Plane Couette Flow at Low Mach Number According to the Kinetic Theory of Gases

Abstract

The thirteen-moment approximation developed by H. Grad for solving the Maxwell-Boltzmann equation is applied to the problem of the relative shearing motion between two infinite, parallel flat plates (plane Couette flow). In order to bring out the molecular effects as directly as possible the problem is linearized by requiring that the Mach number is small compared with unity, and that the temperature difference between the two plates is small compared with ambient temperature. According to the linearized Grad equations the shear stress in this case is given by the usual Navier-Stokes relation for all values of the parameter Re/M, in agreement with R. A. Millikan's postulate. Also the linearized boundary conditions for this problem are identical with the Maxwell slip relations utilized by Millikan, so the same expressions for slip velocity and drag coefficient are obtained. An examination of the drag data obtained by Kuhlthau, Chiang, and Bowyer and Talbot in their rotating-cylinder experiments at low densities shows that the variation of 1/C_DM with Re/M is predicted reasonably well by this theory over a range of Mach numbers from 0.15 to 1.40, in spite of the fact that the theory is supposed to hold only for low Mach numbers.

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