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

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## Abstract

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## 1. Introduction

- A reduction in the number of equations and unknowns;
- Self-adjointness of the equations;
- A full decoupling of the equations;
- Transformation of the equations to a known mathematical type.

- The so-called Clebsch transformation [19,20,23] and related methodologies enabling, in the first instance, Euler’s equation to be reformulated as a generalised Bernoulli equation complemented with two transport equations for the Clebsch potentials. The approach was subsequently generalised to encompass baroclinic flow by Seliger and Witham [24], but still with the restriction that the flow is inviscid and heat conduction absent. Note that the Clebsch transformation has been applied to physical problems beyond fluid mechanics including Maxwell theory in classical electrodynamics [25], the field of Magnetohydrodynamics [26], relativistic dynamical systems [27] and even in relation to quantum theory within the context of (a) the quantisation of vortex tubes Madelung [28], Schoenberg [29], (b) generalised membranes [30] and (c) relativistic quantum vorticity [31].
- The complex variable method, developed in the first half of the 20th century and originally related to problems in plane linear elasticity [32,33]. The method was subsequently adopted by the fluid mechanics community: in the case of 2D Stokes flow ($\mathrm{Re}\to 0$) it has led to solutions based on a complex-valued Goursat representation of the stream function in terms of two analytic functions, which has been generalised incrementally, starting with Legendre [34] and followed by Coleman [35], Ranger [36], and Scholle et al. [37], Marner et al. [38], resulting finally in an exact complex-valued first integral of the 2D unsteady NS equations, based on the introduction of an auxiliary potential field. 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].

## 2. Clebsch Transformation Approach

#### 2.1. The Clebsch Transformation for Inviscid Flows

#### 2.2. A Note on the Global Existence of the Clebsch Variables

#### Derivation of a Clebsch-Like Form by Galilean Invariance and Self-Adjointness

**time translations:**- $$t\to {t}^{\prime}=t+\tau $$
**space translations:**- $$\overrightarrow{x}\to {\overrightarrow{x}}^{\prime}=\overrightarrow{x}+\overrightarrow{s}$$
**rigid rotations:**- $$\overrightarrow{x}\to {\overrightarrow{x}}^{\prime}=\underline{\mathrm{R}}\overrightarrow{x}$$
**Galilei boosts:**- $$\overrightarrow{x}\to {\overrightarrow{x}}^{\prime}=\overrightarrow{x}-{\overrightarrow{u}}_{0}t$$

**Theorem**

**1.**

- Its dynamics are deducible from Hamilton’s Principle (30),
- Equivalence$\overrightarrow{p}=\varrho \overrightarrow{u}$of momentum density and mass flux density is given,

#### 2.3. An Extended Clebsch Transformation for Viscous Flow

**Theorem**

**2.**

**Proof.**

#### 2.4. Axisymmetric Stagnation Flow

## 3. Complex Variable and Tensor Potential Approach

#### 3.1. The Classical Complex Variable Method

#### 3.2. Integration of the Full 2D Navier–Stokes Equations

#### 3.3. Integration of the Dynamic Boundary Condition

#### 3.4. Particular Flow Geometries as Exemplars

#### 3.4.1. Uniaxial Flow: Flow over an Oscillating Plate

#### 3.4.2. Axisymmetric Flow: The Lamb-Oseen Vortex

#### 3.4.3. Steady Film Flow over Topography

#### 3.5. Tensor Potential Approach

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

2/3/4D | two-/three-/four-dimensional |

LSFEM | least square finite element method |

NS | Navier–Stokes |

ODE | ordinary differential equation |

PDE | partial differential equation |

## Appendix A. Proof of the Existence of a Representation with Two Pairs of Clebsch Variables

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**Figure 4.**Schematic of gravity-driven film flow down a corrugated rigid surface inclined at an angle $\alpha $ to the horizontal. The indices i and j in the boundary conditions run from 1 to 2.

**Figure 5.**Streamlines elucidating steady film flow over two different periodically corrugated inclined surfaces: on the left a smoothly varying feature, on the right a feature involving sharp, step changes.

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Scholle, M.; Marner, F.; Gaskell, P.H. Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances. *Water* **2020**, *12*, 1241.
https://doi.org/10.3390/w12051241

**AMA Style**

Scholle M, Marner F, Gaskell PH. Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances. *Water*. 2020; 12(5):1241.
https://doi.org/10.3390/w12051241

**Chicago/Turabian Style**

Scholle, Markus, Florian Marner, and Philip H. Gaskell. 2020. "Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances" *Water* 12, no. 5: 1241.
https://doi.org/10.3390/w12051241