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Benney's long-wave expansion

After Benney [1] we expand Eqs. (1)-(3) by the long-wave parameter $\mu \sim \partial_x \stackrel{\mathrm{def}}{=}\mu\partial_{x_{1}}$, as most liquids have so large surface tension that $\mu \propto W^{-1/2} \ll 1$ safely holds. This expansion[2,3,4] allows to eliminate $\mathbf {u}$ and leads to an evolution equation of the surface $h$ alone, in the form

\begin{displaymath}
\partial_t {h} + \partial_x {Q} = 0, \qquad
Q = \int_0^h u\,dz
= Q_0 + \mu Q_1 + \mu^2 Q_2 + \cdots .
\end{displaymath} (4)

In particular, the result up to $Q_1$, i.e. the long-wave equation of Gjevik[2], reads as
\begin{displaymath}
\partial_t h + \frac{2}{3} \partial_x \left[
h^3
+ \left(...
...ha \right) \partial_x h
+ W h^3 \partial_x ^3 h
\right] = 0.
\end{displaymath} (5)




Author: OOSHIDA Takeshi
ooshida@damp.tottori-u.ac.jp 2000-08-24