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 $\varphi (x,t)$ reads^[2]^[3]^[4]

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

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

References

^ 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.

[1] 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.

[2] Smith, R. (1978). Asymptotic solutions of the Erdogan-Chatwin equation. Journal of Fluid Mechanics, 88(2), 323-337.

[3] 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.

[4] 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.

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