Burnett equations
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]
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Series expansion
[edit]Series expansion approach
[edit]The series expansion technique used to derive the Burnett equations involves expanding the distribution function in the Boltzmann equation as a power series in the Knudsen number :
Here, represents the Maxwell-Boltzmann equilibrium distribution function, dependent on the number density , macroscopic velocity , and temperature . The terms etc., are higher-order corrections that account for non-equilibrium effects, with each subsequent term incorporating higher powers of the Knudsen number .
Derivation
[edit]The first-order term 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 . 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:
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
[edit]The Onsager-Burnett Equations, commonly referred to as OBurnett, which form a superset of the Navier-Stokes equations and are second-order accurate for Knudsen number.[5]
Eq. (1)
Eq. (2) [6]
Derivation
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Starting with the Boltzmann equation
See also
[edit]- Fluid dynamics
- Lars Onsager
- Non-dimensionalization and scaling of the Navier–Stokes equations
- Stokes equations
- Chapman–Enskog theory
- Navier-Stokes equations
References
[edit]- ^ "No text - Big Chemical Encyclopedia".
- ^ Singh, Narendra; Agrawal, Amit (2014). "The Burnett equations in cylindrical coordinates and their solution for flow in a microtube". Journal of Fluid Mechanics. 751: 121–141. Bibcode:2014JFM...751..121S. doi:10.1017/jfm.2014.290.
- ^ Agrawal, Amit; Kushwaha, Hari Mohan; Jadhav, Ravi Sudam (2020). "Burnett Equations: Derivation and Analysis". Microscale Flow and Heat Transfer. Mechanical Engineering Series. pp. 125–188. doi:10.1007/978-3-030-10662-1_5. ISBN 978-3-030-10661-4.
- ^ Burnett, D. (1936). "The Distribution of Molecular Velocities and the Mean Motion in a Non-Uniform Gas". Proceedings of the London Mathematical Society. s2-40 (1): 382–435. doi:10.1112/plms/s2-40.1.382.
- ^ Jadhav, Ravi Sudam; Agrawal, Amit (December 23, 2021). "Shock Structures Using the OBurnett Equations in Combination with the Holian Conjecture". Fluids. 6 (12): 427. Bibcode:2021Fluid...6..427J. doi:10.3390/fluids6120427.
- ^ Agarwal, Ramesh K.; Yun, Keon-Young; Balakrishnan, Ramesh (October 1, 2001). "Beyond Navier–Stokes: Burnett equations for flows in the continuum–transition regime". Physics of Fluids. 13 (10): 3061–3085. Bibcode:2001PhFl...13.3061A. doi:10.1063/1.1397256.
Further reading
[edit]- García-Colín, L.S.; Velasco, R.M.; Uribe, F.J. (August 2008). "Beyond the Navier–Stokes equations: Burnett hydrodynamics". Physics Reports. 465 (4): 149–189. Bibcode:2008PhR...465..149G. doi:10.1016/j.physrep.2008.04.010.
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