On the use of potential fields in fluid mechanics

In classical fluid mechanics, potential fields have been employed to enable the integration of the equations of motion. As is well known, Bernoulli's equation is obtained as a first integral of Euler's equations in the absence of vorticity and viscosity if the velocity vector is perceived as the gradient of a scalar potential. The so-called Clebsch transformation 1] involving three scalar potentials allows for a further extension to flows with non-vanishing vorticity; the resulting equations turn out to be self-adjoint, allowing for a variational formulation. All attempts in classic literature, however, are restricted to inviscid flows and the finding of a potential representation enabling the integration of the Navier-Stokes equations remains desirable. Progress on this topic was reported by 3, 4] who constructed a first integral of the two-dimensional incompressible Navier-Stokes equations by making use of an auxiliary potential field and a representation of the fields in terms of complex coordinates. The new formulation proved to be useful in numerical applications and moreover, replacing the scalar potential by a tensor potential, the theory can be successfully generalised to encompass three-dimensional Navier-Stokes flow. Related to the first integral a finite element method was presented in 2] based on a formulation involving the velocities and the first order derivatives of the introduced potential. This way the dynamic boundary condition could be incorporated elegantly and the system of equations fitted into the first order system least-squares methodology. However, a promising alternative approach results if one considers the streamfunction and a slightly modified potential field as independent variables. This new approach involves Laplacian operators rather than mixed derivatives and allows for a convenient embodiment of the Neumann conditions on the streamfunction that is in contrast to the original stream function / potential formulation 4]. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)