Burnett equations explained

In continuum mechanics, a branch of mathematics, the Burnett equations are 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

Series expansion approach

The series expansion technique used to derive the Burnett equations involves expanding the distribution function

f in the Boltzmann equation as a power series in the Knudsen number Kn :f(r,c,t) = f ^{(0)} (c|n,u,T) \left[ 1 + K _{n} \phi ^{(1)} (c|n,u,T) + K ^{2} _{n} \phi ^{(2)} (c|n,u,T) + … \right]

Here,

f ^{(0)} (c|n,u,T) represents the Maxwell-Boltzmann equilibrium distribution function, dependent on the number density n , macroscopic velocity u , and temperature T . The terms \phi ^{(1)} , \phi ^{(2)} , etc., are higher-order corrections that account for non-equilibrium effects, with each subsequent term incorporating higher powers of the Knudsen number Kn .

Derivation

The first-order term

f ^{(1)} in the expansion gives the Navier-Stokes equations, which include terms for viscosity and thermal conductivity. To obtain the Burnett equations, one must retain terms up to second order, corresponding to \phi ^{(2)} . The Burnett equations include additional second-order derivatives of velocity, temperature, and density, representing more subtle effects of non-equilibrium gas dynamics.

The Burnett equations can be expressed as:

u _{t} + ( u ⋅ \nabla) u + \nabla p = \nabla ⋅ (\nu \nabla u ) + higher-orderterms

Here, the "higher-order terms" involve second-order gradients of velocity and temperature, which are absent in the Navier-Stokes equations. These terms become significant in situations with high Knudsen numbers, where the assumptions of the Navier-Stokes framework break down.

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 .