The Fundamental Differential Equations and the Boundary Conditions for High Speed Slip-Flow, and their Application to Several Specific Problems

Citation

Schamberg, Richard (1947) The Fundamental Differential Equations and the Boundary Conditions for High Speed Slip-Flow, and their Application to Several Specific Problems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/F44D-FG36. https://resolver.caltech.edu/CaltechETD:etd-12272004-161736

Abstract

The differential equations of motion and the associated boundary conditions for the slip-flow regime of fluid mechanics are derived from the point of view of the kinetic theory of non-uniform gases. The slip-flow regime comprises the flow of gases whose molecular mean free path is smaller than but not negligible relative to the macroscopic dimension characterizing the gas flow. A systematic review is presented of the methods of Hilbert and Burnett for obtaining a successive approximation solution to the Boltzmann integro-differential equation. The complete second approximation to the molecular velocity distribution function is calculated for the molecular model of Maxwell. This molecular distribution function is employed for the derivation of the macroscopic differential equations of motion and the associated boundary conditions. It is shown that the same number of boundary conditions are required for slip flows as for gas-dynamical flows, although the differential equations of motion for slip flows are of higher order than those of continuum gas-dynamics. Expressions for the second approximations to the slip velocity and temperature jump are obtained. The general equations obtained are applied to three specific problems: the propagation of sound waves in rarefied gases, high-speed Couette flow of a rarefied gas, and slip-flow between concentric cylinders in relative rotary motion. It is found that the rarefaction of a gas increases the damping of sound waves, whereas the propagation speed differs from the ordinary adiabatic sound velocity by less than two percent. The Couette flow solution indicates that the slippage of gas and the temperature discontinuity at a solid boundary may reduce the gas-dynamical friction coefficient and heat transfer, respectively, by ten percent under approximate conditions. When applied to the flight of aircraft through the rarefied atmosphere, the theory presented is applicable to an altitude range from 100,000 to 300,000 feet.

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