Erdogan–Chatwin equation

In fluid dynamics, Erdogan–Chatwin equation is 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 equation in 1967.^[1][2][3][4] The equation for the scalar field ${\displaystyle c(x,t)}$ reads^[5]

${\displaystyle {\frac {\partial c}{\partial t}}={\frac {\partial }{\partial x}}\left\{D\left[1+{\frac {\gamma g^{2}h^{8}\alpha ^{2}}{\nu ^{2}D^{2}}}\left({\frac {\partial c}{\partial x}}\right)^{2}\right]{\frac {\partial c}{\partial x}}\right\},}$

where

${\displaystyle D}$	is the diffusion coefficient for the scalar ${\displaystyle c}$;
${\displaystyle \gamma }$	is a numerical factor, which in planar problems assumes the value ${\displaystyle 2/2835}$;
${\displaystyle g}$	is the gravitational acceleration;
${\displaystyle 2h}$	is the width of the fluid layer in which dispersion is occurring;
${\displaystyle \alpha }$	is the volumetric expansion coefficient defined by ${\displaystyle \alpha =-\rho ^{-1}\partial \rho /\partial c}$ with ${\displaystyle \rho }$ being the fluid density;
${\displaystyle \nu }$	is the kinematic viscosity of the fluid.

Suppose ${\displaystyle L}$ is the characteristic length scale for ${\displaystyle c}$, then the characteristic time scale is given by ${\displaystyle L^{2}/D}$. And suppose ${\displaystyle C}$ is a characteristic value for ${\displaystyle c}$. Then, we introudce the non-dimensional variables

${\displaystyle \tau ={\frac {t}{L^{2}/D}},\quad \xi ={\frac {x}{L}},\quad \varphi ={\frac {c}{C}}}$

then the Erdogan–Chatwin equation becomes^[5]

${\displaystyle {\frac {\partial \varphi }{\partial \tau }}={\frac {\partial }{\partial \xi }}\left\{\left[1+\gamma Ra^{2}\left({\frac {\partial \varphi }{\partial \xi }}\right)^{2}\right]{\frac {\partial \varphi }{\partial \xi }}\right\},}$

where ${\displaystyle Ra=gh^{4}\alpha C/\nu D}$ is a Rayleigh number. For ${\displaystyle Ra\ll 1}$, the equation reduces to the linear heat equation, ${\displaystyle \varphi _{\tau }=\varphi _{\xi \xi }}$ and for ${\displaystyle Ra\gg 1}$, the equation reduces to ${\displaystyle \varphi _{\tau }=3\gamma Ra^{2}\varphi _{\xi }^{2}\varphi _{\xi \xi }}$.