Derivation of stable Burnett equations for rarefied gas flows

Abstract

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.

One-step replica-symmetry-breaking phase below the de Almeida–Thouless line in low-dimensional spin glasses

“…V above). Other results for a range of dimensions were obtained from high-temperature series [15] that studied the SG susceptibility; evidence of an AT line was found for d ≥ 6, but for d = 5 most Pade approximants showed no divergence of j χ ij . Clearly these results agree with ours to some degree.…”

“…A number of simulations (in both the nearest-neighbor d-dimensional and one-dimensional power law models; see for example Refs. [12][13][14]), and also high-temperature series expansions [15], found no divergence of the correlation length in a magnetic field in low dimensions (d < 6, and for corresponding power-laws in one dimension).…”

On the accuracy of macroscopic equations for linearized rarefied gas flows

“…We also note that another form of Burnett equations has also been proposed recently, based on Onsager's reciprocity principle, where the VDF is expanded as the function of thermodynamics forces and their corresponding fluxes [62]. However, in the linearized case, they are reduced to the linearized NS equations, and hence cannot describe the RBS spectra in the kinetic regime.…”

Improved theory for shock waves using the OBurnett equations

“…In the derivation of the OBurnett equations (Singh et al 2017), the Onsager-consistent distribution function is cast in terms of thermodynamic forces and fluxes (Mahendra & Singh 2013;Singh & Agrawal 2016;Agrawal et al 2020) and constructed carefully so that it is consistent with the Onsager symmetry principle (Onsager 1931a,b) and the H-theorem. This particular form of the distribution function also satisfies the linearized Boltzmann equation and the collision invariance property and which is then utilized to evaluate the Burnett-order constitutive relationships for the stress tensor and the heat flux vector.…”

“…Another recent approach, known as the Onsager-consistent approach, has been proposed recently (Singh & Agrawal 2016;Agrawal, Kushwaha & Jadhav 2020) wherein Onsager's symmetry principle forms the basis for the derivation of the particle distribution function. Utilizing this form of the distribution function, the Burnett-like equations, known as the Onsager-Burnett (OBurnett) equations (Singh et al 2017), and the Grad-like equations, known as the Onsager-13 (O13) moment equations (Singh & Agrawal 2016), were derived. More details about this approach are given in § 2.…”