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]

Part of a series on
Continuum mechanics
${\displaystyle J=-D{\frac {d\varphi }{dx}}}$ Fick's laws of diffusion
Laws Conservations Mass Momentum Energy Inequalities Clausius–Duhem (entropy)
Solid mechanics Deformation Elasticity linear Plasticity Hooke's law Stress Strain Finite strain Infinitesimal strain Compatibility Bending Contact mechanics frictional Material failure theory Fracture mechanics
Fluid mechanics Fluids Statics · Dynamics Archimedes' principle · Bernoulli's principle Navier–Stokes equations Poiseuille equation · Pascal's law Viscosity (Newtonian · non-Newtonian) Buoyancy · Mixing · Pressure Liquids Adhesion Capillary action Chromatography Cohesion (chemistry) Surface tension Gases Atmosphere Boyle's law Charles's law Combined gas law Fick's law Gay-Lussac's law Graham's law Plasma
Rheology Viscoelasticity Rheometry Rheometer Smart fluids Electrorheological Magnetorheological Ferrofluids
Scientists Bernoulli Boyle Cauchy Charles Euler Fick Gay-Lussac Graham Hooke Newton Navier Noll Pascal Stokes Truesdell
v t e

The series expansion technique used to derive the Burnett equations involves expanding the distribution function ${\displaystyle f}$ in the Boltzmann equation as a power series in the Knudsen number ${\displaystyle Kn}$:

${\displaystyle f(r,c,t)=f^{(0)}(c|n,u,T)\left[1+K_{n}\phi ^{(1)}(c|n,u,T)+K_{n}^{2}\phi ^{(2)}(c|n,u,T)+\cdots \right]}$

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

The first-order term ${\displaystyle f^{(1)}}$ 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 ${\displaystyle \phi ^{(2)}}$. 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:

${\displaystyle \mathbf {u} _{t}+(\mathbf {u} \cdot \nabla )\mathbf {u} +\nabla p=\nabla \cdot (\nu \nabla \mathbf {u} )+{\text{higher-order terms}}}$

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.

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) ${\displaystyle {\sqrt {\tau }}{\frac {du^{s}}{ds}}-{\frac {9}{8}}\alpha _{1}u^{*}({\frac {du^{*}}{ds}})^{2}={\frac {\tau }{u^{*}}}-\tau _{0}-1+u^{*}}$

Eq. (2) ${\displaystyle {\frac {45}{16}}{\sqrt {\tau }}{\frac {d\tau }{ds}}+{\frac {9}{4}}\gamma _{1}\tau ({\frac {du^{*}}{ds}})^{2}-{\frac {9}{4}}\Psi u^{*}{\frac {d\tau }{ds}}{\frac {du^{*}}{ds}}={\frac {3}{2}}(\tau -\tau _{0})-{\frac {1}{2}}(1-u^{*})^{2}-\tau _{0}(1-u^{*})}$^[6]

This section needs expansion with: the derivation should be completed.. You can help by adding to it. (July 2024)

Starting with the Boltzmann equation ${\displaystyle {\frac {\partial {f}}{\partial {t}}}+c_{k}\partial {f}{x_{k}}+F_{k}\partial {f}{c_{k}}=J(f,f_{1})}$

• Fluid dynamics
• Lars Onsager
• Non-dimensionalization and scaling of the Navier–Stokes equations
• Stokes equations
• Chapman–Enskog theory
• Navier-Stokes equations

• 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.

This article needs additional or more specific categories. Please help out by adding categories to it so that it can be listed with similar articles. (July 2024)

This mathematics-related article is a stub. You can help Wikipedia by expanding it.

• v
• t
• e