Steady-state heat conduction in a gas undergoing rigid-body rotation. Comparison of Navier–Stokes–Fourier and bivelocity paradigms

This paper proposes a seemingly unequivocal experimental and/or molecular dynamics simulation test of the viability of the compressible Navier–Stokes–Fourier (NSF) equations for gases in the continuum region, namely for near-zero Knudsen (Kn) numbers. While experimental gas kineticists have long known of the inadequacy of the NSF equations for rarefied gases (i.e., noncontinua), for which Kn is no longer small, it is nevertheless believed by fluid mechanicians that the NSF equations remain valid for gaseous continua. The author is, however, unaware of the existence of any unequivocal experimental (or simulation) data to support this view. Indeed, based upon recent work by the author and others on the subject of bivelocity hydrodynamics [Brenner, H. (2013). Proposal of a critical test of the Navier–Stokes–Fourier paradigm for compressible fluid continua. Physical Review E 87, 013014]; [Brenner, H., Dongari, N., & Reese, J. M. (2013). A molecular dynamics test of the Navier–Stokes–Fourier paradigm for compressible gaseous continua. arXiv:1301.1716 [physics.flu-dyn]], ample reasons exist for believing that the NSF equations may not, in fact, be valid for compressible continua. Given the fundamental role played by the NSF equations in both theoretical and applied physics, it would obviously be well if the assumption of the viability of the compressible NSF equations was put to a variety of experimental tests, particularly if the interpretations of those tests were seemingly unequivocal as a consequence of the simplicity of their interpretation and ease of execution. This paper adds one such test to that cited above. It involves contemplating a gaseous continuum confined in the annular space between two co-axial circular cylinders rotating steadily at the same angular velocity while, at the same time, both cylinder walls are maintained at a common temperature, say Tc. In such circumstances the NSF equations, assumed to govern the resulting rigid-body rotation of the fluid, trivially predict the gas’s temperature to be uniform at the value Tc throughout the entire annular body of gas between the cylinders. The proposed test consists of measuring the temperature distribution so as to establish if, in fact, this predicted temperature uniformity is actually observed in practice — and, if not, of establishing whether the observed temperature distribution varies with such experimentally-controllable parameters as the cylinders’ angular velocity or the gas’s mean pressure. Was the temperature distribution found to be nonuniform the compressible NSF equations for continua would have to be abandoned as physically unsound. Anticipating that outcome we have, for the prescribed experimental arrangement and protocol, solved the corresponding bivelocity equations in order to establish if this model accords better with the test data. For, in contrast with NSF behavior, the bivelocity temperature distribution is predicted to be nonuniform.