Kinetic Theory of Linear Shear Flow

Abstract

The flow of a monatomic gas between two parallel plates kept at the same temperature and moving in opposite directions is studied. The relative velocity of the plates is much smaller than the speed of sound. The deviation from the equilibrium distribution, φ(c, x), satisfies the linearized Boltzmann equation. The customary boundary conditions are adopted in which a fraction of the molecules is specularly reflected and the rest emitted with a Maxwellian distribution characteristic of the plate. The method consists of setting φ = φ+ for cx > 0 and φ = φ− for cx < 0 so that positive and negative velocities are distinguished. We take φ± = a0±(x)cz + a1±(x)czcx. The space functions are determined by taking half-range velocity moments of the Boltzmann equation. Explicit results for the distribution function, flow velocity and stress are given for a general law of force. Numerical results are worked out for hard sphere molecules. The method treats both microscopic boundary conditions and conservation laws exactly. Precise results are obtained both for the low-pressure region and for the high-pressure coefficient of viscosity. The region of slip flow is analyzed. Maxwell's slip condition is remarkably close to the condition obtained here from the kinetic theory. In this region of pressures, the deviations from the velocity profile of the hydrodynamic slip flow theory are everywhere very small, and are completely negligible at distances greater than ⅛ of a mean free path from a plate.