Two-Dimensional Stratified Flow Over a Dipole and Determination of the Divided Curve

Abstract

In the study on two-dimensional stratified flow in a channel, Dube (2002) and Yih (1960) proposed that the two-dimensional stratified flow over a barrier in a channel can be investigated by taking a suitable combination of sources, sink and doublets in place of barrier. Trustrum (1964) and then Dube (2023 & 2025) considered, however independently, the problem of two-dimensional channel flow over a barrier by applying an Oseen-type approximation to the general flow and discussed the Long’s hypothesis. In the study, she simply remarked that in the case of the flow over a diploe (in place of the barrier), if the axis of the dipole be parallel to the direction of the uniform stream at infinity then practically there is no far-upstream influence (or the influence is negligibly small). However, if the axis be perpendicular to the uniform stream at infinity, the far-upstream influence of the dipole is not negligible.

Drazin and Moore (1967) and then Dube (2025) used the technique of diffraction theory to obtain the solution of Long’s linearized equation for the steady flow of an incompressible inviscid fluid of variable density over an obstacle in an infinite channel. They extended the consideration to the flow over a dipole placed at the bottom of the channel with its axis parallel to the direction of the uniform stream by using some transformations and remarked in the same way as used to describe the flow over any obstacle whose shape happens to coincide with a streamline. But it was pointed out that a grave difficulty arises in making the shape of the obstacle to coincide exactly with the streamline.

In the study of Drazin and Moore (1967) and then Dube (2025) on the flow past an obstacle it was indicated that blocking may arise depending on the pressure condition at infinity, the value of the Froude number and the size of the obstacle. Blocking may occur also on the case of the flow over a dipole depending on the pressure at infinity and the strength of the dipole. So, it is our aim to study here and find a relation between the pressure at infinity, Froude number and the strength of the dipole to be placed at the bottom of the channel with its axis parallel to it and directed against the uniform flow.

If the pseudo-elastic U₀ at infinity on negative side (at x  ) be not large enough, then there

is apparently a possibility that a layer of the stratified fluid in the lower region of the channel may not be able to cross that dipole. This will result in what may be called the clocking of the incoming fluid by the dipole. Tis leads rather to a contradiction to the remark by Trustrum (1954) that if the axis of the dipole be parallel to the channel wall then there is no possibility of blocking. So, we propose here to restudy the problem of stratified flow over a dipole placed at the bottom of an infinite channel with its axis parallel to the uniform flow at infinity. An attempt is also made, to find analytically the relation between the pressure condition at infinity (on the negative side) and the strength of the dipole for the non-occurrence of blocking.