Abstract:
Many boundary value problems occur in a natural way while studying fluid flow
problems in a channel. The solutions of two such boundary value problems are obtained
and analysed in the context of flow problems involving three layers of fluids of different
constant densities in a channel, associated with an impermeable bottom that has a small
undulation. The top surface of the channel is either bounded by a rigid lid or free to
the atmosphere. The fluid in each layer is assumed to be inviscid and incompressible,
and the flow is irrotational and two-dimensional. Only waves that are stationary with
respect to the bottom profile are considered in this paper. The effect of surface tension
is neglected. In the process of obtaining solutions for both the problems, regular
perturbation analysis along with a Fourier transform technique is employed to derive
the first-order corrections of some important physical quantities. Two types of bottom
topography, such as concave and convex, are considered to derive the profiles of the
interfaces. We observe that the profiles are oscillatory in nature, representing waves
of variable amplitude with distinct wave numbers propagating downstream and with no
wave upstream. The observations are presented in tabular and graphical forms.
