A first integral of Navier-Stokes equations and its applications

作者：Scholle Markus^*; Haas Andre; Gaskell Philip H

来源：Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, 2011, 467(2125): 127-143.

DOI：10.1098/rspa.2010.0157

• fluid dynamics
• potentials
• integrability
• variational methods
• mathematical foundations
• lubrication

Although it is well known that Bernoulli's equation is obtained as the first integral of Euler's equations in the absence of vorticity and that in the case of non-vanishing vorticity a first integral of them can be found using the Clebsch transformation for inviscid flow, generalization of the procedure for viscous flow has remained elusive. Accordingly, in this paper, a first integral of the Navier-Stokes equations for steady flow is constructed. In the case of a two-dimensional flow, this is made possible by formulating the governing equations in terms of complex variables and introducing a new scalar potential. Associated boundary conditions are considered, and an extension of the theory to three dimensions is proposed. The capabilities of the new approach are demonstrated by calculating a Reynolds number correction to the laminar shear flow generated in the narrow gap between a flat moving and a stationary wavy wall, as is often encountered in lubrication problems. It highlights the first integral as a suitable tool for the development of new analytical and numerical methods in fluid dynamics.

• 出版日期2011-1-8