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Burnett equations

From Wikipedia, the free encyclopedia

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

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Series expansion approach

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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

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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

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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

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References

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  1. ^ "No text - Big Chemical Encyclopedia".
  2. ^ 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.
  3. ^ 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.
  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.
  5. ^ 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.
  6. ^ 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

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