Thin Unsteady Heavy Jets (Classic Reprint)

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Excerpt from Thin Unsteady Heavy Jets The hydrodynamic theory of jets attempts to determine the surfaces of a jet and the detailed velocity distribution within a jet. The main difficulty stems from the nonlinear boundary conditions to be satisfied on the unknown jet surfaces. When the flow is restricted to be two-dimensional and steady, and surface tension is neglected, this difficulty can be overcome by the method of conformal mapping and the surfaces and flow can be found exactly. This determination is based on the exact equations of hydrodynamics, omitting viscosity. In order to treat more general problems (unsteady, three-dimensional, or including gravity) the simpler hydraulic theory is usually employed. In this theory one gives up the attempt to find the detailed velocity distribution within the jet, but instead assumes that the velocity is constant on each cross-section of the jet. Since this assumption is incompatible with the exact equations of hydrodynamics, it is necessary to use different, approximate, equations based on the conservation laws of mechanics. These equations can be solved and yield the shape, thickness and speed of the jet, approximately. (This method can also be extended to take account of surface tension, as will be shown below.) Two questions which immediately arise are: "What is the relationship between the two theories?" and "How can the results of the hydraulic theory be improved?" In this paper we answer these questions by presenting a method of solution of the hydrodynamic problem as a series in powers of the jet thickness divided by some other typical length of the jet. The first term in this solution is found to be the solution given by the hydraulic theory, thus answering the first question. The higher order terms in the series yield corrections to the hydraulic theory, thus answering the second question. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully, any imperfections that remain are intentionally left to preserve the state of such historical works.