Open AccessXIV. Some applications of conformal transformation to problems in hydrodynamics
Publication year - 1915
Publication title -
philosophical transactions of the royal society of london. series a, containing papers of a mathematical or physical character
Language(s) - English
Resource type - Journals
eISSN - 2053-9258
pISSN - 0264-3952
DOI - 10.1098/rsta.1915.0014
Subject(s) - conformal map , polygon (computer graphics) , bounded function , boundary (topology) , plane (geometry) , interpretation (philosophy) , transformation (genetics) , variable (mathematics) , mathematics , representation (politics) , constant (computer programming) , geometry , mathematical analysis , computer science , chemistry , telecommunications , biochemistry , frame (networking) , politics , political science , law , gene , programming language
The problem of the conformal representation of the part of the plane of a variablez , which is bounded by a rectilineal polygon, upon the half-plane of a variablew bounded by the real axis, is solved (save for an integration) by the well-known transformation of Schwarzdz = CII (w —ar )-ar /πdw , whereC ,a 1 a 2 , &c., are real constants, and π—α 1 , π—α 2 , &c., are the internal angles of the rectilineal polygon. A more difficult problem is that of the conformal representation upon the half-plane ofw of a region in thez plane whose boundary is partly curved; it is with this problem that the present paper is concerned, always however with a view to interpretation of results in terms of the two-dimensional flow of liquid in regions having particular types of boundary.