Buckmaster equation

Last updated August 07, 2025

In mathematics, the Buckmaster equation is a second-order nonlinear partial differential equation, named after John D. Buckmaster, who derived the equation in 1977.^[1] The equation models the surface of a thin sheet of viscous liquid. The equation was derived earlier by S. H. Smith and by P Smith,^[2]^[3] but these earlier derivations focused on the steady version of the equation.

The Buckmaster equation is

u_{t}=(u^{4})_{xx}+\lambda (u^{3})_{x}

where $\lambda$ is a known parameter.

References

↑ Buckmaster, J. (1977). Viscous sheets advancing over dry beds. Journal of Fluid Mechanics, 81(4), 735–756.
↑ Smith, S. H. (1969). A non-linear analysis of steady surface waves on a thin sheet of viscous liquid flowing down an incline. Journal of Engineering Mathematics, 3(3), 173–179.
↑ Smith, P. (1969). On steady long waves on a viscous liquid at small Reynolds number. Journal of Engineering Mathematics, 3(3), 181–187.

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[1] Buckmaster, J. (1977). Viscous sheets advancing over dry beds. Journal of Fluid Mechanics, 81(4), 735–756.

[2] Smith, S. H. (1969). A non-linear analysis of steady surface waves on a thin sheet of viscous liquid flowing down an incline. Journal of Engineering Mathematics, 3(3), 173–179.

[3] Smith, P. (1969). On steady long waves on a viscous liquid at small Reynolds number. Journal of Engineering Mathematics, 3(3), 181–187.

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