Flow of a Rarefied Gas between Parallel and Almost Parallel Plates

Rarefied gas flows in ultra-thin film slider bearings are studied in a wide range of Knudsen numbers. The generalized Reynolds equation first derived by Fukui and Kaneko [1], [2] has been extended by allowing for bounding surfaces with different physical structures, as an issue of relevance for applications. Since the solution of this equation requires that the Poiseuille and Couette flow rates between two parallel plates have to be accurately calculated in advance, we have used our recent results on Poiseuille flow [3], [4] and new results on Couette flow to evaluate the lubrication characteristics. FLOW OF A RAREFIED GAS BETWEEN TWO PARALLEL PLATES: THE POISEUILLE-COUETTE PROBLEM Let us consider two plates separated by a distance h and a gas flowing parallel to them, in the x direction, due to a pressure gradient. The lower boundary (placed at z h 2) moves to the right with velocity U , while the upper boundary (placed at z h 2) is fixed. Both boundaries are held at a constant temperature To. If the pressure gradient is taken to be small as well as the velocity U , it can be assumed that the velocity distribution of the flow is nearly the same as that occurring in an equilibrium state. This means that the Boltzmann equation can be linearized about a Maxwellian f0 by putting [5]: f f0 1 h (1) where f x z c is the distribution function for the molecular velocity c expressed in units of 2RTo 1 2 (R being the gas constant), z is the coordinate normal to the plates and h z c is the small perturbation upon the basic equilibrium state. If one assumes the linearized BGK model for the collision operator [6], the Boltzmann equation reads [7]: 1 2 cz ∂Z ∂ z 1 θ π 1 2 ∞ ∞ e c 2 z1 Z z cz1 dcz1 Z z cz (2) where by definition Z z cz π 1 ∞ ∞ ∞ ∞ e c 2 x cy cxh z c dcx dcy k 1 p ∂ p ∂x 1 ρ ∂ρ ∂x with p and ρ being the gas pressure and density, respectively, and θ is the collision time. Consequently, the integral equation for the bulk velocity of the gas can be written as follows: q z π 1 2 ∞ ∞ e c 2 z1 Z z cz1 dcz1 (3) From Eq. (2) we obtain in integral form [3], [8]: