View source for Burnett equations

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 equation]]s do not perform well.<ref>{{cite web | url=https://chempedia.info/page/181144203140177134020048169072027076118072084254/ | title=No text - Big Chemical Encyclopedia }}</ref><ref>{{cite journal | url=https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/abs/burnett-equations-in-cylindrical-coordinates-and-their-solution-for-flow-in-a-microtube/F89FC9C029F70CC2669E61C6E5DB0A6B | doi=10.1017/jfm.2014.290 | title=The Burnett equations in cylindrical coordinates and their solution for flow in a microtube | date=2014 | last1=Singh | first1=Narendra | last2=Agrawal | first2=Amit | journal=Journal of Fluid Mechanics | volume=751 | pages=121–141 | bibcode=2014JFM...751..121S }}</ref><ref>{{cite book | chapter-url=https://link.springer.com/chapter/10.1007/978-3-030-10662-1_5 | doi=10.1007/978-3-030-10662-1_5 | chapter=Burnett Equations: Derivation and Analysis | title=Microscale Flow and Heat Transfer | series=Mechanical Engineering Series | date=2020 | last1=Agrawal | first1=Amit | last2=Kushwaha | first2=Hari Mohan | last3=Jadhav | first3=Ravi Sudam | pages=125–188 | isbn=978-3-030-10661-4 }}</ref>

They were derived by the English mathematician D. Burnett.<ref>{{cite journal|title=The Distribution of Molecular Velocities and the Mean Motion in a Non-Uniform Gas|first1=D.|last1=Burnett|journal=Proceedings of the London Mathematical Society |year=1936 |volume=s2-40|issue=1|pages=382–435|doi=10.1112/plms/s2-40.1.382}}</ref>

{{continuum mechanics}}

==Series expansion==

=== Series expansion approach ===
The series expansion technique used to derive the Burnett equations involves expanding the distribution function <math>
f</math> in the Boltzmann equation as a power series in the Knudsen number <math>
Kn</math>:

<math>
f(r,c,t) = f^{(0)}(c|n,u,T) \left[1 + K_n \phi^{(1)}(c|n,u,T) + K_n^2 \phi^{(2)}(c|n,u,T) + \cdots \right]</math>

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

=== Derivation ===
The first-order term <math>
f^{(1)}</math> in the expansion gives the [[Navier–Stokes equations|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 <math>
\phi^{(2)}
</math>. 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:

<math>
\mathbf{u}_t + (\mathbf{u} \cdot \nabla)\mathbf{u} + \nabla p = \nabla \cdot (\nu \nabla \mathbf{u}) + \text{higher-order terms}
</math>

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 are
second-order accurate for [[Knudsen number]].<ref>{{Cite journal|title=Shock Structures Using the OBurnett Equations in Combination with the Holian Conjecture|first1=Ravi Sudam|last1=Jadhav|first2=Amit|last2=Agrawal|date=December 23, 2021|journal=Fluids|volume=6|issue=12|pages=427|doi=10.3390/fluids6120427|bibcode=2021Fluid...6..427J |doi-access=free }}</ref>

Eq. (1)
<math> 
 \sqrt{\tau} \frac{du^s}{ds} - \frac{9}{8} \alpha_1 u^* (\frac{du^*}{ds})^2 = \frac{\tau}{u^*} - \tau_0 -1 + u^*   
</math>

Eq. (2) 
<math>
\frac{45}{16} \sqrt{\tau} \frac{d \tau}{ds} + \frac{9}{4} \gamma_1 \tau (\frac{du^*}{ds})^2 -  \frac{9}{4} \Psi u^* \frac{d \tau}{ds} \frac{du^*}{ds} = \frac{3}{2} (\tau - \tau_0) - \frac{1}{2} (1-u^*)^2 - \tau_0(1-u^*)
</math><ref>{{Cite journal|url=https://aip.scitation.org/doi/10.1063/1.1397256|title=Beyond Navier–Stokes: Burnett equations for flows in the continuum–transition regime|first1=Ramesh K.|last1=Agarwal|first2=Keon-Young|last2=Yun|first3=Ramesh|last3=Balakrishnan|date=October 1, 2001|journal=Physics of Fluids|volume=13|issue=10|pages=3061–3085 |doi=10.1063/1.1397256|bibcode=2001PhFl...13.3061A }}</ref>

==Derivation==
{{expand section|the derivation should be completed.|date=July 2024}}
Starting with the [[Boltzmann equation]]
<math> \frac{\partial{f}}{\partial{t}} + c_k \partial{f}{x_k} + F_k \partial{f}{c_k} = J(f, f_1) </math>

==See also==
*[[Fluid dynamics]]
*[[Lars Onsager]]
*[[Non-dimensionalization and scaling of the Navier–Stokes equations]]
*[[Stokes flow|Stokes equations]]
*[[Chapman–Enskog theory]]
*[[Navier-Stokes equations]]

==References==
{{reflist}}

==Further reading==
*{{cite journal|title=Beyond the Navier–Stokes equations: Burnett hydrodynamics|first1=L.S.|last1=García-Colín|first2=R.M.|last2=Velasco|first3=F.J.|last3=Uribe|journal=Physics Reports|volume=465|issue=4|date=August 2008|pages=149–189 |doi=10.1016/j.physrep.2008.04.010|bibcode=2008PhR...465..149G }}

{{improve categories|date=July 2024}}
[[Category:Continuum mechanics]]


{{Math-stub}}