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Foundations of the Navier-Stokes boundary conditions in fluid mechanics

Badur J., Karcz M., Lemański M., Nastałek L.

Transactions of the Institute of Fluid-Flow Machinery

2011

nr 123

3-55

Until recently it was believed that Navier’s boundary condition could be given as a rigorous foundation for slip phenomena. Due to the latest measurements in the mass flow rate of a gas flowing through nano- and microchannels, several discrepancies in the mathematical modelling have been found. Thus, in the literature, the opinion persists for the Navier slip condition to be correct only under certain circumstances, particularly those restricted to the first order boundary conditions. One of many ways to eliminate this discrepancy, which is extensively employed in the contemporary literature, is to develop a variety of the so-called second order boundary conditions. This path, however, seems incorrect since it lacks consistency between the bulk stress tensor and its boundary representation. In the paper we propose to replace the classical Navier slip condition with the new, more general Navier-Stokes slip boundary condition. Instead of the usual method of consideration, the boundary condition is presented as following from the mass and momentum balances within a thin, shell-like moving layer. Owing to this, the problem of consistency between the internal and external friction in a viscous fluid is solved within the framework of new layer balances, and a proper form of constitutive relations for friction and mobility forces. Finally, the common features of the Navier, Stokes, Maxwell and Reynolds concepts of a boundary slip layer are compared and revalorized. The classifications of different mobility mechanisms, important for flows in nano-, microchannels are also discussed.