Erdogan–Chatwin equation

In fluid dynamics, Erdogan–Chatwin equation refers to a nonlinear diffusion equation for the scalar field, that accounts for shear-induced dispersion due to horizontal buoyancy forces. The equation was named after M. Emin Erdogan and Phillip C. Chatwin, who derived the equaiton in 1967.^[1] The equation for the scalar field ${\displaystyle \varphi (x,t)}$ reads^[2][3][4]

${\displaystyle \varphi _{t}=(\varphi _{x}+a\varphi _{x}^{3})_{x},}$

where ${\displaystyle a}$ is a positive constant. For ${\displaystyle a\ll 1}$, the equation reduces to the linear heat equation, ${\displaystyle \varphi _{t}=\varphi _{xx}}$ and for ${\displaystyle a\gg 1}$, the equation reduces to ${\displaystyle \varphi _{t}=3a\varphi _{x}^{2}\varphi _{xx}}$.