Burnett equations explained

In continuum mechanics, a branch of mathematics, the Burnett equations is a set of higher-order continuum equations for non-equilibrium flows and the transition regimes where the Navier–Stokes equations do not perform well.^{[1] [2] [3]}

They were derived by the English mathematician D. Burnett.^[4]

Series expansion

f(r,c,t) = f ^{0(c|} n,u,T ) 1 + Kn \phi ^{1} (c|n,u,T) + Kn ^{2} \phi ^{2(c|n,u,T)} + …

Extensions

The Onsager-Burnett Equations, commonly referred to as OBurnett, which form a superset of the Navier-Stokes equations and aresecond-order accurate for Knudsen number.^[5]

Eq. (1)

\sqrt{\tau} (du ^{s})/(ds) - (9)/(8) \alpha _{1} u ^{*} ( (du ^{*})/(ds) ) ^{2} = (\tau)/(u ^{*}) - \tau _{0} -1 + u ^{*}

Eq. (2)

(45)/(16) \sqrt{\tau} (d \tau)/(ds) + (9)/(4) \gamma _{1} \tau ( (du ^{*})/(ds) ) ^{2} - (9)/(4) \Psi u ^{*} (d \tau)/(ds) (du ^{*})/(ds) = (3)/(2) (\tau - \tau _{0)} - (1)/(2) ( 1- u ^{*)} ^{2} - \tau ^{*)} _{0 ( 1- u} ^[6]

Derivation

Starting with the Boltzmann equation

(\partial{f)/() } + c_k \partial + F_k \partial = J(f, f_1)

See also

• Fluid dynamics
• Lars Onsager
• Non-dimensionalization and scaling of the Navier–Stokes equations
• Stokes equations
• Chapman–Enskog theory
• Navier-Stokes equations

Further reading

• Beyond the Navier–Stokes equations: Burnett hydrodynamics. L.S.. García-Colín. R.M.. Velasco. F.J.. Uribe. Physics Reports. 465. 4. August 2008. 149–189 . 10.1016/j.physrep.2008.04.010. 2008PhR...465..149G .

Notes and References

1. Web site: No text - Big Chemical Encyclopedia .
2. The Burnett equations in cylindrical coordinates and their solution for flow in a microtube . 10.1017/jfm.2014.290 . 2014 . Singh . Narendra . Agrawal . Amit . Journal of Fluid Mechanics . 751 . 121–141 . 2014JFM...751..121S .
3. Book: https://link.springer.com/chapter/10.1007/978-3-030-10662-1_5 . 10.1007/978-3-030-10662-1_5 . Burnett Equations: Derivation and Analysis . Microscale Flow and Heat Transfer . Mechanical Engineering Series . 2020 . Agrawal . Amit . Kushwaha . Hari Mohan . Jadhav . Ravi Sudam . 125–188 . 978-3-030-10661-4 .
4. The Distribution of Molecular Velocities and the Mean Motion in a Non-Uniform Gas. D.. Burnett. Proceedings of the London Mathematical Society . 1936 . s2-40. 1. 382–435. 10.1112/plms/s2-40.1.382.
5. Shock Structures Using the OBurnett Equations in Combination with the Holian Conjecture. Ravi Sudam. Jadhav. Amit. Agrawal. December 23, 2021. Fluids. 6. 12. 427. 10.3390/fluids6120427. 2021Fluid...6..427J . free .
6. Beyond Navier–Stokes: Burnett equations for flows in the continuum–transition regime. Ramesh K.. Agarwal. Keon-Young. Yun. Ramesh. Balakrishnan. October 1, 2001. Physics of Fluids. 13. 10. 3061–3085 . 10.1063/1.1397256. 2001PhFl...13.3061A .