Marner F., Gaskell P.H., Scholle M. / Ôèçè÷åñêàÿ ìåçîìåõàíèêà 17 3 (2014) 124 130 124 ÓÄÊ 532.517.2 On a potential-velocity formulation of NavierStokes equations F. Marner, P.H. Gaskell 1 , and M. Scholle Heilbronn University, Institute for Automotive Technology and Mechatronics, Heilbronn, D-74081, Germany 1 School of Engineering and Computing Sciences, Durham University, Durham, DH1 3LE, UK Computational methods in continuum mechanics, especially those encompassing fluid dynamics, have emerged as an essential inves- tigative tool in nearly every field of technology. Despite being underpinned by a well-developed mathematical theory and the existence of readily available commercial software codes, computing solutions to the governing equations of fluid motion remains challenging: in essence due to the non-linearity involved. Additionally, in the case of free surface film flows the dynamic boundary condition at the free surface complicates the mathematical treatment notably. Recently, by introduction of an auxiliary potential field, a first integral of the two- dimensional NavierStokes equations has been constructed leading to a set of equations, the differential order of which is lower than that of the original NavierStokes equations. In this paper a physical interpretation is provided for the new potential, making use of the close relationship between plane Stokes flow and plane linear elasticity. Moreover, it is shown that by application of this alternative approach to free surface flows the dynamic boundary condition is reduced to a standard DirichletNeumann form, which allows for an elegant numeri- cal treatment. A least squares finite element method is applied to the problem of gravity driven film flow over corrugated substrates in order to demonstrate the capabilities of the new approach. Encapsulating non-Newtonian behaviour and extension to three-dimensional problems is discussed briefly. Keywords: fluid dynamics, complex methods, potentials, integrability, Airy stress function, least squares finite elements © Marner F., Gaskell P.H., Scholle M., 2014 1. Motivation 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 u is perceived as the gradient of a scalar potential. The so-called Clebsch trans- formation [1, 2] allows for a further extension to flows with non-vanishing vorticity. A similar methodology has recently been reported by Scholle et al. [3] for the case of two-di- mensional incompressible viscous flow by making use of a representation of the fields in terms of complex coordinates. Besides a reduction of differential order the formulation of integrated NavierStokes equations allows for a convenient embodiment of the dynamic boundary condition as a DirichletNeumann condition on the potential field in the case of free surface flows. Initially the integration procedure is motivated and per- formed from a formal mathematical point of view in which a scalar potential field is introduced as an auxiliary vari- able to make the field equations integrable. Meanwhile, the question of physical interpretation of this naturally occur- ring potential motivates a review of complex methods in the field of fluid dynamics in which the exploration of the close relationship between plane Stokes flow and plane lin- ear elasticity proves to be illuminating. In the case of Stokes flow the new potential velocity formulation in complex form can be shown to reproduce the well-known KolosovMus- khelishvili formulae [46] of plane linear elasticity, sug- gesting the potential to be a function of integrated stresses. A short review of the first integral of NavierStokes equations [3] is provided in Sect. 2.1, followed by the deri- vation of the potential representation of the dynamic bound- ary condition in Sect. 2.2. Section 3 is devoted to the analy- sis and interpretation of the potential variable mentioned above. In Sect. 4 a least-squares finite element method, used to solve the fully non-linear problem of gravity-driven thin film flow over corrugated topography, is presented; it shows the general and convenient numerical applicability of the new formulation. Section 5 provides a summary and for- ward look. 2. Two-dimensional potential-velocity formulation for steady incompressible NavierStokes equations Scholle et al. [3] developed an integration procedure for the case of two-dimensional incompressible viscous flow by making use of a representation of the fields in terms of complex coordinates. For convenience the essentials of the methodology are reviewed below.