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.
Highlights
In various branches of physics, potentials continue to be used as additional auxiliary fields for the advantageous reformulation of one or more governing equations
The mathematical derivation is completed by the specification of appropriate boundary conditions, which take the form of no-slip/no-penetration conditions at solid walls, inflow and outflow conditions and, in the case of film or multiphase flows, kinematic and dynamic boundary conditions at a free surface or internal interface
While at the free surface, in addition to a kinematic boundary condition, two dynamic conditions are imposed resulting as inhomogeneous Dirichlet conditions for φ1 and φ2 from Equation (114) by decomposition into real and imaginary parts; these depend on the surface tension, the curvature and the potential energy density
Summary
In various branches of physics, potentials continue to be used as additional auxiliary fields for the advantageous reformulation of one or more governing equations. A further generalisation to 3D viscous flow has been achieved only recently using a tensor potential in place of the complex potential field employed in two-dimensions [39] Both approaches have been subject to limitations on their usage: the Clebsch transformation originated for the case of inviscid flow (Re → ∞) and the complex variable method for that of 2D Stokes flow (Re → 0). It would be remiss and incomplete not to point out that in the field of fluid mechanics other approaches to the use of potentials for solving the equations of motion exist that have not been considered in the present text. In this sense and as instructive examples, the reader is referred to the work of Papkovich and Neuber [40], Lee et al [41], Greengard and Jiang [42]
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