Part I. Cylindrical Couette Flow in a Rarefied Gas According to Grad's Equation. Part II. Small Perturbations in the Unsteady Flow of a Rarefied Gas Based on Grad's Thirteen Moment Approximation

Citation

Ai, Daniel Kwoh-i (1961) Part I. Cylindrical Couette Flow in a Rarefied Gas According to Grad's Equation. Part II. Small Perturbations in the Unsteady Flow of a Rarefied Gas Based on Grad's Thirteen Moment Approximation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/9P2N-HF23. https://resolver.caltech.edu/CaltechETD:etd-12092005-131707

Abstract

Part I

Grad's thirteen moment method is applied to the problem of the shear flow and heat conduction between two concentric, rotating cylinders of infinite length. In order to concentrate on the effects of curvature the problem is linearized by requiring that the Mach number is small compared with unity, and that the temperature difference between the two cylinders is small compared with the mean temperature. The solutions of the linearized Grad equations show a qualitatively correct transition of the cylinder drag from free-molecule flow to the classical Navier-Stokes regime. However the magnitude of the curvature effect on the drag in rarefied flow is not given correctly, because Grad's distribution function ignores the wedge-like domains of influence of the two cylinders.

The solution obtained for the heat transfer rate is physically unrealistic in the free-molecule flow limit, and this result is produced by a cross-coupling between the normal stresses and the radial heat flux imposed by Grad's distribution function. In this simple problem the difficulty can be eliminated by taking the normal stresses to be identically zero and employing a truncated moment method. However, in general this device cannot be utilized in problems involving curved solid boundaries, or when dissipation is considered. One concludes that the choice of the distribution function to be employed in Maxwell's moment equations is dictated by the requirements imposed in the limiting case of highly rarefied gas flows, as well as in the Navier-Stokes regime.

Part II

In this paper, the unsteady one-dimensional flow of a compressible, viscous and heat conducting fluid is treated, based on linearized Grad's thirteen moment equations. The fluid, initially at rest, is set into motion by some small external disturbances. Our interest is to examine the nature of all the responses. The fluid field extends to infinity in both directions; thus no length is involved, and also there is no solid wall boundary existing in the problem. The nature of the external disturbances is restricted to having a unit impulse in the momentum equation and a unit heat addition in the energy equation. The disturbances are located on an infinite plane normal to the flow direction; and the responses induced correspond to fundamental solutions of the problem. The method of Laplace transforms is applied, and the inverse transforms of all quantities are obtained in integral form. Because of the complicated expressions of the integrands involved, we consider only certain limiting cases which correspond to small and large times from the start of the motion, compared to the average time between molecular collisions. In order to study these limiting cases, it is essential to understand the behavior of the integrand in the complex plane; hence all singularities and branch points are obtained.

When t is small, the integrand is expanded in powers of t to obtain a wave front approximation. All discontinuities are propagated along the characteristics of the linearized system, and a damping term also appears.

At large values of time, the integrand gets its main contribution around the branch points, and these solutions are identical to those obtained from the Navier-Stokes equation. The fundamental solution of the one-dimensional unsteady flow, idealized as it seems to be, offers itself as a tool to understand other related problems. The piston problem, as well as the normal quantities in Rayleigh's problem (e.g., normal velocity, normal stress, and thermodynamical quantities), are governed by the same set of equations. Hence, certain parts of the fundamental solutions can be applied directly to these problems. The limiting forms of the normal quantities in Rayleigh's problem are expected to be worked out in another paper in the near future.

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