Analysis of Burnett Stresses and Entropy Generation for Pressure-Driven Plane Poiseuille Flow

Abstract

In this paper, we undertake an analytical study of stresses (augmented and Onsager–Burnett) and entropy generation for the plane Poiseuille flow problem, and their variation with Knudsen number. The gas flow is assumed to be 2D laminar, fully developed, compressible, and isothermal; these assumptions make the problem amenable to analytical treatment. The variation of stresses and entropy generation has been analyzed over a large range of Knudsen number. The magnitude of stresses and entropy generation at a particular position in the channel has been considered. It is found that the augmented and OBurnett normal stresses are of opposite signs to the corresponding Navier–Stokes stresses, while the magnitude of the net normal stress increases with Knudsen number. The magnitude of the augmented Burnett shear stress is insignificant as compared to the augmented Burnett normal stresses. A close match between the augmented and OBurnett normal stresses has been found at low Knudsen number. However, an opposite variation has been observed between the augmented and Onsager shear stresses at high Knudsen number. A good comparison of the normalized mass flow rate with the reported value in the literature helps to validate our analysis. A minimum in the variation of normalized entropy generation against the Knudsen number (Kn) is observed at Kn close to unity, and is being reported for the first time. The magnitude of net entropy generation from the summation of Navier–Stokes and augmented Burnett stresses is found to be positive, even in the transition regime of gas flow. Further, an appearance of minimum or maximum in normalized net shear stress versus Knudsen number, depending upon the lateral position in the microchannel, has also been observed. Altogether, this analysis supports the validity of the Navier–Stokes equation with modified constitutive expression, even for higher Knudsen numbers. Moreover, the significant terms of Burnett stress are pointed out by the analysis, which can help in developing reduced-order model for these equations.