Macfarlane quaternion formalism of ideal fluids

In this study, a significant advancement in the theoretical formulation of fluid dynamics is presented by introducing a novel algebraic structure for modeling ideal fluids. Although analogies between fluid dynamics and electromagnetism are well-documented previously by several formalisms, this work provides a comprehensive framework based on Macfarlane quaternions to express Maxwell-type fluid equations with enhanced structural clarity. It is demonstrated that this formalism elegantly unifies fluid potentials and fields into unique quaternionic entities, leading to highly compact representations of the governing equations for both compressible and incompressible flows. Significant outcomes also include the derivation of fluid wave equations directly from the quaternionic potentials. The versatility of the framework is further demonstrated by its extension to incorporate non-isentropic effects. For theoretical and computational analysis, explicit expressions of the equations are provided by establishing 2 × 2 and 4 × 4 matrix representations of Macfarlane quaternions.