Scaled Navier-Stokes-Fourier Equations for Gas Flow and Heat Transfer Phenomena in Micro- and Nanosystems

A new hydrodynamic model is proposed in order to model critical phenomena in gas flows at the micro- and nanoscale. A scaling is applied to the conventional Navier-Stokes-Fourier equations, mathematically equivalent to using an “effective” viscosity and an “effective” thermal conductivity in the original linear constitutive relations. Expressions for this “effective” viscosity and this “effective” thermal conductivity are obtained from two ideal half-space flow problems: Kramer’s problem, and the temperature jump problem. Our model ensures the correct viscous stress is maintained in the region of the wall in isothermal flow (or the correct heat flux in the pure heat transfer situation); it is only the relationships between stress and the corresponding near-wall strain-rate, and between heat flux and the near-wall temperature gradient, that are altered. The advantage of our model over the traditional linear hydrodynamic model is that the non-equilibrium flow in the Knudsen layer is described. Its advantage over higher-order hydrodynamic models for rarefied gas flows is that no additional boundary conditions are required (although there are minor changes in the slip/jump coefficients), so modifications of current CFD codes to incorporate this new model would be minimal. As an application example, we solve for the velocity profiles and drag force on a micro-sphere moving in a gas at different Knudsen numbers (Kn). For this problem, our model gives excellent results for Kn<0.1 and accptable results up to Kn = 0.25: this is considerably better that the tradition Navier-Stokes model with non-scaled constitutive relations.