Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances

Abstract

The use of potential fields in fluid dynamics is retraced, ranging from classical potential theory to recent developments in this evergreen research field. The focus is centred on two major approaches and their advancements: (i) the Clebsch transformation and (ii) the classical complex variable method utilising Airy’s stress function, which can be generalised to a first integral methodology based on the introduction of a tensor potential and parallels drawn with Maxwell’s theory. Basic questions relating to the existence and gauge freedoms of the potential fields and the satisfaction of the boundary conditions required for closure are addressed; with respect to (i), the properties of self-adjointness and Galilean invariance are of particular interest. The application and use of both approaches is explored through the solution of four purposely selected problems; three of which are tractable analytically, the fourth requiring a numerical solution. In all cases, the results obtained are found to be in excellent agreement with corresponding solutions available in the open literature.

Physical and Mathematical Fluid Mechanics

“…An extremely attractive further development of the tensor potential method would be the mapping of the entire problem to a matrix-algebra framework based on quaternions or Dirac matrices with the goal of developing highly efficient methods of solution. Having demonstrated the benefits of probabilistic methods for the study of the deterministic Navier-Stokes equation, Cruzeiro [19] envisages the development of novel numerical methods in the future. Finally, the tragic incident reported by Cui et al [20] shows the need to detect and prevent such incidents in time with improved prediction models.…”

“…It also closely refers to kinetic models in statistical physics. Cruzeiro [19] presents a selective review about this research field, regarding the velocity solving the deterministic Navier-Stokes equation as a mean time derivative taken over stochastic Lagrangian paths and obtaining the equations of motion as critical points of an associated stochastic action functional, involving the kinetic energy computed over random paths. Different related probabilistic methods are discussed.…”

Symmetries and Related Physical Balances for Discontinuous Flow Phenomena within the Framework of Lagrange Formalism

“…, subsequently called the dual representation, proves to be invariant with respect to Galilei boosts if all fields ψ i are assumed to be likewise invariant. Thus, the essence of the dual representation is that Galilei boosts become manifest as pure geometrical transformations without the need to combine them with a re-gauging of potentials [18]. Since the conventional representation (ψ i ,ψ i , ∇ψ i ) is obviously strictly invariant w.r.t.…”

“…Further consequences of the invariance w.r.t. the Galilei group, especially regarding the representation of the velocity field by means of potentials, are reported in the review article [18].…”

Comment on Clebsch’s 1857 and 1859 papers on using Hamiltonian methods in hydrodynamics

“…Yoshida 2009. Grad and Rubin 1958, Mendes et al 2005, Jackiw, Nair and So-Young Pi 2000, Jackiw and Polychronakos 2000, Ghosh 2002, Deser, Jackiw and Polychronakos 2001, Balkovsky 1994. Scholle, Marner and Gaskell 2020…”