Derivation of stable Burnett equations for rarefied gas flows

A set of constitutive relations for the stress tensor and heat flux vector for the hydrodynamic description of rarefied gas flows is derived in this work. A phase density function consistent with Onsager's reciprocity principle and $H$ theorem is utilized to capture nonequilibrium thermodynamics effects. The phase density function satisfies the linearized Boltzmann equation and the collision invariance property. Our formulation provides the correct value of the Prandtl number as it involves two different relaxation times for momentum and energy transport by diffusion. Generalized three-dimensional constitutive equations for different kinds of molecules are derived using the phase density function. The derived constitutive equations involve cross single derivatives of field variables such as temperature and velocity, with no higher-order derivative in higher-order terms. This is remarkable feature of the equations as the number of boundary conditions required is the same as needed for conventional Navier-Stokes equations. Linear stability analysis of the equations is performed, which shows that the derived equations are unconditionally stable. A comparison of the derived equations with existing Burnett-type equations is presented and salient features of our equations are outlined. The classic internal flow problem, force-driven compressible plane Poiseuille flow, is chosen to verify the stable Burnett equations and the results for equilibrium variables are presented.

Figure 1

Schematic of plane Poiseuille flow driven by an external force $a$.

Figure 2

(a) Pressure, (b) velocity, and (c) temperature distribution in the cross stream direction in force-driven compressible plane Poiseuille flow at $\text{Kn}=0.025$. In the case of velocity and temperature variation, the results of NSF equations, Burnett equations, and stable Burnett equations overlap.

Figure 3

(a) Pressure, (b) velocity, and (c) temperature distribution in the cross stream direction in force-driven compressible plane Poiseuille flow at $\text{Kn}=0.05$. In the case of velocity and temperature variation, the results of NSF equations, Burnett equations, and stable Burnett equations overlap.

Figure 4

(a) Pressure, (b) velocity, and (c) temperature distribution in the cross stream direction in force-driven compressible plane Poiseuille flow at $\text{Kn}=0.1$. In the case of velocity and temperature variation, the results of NSF equations, Burnett equations, and stable Burnett equations overlap.