IX. 



ox DISCOXTIXUOUS MOTIOXS IX LIQUIDS. 



By Prof. A. Oberbeck. 



It is customary to designate by the term discontinuous liuid mo- 

 tions, those phenomena ofmovemeutin which the velocity is not tlirough- 

 out the whole space tilled with the fluid a continuous tuuctiou ot the 

 location. Therefore iu such movements there occur surlaces within 

 the fluid that separate from each other regions within whu-h the veloci- 

 ties differ from each other by tinite quaudties. Ti»e fundamental luin- 

 <iil)le8 of the theory of these motions were first given by Helmholtz.t 



If we assume that a velocity i)otential {qj) does exist for so called 

 steady- fluid motions then the hydro-dynamic differential equations can 

 be summarized iu the one equation, 



''=<^-mj^m<'i1 



Now Helmholtz has shown that the pressure j> and consequently the 

 velocity can be discontinuous fuuctions of the coordinates and that 

 there are a great number of phenomena of motion for which the assump- 

 tion of a discontinuous function is necessary. Especially has this theory 

 been applied by Helmholtz and by Kirchhott" to fluid jets,! and the 

 boundaries of free jets can be given under the following assumptions: 



[a) That uo accelerating force acts upon the fluid. 



[b) That the movement is steady. 



[c) That the movement depends only upon two variables, j^-aud ?/, and 

 is therefore everywhere parallel to a fixed plane. 



If in other cases, for instance for jets that are symmetrical about au 

 axis or that are under the influence of the accelerating force of gravity, 



* Read at the session of the Physical Society' in Berlin, May 11, 1877. TraDslated 

 from Wiedemann's Annalen der Physilc und Chemie, 1877, vol. ii, p. 1-16. 



t See the Berlin Monatsberichte, 1868, p. 215 [or No. II of this series of Translations.] 

 t See Crelle's Journal vol. Lxx, p. 289-299, [and Nos. Ill and VIII of this collection of 

 Translations.] 



139 



