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.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.Smith, 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. The equation for the scalar field reads

where is the diffusion coefficient for the scalar ; is a numerical factor, which in planar problems assumes the value ; is the gravitational acceleration; is the width of the fluid layer in which dispersion is occurring; is the volumetric expansion coefficient defined by with being the fluid density; is the kinematic viscosity of the fluid.

Suppose is the characteristic length scale for , then the characteristic time scale is given by . And suppose is a characteristic value for . Then, we introudce the non-dimensional variables

then the Erdogan–Chatwin equation becomes

where is a Rayleigh number. For , the equation reduces to the linear heat equation, and for , the equation reduces to .

See also Ostroumov flow

References