Interdisciplinary Applied Mathematics

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4-2.2 A Unified Flow Model

In this section we will develop a unified flow model that predicts velocity profiles, pressure drop, and mass flowrate in channels, pipes, and arbitrary aspect ratio rectangular ducts in the entire Knudsen regime. Our approach is divided into two main steps: First, we will analyze the nondimensional velocity profile to identify the shape of the velocity distribution. Then, we will obtain the magnitude of the average velocity, and hence obtain a prediction for the flowrate. The proposed unified model also predicts Knudsen’s minimum.

Velocity Scaling

From the DSMC results and solutions of the linearized Boltzmann equation, it is evident that the velocity profiles remain approximately parabolic for a large range of Knudsen number (see Figure 4.19). This is also consistent with the analysis of the Navier-Stokes and Burnett equations in long channels, as documented in Section 4.2.1. Based on this observation, we model the velocity profile as parabolic in the entire Knudsen regime, with a consistent slip condition. We write the dimensional form for velocity distribution in a channel of height h,

where F ((dP/dx), pc, h, A) shows functional dependence of velocity on the pressure gradient, viscosity, channel height, and local mean free path. The temperature is assumed to be constant, and therefore the dynamic viscosity is also    a    constant.    Here    Us    is the    velocity    slip, and it    satisfies    equation

(2.43), which after substitution in the above equation (assuming Uw = 0) yields

Ф2 + Ф +

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