Buckmaster equation explained

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} + λ (u ^{3 )} _{x}

where

λ is a known parameter.

Notes and References

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.