Analytical solution of plane Couette flow in the transition regime and comparison with Direct Simulation Monte Carlo data

Derivation of extended-OBurnett and super-OBurnett equations and their analytical solution to plane Poiseuille flow at non-zero Knudsen number

“…Therefore, instead of getting bogged down by the unavailability of accurate boundary conditions, treating the problem as an initial value problem has been suggested in the literature (Singh et al. 2014, 2017; Jadhav et al. 2017; Rath et al.…”

“…Bao & Lin (2008) used the augmented Burnett equations to numerically solve the plane Poiseuille flow problem by employing a relaxation method on the boundary values, in line with that previously used by Lockerby & Reese (2003) to present comparisons of the velocity and pressure fields against experimental and DSMC data. More recently, the linearized form of the O(Kn 3 ) super-Burnett equations (Shavaliyev 1993) was solved analytically for the Couette flow problem in the transition regime (Singh et al 2014). It should also be noted that, despite widespread use and success, the above-mentioned Burnett variants are associated with several limitations (Bobylev 1982;Shavaliyev 1993;Uribe & Garcia 1999;García-Colín et al 2008;Dadzie 2013) which have prompted the development of additional Burnett-like variants.…”

“…The first category of higher-order transport equations, namely that of the Burnett-type equations, originated from the original Burnett equations (Burnett 1936). However, due to their complicated and cumbersome nature, the original Burnett equations did not gain widespread adoption; indeed, numerical studies using the original Burnett equations were only recently conducted for the Couette flow problem (Lockerby & Reese 2003; Singh, Gavasane & Agrawal 2014). In order to make the original Burnett equations more tractable, Chapman & Cowling (1970) presented an alternate form, namely the conventional Burnett equations, wherein the material derivatives appearing in the second-order distribution function were replaced by the spatial gradients described by the Euler equations.…”

“…More recently, the linearized form of the () super-Burnett equations (Shavaliyev 1993) was solved analytically for the Couette flow problem in the transition regime (Singh et al. 2014). It should also be noted that, despite widespread use and success, the above-mentioned Burnett variants are associated with several limitations (Bobylev 1982; Shavaliyev 1993; Uribe & Garcia 1999; García-Colín et al.…”

Heat flux correlation for high-speed flow in the transitional regime

“…In addition, these equations involve second-order (Burnett type) and third-order (super-Burnett type) derivatives of flow quantities such as density, temperature and velocity. This poses challenges for numerical solution techniques, and analytical solutions are generally not possible with few exceptions Singh, Gavasane & Agrawal 2014). Despite these challenges, these hydrodynamics models have been successfully applied to low-speed rarefied problems, for which a number of important rarefied phenomena have been captured (Zohar et al 2002;He, Tang & Pu 2008;Gu, Emerson & Tang 2009;Agrawal & Dongari 2011;Rana, Torrilhon & Struchtrup 2013;Singh, Dongari & Agrawal 2013;Akintunde & Petculescu 2014;Khalil, Garzó & Santos 2014;Rahimi & Struchtrup 2014;, and have also been applied to hypersonic flows (Agarwal et al 2001), for which a simplified set of conventional Burnett equations was recently proposed specifically for rarefied hypersonic flows (Zhao, Chen & Agarwal 2014).…”