ ˙ ^ 2 ; (ii) ratios of entropy flow rates Π entrop = F S , 1 / F S , 2 or magnitudes of entropy fluxes Π ^ entrop = | | j S 1 | | / | | j S 2 | | ; and (iii) the ratio of a fluid velocity to that of a carrier of information Π info = U / c . Given that all phenomena involving work against friction, dissipation, spreading, chemical reaction, mixing, separation, or the transmission of information are governed by the second law of thermodynamics, these are more appropriately analyzed directly in terms of competing entropic phenomena and the dominant entropic regime, rather than indirectly using ratios of forces. This work presents the entropic dimensionless groups derived for a wide range of diffusion, chemical reaction, dispersion, and wave phenomena, revealing an entropic interpretation for many known dimensionless groups and many new dimensionless groups.