Reformulation of fluid equations with macfarlane quaternions

The use of multidimensional algebraic structures in physical theories particularly the application of mathematical tools such as quaternions to various physical systems constitutes a significant area of research. Quaternions appear in different forms in the literature and are notable for both their advantages and certain limitations in physical models. This thesis investigates selected applications of Macfarlane quaternions, defined as a type of Pauli algebra, in physical theories. It begins with a brief overview of the historical development of quaternions and a detailed analysis of their algebraic properties. The mathematical structure of Macfarlane quaternions is then examined, and their 2 × 2 and 4 × 4 mmatrix representations are constructed. As a physical application, a quaternionic formulation of Maxwell's equations is successfully developed. Lorentz transformations are also derived within this algebraic framework, and Maxwell's equations are reformulated in the context of special relativity. The equations of ideal fluid dynamics are subsequently re-expressed using the same structure. Suitable operators are defined, and the equations are presented in a 4 × 4 matrix form. Finally, an analogy between electromagnetism and fluid mechanics is established, and Maxwell-type equations are compactly derived for both incompressible and compressible fluids.