Erdogan–Chatwin equation

{{Short description|Fluid dynamics 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.Erdogan, M. E., & Chatwin, P. C. (1967). The effects of curvature and buoyancy on the laminar dispersion of solute in a horizontal tube. Journal of Fluid Mechanics, 29(3), 465-484. The equation for the scalar field $\backslash varphi(x,t)$ readsSmith, R. (1978). Asymptotic solutions of the Erdogan-Chatwin equation. Journal of Fluid Mechanics, 88(2), 323-337.Barton, N. G. (1976). The dispersion of a buoyant solute in laminar flow in a straight horizontal pipe. Part 1. Predictions from Erdogan & Chatwin's (1967) paper. Journal of Fluid Mechanics, 74(1), 81-89.Smith, R. (1982). Similarity solutions of a non-linear diffusion equation. IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications), 28(2), 149-149.

:$\backslash varphi\_t\; =\; (\backslash varphi\_x+a\backslash varphi\_x^3)\_x,$

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