Flow of a multilayer ideal incompressible and heavy fluid past a body

Abstract

The two-dimensional steady flow of a layered fluid past a body with discontinuous stratification is disucssed. The number of layers in finite, and the channel which has a horizontal floor is open. To study the flow behind the body, a hypothesis on the possibility of approximating the velocity profile at the body boundary by that which arises in weightless flow (see /1,2/) is postulated. A boundary value problem for a second-order elliptic equation in combined Euler-Lagrange variables is formulated. The problem is formulated in a rectilinear band with a separation, and under the conditions of consistency, on a finite number of parallel straight lines which correspond to the separation boundary. The introduction of a measure which gives rise to a monotonic density distribution in a non-perturbed flow, makes it possible to reduce the boundary value problem to the symmetrization of Fredholm-type kernels. The linearized equation is solved by Fourier methods. The results obtained in /3/ are amplified : it is shown that for any specified Froude number, the corresponding homogeneous integral equation has only a finite number of positive eigenvalues to which the oscillation modes correspond. It is also shown that if the flow velocity is close to one of a denumerable set of propagation velocities of long-wave modes, the corresponding harmonic becomes stronger because of the resonance.