Couette Flow With Heat Transfer in the Whole Range of the Knudsen Number

Abstract

The steady-state one-dimensional flow of a gas between two parallel plates moving with relative velocity and maintained at different temperatures is examined on the basis of the nonlinear Bhatnagar-Gross-Krook (BGK) model equation, subject to Maxwell diffuse boundary conditions. The computational scheme is based on the discrete velocity method. Results are provided for the bulk quantities of the flow in the whole range of the Knudsen number from the free molecular through the transition and slip regimes all the way up to the hydrodynamic limit. Several interesting issues related to the combined momentum and heat transfer effects are examined including the presence of a heat flux parallel to the plates, induced by the shear flow. This is a cross effect, which is absent in the Navier-Stokes-Fourier regime. The validity of the results is confirmed in several ways including the recovery of the analytical solutions at the free molecular limit.

Navier-Stokes-Fourier analytic solutions for non-isothermal Couette slip gas flow

“…The procedure for solving the system of NSF governing eqs. (1) and 3, and boundary conditions (5) and (6) is the same for the slip and continuum regimes.…”

“…The problem is solved with the same procedure as the previous one, where the walls were on different temperature. The only difference when the bottom wall is adiabatic is in the temperature boundary conditions (6). There is no temperature jump at the bottom wall, and if the temperature is scaled by the top wall temperature, the boundary conditions for the bottom and the top wall are respectively: The velocity boundary conditions are given by the same equations (20) and (21) as in the previous case where the walls were on different temperature.…”

“…Isothermal Couette flow is solved using microscopic approach mostly, either by solving the kinetic Boltzmann equation [3][4][5][6][7] or by the direct simulation Monte-Carlo (DSMC) [3,4,[7][8][9][10][11][12]. Also there are results with macroscopic approach by solving Navier--Stokes-Fourier (NSF) [3] or Burnett equations [8] numerically.…”

“…The velocity and temperature fields for flow with isothermal as well as with constant but different walls temperatures by non-linear Bathangar-Gross-Krook equation in the whole range of Knudsen numbers were given in [6]. Also, the velocity and temperature fields for different walls temperatures obtained by DSMC are shown in [11,12].…”