Nonlinear transport coefficients and boundary-layer equations

Generalized hydrodynamics and microflows

In this paper, a mathematical model within the framework of generalized hydrodynamics is developed for the description of flows in microsystems where the Knudsen number is large and the aspect ratio [(width)/(length)] is not so small. The model is based on a set of empirical generalized hydrodynamic equations, which are fashioned from the steady-state generalized hydrodynamic equations derived from the Boltzmann equation in a manner consistent with the laws of thermodynamics. The constitutive equations used for the model are highly nonlinear, unlike the Newtonian law of viscosity and the Fourier law of heat conduction, but they are thermodynamically consistent. In the absence of heat conduction, the model yields exact solutions for the velocity components and a nonlinear differential equation for the pressure distribution in the rectangular microchannel. The Langmuir adsorption model for surface-gas interaction is used for boundary conditions for the velocity. The calculated flow rate exhibits a Knudsen minimum with respect to the Knudsen number. The longitudinal velocity profile is also non-Poiseuille. The differential equation for pressure distribution is also solved approximately in order to obtain an analytic formula for the flow rate, which exhibits a Knudsen minimum. The formula, although approximate, provides considerable insights into the Knudsen flow phenomena in microchannels.