6.5 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Burnett equations | 1/1 | https://en.wikipedia.org/wiki/Burnett_equations | reference | science, encyclopedia | 2026-05-05T12:04:25.352895+00:00 | kb-cron |
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. They were derived by the English mathematician D. Burnett.
== Series expansion ==
=== Series expansion approach === The series expansion technique used to derive the Burnett equations involves expanding the distribution function
f
{\displaystyle f}
in the Boltzmann equation as a power series in the Knudsen number
K
n
{\displaystyle \mathrm {Kn} }
:
f
(
r
,
c
,
t
)
=
f
(
0
)
(
c
|
n
,
u
,
T
)
[
1
+
K
n
ϕ
(
1
)
(
c
|
n
,
u
,
T
)
+
K
n
2
ϕ
(
2
)
(
c
|
n
,
u
,
T
)
+
⋯
]
{\displaystyle f(r,c,t)=f^{(0)}(c|n,u,T)\left[1+\mathrm {Kn} \phi ^{(1)}(c|n,u,T)+\mathrm {Kn} ^{2}\phi ^{(2)}(c|n,u,T)+\cdots \right]}
Here,
f
(
0
)
(
c
|
n
,
u
,
T
)
{\displaystyle f^{(0)}(c|n,u,T)}
represents the Maxwell-Boltzmann equilibrium distribution function, dependent on the number density
n
{\displaystyle n}
, macroscopic velocity
u
{\displaystyle u}
, and temperature
T
{\displaystyle T}
. The terms
ϕ
(
1
)
,
ϕ
(
2
)
,
…
{\displaystyle \phi ^{(1)},\phi ^{(2)},\dots }
are higher-order corrections that account for non-equilibrium effects, with each subsequent term incorporating higher powers of the Knudsen number.
=== Derivation === The first-order term
f
(
1
)
{\displaystyle 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
ϕ
(
2
)
{\displaystyle \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
⋅
∇
)
u
+
∇
p
=
∇
⋅
(
ν
∇
u
)
+
higher-order terms
{\displaystyle \mathbf {u} _{t}+(\mathbf {u} \cdot \nabla )\mathbf {u} +\nabla p=\nabla \cdot (\nu \nabla \mathbf {u} )+{\text{higher-order terms}}}
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.
== Derivation ==
Starting with the Boltzmann equation
∂
f
∂
t
+
c
k
∂
f
x
k
+
F
k
∂
f
c
k
=
J
(
f
,
f
1
)
{\displaystyle {\frac {\partial {f}}{\partial {t}}}+c_{k}\partial {f}{x_{k}}+F_{k}\partial {f}{c_{k}}=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
== References ==
== Further reading == 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.