kb/data/en.wikipedia.org/wiki/Burnett_equations-0.md

---
title: "Burnett equations"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Burnett_equations"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T12:04:25.352895+00:00"
instance: "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.