Abstract: The effect of small non-parallelism of the base flow
on the stability of slightly curved mixing layers is analyzed in the
present paper. Assuming that the instability wavelength is much
smaller than the length scale of the variation of the base flow we
derive an amplitude evolution equation using the method of multiple
scales. The proposed asymptotic model provides connection between
parallel flow approximations and takes into account slow
longitudinal variation of the base flow.
Abstract: Method of multiple scales is used in the paper in order
to derive an amplitude evolution equation for the most unstable mode
from two-dimensional shallow water equations under the rigid-lid
assumption. It is assumed that shallow mixing layer is slightly curved
in the longitudinal direction and contains small particles. Dynamic
interaction between carrier fluid and particles is neglected. It is
shown that the evolution equation is the complex Ginzburg-Landau
equation. Explicit formulas for the computation of the coefficients of
the equation are obtained.