# Hydrograph methods applied to solve certain problems on the flow of jets

Rangachari, Sanjiv (1964) Hydrograph methods applied to solve certain problems on the flow of jets. MSc(R) thesis, University of Glasgow.

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## Abstract

The present thesis discusses only problems concerning incompressible and inviscid fluids. There are various techniques used to solve hydrodynamical problems in which conformal mapping plays an important part. In Chapter II two problems have been separately discussed. The first deals with an inviscid incompressible fluid escaping in the form of a jet from an infinite chamber through a slit and impinging normally on a wall. The second deals with the flow through a finite Borda mouthpiece. Here the inviscid incompressible fluid flows out of the reservoir through the mouthpiece to form a jet which is bounded by the free streamlines. It is shown that from a mathematical standpoint Levy and Hachemeister were dealing with the same problem and consequently these two problems are included in one chapter. In Chapter III, the work has been extended by combining together the main physical features of the two problems of Chapter II. The new problem has then been solved by Schwart-Christoffel transformations. The transformation technique while mathematically very elegant suffers from a serious drawback for it is limited to a potential flow satisfying Laplace's equation. Thus this method cannot be used to solve problems of compressible fluids. The hodograph technique not only solves the problems for incompressible fluids but has in other connections been adapted to solve problems involving compressible fluids. The application of the hodograph method to incompressible flow in this thesis may serve as an introduction to its use in the treat-ment of compressible flow. In Chapter IV first Levy's problem, discussed in Chapter II, has been solved by hodograph methods. Then the Hachemeister problem has been solved. In both these problems there is a "notched hodograph" which requires an extension of Mackie's work. The notch in both the problems gives rise to a singular integral equation. Since the same singular integral equation is obtained for both problems we confirm here also that these two problems are mathematically identical. The singular integral equations have been solved analytically by an extension of method given by Mikhlin and verified by comparison with results of Chapter II. In Chapter V. the problem discussed in Chapter Mhos been investigated by the hodograph method. There are now two "notches" instead of one. Due to these two notches two simultaneous integral equations are obtained. An analytical solution for these equations has not so far been found. The pattern of streamlines in this problem has some interest.

Item Type: Thesis (MSc(R)) Masters Adviser: D C Pack Fluid mechanics, Applied mathematics 1964 Enlighten Team glathesis:1964-73760 Copyright of this thesis is held by the author. 14 Jun 2019 08:56 14 Jun 2019 08:56 https://theses.gla.ac.uk/id/eprint/73760