Incipient Instability Criterion of Two Confined Superposed Fluids

The problem of stability for the interface of two superposed fluids is determined by several parameters, for example, gravity, density difference, interfacial tension, depth of flow field, and shape of boundary constraint. The present paper studies the effects of boundary shapes on the incipient stability criterion of the interface between two fluid layers with different densities confined within a vertical container. When the interface is disturbed, its shape is easily destroyed by unsymmetric disturbances or due to the sloshing mode of disturbances. In the present study, two different wave-like boundaries are transformed to circular contours by conformal mapping. As the concept that some proeprties of any successive transformations still remain unchanged is applied, we obtain the eigenvalues for the incipient stability criteria under the constraints of wave-like contours.