A first Integral of NavierStokes equations for two‐dimensional flows

As wellknown, Bernoulli's equation is obtained as the first integral of Euler's equations in the absence of vorticity. Even in case of non‐vanishing vorticity, a first integral from Euler's equations is obtained by using the so called Clebsch transformation [1] for inviscid flows. In contrast to this, a generalisation of this procedure towards viscous flows has not been established so far. In the present paper a first integral of Navier‐Stokes equations is constructed in the case of two‐dimensional flow by making use of an alternative representation of the fields in terms of complex coordinates and introducing a potential representation for the pressure. The associated boundary conditions are also considered. The first integral is a suitable tool for the development of new analytical methods and numerical codes in fluid dynamics. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)